How can changing sodium, chloride or albumin alter pH when none of them is simply donating H⁺?

Put the blood-gas report aside for a moment. Imagine the blood itself: water containing charged ions, bicarbonate, CO₂ and proteins able to bind and release H⁺.

We need to understand what happens among those substances before deciding which factors are dependent or independent.

Some ions keep their charge; buffers can change theirs

Start with ordinary sodium chloride. When it dissolves:

NaCl → Na⁺ + Cl⁻

Sodium carries positive charge. Chloride carries negative charge. Across the pH range encountered in blood, they remain essentially in those forms. Their concentrations can change because we add them, remove them, dilute them or move them between compartments, but they do not meaningfully bind or release H⁺ to change their charge state.

These are what Stewart calls strong ions.

“Strong” does not mean powerful. It means that, at physiological pH, they are essentially completely dissociated.

There is a slightly confusing bit of terminology here. The opposite of a strong ion is not a “weak ion”. The useful contrast is with a weak-acid or weak-base system, whose charge changes as H⁺ is released or bound.

Now return to our simple buffer from Part 1:

HA ⇌ H⁺ + A⁻

A⁻ carries negative charge. If it binds H⁺:

A⁻ + H⁺ → HA

that negative charge disappears. The H⁺ has not vanished; it is now attached to the buffer. If HA releases H⁺ again:

HA → H⁺ + A⁻

the negative charge returns.

So unlike chloride, the amount of negative charge carried by this buffer changes according to how much H⁺ it has bound.

Albumin behaves in this way. It is not literally one HA molecule; it is a large protein containing many groups that can pick up and release H⁺. As more H⁺ binds, albumin becomes less negatively charged. As H⁺ is released, it becomes more negatively charged. Phosphate behaves similarly. Bicarbonate belongs to the CO₂ buffer system we already know:

H⁺ + HCO₃⁻ ⇌ H₂CO₃ ⇌ CO₂ + H₂O

Lactate deserves one clarification. Lactic acid is chemically a weak acid, but at blood pH almost all of it is already present as lactate⁻. Within the physiological range it therefore behaves as a strong anion in Stewart’s accounting. The first distinction is therefore quite simple:

Strong ions retain their charge.

Buffers can change their charge by binding or releasing H⁺.

Where is the rest of the charge?

Now count the charge carried by the strong ions. The strong positive side includes sodium, potassium, calcium and magnesium. The strong negative side includes chloride, lactate and other strong anions. Suppose, purely for illustration, that the strong positive charges total:

145 mEq/L

and the strong negative charges total:

105 mEq/L

Subtract one from the other:

145 − 105 = 40 mEq/L

That difference is the strong ion difference, or SID.

Have we just discovered 40 mEq/L of unbalanced positive electrical charge floating around in plasma? No. We have only counted one group of charged substances.

Bicarbonate also carries negative charge. So do albumin and phosphate. Those account for most of the difference left when we count the strong ions alone.

Approximately:

SID ≈ bicarbonate negative charge + weak-acid negative charge

The SID is not an electrically empty gap. It represents strong positive charge that is balanced by negative charge elsewhere in the solution. The whole solution remains essentially electrically neutral. This is electroneutrality.

That is a physical constraint rather than something the body actively regulates. A large separation of charge in bulk fluid would generate enormous electrical forces.

SID counts charge, which is why it is properly expressed in equivalents. Na⁺ carries one positive charge, so 1 mmol of sodium contributes 1 mEq of positive charge. Ca²⁺ carries two, so 1 mmol of calcium ions contributes 2 mEq.

Sodium and chloride dominate the strong-ion concentrations, which makes Na−Cl useful at the bedside. But Na−Cl is not the complete SID. It leaves out potassium, calcium, magnesium, lactate and the other strong ions.

There is another scale worth remembering from Part 1. At pH 7.40, free H⁺ is only about:

40 nanomol/L or 0.00004 mmol/L

SID, bicarbonate and buffer charge are measured in tens of milliequivalents per litre.

So if SID differs by several mEq/L, that cannot be balanced by accumulating several mmol/L of free H⁺. The free H⁺ concentration is far too small. Most of the milliequivalent-scale differences in charge lie in bicarbonate and the other buffers. A tiny change in free H⁺ can still produce a substantial change in pH.

How does SID connect to H⁺?

This is where it is easy to accidentally invent a causal sequence. For the moment, do not imagine anything being infused or removed. Instead compare two solutions at equilibrium. Give them the same temperature, the same PCO₂, and the same total amount of albumin and phosphate per litre.

Suppose one solution has more free H⁺. What else must be different?

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It was a long article. I understood some of it, recognised that there was something important in the parts I could not follow, and made it my aim to understand it properly one day.

That article was a tricky place to begin. Acid–base teaching often starts several steps into the explanation, with equations whose ingredients we have never quite understood. So I want to start from the beginning. No chemistry knowledge assumed.

What is an acid? What is a buffer? What does pH actually measure? Why do bicarbonate and carbon dioxide appear together? What is base excess telling us?

By the end of this first episode, we should be able to look at a blood gas and identify the main acid–base processes using the chemistry we have built along the way. Part 2 will take us further: why do those numbers have the values they do, and what can they reveal about the patient? Basic to advanced, step by step.

What an acid actually does

An ion is an atom, or a group of atoms, carrying an electrical charge. Sodium in blood is positively charged and written Na⁺. Chloride is negatively charged and written Cl⁻. Positive ions are called cations; negative ions are anions.

The ion at the centre of acid–base physiology is H⁺, the hydrogen ion. You will also see it called a proton. Molecules can pick up and release H⁺. That exchange is the starting point for understanding acids and bases.

An acid can donate H⁺. A base can accept it.

Imagine a simple acid consisting of a hydrogen attached to the rest of a molecule. We can call the whole molecule HA. When it releases H⁺, the remaining part, A⁻, carries a negative charge:

HA ⇌ H⁺ + A⁻

The double arrow means that the reaction can run in either direction. A⁻ can take the H⁺ back and become HA again. It is therefore a base: specifically, the conjugate base of HA. The terminology sounds more complicated than the idea. It is simply the same chemical pair, with or without the H⁺ attached.

Blood contains many substances that can take up H⁺. Bicarbonate can. So can groups on proteins such as haemoglobin and albumin. That means adding acid to blood does not leave every added H⁺ floating freely in the plasma.

Buffers

Suppose more H⁺ enters a solution containing HA and A⁻. Some A⁻ takes it up and becomes HA. If H⁺ is removed, some HA releases it again. The mixture therefore limits the change in free H⁺. This is what a buffer does.

The reactions do not stop when the system reaches equilibrium. Molecules continue picking up and releasing H⁺ in both directions. At equilibrium, those opposing reactions balance so that the overall proportions remain stable.

This also explains something that can initially seem odd. If an acid splits into H⁺ and its conjugate base:

HA ⇌ H⁺ + A⁻

why does the A⁻ not simply take up all the H⁺ again?

Some of it does. The forward reaction continues too, releasing H⁺ as other molecules take it back up. The system settles at a balance in which some HA remains intact and some H⁺ and A⁻ remain separate.

The amount of free H⁺ at that balance determines the pH.

Buffers also have limits. As more of the available base takes up H⁺, less remains available to buffer the next addition.

And binding a proton does not remove it from the body. Buffers contain the disturbance while other processes deal with the source and disposal of acid.

Some acids release H⁺ almost completely when dissolved in water. These are called strong acids. Hydrochloric acid is an example. Under comparable conditions, weak acids retain a larger proportion of their H⁺.

“Strong” does not mean concentrated. Strength describes how completely an acid dissociates; concentration describes how much is present.

For now, that is enough chemistry.

What pH tells us

pH tells us about the amount of freely available H⁺. Strictly, pH reflects hydrogen-ion activity, which accounts for how ions behave in their surroundings. Clinically, it is reasonable to think of it as reflecting free H⁺ concentration.

The scale runs backwards:

more H⁺ → lower pH

It is also logarithmic. A fall of one whole pH unit represents a tenfold increase in hydrogen-ion activity.

At pH 7.40, the H⁺ concentration is approximately 40 nanomoles per litre. At pH 7.10, it is about 80. So a fall of only 0.30 on the blood-gas report represents roughly a doubling of free H⁺.

That also shows how tiny the free H⁺ concentration is compared with the millimole-per-litre concentrations of electrolytes and buffers. Most H⁺ added to the body does not remain freely dissolved. It is taken up by buffers.

The pH therefore tells us the current concentration of free H⁺. It does not tell us how much acid has entered the body.

For an arterial sample measured at 37°C, the normal reference interval is approximately 7.35–7.45. Below this is acidaemia. Above it is alkalaemia. These terms describe the state of the blood at the moment it was sampled.

An acidosis is a process tending to lower pH. An alkalosis tends to raise it.

A patient can have more than one process at the same time. If an acidosis and an alkalosis oppose each other closely enough, the measured pH may even fall within the normal range.

A normal pH therefore does not necessarily mean normal acid–base physiology.

How carbon dioxide changes pH

Cells continuously produce carbon dioxide. CO₂ does not contain hydrogen, so it cannot simply donate H⁺ itself. Instead, it reacts with water to form carbonic acid:

CO₂ + H₂O ⇌ H₂CO₃

Carbonic acid can then release H⁺:

H₂CO₃ ⇌ H⁺ + HCO₃⁻

This maps directly onto the simple acid reaction we used earlier:

HA ⇌ H⁺ + A⁻

Here, HA = H₂CO₃ and A⁻ = HCO₃⁻.

Bicarbonate is therefore the conjugate base of carbonic acid.

Because carbonic acid exists only in a relatively small amount and exchanges rapidly with dissolved CO₂, we usually hide the carbonic-acid step in the middle and shorten the whole sequence to:

CO₂ + H₂O ⇌ H⁺ + HCO₃⁻

Now suppose CO₂ rises. More CO₂ enters this system, so the balance shifts towards more carbonic acid and then towards more H⁺ and bicarbonate.

That gives us an initially surprising combination:

CO₂ ↑

H⁺ ↑

pH ↓

HCO₃⁻ ↑

If bicarbonate is a base, why does it not simply mop up all the extra H⁺?

Some of it does. The reaction is reversible:

H⁺ + HCO₃⁻ ⇌ H₂CO₃ ⇌ CO₂ + H₂O

But, just as with our simple HA example, the reaction does not run completely in either direction. It settles at a new equilibrium.

With more CO₂ present, that new equilibrium contains more free H⁺ than before, so the pH is lower. It also contains more bicarbonate.

At a pH of 7.40, the concentration of free H⁺ is only about 40 nanomoles/L, or 0.00004 mmol/L. Bicarbonate, by comparison, is present at around 24 mmol/L.

When CO₂ rises, the additional H⁺ generated by the carbonic-acid system does not all remain free. Much is taken up by other buffers, particularly haemoglobin and other proteins, allowing additional bicarbonate to remain.

But buffering does not have to leave zero additional free H⁺. A rise from 40 to 60 nanomoles/L would represent only another 0.00002 mmol/L of free H⁺. That is almost nothing compared with the quantities of bicarbonate and other buffers present, but it is a 50% increase in free H⁺ and therefore produces a substantial fall in pH.

This is why a respiratory acidosis can produce a raised bicarbonate. A high bicarbonate does not automatically mean metabolic alkalosis. Its meaning depends on the CO₂ and the rest of the system.

How the body handles an acid–base disturbance

Buffers act immediately. The lungs and kidneys then alter what the body retains or removes.

Ventilation controls CO₂. Retaining more CO₂ tends to increase free H⁺ and lower pH. Removing more CO₂ tends to reduce free H⁺ and raise pH.

The kidneys adapt more slowly. They reclaim filtered bicarbonate and excrete acid, much of it carried in the urine as ammonium or bound to urinary buffers such as phosphate.

The details are complicated, but the principle is simple: the kidneys alter acid excretion and bicarbonate balance over hours to days.

This gives us several very different ways to disturb acid–base physiology.

A patient may retain CO₂ because ventilation is inadequate. They may generate ketoacids. They may lose bicarbonate through diarrhoea. They may lose gastric acid through vomiting. Renal dysfunction may limit acid excretion. Intravenous fluids may change the chemical composition of the plasma.

The measured pH is the final result of all those processes acting together.

Henderson–Hasselbalch earns its place

We have seen how a rise in CO₂ can increase both free H⁺ and bicarbonate. The next step is to put numbers to that relationship.

We can start with the reaction:

H₂CO₃ ⇌ H⁺ + HCO₃⁻

At a given temperature and comparable solution conditions, chemical equilibrium fixes a relationship between these three substances. Carbonic acid is also linked to dissolved CO₂. We can express the combined relationship as:

[H⁺] ∝ dissolved CO₂ / [HCO₃⁻]

The square brackets mean concentration; ∝ means “is proportional to”. In words:

more CO₂ relative to bicarbonate → more free H⁺

more bicarbonate relative to CO₂ → less free H⁺

Henderson developed the underlying relationship early in the twentieth century. Hasselbalch subsequently expressed it using the logarithmic pH scale.

For blood at approximately 37°C:

pH ≈ 6.1 + log₁₀[HCO₃⁻ ÷ (0.225 × PCO₂ in kPa)]

or, using mmHg:

pH ≈ 6.1 + log₁₀[HCO₃⁻ ÷ (0.03 × PCO₂ in mmHg)]

Here bicarbonate is in mmol/L, numerically the same as mEq/L for this ion. PCO₂ is the partial pressure of CO₂; the factor 0.225 or 0.03 converts it to an approximate dissolved concentration in the corresponding pressure units.

The ratio does not replace H⁺ as the thing determining pH. It describes the free H⁺ concentration when the system reaches equilibrium.

You do not need to calculate the equation routinely at the bedside. Its value is in understanding what the numbers mean.

Take a bicarbonate of 24 mmol/L and a PCO₂ of about 5.3 kPa (40 mmHg). Dissolved CO₂ is approximately 1.2 mmol/L. The bicarbonate-to-CO₂ ratio is about 20:1, giving a pH close to 7.40.

The two parts of the ratio are chemically linked, but they are not forced to change in the same proportion.

Suppose PCO₂ acutely doubles to about 10.7 kPa (80 mmHg). The amount of dissolved CO₂ approximately doubles. Bicarbonate rises too, but much less, typically only by a few mmol/L in an acute respiratory acidosis.

For example:

HCO₃⁻ 24 mmol/L / dissolved CO₂ 1.2 mmol/L ≈ 20

might become approximately:

HCO₃⁻ 28 mmol/L / dissolved CO₂ 2.4 mmol/L ≈ 12

The ratio has fallen. That lower ratio corresponds to a higher free H⁺ concentration and therefore a lower pH.

Now consider a different change: halve both of our starting values.

Bicarbonate is 12 mmol/L and PCO₂ is about 2.7 kPa (20 mmHg). The ratio remains about 20:1, so pH is still about 7.40. But that gas is not normal. We will come back to it.

Henderson–Hasselbalch is fully quantitative: given bicarbonate and PCO₂, it lets us calculate the corresponding pH. What it does not establish, by itself, is why bicarbonate has the value it does.

Stewart’s framework approaches the same chemistry using different inputs. With consistent assumptions, the two approaches give compatible answers. We will explore what that different view adds in Part 2.

What the blood-gas machine actually measures

The blood-gas analyser directly measures pH and PCO₂. Bicarbonate is usually calculated from those measurements using the known relationship between them. Base excess is calculated as well.

That means pH, PCO₂, bicarbonate and base excess are not four independent measurements confirming the same conclusion. Some are derived from others.

The chemistry remains useful. We just need to understand what each number represents.

Why I like base excess

Base excess tries to isolate the metabolic component of the disturbance.

Imagine taking a blood sample and bringing its PCO₂ to the standard value of 5.33 kPa, or 40 mmHg, at 37°C.

How much strong acid or base per litre would need to be added to bring the pH to 7.40?

That amount defines the base excess for the blood sample.

If acid would be required, the base excess is positive. If base would be required, it is negative.

The version most useful clinically is standard base excess, or SBE. It estimates the metabolic disturbance across the whole extracellular fluid (ECF) volume, using a haemoglobin concentration of 50 g/L (5 g/dL)—approximately what we would get if the blood’s haemoglobin were distributed throughout that volume.

So:

SBE +6 mmol/L (+6 mEq/L) suggests a net metabolic alkalinising effect.

SBE −8 mmol/L (−8 mEq/L) suggests a net metabolic acidifying effect.

Zero is the reference point, and that is the main reason I find BE easier to work with than bicarbonate. Both have normal ranges, but BE already expresses the metabolic result as a positive or negative deviation from zero. There is no need to subtract a reference value of around 24 mmol/L (24 mEq/L) before starting the bedside arithmetic. Positive and negative contributions can be placed alongside one another and added. We will use that advantage in Part 2.

SBE is a net result. A strongly acidifying process and a strongly alkalinising process may partly cancel one another, so a nearly normal SBE does not necessarily mean nothing is happening. An abnormal SBE can also reflect renal compensation for a sustained respiratory disturbance, rather than a separate primary metabolic disorder.

And a base deficit is not automatically a prescription for bicarbonate. We still need to understand what produced it.

The first reading of a blood gas

Before interpreting the numbers, establish the context. Is the sample arterial or venous, and when was it taken? What ventilation, oxygen therapy and treatment were being given? If the results do not fit the patient, ask how the sample was taken and whether it needs repeating.

Then read pH, PCO₂ and the metabolic component together. Traditionally, acid–base disturbances are divided into respiratory and metabolic processes. A primary change in PCO₂ is respiratory. Bicarbonate cannot be interpreted on its own: it also rises during respiratory acidosis, with further renal adaptation over time.

The four basic patterns are:

Compensation means that the body responds in a direction that tends to return pH towards normal. For example, metabolic acidosis stimulates ventilation. CO₂ falls.

That fall in CO₂ is expected. It does not automatically mean that the patient has a second respiratory disorder. The question is whether the response is of the expected size.

Is the compensation appropriate?

Empirical compensation rules tell us roughly what happens when there is a single primary disturbance. For metabolic acidosis, the best-known is Winter’s formula.

Using bicarbonate in mmol/L:

Expected PCO₂ in kPa ≈ 0.2 × bicarbonate + 1.07, ±0.27 kPa

In mmHg:

Expected PCO₂ ≈ 1.5 × bicarbonate + 8, ±2 mmHg

These are clinical approximations, not laws of physics.

If the measured PCO₂ is higher than expected, there is likely to be an additional respiratory acidosis. If it is lower, there is likely to be an additional respiratory alkalosis.

Now return to our earlier example:

bicarbonate 12 mmol/L

PCO₂ about 2.7 kPa, or 20 mmHg

pH about 7.40

Winter’s formula predicts a PCO₂ of about 3.5 kPa, or 26 mmHg, with an expected range of roughly 3.2–3.7 kPa (24–28 mmHg).

The actual PCO₂ is considerably lower. This patient has metabolic acidosis with an additional respiratory alkalosis. The two processes oppose one another closely enough to leave the pH near 7.40.

Calling the blood gas normal would miss both processes. Calling it simply a compensated metabolic acidosis would miss the additional respiratory disorder.

What we can now say — and what we can’t

We can now read pH, CO₂ and the metabolic component together, assess compensation and recognise important disturbances that a normal pH can conceal. Standard base excess gives us a useful estimate of the net metabolic result.

But suppose the SBE is −10 mmol/L (−10 mEq/L). What accounts for that −10? One process may dominate, or a larger acidifying effect may be partly concealed by an opposing alkalinising one.

And why can a chloride-rich fluid produce acidosis when chloride contains no H⁺?

In Part 2, for paid subscribers, we will follow what happens when the composition of blood changes, and see how Stewart’s framework helps us understand the result. Along the way, we will examine two explanations common in the literature: that sodium bicarbonate works because of its sodium, while bicarbonate itself is irrelevant; and that changes in strong ions cause acidosis by making water dissociate and release H⁺. But do they explain what actually happens in the blood?

We will take those explanations apart carefully, then bring the chemistry to the bedside. Using the anion gap and Story’s simple arithmetic, we will investigate the metabolic result, uncover effects that partly cancel one another, and decide what needs measuring next.

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It sounds straightforward until you ask what that load actually is.

Arterial pressure? Systemic vascular resistance? Aortic impedance? Effective arterial elastance? Ventricular wall stress?

All of these have been used to describe afterload. They are related, but they are not the same thing.

Consider two patients with a systolic arterial pressure of 150 mmHg. One has a normal-sized, vigorous left ventricle and is entirely comfortable. The other has a dilated failing ventricle, a low stroke volume and pulmonary oedema.

The pressure is the same. The mechanical problem facing the myocardium is not.

To understand why, we have to start one step earlier than afterload itself.

What is the ventricle actually doing?

The heart is an energy transducer.

Chemical energy derived from metabolism is converted into mechanical energy by the myocardium and transferred to the blood.

At the beginning of systole the ventricle contracts with both valves closed. Pressure rises during isovolumetric contraction, but no blood is ejected and no external stroke work is yet being performed.

Once left ventricular pressure exceeds aortic pressure, the aortic valve opens. Blood begins to move and the ventricle performs work on the circulation.

The usual description is that the ventricle now has to “push against” arterial pressure. In a mechanical sense, that is true. Pressure exerts force. The ventricle must generate enough pressure to open the aortic valve and, during ejection, must continue generating sufficient wall tension to displace blood into an already pressurised arterial system. Moving a volume of blood against that pressure requires work.

But pressure is only part of the story.

The arterial pressure faced by the ventricle represents mechanical energy already present within the arterial system. During ejection, the ventricle transfers additional energy into that system. Some is stored elastically as the arteries distend, some contributes to accelerating blood, and some is irreversibly dissipated as blood flows through vascular resistance.

That distinction helps explain why the heart has to keep supplying energy beat after beat.

Imagine an inflated balloon with a small hole in it.

The pressure inside the balloon creates a real load on the pump. To force more air into the balloon, the pump has to work against that existing pressure.

But if there were no hole, the pump would not need to keep running simply because the balloon was pressurised. Once inflated, the balloon could remain under tension with energy stored in its stretched wall.

Add a hole and the situation changes. Air continually escapes. If the balloon is to remain at the same pressure, the pump must keep replacing what is being lost.

The circulation has a similar energetic problem, with one crucial difference: blood is not escaping from the vascular system. Mechanical energy is.

As blood flows through vessels, viscous forces irreversibly dissipate mechanical energy, predominantly as heat. Resistance describes the pressure-flow relation associated with that dissipation. If flow is to continue, the energy being lost must continually be replaced.

The heart supplies it.

So two related things are happening during ventricular ejection.

The ventricle is ejecting into an arterial system that is already pressurised, and that pressure contributes directly to the mechanical load it must meet.

At the same time, flow through the vascular system continually dissipates mechanical energy, requiring ongoing cardiac energy transfer if the circulation is to maintain its pressure-flow state.

Pressure itself is not what consumes the energy. A vascular system can remain pressurised without flow. The continuous energetic cost appears when blood moves through a dissipative pathway.

For a simple resistive load:

pressure loss = flow × resistance

and the rate at which mechanical energy is dissipated is:

power loss = pressure loss × flow

So, for the same flow, greater resistance means greater energy dissipation and therefore a greater cardiac power requirement.

If the ventricle has sufficient reserve, it may meet that requirement. Ventricular work rises, arterial pressure may rise, and flow can remain relatively well preserved.

If it cannot, flow falls.

The eventual pressure and flow are not dictated by resistance alone. They emerge from the interaction between the ventricle and the vascular system.

This also explains why a larger pressure gradient need not mean greater flow. If resistance increases, the pressure gradient across the circulation may become larger while flow falls. The larger gradient reflects the greater loss of mechanical energy along the pathway; it is not an independent source forcing the blood through it.

The ventricle therefore does more than simply “push against pressure”. It transfers energy into a pressurised, flowing and elastic arterial system — overcoming the existing pressure load while continually replacing the energy dissipated during flow.

The arterial system is more than a resistor

Resistance is only one component of the load faced by the ventricle.

The arterial circulation is pulsatile and elastic. During systole, some of the energy transferred by the ventricle is stored as the arteries distend. During diastole, some of that stored elastic energy is released again. Energy is also required to accelerate blood, and pressure and flow waves travel through the arterial tree and are reflected back from peripheral sites.

The ventricle therefore ejects into a system characterised by resistance, compliance, inertial effects, characteristic impedance and wave reflection. The timing of these effects has implications. A reflected wave arriving during systole does not present the same ventricular burden as one returning later in diastole.

Resistance predominantly describes dissipative loading.

Compliance describes elastic storage.

