Fundamental Principles of Hemodynamics #1: Hemodynamic Ohm’s Law in Circulation

1. Core Concept

The circulation can be understood using the hemodynamic equivalent of Ohm’s law:

ΔP = Q × R

or

Q = ΔP / R

Where:

• Q = blood flow
• ΔP = pressure gradient between upstream and downstream compartments
• R = vascular resistance

This is analogous to the electrical equation:

I = ΔV / R

Where electrical current is driven by a voltage gradient and opposed by electrical resistance.

In cardiovascular physiology, blood flow is not determined by pressure alone. It is determined by the pressure difference across a vascular bed divided by the resistance of that bed. This pressure-flow-resistance relationship forms the basis for interpreting systemic vascular resistance, pulmonary vascular resistance, venous return, transpulmonary gradient, Fontan physiology, and shunt circulation [1–3].

2. Pressure Gradient, Not Absolute Pressure, Drives Flow

A central principle in clinical hemodynamics is that flow depends on the difference between upstream and downstream pressure.

For the systemic circulation:

Cardiac output ≈ (MAP − RAP) / SVR

Where:

• MAP = mean arterial pressure
• RAP = right atrial pressure / central venous pressure
• SVR = systemic vascular resistance

Therefore, an apparently acceptable MAP does not always indicate adequate systemic perfusion. If downstream venous pressure is elevated, the effective systemic perfusion gradient is reduced. The clinically relevant gradient is:

MAP − RAP

not MAP alone.

For the pulmonary circulation:

Pulmonary blood flow ≈ (mPAP − LAP) / PVR

Where:

• mPAP = mean pulmonary arterial pressure
• LAP = left atrial pressure or pulmonary venous atrial pressure
• PVR = pulmonary vascular resistance

The pressure gradient across the pulmonary vascular bed is the transpulmonary gradient:

TPG = mPAP − LAP

Thus:

PVR = TPG / Qp

This framework is especially important when evaluating pulmonary hypertension, Fontan candidacy, postoperative cavopulmonary circulation, and residual pulmonary artery or pulmonary venous obstruction [3,4].

3. Application to Normal Biventricular Circulation

In normal biventricular circulation, the right and left ventricles are arranged in series.

The right ventricle generates flow through the pulmonary vascular bed:

Qp = ΔPpulmonary / PVR

The left ventricle generates flow through the systemic vascular bed:

Qs = ΔPsystemic / SVR

In the absence of shunts:

Qp = Qs

However, the two vascular beds operate under very different pressure-resistance conditions.

The systemic circulation normally has:

• High vascular resistance
• High arterial pressure
• A high-pressure systemic ventricle

The pulmonary circulation normally has:

• Low vascular resistance
• Low arterial pressure
• A low-pressure subpulmonary ventricle

This difference is fundamental to congenital heart surgery because many congenital lesions and surgical reconstructions alter the relationship between ventricles, vascular beds, pressure gradients, and resistance.

4. Systemic Vascular Resistance

Systemic vascular resistance represents the opposition to flow through the systemic arterial and arteriolar circulation.

Conceptually:

SVR = (MAP − RAP) / Cardiac Output

Clinically, SVR rises with vasoconstriction and falls with vasodilation.

Examples of increased SVR include:

• Hypothermia
• Vasopressor excess
• Systemic vasoconstriction
• Pain or agitation
• Low cardiac output with compensatory vasoconstriction

Examples of decreased SVR include:

• Vasodilatory shock
• Sepsis
• Anesthetic vasodilation
• Inflammatory states after cardiopulmonary bypass
• Excessive vasodilator therapy

In congenital cardiac patients, changes in SVR can strongly influence:

• Shunt direction
• Systemic oxygen delivery
• Pulmonary-to-systemic flow ratio
• Ventricular loading conditions
• Diastolic pressure and coronary perfusion
• Balance between systemic and pulmonary circulations

A reduction in SVR may increase right-to-left shunting in lesions such as Tetralogy of Fallot or reduce systemic perfusion in parallel single-ventricle circulation. Conversely, excessive SVR may increase ventricular afterload and impair cardiac output.

5. Pulmonary Vascular Resistance

Pulmonary vascular resistance represents the opposition to flow through the pulmonary vascular bed.

