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 the circulation cannot accommodate volume. It clearly can.

The problem is that “capacitance” is used a lot while meaning different things to different people. It sounds precise, but in cardiovascular physiology it often lacks a stable, agreed meaning.

In electrical circuits, capacitance has a clear definition: how much charge is stored for a given voltage.

In hydraulic terms, the closest equivalent would be how much volume changes for a given change in pressure.

But in haemodynamics we already have the correct word for that.

Compliance.

Compliance is:

dV / dP

If a vascular compartment accepts a large change in volume with only a small change in pressure, it is compliant. If a small change in volume produces a large change in pressure, it is stiff.

If we mean the opposite relationship — how much pressure changes for a given change in volume — then the correct word is:

Elastance.

Elastance is:

dP / dV

And if we mean how much volume a compartment can contain, that is not capacitance either.

That is:

Capacity.

A rigid one-litre bucket has a capacity of one litre. But it does not have high compliance. If it is closed and truly rigid, it cannot accept extra volume without a large rise in pressure.

Capacity and compliance are not the same thing.

This is where “vascular capacitance” becomes a problem.

Sometimes it is used to mean compliance.

Sometimes it is used to mean capacity.

Sometimes it is used to mean venous tone.

Sometimes it is used to mean unstressed volume.

Sometimes it is used to mean the broader ability of the venous system to contain blood without increasing the effective filling state.

Those are not the same thing.

So when someone says “venous capacitance increased”, what do they actually mean?

Did the venous system become more compliant?

Did elastance fall?

Did venous tone decrease?

Did the available containing volume increase?

Did unstressed volume increase?

Did blood redistribute into regions that could accommodate volume more easily?

Did the same circulating volume now generate a lower effective filling state?

Those are different mechanisms with different consequences.

This matters because haemodynamics is already full of shorthand that sounds causal but is often only descriptive. “Capacitance” is one of those terms. It is technical enough to sound explanatory, but vague enough to hide the actual mechanism.

A better approach is to say what we actually mean.

If we mean volume-containing size, say capacity.

If we mean volume change for pressure change, say compliance.

If we mean pressure change for volume change, say elastance.

If we mean venous tone, say venous tone.

If we mean unstressed volume, say unstressed volume.

If we mean redistribution of blood into a more accommodating vascular region, say that. This is when a broader phrase can be useful as long as we know what we mean. I use vascular accommodation. This is the distributional consequence of the vascular elastic state: the ability of the circulation, especially the venous circulation, to contain blood volume without presenting it to the heart as effective filling volume. It should not be treated as a single physical property. It describes the net behaviour of the system. The next question should always be:

what mechanism produced it?

That is the problem with “capacitance”. It can sound like a mechanism when it often isn’t one. It’s generally blurring capacity, compliance, elastance, tone, unstressed volume and accommodation into one word, so it is probably not helping us think clearly.

Use the mechanism when you know it.

Use broader language only when you mean broader behaviour. And use a term that people will precisely understand

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.