Characteristic impedance describes aspects of the immediate pulsatile load encountered as blood is accelerated into the proximal arterial system.

Wave reflection alters the pressure and flow environment seen by the ventricle during ejection.

Together they contribute to arterial load.

The ventricle does not eject into an SVR. It ejects into a pulsatile elastic arterial tree.

Yet even arterial load is not the end of the story.

It is the external challenge presented to the ventricle. It does not tell us the mechanical load actually borne by the myocardium.

For that, we need the ventricle itself.

The hill, the bicycle and the rider

Imagine riding a bicycle uphill.

The hill is the external load. Make the hill steeper and the cyclist has to generate more power to maintain the same speed.

But the hill alone does not determine how difficult the climb feels.

It also depends on the rider. A powerful cyclist may climb a gradient comfortably that overwhelms someone weaker.

And it depends on the bicycle.

Put the same rider on the same hill with terrible gearing and the force required at the pedals changes dramatically even though the hill has not changed at all.

The cardiovascular equivalents are useful.

• The hill is the arterial load.

• The rider is the ventricle’s force- and power-generating capability.

• The bicycle and gearing are the ventricle’s geometry and mechanical advantage.

The complete problem is the interaction between all three.

In the context of afterload, it exposes something more fundamental: the external arterial load and the internal myocardial load are not synonymous.

At the myocardial level, afterload is most rigorously represented by the wall stress developed during ventricular ejection. To keep the cycling analogy, this corresponds to the muscular force the rider has to generate at the pedals.

The arteries create the external challenge. The ventricle must generate sufficient pressure to eject into that load; transmural pressure and ventricular geometry determine how this is translated into myocardial wall stress.

Geometry changes the meaning of pressure

Arterial pressure is a real mechanical load, but the myocardium does not experience pressure simply as a number in mmHg. It experiences wall stress.

Pressure inside the ventricle pushes outward on the inner surface. The myocardium has to generate tension within the wall to contain that pressure and, during systole, to shorten the chamber and eject blood.

How much tension is required depends strongly on ventricular geometry.

A useful way to picture this is to think about curvature.

Wall tension acts mainly along the myocardium. Because the ventricular wall is curved, part of that tension acts inward and helps oppose the pressure pushing the wall outward. A tightly curved wall does this efficiently.

As the ventricle dilates, the wall becomes less tightly curved. The same amount of tension now produces less inward restraining force. More tension is therefore required to contain the same intracavitary pressure.

A hammock gives the same intuition. If a weight is supported by a rope hanging in a deep curve, the tension in the rope has a substantial upward component. Pull the rope almost flat and the geometry becomes much less favourable: enormous tension is required to support the same weight.

Nothing about the weight has changed. The geometry through which the tension acts has.

A dilated ventricle has the same mechanical disadvantage. Its wall is less curved, so more myocardial tension is required to generate and contain the same ventricular pressure.

Wall thickness changes the problem again. A thicker ventricular wall provides more myocardial tissue across which that tension can be distributed. The stress carried by each unit of myocardium is therefore lower.

These relationships are captured approximately by Laplace:

wall stress ∝ transmural pressure × chamber radius / wall thickness

The real ventricle is thick-walled, three-dimensional and continuously changing shape, so this is not an exact equation for ventricular mechanics. The underlying relationships are robust:

• higher transmural pressure increases wall stress;

• a larger chamber radius increases wall stress;

• greater wall thickness reduces wall stress.

Consider two ventricles generating exactly the same systolic pressure.

One has a relatively small cavity and a thick wall.

The other is dilated and relatively thin-walled.

The arterial pressure may be identical, but the dilated ventricle must develop much greater myocardial wall stress to generate it.

120 mmHg is not the same myocardial load for every ventricle.

This is the “gearing” in the cycling analogy. The external hill may be unchanged, but the mechanical advantage with which the muscle meets that load has deteriorated.

Ventricular dilatation can therefore become part of the mechanics of failure rather than merely a marker of it.

Poor ejection increases end-systolic volume. Radius increases. Greater wall stress is then required to generate the same pressure. The force and energy requirements of subsequent contraction rise, making ejection still harder.

The gearing has become worse.

The wall sees the pressure across it

There is one more part of the wall-stress equation that deserves careful attention.

The pressure that distends the ventricle is not simply the pressure inside it.

Pressure also acts on the outside of the ventricular wall.

Intracavitary pressure pushes outward. Pressure surrounding the heart pushes inward.

The myocardium therefore has to contain the difference between them:

transmural pressure = intracavitary pressure − surrounding pressure

Suppose LV systolic pressure is 120 mmHg and surrounding pressure is close to zero.

The ventricular wall is exposed to approximately 120 mmHg of net outward distending pressure.

Now suppose intrathoracic and pericardial pressure rise to +20 mmHg while LV systolic pressure remains 120 mmHg.

The pressure inside is still pushing outward with 120 mmHg.

But the outside is now pushing inward with 20 mmHg.

The net distending pressure across the wall is only:

120 − 20 = 100 mmHg

The external pressure is supplying some of the inward force required to oppose the pressure inside the chamber. The myocardium therefore needs less wall tension to contain the same intracavitary pressure.

It is not that external pressure is somehow making the myocytes contract or supplying contractile energy. It is mechanically unloading the wall by reducing the net pressure trying to distend it.

This is why positive intrathoracic pressure can reduce left ventricular afterload.

If intrathoracic pressure rises, LV transmural systolic pressure falls. For the same ventricular geometry, lower transmural pressure means lower wall stress.

Positive pressure ventilation simultaneously changes venous return, right ventricular loading, pulmonary vascular behaviour and ventricular interaction, so “PEEP reduces afterload” should not be interpreted as a statement that positive pressure is globally beneficial to the circulation. It is specifically a statement about LV transmural loading.

Wall stress also changes continuously during systole. Pressure changes, the ventricular radius decreases as blood is ejected, and the wall thickens as the myocardium contracts. There is therefore no single fixed myocardial afterload throughout the beat.

That is another reason arterial pressure alone is an incomplete description.

At the arterial level we can talk about the load presented to the ventricle. At the myocardial level, the mechanically relevant quantity is the stress within the ventricular wall required to meet that load.

And that stress depends not just on the pressure in the artery, but on the pressure across the ventricular wall and the geometry of the ventricle carrying it.

Why SVR is not afterload

Systemic vascular resistance is often used almost interchangeably with afterload.

It should not be.

SVR is usually calculated as:

SVR = (MAP − RAP) / cardiac output

It is a pressure–flow ratio derived from the haemodynamic state that has already emerged. It is not ventricular wall stress, does not describe ventricular geometry, and does not capture arterial compliance, characteristic impedance or wave reflection. Nor is it a direct measurement of vascular tone.

Suppose cardiac output falls from 5 to 2.5 L/min while MAP falls from 100 to 80 mmHg. Ignoring RAP for simplicity, calculated SVR rises from 20 to 32 — an increase of 60%.

The circulation now has a “high SVR”, even though the calculation alone cannot tell us that vasoconstriction caused the fall in flow. The value has risen partly because flow fell more than pressure did.

This is the same causal trap that appears throughout haemodynamics. Rearranging:

ΔP = flow × resistance

into:

resistance = ΔP / flow

does not turn the calculated resistance into an independently measured cause of the pressure and flow used to calculate it.

Arterial resistance remains a real component of arterial load because resistance dissipates mechanical energy during flow.

But:

resistance contributes to arterial load

is not the same statement as:

SVR is afterload.

Calling SVR “afterload” substitutes a calculated whole-circulation pressure–flow relation for the mechanical burden experienced by the ventricle.

Effective arterial elastance: compressing the arterial system into one number

The full arterial load is difficult to describe at the bedside.

Effective arterial elastance, Ea, is a lumped descriptor of that load and is commonly approximated as:

Ea ≈ end-systolic pressure / stroke volume

The balloon analogy helps here too.

Imagine pumping a fixed amount of air into an elastic balloon that has a leak.

The pressure reached by the end of the inflation depends on several things at once.

A stiff balloon develops a larger pressure rise for the same added volume.

A tighter leak allows less air to escape during inflation, so pressure remains higher.

If you inflate the balloon more quickly, or start the next inflation before much pressure has fallen, the final pressure will also be higher.

Measure only:

final pressure / volume pumped in

and you do not know which of those mechanisms produced the result.

You have compressed their combined effect into a single number.

Ea does something similar for the arterial circulation.

For a given stroke volume, the end-systolic pressure reached depends on the properties and timing of the arterial system into which that volume was ejected. Arterial compliance affects how much pressure rises as volume enters the arteries. Peripheral resistance affects how much blood runs off from the arterial compartment during systole. Heart rate and systolic timing affect how much pressure and stored elastic energy remain from one beat to the next.

Their combined effects are therefore reflected in the relationship between end-systolic pressure and stroke volume.

A higher Ea means that, in the resolved state, a greater end-systolic pressure was associated with each unit of stroke volume ejected.

That does not mean Ea directly measures resistance, compliance or any other single arterial property. Nor does it fully describe the pulsatile arterial load. Characteristic impedance, wave reflections and the detailed pressure-flow waveform can differ substantially between arterial systems with similar Ea.

The word elastance can also be misleading. Ea has the units of an elastance — pressure divided by volume — but it is not simply the physical stiffness of the arterial tree. It is an effective, reduced-order representation of the arterial load as seen by the ventricle.

Its real value appears when it is compared with the ventricular side of the system.

Ea summarises the external arterial load.

Ees summarises ventricular end-systolic contractile properties.

Their relationship gives a compact description of how well the ventricle is matched to the load it is ejecting into.

Which brings us back to the cyclist.

A steeper hill does not necessarily make the cyclist slow down

Another common simplification is:

afterload increases → stroke volume falls

That is not a general law.

Increase the gradient of a hill and a strong cyclist may simply produce more force and power while maintaining the same speed.

A ventricle with sufficient reserve can do the same.

Increase arterial load and the ventricle can generate greater pressure, perform more stroke work and maintain stroke volume.

The cost is higher myocardial work and greater energy expenditure.

A fall in stroke volume occurs when the ventricle cannot adequately meet the increased mechanical demand.

Then ejection becomes less complete. End-systolic volume rises. Stroke volume falls. Residual ventricular volume feeds into the next filling cycle, and filling pressure may rise.

The same arterial load can therefore produce radically different haemodynamic states depending on the ventricle facing it.

This is the essence of ventriculo-arterial coupling.

A useful reduced-order representation compares effective arterial elastance, Ea, with ventricular end-systolic elastance, Ees. Ees describes the force-generating properties of the ventricle at end systole and is commonly used as a relatively load-independent index of contractile state. Their relationship describes how well ventricular capability is matched to arterial load.

The mathematics is less important here than the idea.

There is no clinically meaningful account of a hill without considering the cyclist climbing it.

A high arterial load can coexist with excellent cardiac output when ventricular reserve is substantial.

A much lower arterial load can overwhelm a weak, dilated or mechanically disadvantaged ventricle.

The pathological state is often better described as afterload mismatch than simply “high afterload”: the load is excessive relative to the capacity of that ventricle to meet it.

Why reducing arterial pressure can increase cardiac output

Once afterload is viewed this way, several apparently paradoxical clinical observations become straightforward.

A patient with severe LV systolic dysfunction may be maintaining a high end-systolic volume because the ventricle cannot adequately eject against the arterial load it faces.

Reduce that load and the same ventricle can eject further.

End-systolic volume falls.

Stroke volume rises.

Cardiac output may rise.

Upstream filling pressure can fall.

And arterial pressure may fall at the same time.

So:

MAP ↓
stroke volume ↑
cardiac output ↑

is entirely possible.

There is no contradiction. Pressure and flow are different dependent features of the circulation.

This is why arterial vasodilatation can produce a striking improvement in a patient with load-sensitive LV failure. Reducing the external load allows more of the ventricular mechanical energy to appear as forward stroke work rather than being consumed in generating high wall stress and pressure with poor ejection.

Acute pulmonary oedema makes the interaction particularly visible.

A struggling LV does not empty adequately. End-systolic volume rises. The larger ventricular radius worsens the mechanical disadvantage. Filling pressure rises and that pressure is transmitted backwards into the left atrium and pulmonary circulation.

Arterial vasodilatation reduces the external load.

Positive airway pressure can reduce LV transmural systolic load.

The ventricle may suddenly empty more effectively without any improvement in its intrinsic contractility.

The hill has become easier.

The rider has not become stronger.

Vasopressors act on both sides of the circulation

The same framework prevents another oversimplification: that vasopressors simply “increase afterload”.

A drug such as norepinephrine changes several parts of the coupled circulation at once.

Venoconstriction alters the systemic vascular pressure-volume state and can increase the delivery of filling to the heart.

Arteriolar constriction alters the arterial side and can increase resistive load.

The net effect depends on the operating state of the circulation and on ventricular reserve.

If systemic delivery is inadequate and the ventricle has plenty of power reserve, increasing vascular tone can increase filling and cardiac output despite an increase in arterial load.

If the ventricle is severely impaired and already struggling to eject, additional arterial loading may instead constrain stroke volume.

Neither “norepinephrine increases afterload” nor “norepinephrine improves venous return” is enough to describe the haemodynamics.

It changes the vascular system presented to both sides of the heart. The resulting pressure and flow depend on how the heart responds.

The right ventricle exposes the same problem

The same mistake is made when pulmonary vascular resistance is treated as synonymous with right ventricular afterload.

PVR is an important component of pulmonary vascular load. It is not the whole load.

The right ventricle ejects into a highly compliant, pulsatile pulmonary circulation. Its burden depends not only on pulmonary vascular resistance but also on pulmonary arterial compliance, characteristic impedance, pressure, wave reflection, lung volume and intrathoracic pressure.

The RV is also exquisitely sensitive to geometry.

As it dilates, wall stress rises. Septal position changes. Ventricular interaction becomes increasingly important. A load that a normal RV handles easily may become overwhelming once RV–pulmonary arterial coupling deteriorates.

PVR is no more synonymous with RV afterload than SVR is with LV afterload.

The relevant question is again whether the ventricle is matched to the vascular load it faces.

So what is afterload?

Several different levels have accumulated under one word.

At the arterial level, there is arterial load: the external mechanical burden presented by the vascular system during ejection. Pressure, resistance, compliance, impedance, wave reflection and timing all contribute to it.

At the myocardial level, afterload is most rigorously represented by the wall stress borne during ventricular ejection.

Arterial pressure influences that wall stress.

So does ventricular radius. So does wall thickness. So does surrounding pressure.

And the effect of that wall stress on stroke volume depends on the force- and power-generating capability of the ventricle.

These relationships can be arranged as a simple hierarchy:

That hierarchy also changes the bedside question.

Instead of asking whether “afterload is high”, ask what arterial load the ventricle is facing, how that load is being translated into myocardial stress, and whether the ventricle has enough reserve to meet it.

Look at the arterial pressure, but do not mistake it for the whole load.

Use SVR or PVR if they are helpful descriptions of the current pressure-flow relation, but do not turn them into physical objects resisting ventricular ejection.

Consider ventricular size and wall thickness.

Consider transmural pressure when intrathoracic or pericardial pressure is abnormal.

And watch what happens when the load changes.

• Does stroke volume increase?

• Does the ventricle empty further?

• Do filling pressures fall?

• Does congestion improve?

The response often tells us more about the mechanical problem than a calculated resistance ever could.

The heart does not simply push against pressure. It continually transfers energy into a vascular system that stores some of that energy, returns some of it and dissipates some of it. The arterial circulation sets the external challenge. Ventricular geometry determines how that challenge is translated into myocardial stress. Ventricular capability determines whether the load can be met.

The balloon explains why the pump must keep working.

The hill describes the external challenge.

The bicycle determines the mechanical advantage.

The rider determines whether the climb can be sustained.

Afterload is not the hill. It is the mechanical burden of climbing it.

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A brief extra post outside my usual schedule. I came across this paper while reviewing the recent haemodynamics literature and thought it was too relevant to leave until later. It connects directly with two themes I’ve been exploring here: dependent variables versus causes, and the arguments running through the Venous Return Wars series.

A recent sheep study by Robert Hahn asks what happens to mean systemic filling pressure during fluid loading, adrenergic stimulation and sepsis.

The results are intriguing. But the most interesting part of the paper may be what happens when a calculated haemodynamic variable starts to acquire the status of a measured physiological quantity.

Hahn analysed data from experiments in healthy and septic sheep receiving crystalloid fluid together with phenylephrine, norepinephrine, isoprenaline or dopamine. Mean systemic filling pressure was not measured by stopping flow and allowing systemic pressures to equilibrate. Instead, the study calculated the commonly used mean systemic filling pressure analogue, Pmsa, from cardiac output, mean arterial pressure and central venous pressure.

Read the full open-access paper

The distinction between Pms and Pmsa becomes central to interpreting the results.

What the study found

In the absence of vasoactive drugs, calculated Pmsa was almost identical at baseline in healthy and septic sheep: about 17 mmHg in both groups.

Fluid loading generally increased Pmsa. Phenylephrine combined with fluid produced the most sustained increase during sepsis. Across the entire experiment, however, septic animals had a 23% lower Pmsa than non-septic animals.

The septic circulation was markedly hyperdynamic:

• cardiac output: 12.9 vs 7.7 L/min

• MAP: 74 vs 109 mmHg

• CVP: 5.3 vs 7.6 mmHg

The calculated resistance to venous return was 63% lower in septic animals: 1.41 versus 2.47 mmHg·min/L.

Adrenergic stimulation also produced some striking results. In healthy animals, beta-adrenergic stimulation with isoprenaline and dopamine was associated with substantial increases in calculated Pmsa and cardiac output, while alpha-1 stimulation with phenylephrine increased MAP and CVP but did not produce the same rise in Pmsa.

Taken at face value, the paper appears to show that fluids, sepsis and adrenergic drugs alter mean systemic filling pressure, the pressure gradient for venous return and resistance to venous return.

There is another way to read it.

Pms was never measured

True mean systemic filling pressure is an equilibrium property.

Stop the heart, wait for flow to cease and allow pressures in the systemic circulation to redistribute. The resulting equilibrium pressure reflects the interaction between vascular volume and the pressure–volume characteristics of the systemic circulation.

That experiment was not performed here.

Instead, Pmsa was calculated from:

Pmsa = 0.96 × CVP + 0.04 × MAP + c × CO

where c is an anthropometric coefficient adjusted for the size of the sheep. The equation also assumes a fixed 24:1 venous-to-arterial compliance ratio.

Pmsa is therefore a mathematical transformation of three variables measured during active circulation:

CVP, MAP and cardiac output.

This creates an interesting problem when those same variables change dramatically.

Cardiac output is inside the pressure

Consider the healthy animals given isoprenaline.

Cardiac output increased substantially. Calculated Pmsa also rose.

It is tempting to interpret that result physiologically:

beta stimulation increased stressed volume, which increased mean systemic filling pressure, which increased the gradient for venous return and therefore increased cardiac output.

But cardiac output is already an input to the Pmsa equation.

Increase CO and, all else being equal, Pmsa must increase.

The calculated pressure cannot therefore provide independent evidence that an increase in an underlying equilibrium pressure caused the increase in flow.

The same issue becomes even clearer when the paper calculates the ‘pressure gradient for venous return’:

dVR = Pmsa − CVP

Substituting the equation for Pmsa gives:

dVR = 0.04 × (MAP − CVP) + c × CO

The calculated ‘pressure gradient for venous return’ therefore contains cardiac output within it.

The study then calculates resistance to venous return:

RVR = dVR / CO

Substitution gives:

RVR = 0.04 × (MAP − CVP) / CO + c

So cardiac output appears once in the calculated upstream pressure and then again in the denominator used to calculate resistance.

These are mathematically valid derived quantities. They are not independent measurements of an upstream pressure, a driving gradient and a vascular resistance.

But if resistance changes, hasn’t resistance changed?

If resistance is calculated as a pressure difference divided by flow, then a change in that ratio means the effective resistance of the circulation has changed.

But that does not necessarily mean that the vessels themselves have become intrinsically more or less resistive.

In the circulation, resistance is not produced by a fixed resistor. It emerges from the state of the vascular bed: vessel calibre, smooth-muscle tone, distending pressure, recruitment and derecruitment of vessels, blood viscosity and the tendency of vessels to narrow or close at low pressure.

Change the haemodynamic state and the relationship between pressure and flow can change even if there has been no active vasoconstriction or vasodilatation.

Consider a vascular bed in which increasing pressure distends previously narrow vessels. More flow can now pass for each additional increase in pressure. The calculated pressure-to-flow ratio falls. We can legitimately say that the effective vascular resistance is lower, but it would be wrong to assume that vascular tone must therefore have fallen. The vessels may simply be operating at a different pressure and calibre.

The opposite can occur as pressure falls. Vessels become narrower, some may begin to close, and progressively less of the vascular bed remains available to carry flow. The pressure-to-flow ratio rises even without any active increase in smooth-muscle tone.

Critical closing pressure is an extreme example of the same behaviour. Flow can cease while there is still a positive pressure within the vessel because the vessel is no longer patent. The pressure remaining at zero flow is therefore not being dissipated across a conventional resistance. It is maintaining, or failing to maintain, vessel patency.

So a calculated resistance tells us something real about the current relationship between pressure and flow. It does not, on its own, tell us which physical property of the vascular system produced that relationship.

The same caution applies to Hahn’s calculated resistance to venous return, with an additional problem: the upstream pressure used to calculate the gradient is itself partly calculated from cardiac output.

A fall in calculated RVR therefore means exactly what it says: the model-derived ratio of pressure gradient to flow has fallen.

It does not independently demonstrate that the physical venous circulation has undergone an equivalent fall in some fixed, intrinsic ‘resistance to venous return’.

What Pmsa can still tell us

A model can provide a useful descriptor without being a direct measurement. Pmsa has been compared with other estimates of mean systemic filling pressure and can track haemodynamic changes under some circumstances.

The problem begins when the calculated variable is treated as independent experimental evidence for the causal model from which it was derived.

If we want to know whether isoprenaline genuinely increases the equilibrium pressure of the systemic vascular system, the strongest experiment would be to measure that equilibrium pressure independently while manipulating beta-adrenergic tone.

If measured Pms rose alongside calculated Pmsa, we could then ask why: altered venous tone, redistribution of blood volume, changes in compliance, or some combination of these.

This study cannot make that separation.

Its assumptions may also be least secure under the very conditions being studied. Pmsa assumes a fixed 24:1 venous-to-arterial compliance ratio, while the paper itself acknowledges that sepsis alters arterial compliance and venous vascular behaviour.

The more interesting interpretation

The paper reports that sepsis reduced calculated Pmsa by 23%, the calculated gradient for venous return by 23%, and calculated resistance to venous return by 63%, while cardiac output increased by about 70%.

Those numbers describe how the Pmsa model represents the haemodynamic state.

They do not independently establish that:

• an equilibrium Pms fell by 23%;

• a physical pressure gradient driving venous return fell by 23%;

• or the intrinsic resistance of the venous circulation fell by 63%.

The distinction is especially important when a derived variable is then placed into a causal sequence.

CO rises → calculated Pmsa rises is partly built into the equation.

That result cannot then be turned around without further evidence into:

Pmsa rose → therefore CO rose.

The calculation and the causal explanation are different things.

A useful paper for exactly that reason

Hahn’s study is valuable. The experiments generate large, physiologically interesting changes in flow, arterial pressure and venous pressure under fluid loading, adrenergic stimulation and sepsis.

It also provides an unusually clear example of a broader problem in cardiovascular physiology.

We often begin with an equation that describes relationships between variables. We calculate a new quantity from those variables. We give that quantity the name of a physiological entity. Then, almost imperceptibly, the calculated quantity begins to appear in the causal explanation of the very measurements from which it was constructed.

Pmsa may be a useful estimate of systemic vascular state.

It is not the same thing as stopping the circulation and measuring mean systemic filling pressure.

And a calculated gradient is not evidence, by itself, that the gradient was the independent cause of the flow from which it was calculated.

The equation can describe the haemodynamic state without telling us which way causality runs.

Twenty-five years later, we still move between these meanings almost without noticing.

At the bedside, “preload” may mean CVP. It may mean ventricular filling pressure, chamber volume, venous return or circulating volume. Sometimes it simply means giving fluid.

These are not interchangeable measures of the same thing.

A pressure is not a volume. A chamber volume is not myocardial stretch. Venous return is not preload. And fluid administration is an intervention, not a myocardial state.

Still, we routinely join them together into one of the most familiar stories in cardiovascular physiology:

Give fluid → increase preload → move up the Frank–Starling curve → increase stroke volume → increase cardiac output.

Look at that sequence closely and some awkward questions appear.