Conceptually:

PVR = (mPAP − LAP) / Qp

PVR is affected by:

• Oxygenation
• Carbon dioxide level
• Blood pH
• Lung volume
• Pulmonary vascular tone
• Pulmonary vascular development
• Pulmonary venous pressure
• Pulmonary artery size and architecture
• Surgical distortion or obstruction of the pulmonary arteries

Small increases in PVR may have major clinical consequences, especially in:

• Single-ventricle physiology
• Bidirectional Glenn circulation
• Fontan circulation
• Pulmonary hypertension
• Borderline pulmonary arteries
• Postoperative pulmonary hypertensive crisis
• Pulmonary venous obstruction

Because pulmonary blood flow is inversely related to PVR, even a modest rise in PVR can reduce Qp when the driving pressure is limited. This is particularly important in cavopulmonary circulation, where pulmonary blood flow depends on passive venous-to-pulmonary arterial flow rather than ventricular ejection.

6. Venous Return and the Downstream Side of Flow

Ohm’s law also applies to venous return.

A simplified relationship is:

Venous return ≈ (Mean systemic filling pressure − RAP) / Resistance to venous return

This concept emphasizes that cardiac output is not only determined by ventricular contractility or arterial pressure. It also depends on the pressure gradient that returns blood to the heart [2].

Venous return decreases when:

• RAP/CVP is elevated
• Mean systemic filling pressure is low
• Venous capacitance is increased
• Resistance to venous return is increased
• Intrathoracic pressure is elevated
• Ventricular compliance is impaired

This is clinically relevant after congenital heart surgery because elevated right atrial pressure, elevated Fontan pressure, tamponade physiology, ventricular diastolic dysfunction, positive-pressure ventilation, or pathway obstruction can reduce the effective gradient for venous return.

7. Fontan Physiology: Ohm’s Law Without a Subpulmonary Ventricle

Fontan circulation is one of the clearest clinical examples of hemodynamic Ohm’s law.

In Fontan physiology, there is no subpulmonary ventricle. Pulmonary blood flow is driven passively by the pressure gradient between the systemic venous pathway and the pulmonary venous atrium.

Simplified:

Fontan flow ≈ (CVP − LAP) / PVR

Therefore, Fontan cardiac output depends critically on:

• Low PVR
• Low pulmonary venous atrial pressure
• Low ventricular end-diastolic pressure
• Unobstructed pulmonary arteries
• Unobstructed Fontan pathway
• Competent atrioventricular valve
• Preserved ventricular compliance
• Preserved systolic function
• Favorable respiratory mechanics

The Fontan circulation requires chronically elevated systemic venous pressure to drive pulmonary blood flow through a low-resistance pulmonary vascular bed. This explains why Fontan physiology is vulnerable to even small increases in PVR, pulmonary venous pressure, ventricular filling pressure, or pathway resistance [4,7].

8. Fontan Resistance and Cardiac Output

In biventricular circulation, the right ventricle can increase pressure generation to maintain pulmonary blood flow when pulmonary resistance rises. In Fontan circulation, this compensatory mechanism is absent.

As a result, cardiac output becomes highly sensitive to:

• PVR
• Total cavopulmonary connection resistance
• Pulmonary artery stenosis
• Fontan conduit obstruction
• Elevated ventricular end-diastolic pressure
• Atrioventricular valve regurgitation
• Pulmonary venous obstruction

Computational and physiologic studies show that total cavopulmonary connection resistance can significantly affect resting hemodynamics and exercise capacity in single-ventricle circulation. The Fontan circulation demonstrates markedly greater cardiac output sensitivity to cavopulmonary resistance than normal biventricular circulation [5].

Similarly, modeling of Fontan hemodynamics has shown that PVR plays a major role in regulating cardiac output. This is clinically intuitive: when the only driving pressure for pulmonary blood flow is systemic venous pressure, any increase in pulmonary resistance directly reduces preload to the systemic ventricle [6].

9. Clinical Consequences of Elevated Resistance in Fontan Circulation

In Fontan physiology, elevated PVR or pathway resistance may produce:

• Low cardiac output
• Elevated systemic venous pressure
• Cyanosis if fenestration or venovenous collateral flow is present
• Pleural effusions
• Ascites
• Hepatic congestion
• Protein-losing enteropathy
• Plastic bronchitis
• Exercise intolerance
• Renal dysfunction
• Lymphatic congestion
• Progressive Fontan failure

Evidence from late Fontan failure and post-transplant cohorts supports the clinical importance of pulmonary vascular disease. Elevated pulmonary vascular resistance has been described as a contributor to Fontan failure and may persist or become evident after transplantation [8].