• Why can a profoundly hypovolaemic heart contract so violently when its ventricle is almost empty?

• Why can an empty beating heart on bypass continue to contract at all if stretch is supposed to be what makes the heart pump harder?

• Why can increasing cardiac stimulation fail to increase flow in one circulation, while altering the vascular side of the same circuit suddenly produces a large rise in cardiac output without any further change in heart rate or contractility?

• Why can a ventricle remain small and hyperdynamic even when cardiac output is very high?

• And if Frank–Starling really is the mechanism that converts “more preload” into more cardiac output, what exactly is it doing when the heart already has more than enough contractile capability for the flow being delivered to it?

At the other end of the spectrum, very small increases in cardiac volume can eventually produce large increases in filling pressure. What determines when that happens? How much additional volume can the heart accommodate before pressure rises sharply? And what is Frank–Starling doing as inflow and outflow begin to mismatch?

Robert Anderson was thinking about some of these problems decades ago when he described the heart as having volume reserve and energy reserve. Those ideas never became part of the usual clinical vocabulary, but they provide a useful way of putting preload, Frank–Starling, filling pressure and cardiac output back into the same physiological picture.

First, though, we need to decide what preload actually is.

What is preload?

At its most fundamental, preload describes the mechanical state of the myocardium at the end of filling, immediately before contraction. At the level of the muscle fibre, that means things such as sarcomere length and wall stress. At the level of the whole ventricle, end-diastolic volume is a useful approximation.

Pressure is different.

Right and left atrial pressures are simply that - measurements of filling pressure, not direct measurements of ventricular volume or myocardial stretch. The relationship between pressure and volume depends on the properties of the ventricle, its geometry, its current filling state, the pericardium, intrathoracic pressure, ventricular interaction and other external constraints.

A stiff ventricle may reach a high filling pressure while containing relatively little blood. A compliant ventricle may contain considerably more blood at a lower pressure.

The same filling pressure can therefore be associated with very different end-diastolic volumes, and the same end-diastolic volume can occur at different filling pressures.

This is the first problem with using CVP or RAP as a synonym for preload. Filling pressure tells us something about the pressure associated with filling. It does not uniquely tell us how full the heart is, how stretched the myocardium is or how much additional blood it can accept.

Nor does low preload mean that the heart cannot contract forcefully.

An empty heart can still contract violently

Anyone who has looked after a profoundly hypovolaemic or vasodilated patient has seen the small hyperdynamic ventricle. The cavity may be tiny. Sympathetic activation is intense. The heart rate is high. The ventricular walls move vigorously.

There is very little blood in the chamber, but there is nothing weak about the contraction.

The same point becomes even clearer on cardiopulmonary bypass. Even with the heart emptied, the myocardium can continue to contract forcefully.

Myocardial contraction does not require the ventricle first to be filled or stretched. It is initiated by electrical activation, calcium cycling and cross-bridge interaction. Sympathetic stimulation can increase that activation further.

Stretch does something different. Through length-dependent activation, it modifies the mechanical response of myocardium that is already active.

This distinction is easily lost when Frank–Starling is described as though filling first “switches on” ventricular performance.

It does not.

Activation makes the myocardium contract. Preload modifies what that activated myocardium does.

That becomes important when the heart has far more mechanical capability than the amount of blood being delivered to it allows it to express.

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These statements are not necessarily wrong. The problem is that they are shorthand. They describe what we observe, but they do not tell us what has physically changed inside the circulation.

The same intervention can produce very different haemodynamic effects in different patients. A fluid bolus may increase stroke volume and cardiac output, or produce little additional flow while raising venous pressure and worsening congestion. A vasopressor can raise arterial pressure while cardiac output rises, barely changes or falls. Changes in flow, vascular tone and blood distribution can all contribute to the observed response.

Fluid and vasopressors can therefore produce similar bedside changes while acting on the circulation in fundamentally different ways.

So rather than starting with labels such as preload, afterload or venous return, it is more useful to ask a simpler question.

What has actually changed?

To answer that, we first need to think about something that is usually left implicit: where the blood is.

Vascular pressure-volume behaviour

A vascular pressure-volume relationship describes how much blood a vascular compartment contains at different transmural pressures. Some physiologists use the term capacitance to mean this but I will use pressure-volume behaviour here because capacitance is often used much less precisely. See here for more on this.

It is related to compliance, but it is not the same thing. Compliance is:

C = ΔV / ΔP

where C is compliance, V is volume and P is pressure.

It tells us how much the vessel’s volume changes when its pressure changes. In other words, compliance describes how easily the vessel expands as pressure rises. The full pressure-volume relationship tells us something broader: how much blood the vessel contains at different pressures. That depends on its size, intrinsic stiffness, smooth-muscle tone, the pressure around it and how full it already is.

Two vascular compartments can therefore have similar local compliance yet contain very different volumes at the same pressure. Conversely, a change in vascular tone can alter the amount of blood a vessel contains at a given pressure without being captured adequately by the simple statement that “compliance has fallen”.

This is particularly important in veins. Venous smooth muscle can change the size of the vessel without necessarily changing its compliance very much. So the important question is not simply whether the vein has become more or less compliant, but how its pressure-volume behaviour has changed.

With increased venous tone, less blood can reside in the venous circulation at the same pressure. Looked at the other way around, the same amount of venous blood would generate a higher pressure. The venous pressure-volume relationship has changed.

This gives us a useful distinction between blood volume and vascular pressure-volume behaviour. Blood volume tells us how much blood exists. The pressure-volume behaviour of the vascular compartments helps determine where that blood can reside.

Those two can change separately. We can add blood without changing vascular tone. We can change vascular tone without adding or removing a drop of blood. Either manoeuvre can alter central filling, but by a different physical mechanism.

This also helps explain why the usual image of venoconstriction “pushing blood back to the heart” is incomplete. When venous tone rises, total blood volume has not changed. At the previous pressure, the venous system would now contain less blood. The circulation therefore settles into a new state of volume distribution, pressure and flow.

The redistribution is transient. Once the new vascular configuration has been established, the veins do not continuously squeeze blood forwards. Venoconstriction changes the container; it does not turn the veins into an auxiliary pump.

The obvious next question is where the displaced blood goes.

If blood leaves the veins, where does it go?

Some of it is initially presented more centrally, but it does not follow that it simply accumulates in the heart. That would treat the heart as another passive compliant reservoir. It is not. The heart is a throughput organ, and its output changes when its volume changes.

If more blood reaches the heart and cardiac reserve remains available, end-diastolic ventricular volume rises slightly. This recruits the Frank-Starling matching mechanism: increased fibre length and length-dependent activation increase stroke volume, allowing cardiac throughput to match the greater systemic delivery. A relatively small increase in chamber volume can therefore support a much larger increase in the amount of blood passing through the heart each minute.

The heart is, in that sense, dynamically matched to the blood presented to it. If inflow transiently exceeds outflow, chamber volume rises slightly. The resulting increase in end-diastolic volume recruits a greater stroke volume, which reduces the mismatch between inflow and outflow. Over successive beats a new steady state develops in which mean inflow and mean outflow again match.

The heart does not therefore need to retain all the additional volume presented to it. With adequate reserve, most of the redistributed blood can pass through the cardiac chambers and be transferred to the arterial side of the circulation.

In the simplest model of isolated venoconstriction with preserved cardiac acceptance and otherwise unchanged arterial properties, the new steady state can contain less blood on the venous side, only a small additional volume within the heart, and more blood on the arterial side. Stroke volume and cardiac output rise. Because the arterial circulation is much less compliant than the venous circulation, even a relatively small increase in arterial blood content can accompany a substantial increase in arterial pressure.

That does not mean that arterial pressure rises first and then “drives” the greater cardiac output. The new arterial pressure, arterial volume and flow are coupled features of the new resolved state. Venous redistribution changes cardiac chamber volume, Frank-Starling recruitment changes stroke volume, and the arterial and resistive properties of the circulation determine how the resulting volume and flow are expressed.

More blood is indeed presented to the heart, but a functioning heart does not simply store it. It transfers it onwards. The increase in cardiac chamber volume may be small while the change in throughput is substantial.

That is what happens when the cardiopulmonary system can accept the additional delivery. When it cannot, the picture changes completely.

Delivery has to meet acceptance

The heart and pulmonary circulation have a finite ability to convert additional systemic delivery into additional flow. I use the term cardiopulmonary acceptance to describe that ability.

Acceptance is not a single property. It depends on ventricular filling characteristics, cardiac reserve, systolic capacity, RV and LV loading conditions, the pulmonary circulation, pericardial constraint, ventricular interaction, intrathoracic pressure, valve or pathway obstruction, rhythm and heart rate. All of these can limit the ability of additional systemic delivery to become additional throughput.

If acceptance reserve is good, more systemic delivery produces only a modest change in chamber volume and filling pressure while stroke volume and cardiac output rise. In practical terms, more delivery becomes predominantly more flow.

If acceptance is poor, the same increase in delivery produces little additional throughput. Blood accumulates upstream instead. Right atrial pressure and CVP rise, systemic venous pressure rises, and congestion worsens. More delivery now becomes predominantly more pressure.

The same change in venous tone can therefore produce very different haemodynamic responses in different patients. The systemic intervention may be identical, but the state of the cardiopulmonary circulation is not.

A vasodilated patient with preserved ventricular reserve may respond to venoconstriction with only a small increase in cardiac chamber volume, a substantial increase in stroke volume and little change in right atrial pressure. In a patient with severe ventricular dysfunction, the same increase in systemic delivery may produce little extra throughput. RAP and venous pressure rise, and congestion worsens.

In both cases venous tone has changed the systemic vascular state. What differs is what the circulation can do with the additional delivery.

This is also why right atrial pressure is better thought of as an operating point than a target. It reflects where systemic delivery meets cardiopulmonary acceptance. If delivery rises and flow increases easily, RAP may barely move. If delivery rises but flow cannot, RAP climbs.

Fluid changes something different

A fluid bolus can produce many of the same bedside observations, but its primary physical action is different. It adds volume to the circulation.

There is now more intravascular volume that has to be distributed somewhere within the circulation.

With venoconstriction, total blood volume remains unchanged while venous pressure-volume behaviour changes. With fluid, total blood volume rises.

The distinction can be stated simply:

Fluid changes the contents. Venous tone changes the container.

Both interventions can increase central filling. Both can increase cardiac output. Both can increase arterial pressure. Both can also increase CVP and worsen congestion. Similar haemodynamic phenotypes do not imply similar mechanisms.

What happens to the added fluid again depends on cardiopulmonary acceptance. If acceptance reserve exists, the additional volume increases end-diastolic chamber volume, recruits Frank-Starling and increases cardiac output. If acceptance is limited, relatively little additional flow appears and more of the added volume expresses itself as increased filling pressure and venous congestion.

A fluid bolus therefore has no predetermined haemodynamic destination. It does not simply “go to the heart”, and fluid responsiveness does not tell us that the patient was necessarily short of blood.

A patient with low blood volume may certainly respond to fluid. But a patient with normal blood volume and marked vasodilation may also respond. In that case, the problem is not necessarily too little blood, but where the existing blood resides within the circulation.

This gives a useful way of contrasting hypovolaemia and vasodilation. Hypovolaemia reduces the contents. Vasodilation changes the container so that more blood can reside on the venous side at a given pressure. Both can reduce the systemic delivery presented to the heart.

The reverse also explains why venoconstriction is useful during blood loss. It cannot replace lost blood, but it can shift the venous pressure-volume relationship so that the remaining blood is redistributed. A fall in total volume can therefore be partly compensated by a change in where that volume resides.

This is more physically satisfying than saying that sympathetic activation simply “mobilises venous blood”. It changes the pressure-volume behaviour of the vascular system, and the volume redistributes accordingly.

Stressed volume, elastic state and Pms

Traditional descriptions of venous physiology often divide blood into “stressed” and “unstressed” volume. That language remains useful as a reduced-order model. The problem comes when we start treating those terms as literal anatomical pools.

There is not one bucket of unstressed blood sitting beside another bucket of stressed blood, with venoconstriction transferring volume from one to the other. There is one continuous vascular system containing blood within vessels that have particular geometries, wall properties, external pressures and smooth-muscle states.

Change those properties and the pressure-volume relationship changes. Blood redistributes. The systemic elastic state changes.

So when we say that venoconstriction converts unstressed into stressed volume, the more physical translation is that venoconstriction shifts vascular pressure-volume behaviour and redistributes the existing blood volume.

This also helps place mean systemic filling pressure, Pms, in the right part of the causal hierarchy.

Pms is not the mechanism. It is not an energy source driving flow through a venous resistance. It is a reduced-order descriptor of the systemic vascular elastic state. At no-flow equilibrium, the pressure that remains in the systemic circulation depends on how much blood is present and on the pressure-volume behaviour of the systemic vessels.

The hierarchy is better thought of as:

blood volume + vascular pressure-volume behaviour → systemic elastic state → Pms

Pms can therefore tell us something about the state of systemic delivery, but it does not tell us what cardiac output must be. Flow only emerges when that delivery state interacts with the acceptance of the cardiopulmonary circulation.

That distinction becomes particularly important when we think about vasopressors.

Noradrenaline does more than constrict veins

Noradrenaline changes several cardiovascular properties at once.

On the venous side, increased smooth-muscle tone changes venous geometry and shifts the venous pressure-volume relationship. At the same pressure, the veins now contain less blood, so existing volume is redistributed and systemic delivery increases.

On the arterial side, increased tone raises arterial resistance and changes arterial pressure-volume behaviour. This contributes directly to the rise in arterial pressure and changes the conditions against which the ventricle ejects.

The heart is affected as well. Increased arterial tone can increase ventricular afterload, while beta-1 stimulation may increase contractile performance. A higher arterial pressure may improve coronary perfusion, particularly when baseline pressure is low, and reflex changes in heart rate and autonomic tone may modify the response further.

These effects can pull cardiac output in different directions. Venous redistribution can increase systemic delivery and recruit Frank-Starling. Increased arterial load can limit ejection. Direct cardiac effects may support contractile performance.

In a vasodilated patient with preserved ventricular reserve, increased venous tone increases systemic delivery. Cardiac chamber volume rises only slightly, Frank-Starling recruitment increases stroke volume, and flow increases as blood is transferred to the arterial side. This increase in flow is a major contributor to the rise in arterial pressure. More blood also resides in the arterial circulation, while the increase in arterial tone contributes further to the pressure response.

In a patient with severe ventricular dysfunction, the same increase in systemic delivery may produce little extra throughput. RAP and venous pressure rise, and congestion worsens. Increased arterial load may further limit ejection, so arterial pressure can rise while stroke volume and cardiac output change little or even fall.

The sequence

noradrenaline → venoconstriction → venous return ↑

therefore captures only part of what the drug does.

At steady state, venous return and cardiac output are necessarily the same flow. If cardiac output rises, venous return has risen too. The problem is treating venous return as an independent causal intermediate rather than the same resolved flow viewed from the venous side.

Noradrenaline changes venous tone and pressure-volume behaviour, arterial tone and resistance, ventricular loading and cardiac performance at the same time. The resulting changes in pressure, flow and volume distribution depend on the state of the circulation before the drug is given.

What should we ask instead?

The language of preload and venous return is not useless, but it often jumps too quickly from intervention to outcome.

A better approach is to ask what physical property we have actually changed.

If we give fluid, we have increased total intravascular volume.

If we increase venous tone, we have shifted the venous pressure-volume relationship and redistributed existing blood.

If we give noradrenaline, we have changed several vascular and cardiac properties simultaneously.

The next question is whether systemic delivery was limiting flow in the first place. If it was, increasing delivery may recruit more cardiac throughput. If it was not, the intervention may achieve little.

Then we have to ask whether the cardiopulmonary circulation can accept the additional delivery. Frank-Starling recruitment allows a small increase in chamber volume to match greater delivery with greater stroke volume and throughput; it does not require a sustained rise in filling pressure. Once cardiopulmonary acceptance is constrained, further delivery increasingly produces pressure rather than flow.

This is also why fluid responsiveness should never be equated with hypovolaemia. A positive response only tells us that, under the conditions of the test, increasing systemic delivery increased cardiac throughput. It does not tell us whether the original problem was low blood volume, altered vascular tone and volume distribution, or simply a circulation operating on a recruitable part of its filling-output relationship.

Likewise, venous congestion is not synonymous with excess total body volume. Congestion can arise because there is too much volume, because vascular pressure-volume behaviour has changed, because cardiopulmonary acceptance is poor, or because several of these coexist. At the haemodynamic level, excessive venous pressure tells us that the prevailing systemic delivery state cannot be processed without a pressure penalty.

The physiology can therefore be reduced to three statements.

1. Fluid changes how much blood there is.

2. Venous tone changes venous pressure-volume behaviour and therefore redistributes blood.

3. Cardiopulmonary acceptance determines whether greater delivery becomes predominantly greater flow or greater pressure.

The intervention changes the constraints.

The circulation resolves the result.

The story so far

Guyton created a powerful systems model. The cardiac and vascular function curves met at one operating point, where cardiac output and right atrial pressure, RAP, were determined together. Yet he also described mean systemic filling pressure, Pms, as driving venous return and RAP as opposing it. His diagram showed a coupled equilibrium but his words invited readers to see an upstream pressure pushing against a downstream back pressure.

Guyton stated that the Pms−RAP gradient caused flow.

Flow = (Pms − RAP) / RVR

Levy disagreed. His experimental pump imposed flow and RAP changed in response. He rearranged Guyton’s equation to demonstrate that flow through resistance caused the pressure gradient:

Pms − RAP = flow × RVR

Rothe gave Pms a physical basis: it is the zero-flow pressure signature of the elastic state created by blood volume and vascular accommodation. His overreach was to identify a local venous ‘pivot pressure’ during flow with Pms itself. Brengelmann then showed that no part of the flowing circulation has to remain at Pms. A local pressure that happens to share the same numerical value does not become the source of venous return.

Magder defended Guyton’s pressure gradient language with the image of a bathtub draining through a resistance. His model preserved a genuine clinical insight: the heart cannot substantially increase output when the vascular configuration does not permit it. But he continued to treat Pms−RAP as the pressure gradient driving venous return and his bathtub analogy implied the vascular reservoir provided its own energy. Brengelmann struck back. Gravity powers the bathtub; the heart supplies the work that sustains circulation.

Beard and Feigl broke down the maths to show that Guyton’s equation did not describe a pressure gradient driving flow across a venous resistance. Flow redistributes blood between compliant compartments until pressures, volumes and flow fit together; Pms remains the zero-flow intercept and RVR determines the line’s slope. They showed why the maths could work even though its causal story did not.

These debates raged for decades. Levy, Brengelmann, Beard and Feigl had shown why Guyton and Magder’s physics did not always add up. What they did not leave us was a simple, unified account of how the circulation works. To build one, we need to return to the fundamental physics and ask what happens when the heart puts energy into a closed circulation.

How a flowing state forms

Begin with a circulation in which flow has been stopped. When the heart starts pumping, it transfers blood from its inlet side into the arteries. As arterial volume rises, arterial pressure increases. As venous volume falls, venous pressure decreases. This is what compliance means in practice: moving blood between compartments changes the volume held in each, and that changes its pressure.

As blood flows through the vasculature, resistance dissipates mechanical energy. This produces the familiar fall in pressure from arteries to veins. The pressure in each region also depends on how much blood it contains, its pressure–volume relation and the pressure surrounding it. Vascular tone can change that relation.

At first, the amount entering a compartment may differ from the amount leaving it, so its volume changes. Eventually the average volume in each part of the circulation becomes stable. The same average flow then passes through every part of the loop. The heart continues to add energy and resistance continues to dissipate it, even though the distribution of blood is no longer changing.

Cardiac output and venous return are equal at this point because they are the same flow measured in different places. There is no second motor for venous return. There is one heart supplying energy to one closed circulation. The flow it can sustain depends on the heart, the vasculature and their condition at that moment.

The hierarchy beneath haemodynamics

This physical sequence can be organised into a hierarchy: an order of explanation running from the energy that sustains circulation to the pressures, volumes and flows that result. Physiology runs from the top down. At the bedside, we see the results and reason back towards their causes.

Start with energy. Continuous circulation requires a continuous supply of mechanical work. Its ultimate source is chemical free energy from ATP, which the heart converts into mechanical energy and transfers to the blood with each beat. Without that repeated input, sustained flow cannot continue.

The heart also belongs to the circuit it powers. Blood must enter its chambers, fill them, cross the valves and be ejected through the pulmonary and systemic circulations. The heart can therefore limit flow even though it supplies the energy. What happens depends both on the heart and on the vasculature through which that energy is transmitted.

The next level asks what kind of cardiovascular system receives that energy. A vessel or chamber has characteristic ways of responding when flow or contained volume changes. If the same flow is imposed through two vascular pathways, more energy is dissipated in the one with the higher resistance, and a larger pressure difference appears across it. If the same additional volume enters two vessels, pressure rises less in the more compliant one. The same filling volume produces a lower pressure in a compliant ventricle than in a stiff ventricle. A narrowed valve or pathway makes transferring blood more difficult.

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The equation that refused to die

After decades of debate, the classical explanation of venous return was in serious trouble.

Mean systemic filling pressure, Pms, had no demonstrated anatomical location during flow. Right atrial pressure was not an independent back pressure controlling how much blood returned to the heart. The venous system could store elastic energy and release some of it during transitions, but it could not supply the continuing energy required for steady flow.

The bathtub had provided a memorable picture containing some useful insights. Unfortunately, its gravitational energy source belonged to the bathtub rather than the circulation.

Yet the equation remained:

where Q is systemic blood flow, P_RA is right atrial pressure and R_VR is the so-called resistance to venous return.

Guyton’s equation fitted his experimental data. It still appeared in textbooks. Clinicians used it to think about fluids, vasopressors and cardiac function. Even critics who rejected its usual interpretation accepted that the mathematics was correct.

This presented a difficult question. How could an equation containing a pressure defined when flow had stopped describe a circulation in which blood was still moving?

In 2011, Daniel Beard and Eric Feigl returned to Guyton’s original model and worked through its mathematics. They were not proposing an entirely new explanation. Levy had already corrected the direction of dependency. Rothe had described the passive redistribution of blood between vascular compartments, and Brengelmann had used that redistribution to challenge the back-pressure interpretation of right atrial pressure. Beard and Feigl brought these earlier arguments together and showed how they were already encoded within Guyton’s own model. The equation was valid, but it did not mean what generations of physiologists had assumed.

To understand why, we need to rebuild the model from its component parts.

Starting the pump

Beard and Feigl depicted the systemic circulation using two elastic compartments.

One represented the arteries. The other represented the veins. Each could expand as it contained more blood. Physical resistances separated the compartments, and a pump transferred blood from the venous side back into the arterial side.

Begin with the pump stopped. There is no flow, so the pressure differences across the resistances disappear. Blood redistributes between the arterial and venous compartments until they reach a common equilibrium pressure. That pressure is mean systemic filling pressure.

Now start the pump. Blood is transferred into the arterial compartment faster than it initially leaves. Its contained volume rises, its walls distend and arterial pressure rises. Because total systemic blood volume is fixed, the additional arterial volume comes from the venous side. The veins contain less blood and their pressure falls. Right atrial pressure falls as part of this redistribution.

After a short transition, flow through every part of the system becomes equal again. Blood continues to circulate, but the average volume contained within each compartment is now stable. This is a flowing steady state. Steady does not mean stationary. It means that each compartment receives and loses blood at the same average rate, so its contained volume no longer changes.

The pump has established a new pressure and volume distribution. Arterial pressure lies above the original equilibrium pressure, while venous and right atrial pressures lie below it. The actual pressure difference across the systemic circulation is:

where (Pa) is mean arterial pressure and (Rt) is total systemic vascular resistance.

Both pressures exist while blood is flowing. They lie at identifiable points on the same vascular pathway. The heart supplies the energy, and blood loses mechanical energy as it passes through the intervening resistance.

No compartment maintained at Pms is required.

The zero-flow intercept

Guyton did not measure one flowing state. He imposed several different pump flows and allowed the circulation to settle after each adjustment.

Each flow produced a different distribution of the same total blood volume. At higher flows, more blood resided on the arterial side and less remained in the veins. Arterial pressure rose while right atrial pressure fell.

These were separate steady states of the same vascular system. Total blood volume, vascular resistance and vascular compliance were held constant. Only the imposed flow and the resulting pressure-volume distribution changed.

If flow is plotted on the vertical axis and right atrial pressure on the horizontal axis, the points form a descending line. When flow reaches zero, right atrial pressure reaches Pms. Mean systemic filling pressure is therefore the zero-flow intercept of the relationship.

An illustrative set of values makes this easier to see.

Imagine four steady states:

• At an imposed flow of 0 L/min, RAP is 10 mmHg.

• At 1 L/min, RAP is 8 mmHg.

• At 2 L/min, RAP is 6 mmHg.

• At 3 L/min, RAP is 4 mmHg.

These are four separate steady states. At each flowing state, blood is moving at the same rate through every part of the circuit and the average volume of each compartment is stable.

When flow is zero, right atrial pressure is 10 mmHg. This is the zero-flow intercept, Pms. For every 1 L/min increase in imposed flow, right atrial pressure falls by 2 mmHg as blood is redistributed from the venous compartment to the arterial compartment.

The equation can therefore be written:

At a flow of 2 L/min, right atrial pressure is 6 mmHg. The value 10 mmHg does not represent an upstream pressure acting on the right atrium during that flowing state. It tells us only that right atrial pressure now lies 4 mmHg below the value it would have at zero flow.

Replacing 10 with Pms, and replacing the slope of 2 with R_VR, gives:

Rearranging the same line produces the familiar Guyton equation:

The rearrangement changes how the equation looks. It does not turn the zero-flow intercept into a physical upstream pressure.

Another way to think about it:

For a moment, forget the name Pms and call this intercept P0. The line can then be written:

where K is the slope relating a change in flow to the resulting change in right atrial pressure.

Read this from left to right:

Current right atrial pressure equals its zero-flow value minus the change associated with flow-driven redistribution of blood volume.

There is no suggestion that blood flows from P0. It is simply the value that right atrial pressure would reach if flow returned to zero.

In Guyton’s model, P0 is mean systemic filling pressure. The equation therefore becomes:

A uniform pressure equal to Pms exists physically only after flow has stopped and the systemic pressures have equilibrated. Its value can nevertheless be defined while flow continues. The blood volume and vascular properties that would determine the equilibrium pressure still exist.

A visual analogy:

Imagine two tanks containing a fixed amount of water. A pump maintains one water level above the other. If the pump stops and the tanks are connected, the water will settle at a common level. That future equilibrium level is not currently present in either tank, but it can still be calculated from the total volume of water and the dimensions of the tanks.

Mean systemic filling pressure has the same status during flow. It is the pressure the vascular system would reach if the pump stopped without any change in blood volume or vascular tone. It appears in the equation as a reference value. It is not an upstream pressure within the flowing circulation.

A resistance containing compliance

The denominator presents another puzzle.

Guyton’s original term for it was not resistance to venous return. In 1955, he wrote that it depended on the resistance and capacitance of different parts of the peripheral circulation. He called it the impedance to venous return.

That name contained an important warning. Guyton was not describing a single venous resistor lying between Pms and the right atrium. He was combining the effects of the whole vascular network into one quantity.

He was using impedance broadly. In its modern technical sense, impedance describes the relationship between pulsatile pressure and flow and may include resistance, compliance, inertance, frequency and timing. Guyton’s equation related steady states, so this was not impedance in its complete modern sense. Even so, the term was more informative than the later resistance to venous return, because it signalled that resistance alone was not enough.

Beard and Feigl showed that the quantity later called resistance to venous return was:

Where R_V and R_A are venous and arterial resistance, while C_V and C_A are venous and arterial compliance.

At first sight, this looks like a resistance containing compliance. But physical resistance does not contain compliance. Resistance dissipates mechanical energy as blood flows through a vessel. Compliance describes how much volume a vascular compartment stores as its pressure changes.

The compliance ratio in the equation has no units, so R_VR retains the units of resistance. That does not make it a physical resistor. It is a calculated slope describing how much right atrial pressure changes when imposed flow changes between steady states.

That response depends on the whole vascular system.

When Guyton increased pump flow, blood was transferred into the arterial compartment. Passing the greater flow through the arterial resistance required a larger pressure difference. Because the arterial compartment was compliant, its higher pressure was accompanied by a greater contained volume. Total systemic blood volume was fixed, so this additional arterial volume came from the venous compartment. Venous volume and right atrial pressure therefore fell.

The pump transferred the blood. Resistance helped determine how much the arterial pressure and volume had to rise before the imposed flow could pass through the compartment.

Compliance was essential to this response. If the arterial compartment had been completely rigid, its pressure could have risen without storing additional blood. No extra volume would then have been removed from the veins for arterial storage. If there had been no arterial resistance, no additional pressure difference or arterial expansion would have been required for blood to leave the compartment.

Venous compliance also affected the result. A given loss of venous volume produces a smaller fall in right atrial pressure when the venous compartment is highly compliant than when it is stiff. The slope therefore depends on how compliance is distributed between the arterial and venous sides, as well as on their physical resistances.

This is why R_VR can differ greatly from the actual venous resistance. Under some plausible assumptions, arterial resistance makes the larger contribution.

The term resistance to venous return conceals all of this. It encourages the reader to imagine a resistor situated between Pms and the right atrium. Guyton’s earlier impedance to venous return was better, although still incomplete. The most exact description would be the slope coefficient of the venous return curve: a composite measure of how resistance, compliance and conservation of blood volume relate imposed flow to right atrial pressure.

Once the denominator is understood this way, the equation looks rather less like Ohm’s law across a physical venous pathway.

Turning the equation around

The familiar form of Guyton’s equation reads:

Its arrangement suggests a causal story. A higher upstream pressure, a lower downstream pressure and a resistance between them. The pressure difference appears to produce the flow.

But the same equation can be rearranged:

It now suggests that the pressure gradient is caused by flow through the vascular system.

In Guyton’s experiments, this was the direction of the intervention. The pump imposed flow. Blood moved from the venous compartment into the arterial compartment and right atrial pressure fell below its zero-flow value.

The equation itself contains no arrow of causation. Its arrangement merely encourages us to infer one.

Beard and Feigl made the problem even clearer by writing an equivalent equation from the arterial side of the same model:

where Ct is total systemic compliance.

This equation is just as valid as the venous return equation. Yet if its arrangement were interpreted causally, Pms would become a back pressure opposing arterial delivery. Lowering Pms would appear to increase cardiac output

No one interprets it this way.

Both equations emerge from the same physical resistances, compliant compartments and conservation of blood volume. One is written relative to right atrial pressure and the other relative to arterial pressure. Neither identifies a hidden motor.

If the position of a term within an equation established causality, Pms could be made to promote venous return in one arrangement and oppose arterial delivery in another.

The Guyton equation resembles Ohm’s law because it can be written in the form ΔP = QR. Its terms do not carry the same physical meaning.

The quantity Pms − Pra is not the pressure difference between two locations in the flowing circulation. It is the difference between the current right atrial pressure and its zero-flow reference value.

The quantity Rvr is not the physical resistance connecting those pressures. It is the slope relating flow to the redistribution of pressure and volume across the whole vascular system.

The equation is valid as vascular bookkeeping. It fails when treated as a mechanism of propulsion.

Should the curve still be taught?

Once the equation had been separated from its usual physical interpretation, a further question remained. Was the venous return curve still a useful way to teach the circulation?

The Journal of Physiology debated this directly in 2013. Philip Andrew accepted the central criticisms of Guyton’s interpretation. Flow had been imposed in the original experiments, right atrial pressure was not the independent variable, and Pms − Pra was not the physical pressure gradient for systemic blood flow. He nevertheless argued that Guyton’s graphical analysis remained valuable. The vascular curve could be reinterpreted as a venous pressure curve, showing how right atrial pressure changed when flow was imposed. Its intersection with the cardiac function curve would still identify the flow and right atrial pressure compatible with both the heart and the vascular system.

Beard and Feigl thought the confusion was too deeply embedded to be repaired by renaming the curve. Right atrial pressure remained on the conventional horizontal axis, encouraging it to be read as the independent controller of venous return. The zero-flow intercept still looked like an upstream pressure source, and the slope still looked like a physical resistance leading to the right atrium. They concluded that the curve generated more misunderstanding than insight and should no longer be taught.

I think they were right.

Guyton’s operating-point insight remains important. The heart and vascular system cannot independently select their own flow and pressure. The circulation must settle on a state compatible with cardiac acceptance and ejection, blood volume, vascular accommodation and resistance.

None of this requires a venous return curve.

If cardiovascular physiology were being constructed from first principles today, we would begin with the heart, the blood it moves and the vessels through which it moves. Resistance, compliance, blood volume and conservation would come before mean systemic filling pressure or right atrial pressure. Those pressures would be introduced later as measurements of the circulatory state, rather than as the two ends of a pressure gradient.

If Guyton’s venous return curve wasn’t so ubiquitous, no one attempting to teach the circulation from first principles would use it today.

Its remaining educational value is historical. It shows how a mathematically correct relationship can acquire an incorrect physical explanation, particularly when the arrangement of a graph encourages the reader to mistake a dependent variable for a cause.

Does the equation survive?

Guyton’s equation is not meaningless.

Within its assumptions, it describes a family of steady states in a simplified vascular system. It tells us how imposed flow redistributes a fixed blood volume and how right atrial pressure changes as part of that redistribution. It identifies the zero-flow equilibrium pressure associated with the vascular state. It shows which combinations of flow and right atrial pressure are compatible with the model.

It does not show where the energy for flow originates. It does not identify a pathway running from Pms to the right atrium. It does not make right atrial pressure an independent controller of venous return. Its denominator does not correspond to a resistance located in the veins. Nor can the equation decide what cardiac output the intact circulation will establish.

Beard and Feigl concluded: ‘Guyton’s idea, that venous return (equal to cardiac output) is determined by the pressure difference between mean systemic pressure and right atrial pressure (Pms – Pra) is physically and physiologically wrong’. I agree and would argue that its remaining educational value is historical: it shows how valid mathematics, arranged in a familiar form, can acquire a physical meaning that the underlying model does not support.

There remains one final problem. The Guytonian framework remains clinically persuasive because many of its predictions appear to work. Magder gave that framework its most intuitive modern form through the bathtub analogy. Guyton’s intersecting curves had already shown that the heart and vasculature constrain each other. Magder made the division more explicit: once the heart can accept and replace the blood presented to it, further cardiac power achieves little unless vascular delivery also changes.

This seems to fit clinical experience. Fluids and vasopressors can increase cardiac output. Poor cardiac function produces congestion. Once cardiac function is no longer limiting, additional cardiac power often achieves little.

If Pms does not push blood towards the heart, why does increasing the vascular elastic state so often increase cardiac output? If RAP does not oppose venous return, why does a high RAP so often accompany circulatory failure? Why can the same intervention increase flow in one patient while producing only higher pressures in another?

The critics have shown why the old physical explanation cannot be right. They have been less successful at replacing the clinically useful model built upon it.

Yet the pieces of that replacement have been present throughout this series: Guyton’s coupled circulation, Levy’s dependent variables, Rothe’s elastic state, Brengelmann’s energy analysis, Magder’s clinical constraints and Beard’s surviving equation.

In the final episode, I will bring those pieces together. The aim is no longer to decide whether the heart or the vasculature controls cardiac output. It is to explain how cardiac energy, vascular delivery and cardiac acceptance resolve into one flowing state, why familiar treatments work, and why they sometimes fail.

Dismantling the old model was only the beginning.

The war has cleared the ground. It is time to rebuild the circulation.

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Next: Episode VIII — The Dependent Variable

References

Andrew P. CrossTalk proposal: Guyton’s venous return curves should be taught. J Physiol. 2013;591:5791–5793.

Beard DA, Feigl EO. Understanding Guyton’s venous return curves. Am J Physiol Heart Circ Physiol. 2011;301–H633.

Beard DA, Feigl EO. CrossTalk opposing view: Guyton’s venous return curves should not be taught. J Physiol. 2013;591:5795–5797.

Guyton AC. Determination of cardiac output by equating venous return curves with cardiac response curves. Physiol Rev. 1955;35:123–129.

Levy MN. The cardiac and vascular factors that determine systemic blood flow. Circ Res. 1979;44:739–747.

The appeal of Magder’s bathtub was easy to understand.

The water stored in the tub represented blood held within the compliant venous system. Its depth represented mean systemic filling pressure, Pms. The drain led towards the right atrium, with the difference between Pms and right atrial pressure providing the pressure head for venous return. The heart appeared as a tap, replacing the water that escaped and preserving the state of the reservoir.

Most of the ingredients were real. The veins contain most of the circulating blood. Their walls are elastic. Blood volume distends them and creates stored energy. Open a large vein to atmospheric pressure and blood continues to leave after the heart has stopped, propelled for a time by vascular recoil.

Magder accepted that this flow could not continue indefinitely. As the vascular system emptied, its pressure would fall and the stored energy would be exhausted. The heart was therefore needed to return blood to the reservoir and restore the energy released during drainage.

This gave the heart a role, but a limited one. The venous reservoir determined how much blood could drain. The heart permitted that flow by keeping right atrial pressure low and restored the volume that had been lost.

Brengelmann saw a problem hidden inside this division of labour.

If the heart continuously restored everything leaving the reservoir, the reservoir would remain at the same average volume and pressure. Its elastic walls would remain equally distended.

How, then, could they continuously release energy?

No recoil without emptying

An elastic structure releases stored energy by changing shape.

A stretched spring releases energy as it shortens. An inflated balloon releases energy as it becomes smaller. A distended vein can release energy as its contained volume falls and its wall recoils.

This is exactly what happens when the circulation stops and a vein is opened to atmospheric pressure. The vascular system has been connected to a new, lower-pressure boundary. Blood leaves. Vascular volume falls. The walls recoil and the elastic energy stored within them is released.

The flow is real, but it is transient. As the vessels empty, their pressure falls. Eventually the available energy has been spent and flow stops.

Steady venous return is different. Blood may be flowing rapidly through a vascular compartment while the amount of blood contained within it remains unchanged. If five litres enter each minute and five litres leave, the compartment passes a flow of five litres per minute without losing any volume.

Throughput is not the same as emptying.

In the steady state, average inflow and outflow are equal. The compartment does not become progressively smaller and its walls do not progressively recoil. Blood passes through it, but its stored elastic energy remains stored.

This was the meaning of Brengelmann’s terse statement that there could be “no energy release” without a decrease in volume. The mathematics expresses mechanical work as pressure acting through a change in volume. If the volume does not fall, the elastic wall has not performed net work on the blood.

Magder’s reservoir was continuously drained, but it was also continuously refilled. Something therefore had to return each lost volume element and restore its energy. In the bathtub this work was hidden inside the tap. In the circulation it was performed principally by the heart.

The reservoir still mattered. Its volume and elastic properties determined its pressure state and influenced how the circulation responded to a change in cardiac activity. What it could not do was remain unchanged while also acting as a continuous source of mechanical work.

The physics belong to the bathtub

The bathtub analogy feels natural because its source of energy is visible.

Water in the tub sits above the drain. Gravity acts on it continuously. As water descends, gravitational potential energy is converted into movement and then dissipated by resistance in the outlet.

If the water level is to remain constant, the tap must replace what leaves. But that replacement water must first be returned to the height of the tub. Somewhere outside the picture, a pump, elevated reservoir or pressurised water supply has performed the work needed to restore its gravitational potential energy.

The tap is therefore doing more than keeping an inventory. It is restoring the energy that gravity releases through the drain.

In reality there is no equivalent independent gravitational reservoir in the circulation. Blood does not travel around the systemic circulation because its venous end sits physically above the heart. The heart supplies mechanical energy to a closed vascular system. Its work, together with vascular resistance, compliance and blood volume, establishes the flowing distribution of pressure and volume.

The bathtub gives the venous reservoir a separate gravitational energy source that the circulation does not possess.

Its geometry creates another problem. In the illustrations of high venous return, the heart is placed below the outlet of the tub. Lowering it would increase the hydrostatic pressure measured at heart level. Yet the Guyton curve associates higher flow with a lower right atrial pressure. Correcting all pressures to the same vertical reference would expose the mismatch.

The arterial side has largely disappeared as well. The tap pours cardiac output back into the tub as though it arrives at atmospheric pressure. In the circulation, the left ventricle ejects into a high-pressure arterial system. Blood then passes through a succession of real vascular segments, each with its own resistance, compliance, pressure and contained volume.

The analogy makes another point about the heart.

In Magder’s model, the heart lowers right atrial pressure and replaces the blood that drains from the reservoir. If it is too weak to do this, it limits flow. Once it is no longer the limiting element, however, greater cardiac power cannot make the reservoir drain faster. Flow is then set by the stressed volume and drainage characteristics of the vascular system. The heart permits and restores that flow, but does not determine its magnitude.

Magder supports this by pointing out that the heart contains very little blood compared with the veins and venules. It cannot add much volume to the venous reservoir from its own contents or rapidly increase the reservoir’s elastic pressure.

This captures an important truth. Steady flow depends on what the systemic circulation can deliver and what the heart can accept and eject. If cardiac function is limiting, improving it can increase flow. Once the heart can readily handle the blood presented to it, additional power has little effect unless the vascular state also changes.

The heart’s small contained volume is relevant, but it is not the whole explanation. A pump influences a circuit through the volume it moves over time as well as the volume held inside it at one moment.

The heart continually transfers blood between vascular compartments. This helps establish how much blood resides on the arterial and venous sides, and therefore the pressures observed during flow. It can alter this distribution even when total blood volume, vascular accommodation and true zero-flow Pms remain unchanged.

The bathtub therefore captures a real vascular constraint. Once the heart is no longer limiting, the vascular state restricts how much flow can be sustained. What the analogy does not establish is that the venous reservoir supplies the continuous energy for that flow. The heart provides the energy, while the vascular state constrains the flow that results.

An aggregate without an address

Defenders of the Guytonian account do not all require a single anatomical reservoir maintained at Pms.

Berger, Moller and Takala describe Pms as the aggregate pressure of the entire systemic vasculature. It represents systemic stressed volume acting within the combined compliance of many vascular beds. No single compartment needs to sit at precisely that pressure during flow.

This is a more sophisticated account than the bathtub. It preserves the importance of the vascular elastic state without claiming that Pms must reside in one particular vein.

It also creates a problem for the original gradient.

A spatial pressure difference exists between two locations at the same time. Blood flowing through a vein encounters one local pressure at its upstream end and another at its downstream end. Those pressures and the flow develop together within the operating circulation.

Pms has a different origin. It is the common pressure approached when systemic flow stops and blood redistributes between vascular compartments. If it is instead treated as a weighted description of the whole system during flow, it still lacks a single upstream location from which blood travels to the right atrium.

Magder’s more recent account takes a different route. He distinguishes mean circulatory filling pressure, the equilibrium pressure of the whole circulation at zero flow, from mean systemic filling pressure during flow. Because the systemic veins are so compliant, their pressure changes relatively little, allowing him to place the flowing value of mean systemic filling pressure within that region and retain it as the upstream pressure for venous return.

A local systemic venous pressure is physically real. It can form one end of a real pressure difference across the vascular segments downstream from it. But it is also determined by the volume residing in that region, its compliance, the surrounding pressure and the flow through the connected circulation. It cannot simply inherit the identity of the equilibrium pressure measured after flow stops.

Numerical similarity is not physical identity.

The vascular elastic state is real. Local venous pressures are real. Neither fact demonstrates that the zero-flow pressure Pms persists at an anatomical location during flow and continuously supplies the energy for venous return.

The curve without a reservoir

Brengelmann’s 2019 analysis offered another explanation for the famous venous return curve.

Begin with the circulation at zero flow. Pressures have equilibrated at Pms. Blood is distributed among the arterial and venous compartments according to their volume-containing properties.

Now start the pump.

The pump initially transfers blood into the arterial compartments faster than it leaves them. Their contained volume rises, their walls distend and arterial pressure rises above Pms. Because total blood volume is fixed, this additional arterial volume comes from the venous side. The venous compartments contain less blood and their pressures fall below Pms.

Right atrial pressure falls as part of this redistribution.

Once the redistribution is complete, inflow and outflow again match in every compartment. Flow continues through the new pressure profile while the compartments remain at their new volumes.

Increase pump flow again and the redistribution becomes larger. More volume resides on the arterial side and less on the venous side. Arterial pressure rises further while right atrial pressure falls further. Repeating the process produces the familiar inverse relationship between flow and right atrial pressure.

No compartment has remained at Pms. No reservoir at Pms has drained through a single venous resistance. The curve has emerged from flow, vascular resistance, compliance and conservation of blood volume.

Real pressure differences exist throughout this model. Each connects two neighbouring locations in the flowing circulation. The problem lies specifically with treating Pms and RAP as if they were the two ends of one physical pathway.

Guyton’s original mathematics had described the redistribution correctly. The result could be written so that flow was proportional to Pms minus right atrial pressure. But Brengelmann showed that the same volume accounting could instead be written from the arterial end, making flow proportional to arterial pressure minus Pms.

If the position of a term within an equation established causality, this second arrangement would invite an equally strange conclusion: Pms was a back pressure opposing arterial delivery, and cardiac output could be increased by lowering it.

Neither reading follows from the mathematics.

The denominator traditionally called resistance to venous return creates a similar illusion. It sounds like the resistance of the veins between an upstream Pms reservoir and the right atrium. In Guyton’s model, however, its value depended on arterial as well as venous resistance and on the distribution of compliance across the network. Guyton initially called it an impedance to venous return, although it was not impedance in the usual frequency-dependent sense.

It was an effective parameter describing the behaviour of the whole vascular system, not an anatomical resistor placed downstream from Pms.

The equation was performing vascular bookkeeping. It related flow to the pressure and volume distribution that accompanied it. It did not identify a hidden motor at Pms or a discrete resistance lying between Pms and the right atrium.

What PEEP reveals

Positive end-expiratory pressure provides a useful test of the supposed driving gradient. It raises pressure around the heart and thoracic veins, alters cardiac filling and redistributes blood between the pulmonary, arterial and systemic venous compartments.

Jellinek and colleagues studied patients undergoing testing of implanted defibrillators. Before cardiac arrest, average right atrial pressure was 7.3 mmHg. The systemic venous pressure measured after flow stopped was 10.2 mmHg, giving a Pms–RAP difference of approximately 3 mmHg.

When PEEP was increased from 0 to 15 cmH₂O, right atrial pressure rose to 10.0 mmHg and Pms rose to 12.7 mmHg. The difference between them was almost unchanged.

Stroke volume nevertheless fell by approximately 23%.

The equation could accommodate this result by assigning the fall in flow to an increase in resistance to venous return. But that calculation cannot explain what caused the new state. The central observation remains: substantially different flows were associated with the same Pms–RAP difference. The gradient did not independently determine flow.

Berger and colleagues later demonstrated similar complexity in pigs. Raising PEEP increased both right atrial pressure and directly measured stop-flow Pms. The increase in PEEP was much smaller than in Jellinek’s study, and blood flow fell only slightly. During inspiratory holds, flow from different venous regions fell and recovered at different rates. The systemic veins behaved as a network of compliant compartments that exchanged volume and responded differently, rather than as a single reservoir draining uniformly towards the heart.

PEEP changed cardiac function and the distribution of blood around the circulation. More blood could reside in the systemic veins while less remained in the pulmonary and arterial compartments. The resulting volume changes were expressed as changes in pressure. Pms and right atrial pressure could therefore rise together even while cardiac output fell.

These experiments did not invalidate the mathematical relationship between Pms, right atrial pressure, flow and resistance. They showed its proper status. The gradient described the pressure distribution accompanying the new circulatory state; it did not determine the flow on its own.

Can pulsatility save the reservoir?

The real circulation is not perfectly steady.

Within each heartbeat, inflow and outflow from individual vascular compartments do not match at every instant. Their volumes fluctuate. Elastic walls expand, recoil and exchange energy with the blood.

Berger and colleagues argued that this pulsatility matters. Stressed volume does not disappear when the circulation is flowing. Small changes in venous volume allow elastic energy to be released during emptying, while cardiac inflow restores the volume. These emptying characteristics help constrain the maximum flow that the vascular system can present to the heart.

Brengelmann accepted the volume fluctuations. His objection concerned their net contribution over time.

If a compartment empties slightly during one part of the cardiac cycle and refills during another, it releases energy and then stores it again. Once the compartment returns to the same average volume, the energy released during emptying has been replaced during filling. Averaged over repeated cycles, its net contribution is zero.

A spring can release energy on every cycle only because something compresses it again.

In the steady state, the heart supplies the principal continuous energy input. During exercise, repeated skeletal-muscle contractions can add mechanical energy by compressing veins. The respiratory pump can contribute as well. Because these actions recur while flow continues, they can help sustain cardiac output.

Venous smooth muscle can constrict, but it does not beat. A change in venous tone can reduce vascular accommodation, displace a finite volume and perform work while the circulation moves towards a new state. Once the constriction is maintained and redistribution is complete, it supplies no further hydraulic power.

Repeated skeletal-muscle and respiratory pumping can therefore add energy while flow continues. Venoconstriction can add a finite amount during a transition. Neither turns a venous compartment maintained at a fixed average volume and pressure into a continuous source of energy.

Elastic recoil helps bridge transient imbalances. It buffers pulsatile inflow and outflow. Changes in vascular tone can redistribute volume and alter the operating state. None of this allows a reservoir maintained at Pms to release energy continuously without changing.

What survives the bathtub

Brengelmann’s argument does not make the venous system irrelevant.

Blood volume and vascular accommodation remain central to cardiovascular function. Together they determine the elastic state from which Pms arises. Changes in venous tone can alter that state and redistribute blood towards or away from the heart. The resulting pressures strongly influence cardiac filling, congestion and the operating point of the circulation.

Magder is also right that greater cardiac power does not guarantee greater cardiac output. The heart operates within the constraints imposed by blood volume, vascular accommodation, resistance and its own filling characteristics. A more powerful pump cannot eject blood that the connected system does not allow it to accept.

But constraint is not the same as propulsion.

The vascular system determines the conditions within which the heart works. It stores volume, redistributes it and buffers differences between inflow and outflow. Its elastic pressure state tells us something important about those conditions.

The continuous net energy needed to move blood through systemic resistance still has to be supplied. In the simplified circulation considered here, that source is the heart.

Pms therefore survives as a useful expression of the vascular elastic state. RAP survives as an important part of the resolved cardiovascular condition. Real local pressure differences survive throughout the flowing circulation.

What does not survive is the picture of a reservoir at Pms continuously supplying energy while right atrial pressure independently holds back the resulting flow.

Brengelmann had removed the proposed mechanism without removing the relationship. The bathtub could not survive the energy accounting.

The equation did.

Continued in Episode VII: The Last Equation.

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Original papers

Magder S. Point: The classical Guyton view that mean systemic pressure, right atrial pressure, and venous resistance govern venous return is correct. Journal of Applied Physiology. 2006;101(5):1523–1525. https://doi.org/10.1152/japplphysiol.00698.2006

Brengelmann GL. Counterpoint: The classical Guyton view that mean systemic pressure, right atrial pressure, and venous resistance govern venous return is not correct. Journal of Applied Physiology. 2006;101(5):1525–1526. https://doi.org/10.1152/japplphysiol.00698a.2006

Magder S. Volume and its relationship to cardiac output and venous return. Critical Care. 2016;20:271. https://doi.org/10.1186/s13054-016-1438-7

Brengelmann GL. Letter to the editor: Why persist in the fallacy that mean systemic pressure drives venous return? American Journal of Physiology–Heart and Circulatory Physiology. 2016;311–H1335. https://doi.org/10.1152/ajpheart.00536.2016

Berger D, Moller PW, Takala J. Reply to “Letter to the editor: Why persist in the fallacy that mean systemic pressure drives venous return?” American Journal of Physiology–Heart and Circulatory Physiology. 2016;311–H1337. https://doi.org/10.1152/ajpheart.00622.2016

Brengelmann GL. Venous return and the physical connection between distribution of segmental pressures and volumes. American Journal of Physiology–Heart and Circulatory Physiology. 2019;317(5)–H953. https://doi.org/10.1152/ajpheart.00381.2019

Jellinek H, Krenn H, Oczenski W, Veit F, Schwarz S, Fitzgerald RD. Influence of positive airway pressure on the pressure gradient for venous return in humans. Journal of Applied Physiology. 2000;88(3):926–932. https://doi.org/10.1152/jappl.2000.88.3.926

Berger D, Moller PW, Weber A, Bloch A, Bloechlinger S, Haenggi M, Sondergaard S, Jakob SM, Magder S, Takala J. Effect of PEEP, blood volume, and inspiratory hold maneuvers on venous return. American Journal of Physiology–Heart and Circulatory Physiology. 2016;311–H806. https://doi.org/10.1152/ajpheart.00931.2015

Brengelmann GL. Reply to “Letter to the editor: The venous circulation actively alters flow: a brief evolutionary perspective.” American Journal of Physiology–Heart and Circulatory Physiology. 2021;320(1)–H473. https://doi.org/10.1152/ajpheart.00910.2020

Magder S, Slobod D, Vieillard-Baron A. Physiological and clinical significance of mean circulatory and mean systemic filling pressure. Annals of Intensive Care. 2025;15:187. https://doi.org/10.1186/s13613-025-01595-0

What survived the phantom

Episode IV dismantled a familiar explanation of venous return. Brengelmann showed that Guyton’s curve did not require a pressure source hidden within the veins. Mean systemic pressure (Pms) remained real, but as the equilibrium pressure of the systemic circulation after flow had stopped, not an anatomical pressure shown to drive blood towards the heart.

That left an important question unanswered. If Pms was not pushing blood home from somewhere inside the circulation, how did blood volume and vascular tone influence cardiac output?

Their influence was difficult to deny. Haemorrhage can reduce cardiac output even when the heart itself is healthy. Intravenous fluid or venoconstriction can alter cardiac filling and raise output without any primary change in contractility. Removing Pms as a driving pressure did not make the vascular system irrelevant. It created a need for a better explanation of what the vascular system contributed.

Brengelmann’s analysis could not provide that explanation on its own. The experiments he criticised used pumps that imposed flow and then measured how pressure and volume responded. They showed how a vascular system behaved when its flow had already been chosen, but not how a real heart and circulation arrived at their shared operating point.

Sheldon Magder offered an answer. He retained Guyton’s distinction between cardiac and return functions, but gave the elastic vascular reservoir a central role. In his account, the filled vasculature stored energy and recoiled, returning blood towards the heart. The heart lowered the pressure at its inlet to permit that return, then restored the blood to the arterial circulation.

The idea would find its most memorable expression in a bathtub.

The phantom may have gone, but the reservoir was about to take its place.

The elastic reservoir

At the centre of Magder’s argument lies a fact that no critique of Guyton can deny: filling an elastic vascular system changes its mechanical state. Blood volume distends vessel walls and produces pressure even after the heart has stopped. In the stressed and unstressed volume model discussed in Episode III, the stressed-volume term describes the part of the contained volume associated with that distension. Pms expresses the relationship between this volume and the combined compliance of the systemic vasculature.

The veins dominate that relationship. They contain most of the systemic blood volume because the venous system has a large resting vascular volume and is highly compliant. They can therefore accommodate substantial changes in volume over a relatively small pressure range. Venous smooth-muscle contraction changes the space available to contain blood; intravenous fluid changes the volume that must be contained. Both interventions can alter the systemic elastic state and therefore Pms.

Magder illustrates the energy stored in this state with a simple demonstration. Stop the circulation and open a large vein to atmospheric pressure. The distended vascular walls now have a route through which to release their stored elastic energy. As they recoil, blood flows through the opening even though the heart is no longer contracting. The vessels become progressively less distended, and flow slows until the pressure difference from atmosphere disappears or the vasculature approaches its resting configuration.

The experiment proves that the filled vasculature stores potential energy. It also reveals the limit of that energy. The discharge is finite. As blood leaves the system, vascular volume falls and the elastic pressure falls with it.

Magder carries this picture into the intact circulation. The veins and venules remain distended, so their elastic recoil becomes the upstream source for venous return. Right atrial pressure provides the downstream boundary. The heart keeps that boundary low enough to permit drainage, receives the returning blood and restores it to the arterial side.

It is an appealing division of labour: the reservoir returns the blood; the heart puts it back.

The permissive heart and the bathtub

Magder describes the heart as both permissive and restorative. By emptying its chambers, it lowers the pressure at its inlet and allows blood to enter from the systemic veins. It then returns that volume to the arterial circulation. Once a steady state is established, the amount restored each minute matches the amount returning.

The word permissive can sound as though the heart has been demoted to a passive bystander. That is not Magder’s claim. A failing heart can obstruct the return function and limit the flow achieved by the whole circulation. When filling or ejection is impaired, right atrial pressure rises and the operating cardiac output falls. Greater pump power, however, cannot sustain an output that the vascular system cannot accommodate. Once the heart has lowered its inlet pressure sufficiently, Magder places the remaining limit in the vascular reservoir and the path through which it drains.

He uses a bathtub analogy to explain this division of labour between the heart and vasculature.

The model feels almost self-evident once drawn. Magder strengthens the comparison by pointing out that the heart contains little blood compared with the veins and venules. It cannot suddenly add enough volume to the venous reservoir to increase its elastic recoil pressure substantially. Like a tap filling a large bathtub, it can alter drainage only if inflow exceeds outflow for long enough to raise the water level. A more forceful jet therefore has no immediate effect on the drain; its effect appears only after sufficient volume has accumulated in the reservoir.

If inflow briefly exceeds outflow, the water level rises until drainage catches up. If the drain removes water faster than the tap supplies it, the level falls. At steady state the two flows are equal, yet the tub’s water level, outlet height and drainage resistance appear to decide what that common flow must be.

Several features of the circulation fit neatly into the picture. The amount of fluid matters, as does the size and elasticity of the space containing it. Venoconstriction resembles making the reservoir smaller: the same blood volume must then be accommodated within less space, increasing the vascular system’s elastic pressure. The heart must replace whatever leaves the venous side, and inflow and outflow must match when the circulation reaches equilibrium.

The analogy also offers a clear account of the plateau at the upper end of a venous return curve. When right atrial pressure falls sufficiently, the great veins begin to collapse where their internal pressure approaches the surrounding pressure. Further reductions in right atrial pressure then fail to increase flow. This is a genuine flow limit, although it explains the plateau at very low right atrial pressure rather than the ordinary sloping part of the curve.

I found the bathtub persuasive on first reading. Every part appears to have an obvious cardiovascular counterpart, and the whole model can be understood in seconds. That is a formidable advantage in physiology. It is also why the analogy deserves more than a superficial reading. But while a good picture can clarify a mechanism, it can also lend the mechanism physics that belong to the picture alone.

What Magder gets right

A fair reading of Magder should begin with the strength of his case. The vascular system is not scenery through which the heart happens to pump. It places real limits on the state that the heart can establish.

Increasing blood volume can strengthen the systemic elastic state by increasing the volume associated with vascular distension. Venoconstriction can produce a similar change without adding fluid because a smaller venous space must accommodate the same total blood volume. Both interventions can increase Pms and present a different filling state to the heart.

Neither intervention guarantees a higher cardiac output. A heart already operating near its filling or pumping limit may be unable to accept the altered vascular state. Venous and atrial pressures then rise, while flow changes little. The converse also applies. Increasing contractility may achieve little when the vascular system cannot support a different steady distribution of blood.

This preserves the deepest part of Guyton’s insight. Cardiac output belongs to neither the heart nor the vasculature alone. It emerges from their interaction. Magder is right to resist a pendulum swing from “the veins drive flow” to “only the heart matters”. The circulation contains one energy source, but many constraints.

Right atrial pressure therefore retains clinical meaning. A high value may accompany a heart that cannot transmit the volume presented to it, and it may alert us to an important cardiac constraint. Its value describes part of the resolved circulatory state. Whether it independently opposes the return of blood is a different question.

One reservoir becomes many

The single bathtub is useful because it is simple. The real circulation refuses to remain that simple.

Magder turns to the Krogh model, in which two vascular beds are arranged in parallel. The splanchnic circulation behaves as a large, highly compliant region that contains substantial volume and adjusts relatively slowly. Much of the muscular circulation has lower compliance and a shorter drainage time. The speed with which either region adjusts depends on both its compliance and the resistance through which it empties.

The comparison is no longer one bathtub. Imagine instead a deep, wide bath beside a smaller sink, both supplied in parallel. Direct more of the total flow through the bath and more of the fixed blood volume must reside there. Direct flow through the smaller, faster compartment and the same total volume can support a different overall throughput.

Regional arterial resistance helps determine how flow is divided between these beds. A greater share passing through the slow, compliant splanchnic region commits more blood to that region and reduces the total flow compatible with the fixed vascular volume. Redirecting blood towards a less compliant, faster region can permit a higher total flow without changing total blood volume or the zero-flow Pms.

The same Pms can therefore coexist with different steady flows. That observation does not invalidate the venous return equation, but it changes the meaning of its denominator. “Resistance to venous return” cannot be only the physical resistance of veins running from a Pms reservoir to the right atrium. It is an effective property of the entire network, shaped by regional arterial and venous resistances, compliance, flow distribution and the volume shifts that accompany them.

Magder’s more developed model is richer than the bathtub that introduced it. The elastic state constrains the range of possible flows; regional mechanics influence which flow the circulation reaches. Pms sets an important boundary without supplying a unique answer.

The contradiction and the hidden energy source

A tension runs through Magder’s writing that is easy to miss. His treatment of pressure shifts between a systems description, in which pressure and flow are determined together, and a causal description, in which particular pressures govern the resulting flow. This shift appears at several points in his argument.

The first involves right atrial pressure. Magder recognises that cardiac output, venous return and right atrial pressure are determined together through the interaction between cardiac and vascular function. Yet he also describes right atrial pressure as an independent back pressure opposing venous return. These are different accounts of the same variable. In one, right atrial pressure is part of the state reached by the circulation. In the other, it acts upon the circulation to determine that state.

Guyton had lived with the same contradiction. His 1955 paper stated that right atrial pressure was determined simultaneously with cardiac output, yet repeatedly described it as an opposing pressure. Magder inherits both versions, although the language of opposition carries more of the explanatory weight.

The second contradiction concerns Pms. It is defined as the common equilibrium pressure reached when flow stops and pressures equalise across the systemic circulation. During flow, however, Magder follows Rothe in locating an equivalent pressure within the venous circulation and treating it as the upstream pressure driving venous return. A local venous pressure during flow is a different physical variable. It changes with cardiac function, vascular resistance and the distribution of blood volume, and may lie above or below the zero-flow Pms even when total stressed volume remains unchanged. Numerical similarity does not turn a local pressure in the flowing circulation into the equilibrium pressure of the whole system.

The third contradiction becomes apparent when Magder compares the arterial and venous sides. He describes arterial pressure as a consequence of cardiac output interacting with resistance and arterial compliance. Yet venous pressures are granted a more causal role. Pms drives blood towards the heart, while right atrial pressure opposes it. The same systems logic should apply on both sides. Cardiac activity redistributes blood towards the arteries, raising arterial pressure while reducing venous volume and pressure. Both pressures emerge from the interaction between cardiac activity, vascular properties and the resulting distribution of blood.

The bathtub makes these inconsistencies difficult to see because it brings its own source of energy. Water at the surface possesses gravitational potential energy relative to the drain. Gravity continues to do work as the water descends, while the external supply feeding the tap restores the water to its elevated position. The circulation also stores energy in its distended vascular walls. Opening a vein to atmosphere releases some of it. A vascular compartment can surrender that stored energy, however, only while it recoils and loses volume. During steady flow its average volume and pressure remain constant because the blood leaving is replaced.

The reservoir clearly constrains the state that the circulation can achieve. Whether an unchanged reservoir can continuously supply the work that Magder assigns to it is a different question.

The reservoir has awakened. In the next episode, Brengelmann pulls the plug.

Continued in Episode VI: The Bathtub Menace.

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Original papers

Magder S. Point: The classical Guyton view that mean systemic pressure, right atrial pressure, and venous resistance govern venous return is correct. Journal of Applied Physiology. 2006;101(5):1523–1525. https://doi.org/10.1152/japplphysiol.00698.2006

Magder S. Volume and its relationship to cardiac output and venous return. Critical Care. 2016;20:271. https://doi.org/10.1186/s13054-016-1438-7

Magder S, Slobod D, Vieillard-Baron A. Physiological and clinical significance of mean circulatory and mean systemic filling pressure. Annals of Intensive Care. 2025;15:187. https://doi.org/10.1186/s13613-025-01595-0

The pressure that remained

Rothe had returned the elastic state to the circulation. Mean systemic filling pressure (Pms) was not merely the value obtained after the pump stopped; it was the equilibrium expression of the relationship between blood volume and the vascular system containing it. During flow, arterial pressure lay above this equilibrium value and right atrial pressure lay below it. Somewhere between the two, the local pressure therefore had to pass through the numerical value of Pms.

Rothe called this the pivot pressure. It was usually found among the small veins and venules, where much of the systemic blood volume and vascular compliance resided. When cardiac activity changed, compartments on one side of the pivot gained volume while those on the other side lost it. As a description of redistribution, the pivot was useful.

Rothe then gave the numerical correspondence greater physical significance. If pressure in the small veins approximated Pms during flow, perhaps this was where Pms persisted within the circulation. The equilibrium pressure had acquired an approximate anatomical location. From there, it could become the upstream pressure for venous return, opposed downstream by right atrial pressure.

George Brengelmann challenged precisely this kind of interpretation, but not by dismissing Guyton’s work. He regarded Guyton’s graphical analysis of cardiovascular equilibrium as a major advance and accepted the validity of the experimental relationship between flow and right atrial pressure. Guyton’s own models were not simple reservoirs draining through a single venous resistance; they contained networks of compliant compartments and resistive pathways. The problem arose when the reduced equation was treated as though it were a literal picture of the circulation.

A local venular pressure during flow and the common equilibrium pressure that appears after flow stops belong to different states of the system. They may share the same numerical value without being the same physical quantity. Before Pms could be accepted as a pressure source persisting within the flowing circulation, the experiment from which the equation arose had to be reopened.

The question was no longer whether the number existed. It was what, physically, that number represented.

One steady flow, not two

Brengelmann began with a point that had become obscured by the language of venous return. In a steady circulation, cardiac output and venous return are not two separate flows. They are the same flow measured at different positions around a closed loop. Blood is not steadily leaving the veins at one rate while entering the arteries at another. If that occurred, vascular volumes would continue to change.

The term venous return becomes distinct from cardiac output only during transitions. For a brief period, the amount entering a vascular compartment may differ from the amount leaving it. The difference changes the volume contained within that compartment. Once inflow and outflow become equal again, its volume stops changing and a new steady state has been established.

This did not mean that the cardiac and vascular parts of the circulation could not be studied separately. Guyton’s great insight had been to open the loop conceptually. The cardiac subsystem could be examined by asking what flow the heart produced at different right atrial pressures when its other properties were held constant. The vascular subsystem could be examined by imposing different flows and observing the right atrial pressure associated with each one at a fixed vascular state.

When the two subsystems were reconnected, neither relationship determined the other. The circulation settled at the one flow and right atrial pressure compatible with both. The original experiments had therefore revealed an important vascular relationship. The remaining question was what physical process produced it.

Back to the apparatus

Guyton and his colleagues collected blood from the right atrium and passed it through an external collapsible tube—a Starling resistor—to the inlet of a mechanical pump. The pump delivered the blood into the pulmonary artery, and the left heart returned the same flow to the systemic circulation. An external blood reservoir was connected to the apparatus, but its connection was closed while each venous return curve was recorded, so the volume contained within the peripheral vasculature remained fixed.

The only flowmeter was positioned on the outflow side of the pump. Each data point was recorded after the preparation had reached a new steady state. At that moment, pump output, cardiac output and venous return were necessarily equal. The experiment did not measure venous return as a flow distinct from cardiac output. It measured the steady flow through the vascular subsystem and the right atrial pressure accompanying it.

The Starling resistor allowed that pressure to be varied. It consisted of a thin, collapsible tube that was kept partly compressed. At the point where collapse began, the pressure inside the tube was approximately atmospheric. Its height relative to the right atrium therefore established a hydrostatic pressure difference. Raise the resistor above heart level and right atrial pressure had to settle at a positive value. Lower it below the heart and right atrial pressure could become subatmospheric.

But moving the resistor did not alter right atrial pressure in isolation. Suppose it was raised. The right atrium and nearby great veins now had to contain more blood to establish the higher local pressure. Because total systemic vascular volume was fixed, that blood had to come from elsewhere. The distribution of volume and pressure throughout the vasculature changed, and pump output could not remain at its previous value while this occurred.

The original descriptions do not make it entirely clear whether the investigators manually changed the pump setting after each movement of the resistor or whether increased collapse of the tube impaired pump filling and throttled its output. Brengelmann argued that the distinction did not matter. In either case, the recorded point was the final combination of flow and right atrial pressure compatible with the resistor height, the vascular properties and the fixed amount of blood inside the systemic vasculature.

The legitimate conclusion was therefore precise:

At a fixed systemic vascular volume and vascular state, each steady flow was associated with a particular right atrial pressure.

The experiment did not directly reveal a compartment maintained at Pms, delivering blood through a venous resistance against a downstream back pressure. That hydraulic arrangement was an interpretation placed upon the curve.

The flat segment at very negative right atrial pressures had a separate explanation. Once the great veins partially collapsed, further reductions in right atrial pressure no longer altered upstream pressure or flow.

The missing variable

The crucial constraint was not simply pressure or flow. It was the distribution of a fixed amount of blood across compliant vascular compartments.

Rothe had already provided the necessary framework. Every compartment contained a volume related to its pressure and elastic properties. If the total volume inside the systemic vasculature was fixed, an increase in one region had to be balanced by an equal decrease elsewhere. The different points on a venous return curve therefore represented different distributions of the same total blood volume.

Consider what happened when the Starling resistor was raised. The higher required right atrial pressure meant that the central veins had to contain more blood. During the transition, their inflow briefly exceeded their outflow and their volume increased. That additional blood came from upstream compartments, whose volumes and pressures fell. Pump flow eventually settled at the lower value compatible with the new pressure and volume distribution.

The reverse process is easier to see by imagining an increase in imposed pump flow. At the first instant, the pump delivers blood into the arterial side faster than the arterial compartment can pass it onward. Arterial volume therefore rises. Because arterial compliance is fixed, arterial pressure rises with it.

At the same time, the pump removes blood from the central venous end faster than it is initially replaced from upstream. Central venous volume falls, and right atrial pressure falls with it. A finite amount of blood has been transferred from the downstream venous compartments to the upstream arterial compartments.

As the imposed pump flow redistributed blood, arterial pressure and volume rose while venous pressure and volume fell. The resulting local pressure differences were those required for progressively greater flow through the intervening resistances. Redistribution continued until the inflow and outflow of each compliant compartment were equal again; in Brengelmann’s lumped model, the same total flow then passed through each serial resistive element.

The new steady state therefore contains more blood on the arterial side and less on the venous side. Arterial pressure lies above Pms; peripheral and central venous pressures lie below it. Because the arterial compartment is stiff, a relatively small gain in arterial volume produces a large increase in pressure. The equal loss from the much more compliant venous system produces a smaller fall in pressure.

Pressure did not appear first and then command blood to move. The physical sequence began with pump action and temporary imbalances between compartmental inflow and outflow. Volume and pressure changed together until the network could transmit the imposed flow steadily:

pump action → transient inflow–outflow imbalance → redistribution of volume and pressure → new steady state

Progressively higher imposed flows therefore required progressively different distributions of the same blood volume. More blood resided upstream, less in the central venous compartments, and right atrial pressure fell.

This was Brengelmann’s central point:

The sloped segment appeared because a fixed total volume was redistributed across compliant vascular compartments at different flow rates, not because a reservoir remained at Pms and drained towards the right atrium.

Reconstructing the curve

Brengelmann made the volume accounting clearer with a hypothetical alternative to Guyton’s apparatus. He removed the Starling resistor and allowed blood from the right atrium to spill through an open tube into an external reservoir. The height of the tube set right atrial pressure. A pump drew blood from the reservoir and returned it to the pulmonary artery.

The arrangement looked superficially like a bathtub emptying through a drain. But here the external reservoir was not presented as the source of energy for systemic flow. It served as a visible ledger of how much blood entered or left the peripheral vasculature.

The thought experiment began with the pump stopped and the open tube positioned so that right atrial pressure was equal to the original Pms. With no flow, pressures equilibrated throughout the systemic vasculature, which contained a reference volume that Brengelmann called V₀ .

The pump was then started at a chosen flow while the tube remained at the same height. Right atrial pressure therefore remained at Pms. A flowing pressure profile developed upstream: arterial and intermediate pressures rose above Pms, expanding their compliant compartments. The peripheral vasculature now contained more than V₀ . The additional blood had come from the external reservoir because, during the transition, pump inflow to the circulation exceeded venous outflow back into it.

V₀ was the target systemic vascular volume defining the curve; it was not automatically preserved during every intermediate step. Starting flow while holding right atrial pressure at Pms forced the compliant vasculature to take up additional blood.

Brengelmann then lowered the open tube. Right atrial pressure fell, and venous outflow temporarily exceeded pump inflow. Blood left the vascular compartments and returned to the external reservoir. Once the excess had been removed, the systemic vasculature again contained exactly V₀ . Pump flow and venous outflow were once more equal, but right atrial pressure was now below Pms.

At each successive point, pump output could be increased and the tube then lowered until systemic vascular volume returned to V₀. Brengelmann noted that the order could equally be reversed: the tube height could be changed first and pump output then adjusted to restore V₀. Either sequence recreated the same sloping relationship obtained by Guyton. No compartment maintained at Pms was required. The curve emerged from paired changes in flow and right atrial pressure, real segmental pressure differences, compliant vascular compartments, conservation of blood volume and restoration of the same target vascular volume at each data point.

Looking inside the black box

Brengelmann next examined a conceptual three-compartment model. Its arterial, peripheral venous and central venous compliances were linked by arterial and venous resistive elements. An ideal pump imposed flow, and Brengelmann analysed an equivalent electrical circuit by computer simulation.

The model was not intended to reproduce the full complexity of the circulation. Its purpose was to ask a narrower question: did the familiar venous return equation require a compartment that remained at Pms during flow?

At zero flow, no pressure differences were required across the resistances. Pressures in the three compartments equilibrated, and the common value was Pms. Blood distributed between the compartments according to their compliances. In this state, Pms had a clear physical meaning: it was the equilibrium pressure of the whole vascular system.

Once the pump produced flow, the compartments no longer shared a common pressure. Arterial pressure rose above Pms. Peripheral venous pressure and right atrial pressure fell below it. Volume moved into the arterial compartment and out of the venous compartments until each contained the amount appropriate to its new pressure.

No compartment remained fixed at Pms.

As flow was increased, the pressure profile became steeper, arterial volume increased, venous volumes decreased and right atrial pressure fell. The model generated the familiar near-linear relationship between flow and right atrial pressure, despite containing no reservoir at Pms.

The numerical crossing had not disappeared. Because arterial pressure was above Pms and right atrial pressure below it, every flowing pressure profile still crossed the numerical value of Pms somewhere. But the location of that crossing moved as flow changed. It was a feature of the pressure profile, not a compartment maintained at Pms.

The equation survives

The equations describing Brengelmann’s three-compartment model could still be rearranged into the familiar form:

Here, Req was an effective resistance calculated from the model’s regional resistances and capacitances, corresponding to what Guyton had described more broadly as the impedance to venous return.

The model therefore reproduced not only the experimental curve, but also the equation traditionally used to interpret it.

Pms entered the derivation because total vascular volume was fixed. At zero flow, that volume was distributed across the combined vascular compliances at the common equilibrium pressure Pms. During flow, the same volume was distributed among compartments with different pressures. When this conservation of volume was incorporated into the algebra, Pms appeared as a compact equilibrium reference.

The derivation had not discovered a vessel held at Pms. It had introduced the zero-flow equilibrium pressure while accounting for the same total blood volume under flowing conditions.

The denominator was also more complicated than its name suggested. Once Pms had been assigned to the small veins and venules, RVR invited an equally literal interpretation: a physical venous resistance lying between this upstream pressure and the right atrium, impeding the return of blood to the heart. But the apparent resistance defined by the slope was not simply the resistance of those veins. It was a composite determined by arterial and venous resistances together with the distribution of compliance across the vascular network. Guyton’s own quantitative work recognised this complexity and initially called it an impedance to venous return.

The reduced equation therefore described a genuine steady-state relationship: its slope and intercept were experimentally meaningful. But its form did not prove that the circulation contained the two-pressure, one-resistor circuit suggested by its appearance. The equation survived; the imagined hydraulic circuit did not.

The phantom gradient

None of this meant that pressure gradients were unreal.

During flow, neighbouring vascular segments possessed simultaneously existing local pressures. Blood passed through actual resistive pathways, and pressure fell along those pathways as mechanical energy was dissipated. These local pressure differences were real features of the flowing circulation.

The supposed gradient from Pms to right atrial pressure had a different status. Right atrial pressure was a local pressure in the flowing circulation, whereas Pms was the common equilibrium pressure reached when flow stopped. The equation made them look like opposite ends of a physical pathway, but no compartment maintained at Pms had been identified upstream of the right atrium.

This was unlike water flowing downhill, where the higher and lower elevations exist simultaneously and gravitational potential energy is released as the water descends. In the circulation, the heart supplies the energy and establishes the flowing pressure profile. The local pressure gradients along that pathway are real. The phantom was the idea that one of them began at Pms.

Brengelmann was not denying the reality of local venular pressure, the meaning of Pms at equilibrium, the importance of vascular volume and compliance, or the inverse steady relationship between flow and right atrial pressure. His criticism concerned the progression from one valid observation to a much larger physical claim:

A local venular pressure approximated Pms numerically. That pressure was therefore treated as Pms. Pms was then assumed to persist there during flow, supplying an upstream pressure opposed by right atrial pressure across a resistance to venous return.

Only the first observation followed from the pressure profile.

Rothe’s pivot still had descriptive value. It marked the crossover between vascular regions that gained and lost volume as the circulation moved between states. Its frequent location among veins and venules reflected where much of the vascular volume and compliance resided. But a moving crossover did not locate an equilibrium pressure within the flowing circulation.

The pivot was real as a crossing point. The pressure was real as a local pressure. The phantom was the identity assigned to it.

The energy warning

A second problem was already beginning to appear.

An elastic compartment can release stored energy while it loses volume and its walls recoil. That process can contribute to transient flow during redistribution. But once the compartment has reached a new steady volume and pressure, its walls are no longer shortening and it is no longer releasing additional elastic energy.

A reservoir whose pressure and volume remain fixed cannot continuously power its own outflow. If blood enters it at exactly the rate blood leaves, it becomes a passive conduit through which energy supplied elsewhere is transmitted.

Brengelmann would later make this objection central. For now, it was enough to see that removing a physical Pms reservoir did not make the vascular elastic state irrelevant. Blood volume, compliance and smooth-muscle tone still altered the pressure and volume distributions presented to the heart. Brengelmann’s ideal pump imposed flow; it showed how a fixed vascular volume redistributed at different flow rates, but did not ask how much flow a real heart could establish or sustain at a given vascular state.

Brengelmann had shown why Pms could not simply be placed upstream of venous return. He had not removed the elastic circulation from the argument.

What survived

Guyton’s greatest contribution remained intact.

The cardiac and vascular subdivisions could be studied through their separate open-loop relationships. In the cardiac subsystem, right atrial pressure influenced the flow produced by the heart. In the vascular subsystem, flow influenced the pressure and volume distribution that resulted in right atrial pressure. When the two were connected, the circulation settled at the one operating point compatible with both.

This framework allowed changes in cardiac function, blood volume and vascular properties to be understood as changes in the relationships whose intersection defined the steady state. Brengelmann did not reject that analysis. He rejected the idea that the vascular curve represented blood draining from a reservoir maintained at Pms.

Pms also survived. It remained a meaningful zero-flow equilibrium pressure and an aggregate expression of the relationship between blood volume and vascular accommodation. Brengelmann removed its supposed anatomical location and its role as a continuing source of energy during flow, not its meaning as a descriptor of the systemic elastic state.

But the defence of the reservoir had not yet been heard.

Brengelmann’s model had imposed flow; it had not determined whether a real heart could sustain any chosen flow without a change in vascular state. If Pms was not a pressure source sitting upstream of the right atrium, why did the elastic state of the circulation so clearly constrain the flow that could be sustained?

The heart might supply the energy. But perhaps the reservoir still determined how much blood could return to it.

Continued in Episode V: The Reservoir Awakens.

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Original paper

Brengelmann GL. A critical analysis of the view that right atrial pressure determines venous return. Journal of Applied Physiology. 2003;94:849–859. https://doi.org/10.1152/japplphysiol.00868.2002

From pressure and flow to volume

Levy had taken the venous return equation apart and put it back together in reverse. The same relationship that Guyton had read as a pressure difference determining flow could equally be read as flow determining the pressure difference. Neither arrangement established causality. Guyton himself had acknowledged the deeper implication: in the intact circulation, pressure and flow were both dependent variables.

But that left an obvious question unanswered. Dependent on what?

The equation related venous return to mean systemic filling pressure, right atrial pressure and resistance, but said little about the vascular properties from which those pressures arose. Blood did not occupy a rigid system of pipes. It was distributed through vessels that expanded, recoiled and changed shape. The same total blood volume could therefore produce very different pressures, depending on the vessels containing it.

This was the territory explored most systematically by Carl Rothe. Rather than beginning with the pressure difference said to drive venous return, he examined how the circulation contained blood: where that blood resided, how easily the vessels expanded, and how smooth-muscle activity altered the available vascular space.

The veins were central to this account. Their large resting volume and high compliance mean that they contain most of the systemic blood volume and provide most of the circulation’s ability to accommodate changes in volume. Their importance was therefore not simply that blood passed through them on the way to the heart. Venous elastic and contractile properties helped determine how the blood volume was distributed and what pressures appeared throughout the circulation.

Two ways to redistribute blood

Rothe began with a distinction that sounds obvious but is easily lost. Venous return is a rate of flow, not a parcel of blood being emptied from a reservoir. In a steady circulation, blood does not progressively leave the veins and accumulate in the heart or arteries. Cardiac output and venous return are the same continuous flow viewed at different locations.

Vascular volumes can, however, change while the circulation moves from one steady state to another. Rothe demonstrated this in preparations where cardiac output was imposed by a mechanical pump. Increasing flow altered the distribution of blood within the vasculature: arterial pressure and volume rose, while volume fell in the small and great veins. He then applied the same principle to the intact circulation. If increased cardiac activity succeeded in producing a higher flow, the heart would temporarily transfer a finite volume from the venous side into the arterial system. Once that redistribution was complete, cardiac output and venous return would again be equal—but at a new operating state, with more blood residing on the arterial side and less in the veins.

No active venoconstriction is required. The venous walls already possess passive elastic properties. As the circulation establishes the new state, the veins move to a lower-pressure, lower-volume point on their existing pressure–volume relationship. The recoil occurs during the transition; it is not another pump that continues to propel blood afterwards. The higher flow, where one develops, remains sustained by the heart’s continuing energy input.

Flow can redistribute blood regionally as well. If resistance to inflow into an organ rises, regional flow and downstream vascular pressure may fall. Its distensible venous vessels then contain less blood, leaving more of the fixed total volume elsewhere in the circulation. Vasodilation can have the opposite effect, allowing a vascular bed to contain more. Rothe argued that passive volume changes of this kind could be substantial and could easily be mistaken experimentally for active contraction of the veins.

Active venous redistribution is fundamentally different. Venous smooth-muscle contraction changes the volume-containing properties of the vessel itself. Blood then redistributes through the circulation as volume, pressure and flow settle into a state compatible with the altered vascular compartment.

Both processes therefore move real blood between compartments. The distinction concerns what changes first. In a passive response, the vessels move to a new operating point while their underlying mechanical properties remain the same. In an active response, vascular smooth muscle changes those properties, and a new distribution follows.

The shape of the container

Rothe needed a more exact vocabulary to describe this behaviour. Capacity meant the volume currently contained within a vascular compartment. Compliance described the relationship between a change in contained volume and the associated change in distending pressure:

C = ΔV / ΔP

Compliance was therefore the slope of the volume–pressure relationship over the range being examined. A highly compliant vessel could accommodate a large change in volume with only a small change in pressure. A less compliant vessel produced a larger pressure change for the same change in volume.

Compliance alone was not enough. Two vascular compartments could have the same compliance—the same slope—yet be very different sizes and therefore contain very different volumes at the same pressure. Rothe used vascular capacitance for the whole volume–pressure relationship, including both the resting size of the compartment and how readily it expanded.

The terminology is potentially confusing. Electrical capacitance maps more naturally onto vascular compliance than onto Rothe’s broader definition. I will use vascular accommodation for the broader property: the relationship governing how much blood a vascular compartment can contain at different pressures.

Rothe described the approximately linear portion of this relationship as:

V = Vᵤ + CP

where V is the total contained volume, Vᵤ is unstressed volume, C is compliance and P is transmural pressure.

In this simplified model, unstressed volume represented the resting size of the vascular container: the volume-axis intercept obtained by extending the measured relationship back to zero transmural pressure. Stressed volume was the remaining part of the contained volume associated with elastic distension:

Vₛ = CP

These were calculated components of the same contained blood volume, not two anatomically separate reservoirs. In later teaching, unstressed volume came to be imagined as a hidden reserve waiting to be recruited, while stressed volume was pictured as a separate pool already placed under tension. Rothe’s definitions were more careful. They provided a mathematical way of separating the resting size of the container from the additional volume associated with its distension. All the blood within a pressurised vessel is physically exposed to pressure; stressed volume is a model-derived partition, not the only blood that is literally under stress.

This is why venoconstriction is often described as “recruiting” unstressed volume. When venous smooth muscle contracts, the effective resting size of the vascular compartment becomes smaller. Within Rothe’s model, the same total volume is then represented by a larger stressed-volume term. Blood also genuinely redistributes into other compartments as the circulation establishes a new state. What has not occurred is a literal transfer from an unstressed reservoir into a separate stressed reservoir; the mathematical partition changes because the vascular compartment has changed.

Passive redistribution is different. If increased cardiac activity transfers a finite volume from veins to arteries without changing vascular tone, the venous stressed-volume term falls while the arterial term rises. Vascular accommodation has not changed; blood has simply changed location within the same set of compartmental relationships.

The distinction can therefore be stated simply:

Passive redistribution changes where blood resides without changing vascular accommodation. Active venoconstriction changes vascular accommodation itself—usually by reducing unstressed volume, but sometimes by changing compliance as well.

The pressure of the whole circulation

These relationships existed in every vascular compartment. Mean systemic filling pressure, Pms, described their aggregate state across the systemic circulation. Rothe generally wrote of mean circulatory filling pressure, Pmcf, for the circulation as a whole. Here I use Pms for the corresponding systemic equilibrium pressure relevant to systemic venous return.

The concept predated Rothe. His contribution was to explain its physical meaning and measurement in much greater detail. During normal flow, pressures differ widely: arterial pressure is high, right atrial pressure is low, and pressures within individual organs lie between them. If the heart stops, blood redistributes until these differences disappear and the systemic vasculature approaches a common equilibrium pressure. That pressure is Pms.

In Rothe’s linearised description:

Pₘₛ = ΣVₛ / ΣC

The numerator is total systemic stressed volume and the denominator is total systemic vascular compliance. Pms is therefore not the pressure of one vessel, nor a simple arithmetic average of the pressures measured during flow. It is the equilibrium pressure produced by the interaction between total stressed volume and the combined compliance of the systemic circulation.

This helps explain why the veins matter so much. Most systemic blood volume and most vascular compliance reside on the venous side. The arteries contain less blood and are much stiffer, so even large changes in arterial pressure involve relatively small changes in contained volume. The systemic elastic state is therefore determined primarily—although not exclusively—by the volume–pressure properties of the veins and venules.

Rothe sometimes described Pms as an index of the circulation’s “fullness”. The word is useful only if it is understood as an elastic filling state, rather than as blood volume alone. The same blood volume can produce a lower Pms when it is accommodated within a larger vascular space, or a higher Pms when vascular accommodation is reduced.

Increasing blood volume can therefore raise Pms by increasing total stressed volume. Venoconstriction can also raise Pms without changing total blood volume. By reducing unstressed volume, changing compliance, or both, it alters the relationship between the existing blood volume and the vascular compartment containing it. The same blood volume can therefore equilibrate at a higher pressure. Reducing blood volume or increasing venous accommodation has the opposite effect.

Passive redistribution between arteries and veins need not alter Pms. If total blood volume and vascular accommodation remain unchanged, it changes how stressed volume is distributed during flow rather than its aggregate amount. If the heart is stopped, that arterial–venous distribution is undone as the same total stressed volume spreads across the same total compliance and returns to approximately the same equilibrium pressure.

Rothe had therefore restored something essential to the argument. Pms was not merely a number that appeared after the pump stopped. It was a compact pressure expression of the systemic circulation’s elastic state: the relationship between the contained blood volume and the vascular system accommodating it.

But then he went one step further.

The pivot and the phantom

During flow, pressure falls progressively from the arterial system towards the right atrium. Arterial pressure lies above Pms and right atrial pressure lies below it, so somewhere along the flowing circulation the local pressure must cross the numerical value of Pms.

When the heart stops, arterial pressure falls towards Pms while central venous pressure rises towards it. Compartments whose pressure was initially above Pms lose volume; those initially below it gain volume. Near the point where local pressure already approximated Pms, relatively little changes. Rothe called this the pivot pressure.

The pivot had legitimate descriptive value. It marked the approximate crossover between regions that gained volume and those that lost it as cardiac activity changed. Because much vascular volume and compliance reside in small veins and venules, the crossover often appeared in that part of the circulation. Its precise location was not fixed and could differ between organs or move as flow and vascular tone changed.

Rothe nevertheless gave the numerical resemblance greater significance. Because pressure in the small veins could approximate Pms during flow, he treated Pms as an estimate of their distending pressure and therefore as the upstream pressure for venous return.

That step is difficult to justify.

Was the local venous pressure during flow truly the same physical quantity as Pms—or had Rothe given an equilibrium property an anatomical home it did not possess?

Continued in Episode IV: The Phantom Pressure.

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Original papers

Rothe CF. Reflex control of veins and vascular capacitance. Physiological Reviews. 1983;63(4):1281–1342.

Rothe CF. Mean circulatory filling pressure: its meaning and measurement. Journal of Applied Physiology. 1993;74(2):499–509. https://doi.org/10.1152/jappl.1993.74.2.499

In 1955, Arthur Guyton changed the way physiologists thought about cardiovascular function. The circulation was no longer viewed simply as a heart pumping into a passive network of vessels, but as an integrated system in which the properties of the heart and circulation interacted to determine the final cardiovascular state.

At the centre of this framework was a deceptively simple relationship, which can be expressed in compact notation as:

Venous return was related to the difference between mean circulatory filling pressure and right atrial pressure, divided by what Guyton called the impedance to venous return. This terminology reflected an important insight: the denominator depended not on a single vascular resistance, but on the distribution of resistance and capacitance throughout an elastic vascular network.

The interpretation seemed obvious. Blood flowed from the higher pressure of the systemic circulation toward the lower pressure of the right atrium. Mean systemic pressure promoted venous return, right atrial pressure opposed it, and the difference between them represented the pressure gradient for venous flow.

For more than two decades, this framework shaped how physiologists understood the circulation.

Then, in 1979, Levy returned to Guyton’s equation and asked whether the problem was not the mathematics, but the interpretation placed upon it.

The same equation could tell a very different story…

One circulation, not two

Guyton’s experimental approach was extraordinarily powerful because it allowed the heart and vasculature to be examined separately.

The heart and circulation form a closed loop. Any change in cardiac function alters the circulation, while any change in the circulation alters the conditions presented back to the heart. Understanding the behaviour of each component separately is therefore extremely difficult.

Guyton’s solution was to open the loop experimentally. In different preparations, he replaced either the whole heart or the right heart with a mechanical pump, allowing him to examine the properties of the vascular system separately before recombining the two systems mathematically.

It was an elegant solution, but Levy argued that the success of the experiment had created a conceptual problem. A separation that was useful experimentally had gradually become a separation in the way people thought about the circulation itself.

Cardiac output and venous return began to appear as two competing processes: one generated by the heart, the other generated by the circulation.

But in an intact cardiovascular system there is only one flow. The blood leaving the heart is the same blood returning to it. Cardiac output and venous return are not opposing forces negotiating a compromise; they are the same circulation viewed from different locations.

Explaining a steady-state change in cardiac output by invoking an equal change in venous return was, Levy wrote, a “patent example of circular reasoning”. It was equivalent to explaining a change in total flow by the same change in total flow.

The question was not whether cardiac output controlled venous return or venous return controlled cardiac output.

The question was whether either could be considered the independent controller of the system.

The numerator strikes back

Levy’s most powerful argument came from looking again at the equation itself. Focusing specifically on the systemic circulation, he used mean systemic pressure (Pms)—the pressure to which the systemic vasculature would equilibrate if flow ceased—rather than Guyton’s mean circulatory filling pressure. By this stage, Guyton’s impedance term was also generally expressed as the resistance to venous return (RVR):

The interpretation appeared straightforward. The numerator, Pms − RAP, seemed to be the independent factor. Increase mean systemic pressure and venous return increased. Increase right atrial pressure and venous return decreased.

The pressure gradient appeared to determine flow.

But Levy pointed out something deceptively simple: rearranging an equation does not change the physiology.

The same relationship can also be written:

Now the interpretation looks very different. If resistance remains constant, the pressure gradient is not the variable controlling flow. It is the variable created by flow.

The circulation does not flow because a gradient has appeared between mean systemic pressure and the right atrium. The gradient exists because blood is flowing through a resistance.

This was not a mathematical trick. It was a reminder that equations describe relationships, and the way they are written can subtly influence which variable we imagine to be the cause.

In Guyton’s experiments, this was more than a question of presentation. The venous return curves were not created by observing a passive pressure gradient spontaneously producing flow. They were generated by altering flow through the experimental preparation and measuring the resulting pressure response.

In some preparations, pump output was directly varied and right atrial pressure was measured as the response. In later experiments, Guyton retained the mechanical pump but added a collapsible tube acting as a Starling resistor in an attempt to control right atrial pressure more directly.

This was an ingenious experimental solution, but Levy argued that it did not remove the fundamental problem. Establishing a new steady state still required a change in flow through the circulation. The experiment could therefore be interpreted in the opposite direction: rather than right atrial pressure determining venous return, changing flow altered the distribution of pressure within the vascular system.

The details of this experimental debate would continue for decades, but Levy’s central point was more forceful than mere ambiguity. The curves certainly demonstrated a relationship between pressure and flow, but he argued that their usual interpretation had reversed cause and effect: flow was the experimentally imposed variable, and right atrial pressure was the response.

The numerator had struck back.

Where do pressures come from?

Levy’s argument went beyond rearranging an equation. If right atrial pressure was not simply a back pressure opposing venous return, then a more fundamental question followed: where did the observed pressures come from?

His answer began with the vascular system itself. Blood vessels are not rigid pipes but elastic compartments whose pressures depend on how much blood they contain and on their pressure–volume properties. The heart supplies the energy for circulation and transfers blood from the venous side into the arterial side. As flow increases, arterial pressure rises and the arterial compartment contains more blood. Because the total systemic blood volume in Levy’s model is fixed, that additional arterial volume must come from the venous compartment, causing venous and right atrial pressures to fall.

For Levy, this redistribution explained the familiar inverse relationship between flow and right atrial pressure. Right atrial pressure did not independently limit venous return; it changed because the heart had altered the distribution of blood within an elastic vascular system.

Mean systemic pressure still mattered, but Levy treated it as a zero-flow boundary condition determined by total blood volume and systemic vascular capacitance, not as the source of energy for flow. Flow and resistance determined the arteriovenous pressure difference, while blood volume and the arterial and venous capacitances determined how that difference was distributed into the actual levels of arterial and venous pressure.

This was why Levy treated flow as the independent variable in Guyton’s experiments. The pump imposed the flow, and the vascular system generated the corresponding pressure distribution.

The disappearing gradient

Levy also highlighted a striking consequence of interpreting the venous return equation too literally.

In a typical circulation, mean systemic pressure is only a few millimetres of mercury above right atrial pressure. If Pms is approximately 7 mmHg and right atrial pressure is approximately 2 mmHg, then the pressure gradient for venous return is only around 5 mmHg.

A small increase in right atrial pressure would therefore have a dramatic effect on this gradient.

If Pms were held constant, increasing right atrial pressure by just 5 mmHg would make the gradient disappear completely.

Taken literally, venous return should stop.

But Levy argued that this was a strange way to view the circulation. The heart had generated a much larger pressure difference across the systemic circulation. An arterial pressure of approximately 100 mmHg and a right atrial pressure of 2 mmHg represented a pressure difference almost twenty times larger.

Increasing right atrial pressure by a few millimetres of mercury barely changed this total systemic pressure difference.

To Levy, this revealed the problem with focusing on the numerator of the venous return equation. The small difference between Pms and right atrial pressure was not the energy source responsible for circulating blood around the body. It was a pressure difference created within the systemic circulation as blood flowed through it.

The gradient was real, but Levy argued that its meaning had been misunderstood.

Guyton replies

Levy’s paper contained an unusual postscript. Arthur Guyton had reviewed it, and the editors published his response in full.

What followed was not the rebuttal one might expect.

Guyton wrote that he found himself in “complete agreement” with Levy on almost every conceptual point. He agreed wholeheartedly that venous pressure was a dependent variable. His objection was that Levy had not gone far enough: venous return—and therefore cardiac output—was no more independent than venous pressure.

Levy’s analysis treated flow as the input to the vascular system. This was entirely appropriate in an experiment where the investigator controlled the output of a mechanical pump and measured the resulting pressures. Under those conditions, flow was the manipulated variable and right atrial pressure was the response.

But Guyton argued that this experimental arrangement should not be mistaken for the intact circulation. The heart does not freely select a cardiac output and impose it upon the vasculature. Flow and pressure emerge from the interaction between the functional state of the heart and the mechanical properties of the circulation. In his reply, Guyton identified factors such as cardiac contractility, heart rate, vascular resistance and vascular capacitance as the more fundamental variables; cardiac output, venous return, arterial pressure and venous pressure were all dependent variables determined simultaneously.

This was remarkably close to the systems interpretation contained in his original 1955 paper. The venous return curve could be constructed by temporarily treating right atrial pressure as the independent variable, just as Levy’s vascular function curve could be constructed by temporarily treating flow as independent. Both were legitimate “what if” analyses. Neither assignment described the actual hierarchy of the intact circulation.

The disagreement between Levy and Guyton was therefore narrower than it first appeared. Levy had shown why the venous return curve should not be read as proof that right atrial pressure controlled flow. Guyton accepted that criticism, but insisted that reversing the axes did not make flow the ultimate controller either.

Pressure did not determine flow alone.

Flow did not determine pressure alone.

They were resolved together by the system.

That leaves the obvious question for the next episode: what were the physical properties of the vascular system that helped determine them?

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Original papers

1. Levy MN. The cardiac and vascular factors that determine systemic blood flow. Circulation Research. 1979;44:739–747.

2. Guyton AC. Determination of cardiac output by equating venous return curves with cardiac response curves. Physiological Reviews. 1955;35:123–129.

3. Guyton AC, Lindsey AW, Kaufmann BN. Effect of mean circulatory filling pressure and other peripheral circulatory factors on cardiac output. American Journal of Physiology. 1955;180:463–468.

Some papers change what we know. A few change how we think.

Arthur Guyton’s 1955 papers did something rarer still: they gave cardiovascular physiology a new way of seeing itself.

Before Guyton, physiologists understood many of the individual components of the cardiovascular system. The heart obeyed the Frank–Starling mechanism: as filling increased, the heart pumped more blood. Blood vessels were elastic structures that stored volume, generated pressure and resisted flow. Cardiac output, blood pressure and vascular resistance could all be measured.

Yet the circulation remained difficult to understand as a complete system.

The problem was deceptively simple: what determines cardiac output?

The obvious answer was the heart. After all, the heart was the pump. But Starling’s mechanism immediately created another question. If the heart pumps what it receives, what determines what it receives?

The circulation was a closed loop. The heart determined flow through the circulation, but the circulation determined the conditions presented back to the heart. Each side influenced the other, making it difficult to identify where cause ended and effect began.

Guyton’s insight was to break that circle conceptually.

He separated the circulation into two interacting systems: a cardiac function describing the relationship between filling and output, and a vascular function describing the relationship between the systemic circulation, flow and right atrial pressure.

Study them separately. Then put them back together. Where those two relationships intersected, the entire circulation found its equilibrium.

Cardiac output and venous return were not competing explanations of flow. They were two views of the same circulation seen from opposite sides of the loop.

It was an extraordinarily powerful idea. In a single diagram, Guyton united Starling’s law, vascular properties and cardiac output into one coherent picture. Physiologists could now visualise how changes in blood volume, vascular tone or cardiac function shifted the operating point of the entire circulation.

Few papers have influenced cardiovascular physiology more profoundly.

But every powerful idea carries a hidden danger. Sometimes the way we draw a diagram changes the way we think, and sometimes the language we use to describe a relationship instead becomes a story about cause and effect.

That is where the venous return wars began.

Separating the inseparable

Studying a closed loop creates a fundamental problem: everything affects everything else.

If cardiac output increases, blood redistributes between different parts of the circulation. Pressures change. Volumes change. The conditions for venous return change. Conversely, if venous return changes, cardiac filling changes, the heart responds, and cardiac output changes.

Guyton’s solution was both simple and brilliant. He opened the loop.

In experimental preparations, the heart and systemic circulation were separated. The heart could be replaced by a mechanical pump, allowing the behaviour of the vascular system to be studied independently.

The technical details were ingenious and would later become part of the controversy, but the conceptual aim was clear: study the properties of the circulation independently of the heart.

By changing pump flow and observing the resulting pressures, Guyton could describe the relationship between blood flow through the vascular system and right atrial pressure.

The result was the venous return curve.

At high flows, right atrial pressure was low. At lower flows, right atrial pressure rose. When flow stopped completely, pressure throughout the circulation equilibrated at a single value.

Guyton called this the mean circulatory filling pressure.

This represented something fundamental: the pressure generated by the blood volume contained within the elastic vascular system when flow had ceased.

The circulation was not simply a collection of tubes. It was an elastic container capable of storing energy.

A New Gradient

Mean circulatory filling pressure was not a new observation. Physiologists already knew that when the heart stopped and flow ceased, pressures throughout the circulation equilibrated to a common value.

Guyton’s insight was to recognise that this equilibrium pressure contained important information about the vascular system itself.

The circulation was not simply a passive network of vessels waiting for the heart to pull blood through it. It had its own properties. Blood volume, vascular elasticity and resistance determined the relationship between the vascular system and the flow returning to the heart.

Guyton made the conceptual leap that would define venous return physiology for the next seventy years.

If the circulation had an equilibrium pressure when flow stopped, and the right atrium had a pressure where blood returned to the heart, then the difference between these pressures could be described as a gradient.

The model appeared intuitive: mean circulatory filling pressure on one side, right atrial pressure on the other, with the resistance and capacitance properties of the vascular system lying between them.

Venous return could therefore be expressed as:

VR = (MCFP − RAP) / ZVR

where VR is venous return, MCFP is mean circulatory filling pressure, RAP is right atrial pressure, and ZVR is the impedance to venous return.

It was elegant, powerful, and captured something clinicians recognised intuitively. Increasing blood volume or constricting veins increased mean circulatory filling pressure and shifted the system towards greater flow. Weakening the heart caused blood to accumulate upstream and increased right atrial pressure.

The framework explained why neither the heart nor the circulation alone controlled cardiac output. Guyton had created a model of a coupled system.

The next step would transform this idea into one of the most famous diagrams in physiology.

The most famous graph in physiology

The final step was Guyton’s masterstroke.

He had described two relationships. The first was the familiar cardiac function curve: the relationship between filling pressure and cardiac output described by Starling. The second was the venous return curve: the relationship between flow through the systemic circulation and right atrial pressure.

Individually, each relationship described only half of the system. Together, they created something entirely new.

Guyton placed both curves on the same graph. The point where they crossed represented the only state compatible with both the heart and the circulation. At that point, cardiac output and venous return were equal, and the corresponding right atrial pressure was the value that satisfied both systems.

This was the true power of the model.

Right atrial pressure was not chosen by the heart. Cardiac output was not chosen by the circulation. Both emerged from the interaction between the pumping characteristics of the heart and the physical properties of the vascular system.

Guyton transformed the circulation from a collection of separate components into an integrated system.

Changes that previously seemed complex could now be visualised. Increased blood volume altered the vascular curve. Impaired cardiac function altered the cardiac curve. The new intersection predicted the new cardiovascular state.

It was elegant, intuitive and clinically useful.

Most importantly, it demonstrated something that remains true today: neither the heart nor the circulation independently determines cardiac output. Flow emerges from the interaction between both.

A disturbance in the Force

But every powerful model carries a risk. The clearer an idea becomes, the easier it is to forget what has been simplified.

Guyton’s diagram made a complex interaction visible. It placed mean circulatory filling pressure, right atrial pressure, venous return and cardiac output onto a single set of axes. That was its genius.

It was also its danger.

The eye naturally reads a graph as a causal story. One variable sits on the horizontal axis. Another rises or falls on the vertical axis. A line connects them. Before long, a relationship starts to look like a mechanism.

The venous return curve invited exactly that reading.

Mean circulatory filling pressure on one side. Right atrial pressure on the other. A gradient between them.

The mathematics described a relationship between variables in a coupled system.

The picture looked like a mechanism.

The war begins

Perhaps the most fascinating part of this story is that Guyton seemed to recognise the problem, but never fully escaped it.

In the same paper that introduced his famous curves, he acknowledged that right atrial pressure was not a primary determinant of cardiac output. It was determined simultaneously with cardiac output by the interaction between the heart and vascular system.

That was the systems physiologist speaking.

But elsewhere in the same work, Guyton used a very different language. Right atrial pressure was described as a back pressure. It opposed venous return. Mean circulatory filling pressure promoted venous return and was described as a force tending to push blood toward the right atrium. The difference between them was the pressure gradient for venous return.

That was the language that stuck. And it stuck because it was so easy to understand.

A pressure difference. A resistance. A flow.

The equation looked familiar. The diagram looked familiar. The language invited a simple mental model: blood flowing from mean circulatory filling pressure towards the right atrium, with right atrial pressure acting as the downstream pressure holding it back.

The problem was not that this was mathematically useless. The problem was that it sounded mechanistic.

A model designed to describe equilibrium began to look like a model explaining cause and effect.

Guyton did not invent the later misconception out of thin air, but he did give it much of its vocabulary.

That is why the venous return wars are so interesting. The conflict was not between a brilliant physiologist and confused readers. It was already present inside the original papers: an elegant systems model described in language that made the system look like a simple pressure-driven pipe.

For more than two decades, the gradient shaped how generations of physiologists and clinicians thought about venous return.

Then, in 1979, Matthew Levy looked again at Guyton’s famous curves and asked a different question.

Not:

What determines venous return?

But:

Which variable is actually being determined?

The numerator was about to strike back…

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Ancient texts

1. Guyton AC, Lindsey AW, Kaufmann BN. Effect of mean circulatory filling pressure and other peripheral circulatory factors on cardiac output. Am J Physiol. 1955;180:463–468.

2. Guyton AC. Determination of cardiac output by equating venous return curves with cardiac response curves.Physiol Rev. 1955;35:123–129.

Not because capacitance has no defensible physiological meaning. At its broadest, it describes the pressure–volume relationship of a vascular compartment: how much blood that compartment contains at a given transmural pressure. Transmural pressure is the pressure inside the vessel relative to the pressure surrounding it.

The problem is that “capacitance” rarely keeps this single meaning. Across the haemodynamic literature, the same word is used for several different properties and states. It sounds precise, but the author and reader may be using it to mean entirely different things.

In different sources it may refer to compliance, capacity, vascular tone, the full pressure–volume relationship, the volume of blood currently contained within a vascular compartment, or a redistribution of blood across the circulation.

Those are not interchangeable.

The electrical analogy ought to make the language clearer. Instead, it exposes the ambiguity.

In an electrical circuit, capacitance is the change in stored charge for a change in voltage. The direct hydraulic equivalent is the change in contained volume for a change in pressure. Electrical capacitance therefore maps most clearly onto vascular compliance. Not capacity, tone, vascular volume or volume distribution.

And haemodynamics already has a word for that.

Compliance.

Compliance = ΔV/ΔP

Here, Δ means change. Compliance describes how much the contained volume changes when transmural pressure changes. More precisely, it is the local slope of the pressure–volume relationship. A vascular compartment is highly compliant at an operating point if a relatively large change in volume produces only a small change in pressure.

Vascular pressure–volume relationships are not necessarily straight lines. Their slope can change as the vessel fills, so compliance is not always a single fixed property of a compartment.

If we turn the relationship around and ask how much pressure changes for a given change in volume, the correct term is elastance.

Elastance = ΔP/ΔV

Elastance is the reciprocal framing of compliance. A large pressure change for a small volume change means high elastance; its intuitive correlate is stiffness.

Compliance and elastance describe how pressure and volume change together. They do not tell us how much blood a vessel contains at a given pressure. Two vessels can have the same compliance but contain different volumes at the same pressure.

Capacity is different again.

Capacity is how much volume a compartment can physically contain. It is not its compliance, its tone or the amount of blood currently inside it.

A rigid one-litre container has a capacity of one litre whether it is empty or full. That tells us nothing about its compliance. Once it is full and closed, even a tiny addition of volume would produce a very large rise in pressure. Its capacity may be large while its compliance is effectively negligible.

Capacity and compliance are not the same thing.

Nor is vascular tone another name for either of them.

Tone is the active state of vascular smooth muscle. When a vein constricts, it changes the size and shape of the container.

The vein may then hold less blood at the same pressure even if its compliance has not changed. Put simply, the container has become smaller; it has not necessarily become stiffer.

Compliance describes how much the contained volume changes when pressure changes. It is the slope of the pressure–volume relationship. Tone can shift the whole relationship without changing that slope.

Passive wall stiffness, vessel shape, surrounding pressure and how full the vein already is also affect its pressure–volume behaviour. Some may alter compliance. Others may change how much blood the vein contains at a given pressure without changing compliance.

So “venous capacitance fell” still leaves the mechanism unclear. Did the veins become less compliant? Or did venoconstriction instead make them hold less blood at the same pressure?

Those are different changes.

We must also separate the properties of the container from the blood actually inside it.

Vascular volume is the amount of blood currently present in a vascular compartment. Volume distribution describes where blood resides across the circulation as a whole.

If venoconstriction changes the pressure–volume relationship of one venous region, that region may contain less blood at the new operating state. The blood has not vanished, and venoconstriction has not created new volume. Existing blood must move elsewhere within the closed circulation. That is redistribution.

This is why venoconstriction should not be imagined as an auxiliary heart that continuously pumps blood towards the chest. It changes the container. Blood redistributes as the circulation settles into a new pressure, volume and flow state. Any sustained flow still requires continuing energy transfer from the heart.

Fluids and changes in tone therefore do fundamentally different things. Fluids add vascular volume. A change in venous tone changes venous geometry and pressure–volume behaviour, redistributing volume that is already present. Both may alter pressure and cardiac filling, but they do so by different mechanisms.

Now return to the apparently simple statement:

“Venous capacitance increased.”

What actually happened?

Did compliance increase?

Did elastance fall?

Did venous tone decrease?

Did the whole pressure–volume relationship shift?

Did physical containing capacity change?

Did a vascular compartment simply receive more blood?

Did blood redistribute into a different part of the circulation?

These descriptions can point in similar clinical directions, but they do not describe the same physiology. Without clarification, “capacitance” conceals more than it reveals.

The temptation is to solve this by inventing another umbrella term for the circulation’s overall ability to contain and redistribute blood. That only moves the ambiguity into a new word. When the mechanism can be named directly, it should be.

If we mean the local change in volume for a change in pressure, say compliance.

If we mean the local change in pressure for a change in volume, say elastance.

If we mean how much a compartment can physically contain, say capacity.

If we mean active smooth-muscle state, say vascular tone.

If we mean the full relation between contained volume and transmural pressure, say the vascular pressure–volume relationship.

If we mean how much blood is actually present, say vascular volume.

If we mean where blood resides across the circulation, say volume distribution or redistribution.

“Capacitance” can still be defensible when an author explicitly uses it for the pressure–volume behaviour of a vascular compartment as a whole. But it should not be treated as a self-explanatory scalar property, and it should never substitute for a mechanism that can be stated more precisely.

Haemodynamics already contains too much shorthand that sounds causal while remaining merely descriptive. “Capacitance” is particularly troublesome because it can blur the pressure–volume relationship, its local slope, the active state of the vessel wall, the physical size of the container, the volume of blood inside it and the distribution of blood across the circulation.

Use the word only if you define it.

Use the mechanism whenever you can.

Use the mechanism, not the metaphor.

The explanation seemed obvious. Blood enters an organ from the arterial side and leaves through the venous side. If venous pressure rises, the difference between arterial and venous pressure narrows. The driving pressure falls. Flow falls. Organ perfusion deteriorates.

It was simple, intuitive and clinically useful.

The problem came later, when I started going back to first principles. The more I thought about pressure, flow and energy, the harder it became to accept the usual explanation. Not because venous congestion is harmless. It clearly is not. High venous pressures are associated with renal dysfunction, hepatic congestion, bowel oedema and worse outcomes in many clinical settings.

The problem was more fundamental.

Pressure gradients describe what is happening in a flowing system. They are not little motors that cause flow by themselves.

That sounds like a small distinction, but it changes the whole explanation. If a pressure gradient is part of the state the system has settled into, then saying that organ flow falls because the pressure gradient has fallen does not really explain the mechanism. It risks taking one measured feature of the final state and promoting it into the cause.

This is what bothered me.

If venous congestion reduces organ perfusion by opposing flow from the venous side, then the main effect should be a reduction in total flow through the circulation. Cardiac output should fall, and organ blood flow should fall with it. But venous congestion can coexist with a preserved cardiac output, while some organs still become dysfunctional. That observation is difficult to reconcile with a simple back-pressure model. If total flow is still being maintained, then blood has not simply stopped moving through the system. It has either been redistributed, or the affected organ has become a less favourable pathway for flow.

That was the piece I found hardest to reconcile. If pressure gradients describe the state of the system rather than causing it, then venous congestion cannot be explained by pointing to a smaller gradient and stopping there. We have to explain what changed in the system: global flow, regional distribution, local organ conductance, external constraint, or some combination of these.

Those questions forced me to rethink the whole thing.

Back to first principles

Flow in the circulation requires three conditions. There must be energy available to be dissipated. There must be a continuous conductive pathway. And there must be resistance or impedance through which energy is lost as blood moves.

In the systemic circulation, the available energy is related to the elastic state of the vasculature. That elastic state is often represented by mean systemic pressure, although mean systemic pressure should not be imagined as a literal upstream reservoir pressure sitting somewhere in the body. It is a property of the system: the pressure the systemic circulation would reach if flow stopped and the pressures equilibrated.

The heart must then accept venous return and transfer it into forward output. That acceptance depends on diastolic properties, external constraint, ventricular interaction, systolic reserve and the ability of the heart to convert inflow into ejection. The vascular pathways then determine how energy is dissipated across the organs and systemic circulation.

So flow is not produced by pressure alone. It emerges from the interaction between delivery, acceptance and dissipation.

• Delivery reflects what the circulation can make available.

• Acceptance reflects what the heart can accommodate.

• Dissipation reflects how energy is lost through the vascular pathways.

These are not independent switches. The circulation has to satisfy all of them at once.

What high venous pressure actually means

That is why venous congestion is so easy to misunderstand. A high right atrial pressure is a real pressure. A high central venous pressure is a real clinical finding. But it is not a hand pushing backwards against organ blood flow. It is a pressure state that appears when the circulation cannot accommodate flow in the previous way.

Most often, that means cardiac acceptance has become more constrained. The heart may be unable to accept venous return without a large rise in filling pressure. The right ventricle may be failing. Intrathoracic pressure may be high. The pericardium may be constraining the heart. The left ventricle may be stiff. Ventricular interaction may be important. Or enough intravenous fluid may have been given that the normally compliant right heart has exhausted its volume reserve, accepting further volume only at the cost of a rise in pressure.

Two ways to be congested

There are two very different congestion states.

In the first, acceptance is impaired and global flow falls. The heart cannot accept or transfer venous return effectively. Right atrial pressure rises. Cardiac output falls. Organ perfusion may still be protected for a time by autoregulation, as vascular beds reduce resistance to preserve flow. But that protection has limits. Once the circulation falls below the autoregulatory range, or if autoregulation is impaired, organ blood flow falls because the system can no longer deliver enough flow to that vascular bed.

That situation does not require a separate back-pressure mechanism. The circulation is failing as a whole. Organs receive less blood because there is less forward flow available.

In the second situation, venous pressure rises while global flow is preserved, or may even be high. This is the over-resuscitated state. Cardiac output may have been normal from the start, or initially low because vasodilation had reduced the effective elastic state of the circulation. Fluid loading increases stressed volume and moves the heart up its filling curve. At first, output may increase. But as volume reserve is consumed, further fluid produces more venous pressure than useful additional flow. The circulation is still generating forward output, but from an increasingly congested operating point.

Now the simple pressure-gradient explanation becomes much less useful. If total flow is preserved, reduced perfusion in one organ cannot be explained merely by saying that the venous pressure has risen. The blood has not disappeared. It is still flowing somewhere. The real questions are where it is going, how it is being distributed, and whether the local properties of the affected organ have changed.

The circulation is not one pathway

The body is not one tube. It is a set of parallel vascular beds with very different properties.

The kidney, liver, gut, muscle and skin differ in baseline resistance, vascular tone, venous compliance, autoregulatory capacity, tissue pressure and structural support. Some organs sit in confined spaces. Some are encapsulated. Some are highly sensitive to venous pressure. Some can tolerate venous pressure transmission better, at least initially.

When the constraints of the whole circulation change, flow does not have to fall uniformly. It can be preserved globally but redistributed regionally. That is not paradoxical. It is what a heterogeneous network does. The different organs are not simply experiencing the same reduced gradient with different sensitivity; each vascular bed is resolving a different local pressure-flow state because its resistance, compliance, autoregulation and surrounding tissue pressure are different.

This is one reason venous congestion can be associated with organ dysfunction despite a normal or high cardiac output. Global flow may be acceptable while regional flow has become abnormal. Total flow tells us what the whole circulation is doing. It does not guarantee that every vascular bed is receiving the same share as before.

When congestion changes the organ

Time also matters.

Immediately, organ blood flow can fall if global flow falls. That is straightforward. It can also fall if flow redistributes away from a vulnerable vascular bed. At this early stage, there is no need to invoke oedema or structural injury. The circulation has found a new distribution of flow under altered constraints.

Some organs then develop problems early because venous and capillary congestion alter their local mechanics. The kidney is the obvious example. It is encapsulated, relatively low-compliance and sensitive to changes in interstitial and tubular pressure. A rise in renal venous pressure can increase intrarenal blood volume and capillary pressure. In a confined organ, even small increases in volume can raise tissue pressure. Capillaries and small venous channels are then externally compressed. Microvascular conductance may fall. Filtration may become impaired. The problem is no longer just a number in a pressure equation. The organ itself has become a different conductive pathway.

Later, sustained congestion changes the tissue more obviously. Capillary hydrostatic pressure rises. Filtration increases. Lymphatic drainage may be overwhelmed or impaired. Interstitial oedema develops. As the organ swells within a limited space, interstitial pressure rises. Capillaries, small venous channels and tubules may be compressed. Diffusion distances increase. Local resistance rises and conductance falls further.

At that point, congestion has altered the organ’s internal impedance. Flow may fall because the pathway has changed.

What the evidence shows

This interpretation also makes better sense of the experimental evidence.

Venous outflow obstruction models often show that raising renal venous pressure reduces renal blood flow and GFR, even when systemic arterial pressure and cardiac output are maintained. But the common experimental method is venous constriction, so the model changes the outflow pathway as well as the pressure. These studies are powerful evidence that renal venous congestion can impair renal function. But they are not proof that the mechanism is simply an arithmetic fall in MAP minus venous pressure.

Other findings are more nuanced. Experimental renal venous hypertension does not produce one uniform response in all settings. Effects depend on volume state, neurohumoral tone, baseline renal vascular resistance, whether cardiac output is preserved, and how long congestion persists. Some models show reduced renal blood flow and GFR. Others show preserved or partially preserved filtration despite venous congestion, especially when global haemodynamics are maintained. That variability is not a weakness in the evidence. It is exactly what one would expect if venous congestion acts by changing system constraints and local organ conductance rather than by a single pressure-gradient mechanism.

The abdominal venous congestion models are especially interesting because they attempt to separate congestion from overt forward cardiac failure. When abdominal venous pressure is raised experimentally, cardiac function may remain relatively preserved while renal and hepatic changes develop over time. In one model, renal and hepatic morphological and functional changes occurred despite preserved cardiac function; in another, glomerular hypertension occurred without a concomitant fall in GFR. The pattern is organ-specific and time-dependent, not simply an immediate uniform fall in perfusion caused by a smaller arterial-to-venous pressure difference.

Clinical observations point in the same direction. In heart failure and critical illness, high venous pressures are often associated with worsening renal function. That association matters, but it does not prove the usual back-pressure story. The more coherent mechanism is that impaired cardiac acceptance, often worsened by fluid overload, produces venous and capillary congestion. As tissue pressure rises, whether early in a low-compliance organ or later through interstitial oedema, the microcirculation becomes externally constrained. Capillaries and small venous channels are compressed. Local resistance rises, conductance falls and regional perfusion deteriorates. Flow is reduced because the organ has become a less favourable pathway, not because right atrial pressure has simply reduced an arithmetic pressure gradient.

Venous congestion is part of that constrained state.

The bedside implication

This is why I now think the phrase “perfusion pressure” can mislead us if used carelessly. MAP minus CVP may be a useful descriptor. It may help identify a circulation operating under unfavourable conditions. But it does not explain organ perfusion on its own.

A lower calculated gradient does not tell us whether global flow has fallen, whether flow has redistributed, whether the organ has become more resistive, whether tissue pressure has risen, whether autoregulation has failed, or whether cardiac acceptance has become the dominant constraint.

Those are the questions that matter.

When CVP is high, I now want to know what kind of congestion state I am seeing.

Has global flow fallen because the heart cannot accept and transfer venous return? Has stressed volume been increased to maintain flow at the cost of higher venous pressures? Is the organ vulnerable because it is encapsulated or low-compliance? Is flow being redistributed across a heterogeneous vascular network? Has sustained congestion increased local tissue pressure, oedema and microvascular resistance?

That way of thinking preserves the clinical importance of venous congestion without relying on a misleading back-pressure story.

Venous congestion matters because it tells us the circulation has changed. Acceptance may be constrained. Stressed volume may be excessive. Flow may have redistributed. Local external constraint may be compressing the microcirculation. The system has found a new operating point, and that operating point may be harmful.

So what is venous congestion telling us?

I still think venous congestion is one of the most important causes of organ dysfunction in critical illness.

I just no longer think the usual explanation is good enough.

Sometimes organ flow falls because the whole circulation is failing. Sometimes it falls because congestion has created local external constraint, compressing the microcirculation and making the organ a less favourable pathway for flow.

Both may be accompanied by high venous pressure.

Neither is explained by a pressure gradient pushing blood backwards.

Delivery sets what is available. Acceptance sets what can be accommodated. Resistance determines how energy is dissipated through the available pathways.

Venous congestion matters because that balance has changed.

The pressure gradient is part of the state that results.

It is not the mechanism.

It is one of those statements that is true enough to be useful and simple enough to be dangerous. It gives us a clean mental model: the heart generates pressure, pressure is higher in the arteries than the veins, and blood flows down the gradient.

The equation seems to confirm it:

Q = ΔP / R

Flow equals pressure difference divided by resistance. Pressure difference sits on the right-hand side. Flow is the output. The story almost writes itself.

But the equation does not say what we often make it say.

It describes a relationship between variables in a solved system state. It tells us that, for a given flow through a given resistance, there must be a corresponding pressure difference. It does not tell us that the pressure difference is the independent cause of the flow.

That causal interpretation has to come from somewhere else. It has to come from the physics.

And the physics is more interesting than the shorthand.

What equations do not tell us

The same issue appears every day in clinical haemodynamics.

We write:

CO = (MAP − RAP) / SVR

Cardiac output equals mean arterial pressure minus right atrial pressure, divided by systemic vascular resistance.

Read too quickly, this seems to say that MAP drives cardiac output, RAP opposes it, and SVR sets the degree of obstruction. That reading shapes how clinicians understand what they are doing. Vasopressors are thought to restore perfusion by restoring pressure, when their haemodynamic benefit is better understood as restoring venous return through effects on venous capacitance and stressed volume. A MAP target of 65 is treated as a perfusion guarantee, when it is better understood as a population-derived threshold intended to keep most patients above their autoregulatory lower limit — individually variable, and silent on whether this patient’s organs are actually being perfused above it right now.

But MAP, RAP, SVR and CO are not independent and dependent variables arranged in a neat causal sequence. They are coupled features of the cardiovascular operating state. Change one, and the others may change because the whole system has changed. The equation relates them once the system has resolved into a particular state. It does not specify which variable caused which.

This is not a minor semantic complaint. In haemodynamics, variables that describe the state of the system are constantly mistaken for the things controlling the system. Pressure becomes a driver. Filling pressure becomes preload. Resistance becomes afterload. Central venous pressure becomes a guide to fluid therapy. The equation gives a relationship, and we naively turn it into a mechanism.

Terms in an equation tell us about consistency. They do not automatically reveal causality.

Start with energy

To understand flow, we need to start upstream of the equation.

The principal active source of mechanical energy in the circulation is the heart. It converts metabolic energy into mechanical work and transfers that energy into blood and vascular walls. The vasculature then stores, transmits and dissipates that energy according to its own properties: compliance, resistance, geometry and tone.

But energy input does not automatically become flow.

That is the point the pressure-gradient story tends to hide. When energy enters a constrained fluid system, it has to go somewhere — but where it goes depends on the system. It may be stored as elastic deformation. It may raise pressure without producing movement. It may open a valve. It may recruit a collapsed pathway. It may be dissipated as heat through viscous resistance. Or it may produce bulk flow.

Usually, it does several of these things at once.

So the causal chain is not simply:

ΔP → Q

It is closer to:

Energy input + system properties + pathway state → storage, transmission, dissipation, or pathway change → pressure field and/or flow field

Pressure gradients and flow are not the starting point. They are features of what emerges when energy acts on a constrained system.

The clearest demonstration is what happens when resistance increases. If pressure gradients were the independent driver of flow, a higher gradient should mean more flow. But when resistance rises, flow falls — and the pressure gradient increases as a consequence. The gradient went up while the flow went down. They are not coupled the way the equation implies when read causally.

The pathway has to be open

The left ventricle during isovolumetric contraction is the cleanest way to see this.

During early systole, the ventricle contracts. Myocardial energy is transferred into the chamber-wall-blood system. LV pressure rises rapidly. But there is no forward flow, because the aortic valve is closed.

If pressure simply caused flow, this phase would be hard to explain. Pressure is rising, but nothing is leaving.

The missing condition is pathway state.

With no patent outlet, energy input cannot produce forward flow. It is expressed mainly as pressure rise, wall stress and elastic storage. When LV pressure exceeds aortic pressure, the aortic valve opens — a gating event determined by the pressure threshold. That pressure threshold is important as it changes the state of the pathway.

But opening the pathway is not the same thing as explaining sustained ejection.

Once the valve is open, flow depends on ongoing myocardial energy transfer into an open, impedance-loaded arterial system. LV pressure, aortic pressure and flow then evolve together as coupled features of that operating state. The pressure difference is not an independent engine standing outside the system. It is part of the system’s response to energy input.

So the formulation needs to be precise.

When no patent pathway exists, energy input may produce pressure change without flow. When a patent pathway exists, energy input produces pressure gradients and flow as co-determined features of the operating state. Neither is the independent cause of the other. Both reflect how energy is being stored, transmitted and dissipated through the system.

The LV pressure rise opens the gate. Ongoing myocardial work drives the transfer. Those are not the same thing.

Pressure still matters

None of this makes pressure irrelevant.

Pressure is mechanically important. It opens and closes valves. It determines transmural stress. It influences vessel calibre, collapse, recruitment and filtration. Pressure drops mark where energy is being dissipated through resistance and impedance.

The argument is not that pressure does nothing.

The argument is that pressure is not the original source of energy. It is one of the ways energy appears inside the system — a state variable with mechanical consequences, not an autonomous force generator sitting upstream of flow.

That distinction is often lost because the language of “driving pressure” is so familiar. It is not useless language, but it compresses too much. In a simple passive tube, treating pressure difference as the input is a reasonable experimental setup. But the circulation is not a tube with an imposed pressure difference. It is a closed, elastic, actively energised system in which the pressure field itself is produced dynamically.

The river analogy only gets us so far

The intuition that pressure gradients cause flow is reinforced by an obvious analogy: water flows downhill.

A river flows because gravity acts on water in an elevation field. Water upstream has more gravitational potential energy than water downstream. If a pathway exists, that stored energy is released as flow and dissipated through turbulence and friction.

But the analogy misleads if we forget what is pre-existing.

A dry riverbed can slope before any water flows. The elevation difference is present before the flow begins. The river is releasing stored gravitational potential energy that exists independently of its own operation.

The circulation has no equivalent fixed downhill pressure slope. There is no anatomical pressure gradient from arteries to veins waiting for blood to run down it. The pressure field is generated by cardiac work and vascular elastic storage, moment by moment, in interaction with volume, tone, resistance, compliance and impedance. When effective cardiac work stops, the arterial-venous pressure gradient decays toward equilibrium.

The better comparison is not the fixed riverbed slope — that is the anatomy, the pipework. The better comparison is the pattern of energy loss as water moves through the channel. That pattern only exists during flow. It is not pre-existing geography. And that is closer to what a cardiovascular pressure gradient represents.

Not the engine. The pressure signature of energy distribution and dissipation.

Why this matters at the bedside

This would be an academic distinction if clinicians did not use these variables to make decisions. But we do.

If MAP is treated as the independent driver of cardiac output, then raising MAP becomes synonymous with restoring perfusion. Sometimes that is exactly what is needed. But the reason it helps is not the reason implied by the simple equation. A vasopressor alters venous tone, recruits stressed volume, changes arterial load, modifies cardiac-vascular coupling and affects regional perfusion. The observed MAP is only one visible feature of that changed operating state.

It is entirely possible to raise arterial pressure without improving flow. It is also possible to improve flow with little obvious rise in pressure. Treating the pressure as the driver gives false reassurance when the two diverge.

The same problem appears with CVP. A high CVP is often discussed as if it impedes venous return by reducing the gradient. But RAP is usually not an independent controller — it is the consequence of the relationship between venous delivery and cardiac acceptance. If cardiac acceptance is impaired, RAP rises because the system cannot accept flow without accumulating upstream pressure. The raised RAP is not the original mechanism. It is part of the solved state.

The same logic applies to filling pressures, preload targets and systemic vascular resistance. It applies equally to the x-axis on venous return curves — where plotting RAP as the independent variable implies it controls venous return, when it does not. In each case, a variable that describes the operating state is promoted into a controller. The equation makes that mistake easy. Physiology should make it harder.

Treatment that raises arterial pressure without improving flow is not a theoretical concern. Increased arteriolar tone without corresponding improvement in flow may raise the effective closing pressure of vulnerable vascular beds, allowing regional perfusion to fail despite acceptable global variables. That is the clinical cost of reading equations as causal chains.

The better question

Pressure gradients are indispensable during sustained flow through resistive pathways. They are measurable, useful and often clinically important. No serious model of the circulation can ignore them.

But they are not the driver of circulatory flow.

Flow requires energy input, a conductive pathway and a system through which energy can be stored, transmitted and dissipated. In the circulation, myocardial work and vascular elastic storage generate the pressure field dynamically. The vascular system determines what is possible. Pressure gradients and flow emerge from that interaction.

So when we look at an equation, we should be careful before turning it into a story.

The question is not simply which term sits on which side of the equals sign. The question is whether we are looking at a cause, a constraint, or the resolved state of a coupled system.

Because the equation is not the causal chain.

And the gradient is not the driver.

I now think part of the problem is that we are often taught the variables before we are taught the system.

Blood pressure. Cardiac output. Stroke volume. Heart rate. Systemic vascular resistance. Preload. Afterload. Contractility. Central venous pressure. Mean systemic pressure. Fluid responsiveness. Venous return.

The words become familiar long before the machinery underneath them becomes clear.

This is not because cardiovascular physiology lacks great thinkers. It has been advanced by extraordinary physiologists, clinicians and experimentalists. Starling, Guyton and many others gave us ideas that remain powerful. The problem is what happens when parts of that work are taught too literally, too early, or in isolation from the rest of the system.

A model that explains one relationship becomes a complete story.

An equation becomes a mechanism.

A curve becomes a treatment instruction.

We learn Ohm’s law applied to the circulation. We learn that cardiac output is heart rate multiplied by stroke volume. We learn Starling curves. We learn cardiac function curves. We learn venous return curves. We learn that preload increases stroke volume, afterload opposes ejection, pressure gradients drive flow, and the heart pumps blood around the body.

None of this is useless.

The problem is subtler than that.

The problem is that these ideas are often taught in a way that makes relationships look like levers. A term in an equation starts to look like a controller. A pressure in a model starts to look like a real pressure pushing on the circulation. A curve on a graph starts to look like a direct treatment pathway. A dependent variable starts to look like a cause.

Take cardiac output. The equation is simple:

Cardiac output = heart rate × stroke volume.

It is a useful relationship. But it can also mislead. If cardiac output is heart rate multiplied by stroke volume, it is tempting to think that increasing heart rate must increase cardiac output. Sometimes it does. Often it does not.

Change heart rate and the rest of the system does not stay still. Filling time changes, so stroke volume tends to change in the opposite direction. Within a broad physiological range, cardiac output may therefore change surprisingly little. Ventricular interaction, myocardial oxygen demand and vascular loading may also change. The circulation does not simply obey the arithmetic. It settles into a new state. At very fast or very slow rates, the system fails for different reasons.

The equation remains true. The interpretation was the problem.

The same thing happens with cardiac function curves. We are shown that increasing inotropy shifts the curve upwards, and cardiac output rises. Again, there is truth in this. A stronger ventricle may eject better. But the circulation is not a ventricle in isolation. Output also depends on venous return, vascular tone, stressed volume, ventricular filling, arterial load, impedance, ventricular-arterial coupling and the ability of the heart to accept venous return without excessive rise in pressure.

The curve is useful. But the patient is not the curve.

Starling’s law is another example. It is often taught as if increasing right atrial pressure, or filling pressure, increases cardiac output. In one sense, that is the classic curve: more filling, more stretch, more stroke volume.

But at the bedside, right atrial pressure is usually not an independent handle we can turn. It is part of the state into which the system has settled.

A rising right atrial pressure is usually not evidence that venous return has improved. It is more often evidence that the heart is failing to accept what is being returned. That distinction is not a detail. It changes what fluid means.

This became hard to ignore in intensive care.

I trained in an era when much of haemodynamic resuscitation still aimed to increase cardiac output by giving large volumes of fluid. The logic was simple and fitted the diagrams. Fluid would increase filling pressure. Filling pressure would increase stroke volume. Stroke volume would increase cardiac output. Cardiac output would improve perfusion.

At the time, this felt like applying physiology. But the patients in front of me often seemed to get worse.

They were oedematous, ventilator dependent, vasoplegic, congested and still shocked. More fluid seemed to increase right atrial pressure without restoring useful flow. It raised venous pressure and capillary pressure while worsening organ function. The patient looked more “filled” while the circulation looked less coherent.

That was not just incomplete physiology. In many cases, it was wrong physiology. And wrong physiology can be dangerous.

Aiming for high right atrial pressures was not a sophisticated haemodynamic strategy. It was often a dangerous misunderstanding of what the pressure represented. The pressure was not proof that the circulation had been usefully loaded. It was often the sign that the system could no longer accept what we were giving it.

That realisation changed how I thought about haemodynamics.

The same problem appears in the way venous return is taught.

Guyton’s equation is elegant:

Venous return is related to the difference between mean systemic pressure and right atrial pressure, divided by resistance to venous return.

As an abstraction, this is powerful. It points to something real: the circulation has elastic properties; stressed volume matters; the vasculature is not just passive tubing; the heart and vessels interact.

But if taught too literally, the equation can become deeply misleading.

Mean systemic pressure is sometimes described as if it were a real upstream pressure source pushing blood back to the heart during flow. Right atrial pressure is described as a back pressure opposing venous return. The equation starts to sound like a hydraulic circuit with a hidden pressure reservoir at one end and the right atrium at the other.

But mean systemic pressure does not exist as a directly measurable pressure during ongoing flow. It is an abstract pressure defined under no-flow conditions. It tells us something about the elastic state of the circulation, but it is not a hidden motor sitting upstream of the right atrium.

Nor does the venous system supply energy for flow as though it were a second pump.

That point matters, because it prevents one simplification being replaced by another.

The heart does not simply “drive” the circulation alone. But neither does venous return “drive” cardiac output in isolation. The heart supplies energy. The vasculature stores, distributes and dissipates energy. The observed flow is the result of their interaction.

The circulation is a coupled system.

That sounds obvious, but it is often missing from the way haemodynamics is taught.

Change one part of the circulation and another usually changes with it. Increase vascular tone and you may alter arterial pressure, venous return, stressed volume, right atrial pressure, ventricular loading, cardiac output and regional flow. Give fluid and you may alter stressed volume, venous pressure, cardiac filling, cardiac output, capillary pressure and oedema formation. Improve cardiac function and you may lower right atrial pressure while increasing venous return and cardiac output. Raise downstream pressure and you may preserve global pressure while impairing regional perfusion.

The system re-solves.

That is the missing idea.

Equations and curves are often presented as if we can alter one term while the others politely stay still. Real circulations do not behave like that. The variables we measure are usually descriptions of a resolved system state, not independent causes waiting to be manipulated.

Pressure is a particularly good example.

We use pressure language constantly. Pressure pushes. Back pressure opposes. Afterload pushes back. Filling pressure fills. Venous pressure limits venous return. A pressure gradient drives flow.

Some of this language is useful shorthand. But it also smuggles in a mechanical picture that can become too simple.

Pressure gradients are required for flow in resistive pathways, but they are not free-standing causes detached from the system that generated them. Gradients arise with flow, energy input, elastic storage, dissipation, vascular tone, impedance and boundary conditions. A pressure difference is part of the solved state of the system. It is not an explanation by itself.

The same applies to afterload. We often speak as if the ventricle ejects and afterload pushes back. That is a useful image up to a point. But afterload is not a single pressure object sitting outside the ventricle. It reflects arterial pressure, elastance, impedance, vascular tone, wave reflection, timing and ventricular-arterial coupling. It is not one thing opposing another thing. It is a constraint imposed by the arterial system on ventricular ejection.

The language of drivers and opposition is tempting. It is also often too crude.

This is why haemodynamics becomes confusing. Not because clinicians are stupid. Not because the equations are useless. But because the language often turns system relationships into causal stories.

CVP becomes preload. Fluid responsiveness becomes hypovolaemia. Blood pressure becomes perfusion. Mean systemic pressure becomes a hidden upstream pressure. Right atrial pressure becomes a back pressure. Starling’s law becomes a treatment instruction. Cardiac output becomes what the heart does.

Each step is understandable. Each contains some truth. Each can mislead.

The more I have thought about cardiovascular physiology, the more I have moved away from looking for single drivers. I am more interested now in limits and constraints.

What is the system capable of doing? Where is energy being added? Where is energy being stored? Where is it being dissipated? What limits venous return? What limits cardiac acceptance? What limits ventricular ejection? What is the downstream boundary condition? What regional beds are being sacrificed to preserve global pressure?

And, most importantly:

Why has the system settled here?

Not merely: what is the blood pressure?

Not merely: what is the cardiac output?

Not merely: what is the CVP?

But: what combination of cardiac function, vascular tone, stressed volume, compliance, resistance, impedance, venous pressure, capillary pressure, ventricular interaction and regional perfusion has produced this state?

That is a different way of thinking.

It is not anti-equation. It is not anti-Guyton. It is not anti-Starling. It is not anti-echo, anti-monitoring, or anti-bedside shorthand.

It is anti-literalism.

The equations are useful when we understand what kind of thing they are. Curves are useful when we understand what has been held constant, what has been abstracted away, and what is actually being described. Pressures are useful when we understand whether they represent energy, constraint, consequence, boundary condition, or measurement artefact.

The problem is not that haemodynamics has too many variables.

The problem is that too often we are taught the variables before we are taught the system.

That is why this Substack is called The Dependent Variable. It captures the mistake I kept encountering: taking a measured output of a coupled system and treating it as the thing that controls the system.

I am not going to build this publication as a perfect textbook from chapter one onwards. That would be tempting, but probably fatal. What has helped me most is going back to first principles and asking whether the usual explanations remain coherent when tested against the underlying physics: pressure, flow, energy, resistance, impedance, compliance, capacitance and the behaviour of coupled physical systems. The aim is not to produce a finished doctrine, but to return to familiar haemodynamic ideas from that ground-up perspective.

Each topic has its own details. But the same question will keep returning:

Are we looking at a cause, a constraint, or the resolved state of a coupled system?

That question has changed how I think about haemodynamics.

I hope it changes how you think about it too.

That claim sits at the centre of this project, because it exposes a larger problem in clinical cardiovascular physiology. We often take variables from equations, curves and bedside monitors, then treat them as if they are independent drivers of the circulation. Usually they are better understood as descriptors of the state into which the system has settled.

A term in an equation is not necessarily a controller. A pressure in a model is not necessarily a back pressure in the real circulation. A gradient associated with flow is not necessarily an autonomous force driving flow. A point on a venous return curve is not necessarily an independent input. A measured value at the bedside may be important without being causal in the way we assume.

This publication is called The Dependent Variable because cardiovascular physiology becomes clearer when we ask a simple question:

Are we looking at a cause, a constraint, or the resolved state of a coupled system?

Cardiac output is not an isolated property of the heart. It is the flow that emerges from the interaction between the heart and the circulation. Blood pressure is not perfusion. It is one expression of the mechanical and energetic state of the system. Central venous pressure is not simply preload. It is the pressure at the interface between venous return and cardiac acceptance. Fluid responsiveness is not hypovolaemia. It is a property of where the circulation sits on a functional curve.

None of these variables is useless. They are clues. But they mislead when we treat them as independent mechanisms.

I have often described my work as haemodynamics. That is true, but it is too narrow. Lots of people write about haemodynamics. The more interesting question is how we should think about the cardiovascular system in the first place.

The circulation is not a set of independent numbers. It is a coupled physical system.

Change one part, and another usually changes with it. Increase vascular tone and you may alter arterial pressure, venous return, stressed volume, right atrial pressure, cardiac output, ventricular loading and regional flow. Give fluid and you may increase stressed volume, venous pressure, cardiac filling, cardiac output, capillary pressure and oedema formation. Improve cardiac function and you may lower right atrial pressure while increasing venous return and cardiac output. Raise downstream pressure and you may preserve global flow while impairing regional perfusion.

This is why simple driver language often fails.

The circulation is not controlled by one variable pushing on another. It is constrained by the properties of the system: cardiac power, vascular tone, stressed volume, compliance, capacitance, resistance, impedance, venous return, ventricular interaction, collapsible vessels, downstream pressures and regional vascular beds.

That is why some familiar ideas need to be handled carefully.

Pressure gradients are required for flow in resistive pathways, but they are not free-standing drivers detached from the system that generated them. Guyton’s venous return equation is a powerful abstraction, but it can become misleading if mean systemic pressure is treated as a literal upstream pressure pushing blood back to the heart during ongoing flow. Venous return curves can clarify the interaction between heart and vessels, but they can also mislead if we forget that several of their terms are dependent on the state of the coupled system. Starling’s law describes how the heart responds to filling, but it does not mean the heart alone determines flow.

These are not semantic objections. They affect bedside reasoning.

If we call CVP “preload”, we may treat a downstream pressure as if it directly measures cardiac filling. If we call fluid responsiveness “volume depletion”, we may give fluid because of curve position rather than deficit. If we call blood pressure “perfusion”, we may mistake a pressure state for tissue flow. If we say the heart “generates cardiac output”, we may forget that cardiac output is only possible because the vasculature returns blood to the heart at the same rate. But the opposite error is just as important. The venous system does not supply energy for flow as though it were a second pump. Mean systemic pressure is not a hidden motor sitting upstream of the right atrium. It is an abstract description of the elastic state of the circulation under defined conditions. The heart supplies energy; the vasculature stores, distributes and dissipates it; the observed flow is the result of their interaction.

The heart and vessels do not act in sequence.

They interact.

The numbers we measure are outputs of that interaction. They are not meaningless, but they are not self-explanatory.

The same problem appears in fluid physiology. Sepsis is often described as if vasodilatation and capillary leak automatically produce hypovolaemia. Oedema is described as “fluid in the wrong place”, as though the interstitial compartment can expand while the circulation remains meaningfully empty at steady state. Albumin is sometimes discussed as if it pulls fluid back into the circulation by a simple oncotic trick, rather than acting within a dynamic system of capillary pressure, endothelial permeability, glycocalyx function, lymphatic return and venous pressure. These are not separate topics. They are the same problem: dependent variables, simplified language and system behaviour being mistaken for direct mechanisms.

That is the recurring theme of The Dependent Variable: not replacing one slogan with another, but making the machinery underneath the slogan visible.

If there is a single promise behind The Dependent Variable, it is this:

We will take the familiar language of cardiovascular physiology and ask what it actually means.

Not what we casually use it to mean.

Not what it implies in a teaching diagram.

Not what it seems to mean when we are rushing at the bedside.

What it actually means.

Because in cardiovascular physiology, the most important question is often not whether a variable is high or low.

It is whether we have mistaken the dependent variable for the cause.