Hemodynamic profiling using cardiac index and systemic vascular resistance also demonstrates that Fontan outcomes are not determined by a single pressure value. Rather, risk reflects the interaction between cardiac output, SVR, PVR, ventricular function, venous pressure, and pulmonary vascular reserve [9].

10. Shunt Circulation

Ohm’s law also explains systemic-to-pulmonary shunt physiology.

For a systemic-to-pulmonary shunt:

Shunt flow ≈ (Systemic arterial pressure − Pulmonary arterial pressure) / Shunt resistance

Shunt flow increases when:

• Systemic arterial pressure rises
• Pulmonary arterial pressure falls
• PVR decreases
• Shunt resistance decreases
• Shunt diameter is larger

Shunt flow decreases when:

• Systemic arterial pressure falls
• Pulmonary arterial pressure rises
• PVR increases
• The shunt becomes narrowed, kinked, thrombosed, or compressed
• Distal pulmonary artery resistance is elevated

This principle applies to:

• Modified Blalock–Taussig shunt
• Central shunt
• Ductal stent
• RV–PA conduit
• Hybrid palliation
• Systemic-to-pulmonary collateral flow

Excessive pulmonary blood flow may cause:

• Pulmonary overcirculation
• Systemic hypoperfusion
• Low diastolic pressure
• Coronary steal
• Respiratory failure
• Metabolic acidosis
• Feeding intolerance
• Ventricular volume overload

Inadequate shunt flow may cause:

• Cyanosis
• Low pulmonary blood flow
• Hypoxemia
• Shunt thrombosis
• Hemodynamic collapse

The goal is not simply “more pulmonary blood flow.” The goal is balanced flow.

11. Qp/Qs and Congenital Heart Disease

In congenital heart disease, abnormal communications between systemic and pulmonary circulations alter pressure-flow relationships.

The magnitude and direction of shunting depend on:

• Size of the communication
• Pressure gradient across the defect
• SVR
• PVR
• Ventricular compliance
• Atrioventricular valve function
• Pulmonary venous pressure
• Systemic outflow obstruction
• Pulmonary outflow obstruction
• Downstream vascular resistance

Examples:

Large VSD

In a large ventricular septal defect, shunt magnitude is strongly influenced by the balance between SVR and PVR. When PVR falls after birth, left-to-right shunting increases, producing pulmonary overcirculation and left heart volume loading.

PDA

In a patent ductus arteriosus, systemic-to-pulmonary flow depends on the aortic-to-pulmonary arterial pressure gradient and ductal resistance. A large, low-resistance PDA can produce excessive pulmonary blood flow and systemic diastolic runoff.

ASD

In atrial septal defect, shunt flow is often determined more by ventricular compliance than by pressure alone. Greater right ventricular compliance favors left-to-right atrial shunting and right heart volume loading.

Single-Ventricle Parallel Circulation

In single-ventricle physiology before cavopulmonary palliation, systemic and pulmonary blood flow compete in parallel. Qp/Qs depends on the relative resistance of the pulmonary and systemic circuits. A fall in PVR may increase Qp at the expense of systemic blood flow.

Thus, the same anatomic defect may produce very different physiology depending on resistance, compliance, and downstream pressure.

12. Clinical Interpretation at the Bedside

The equation:

Q = ΔP / R

provides a practical framework for bedside interpretation.

When flow is low, ask:

1. Is upstream pressure too low?

Examples:

• Low MAP
• Poor ventricular systolic function
• Hypovolemia
• Inadequate preload
• Excessive vasodilation

2. Is downstream pressure too high?

Examples:

• Elevated RAP/CVP
• Elevated LAP
• Elevated ventricular end-diastolic pressure
• Pulmonary venous obstruction
• Tamponade physiology
• High intrathoracic pressure

3. Is resistance too high?

Examples:

• Elevated SVR
• Elevated PVR
• Pulmonary artery stenosis
• Shunt obstruction
• Fontan pathway resistance
• Pulmonary vascular disease

4. Is the pathway anatomically obstructed?

Examples:

• Narrowed Fontan conduit
• Branch pulmonary artery stenosis
• Shunt thrombosis
• Aortic arch obstruction
• Pulmonary venous obstruction
• Residual outflow tract obstruction

This approach is useful in postoperative congenital heart surgery, where low cardiac output, cyanosis, high venous pressure, pulmonary overcirculation, or poor systemic perfusion may result from different combinations of pressure gradient and resistance.

13. Important Limitations

Hemodynamic Ohm’s law is a conceptual model, not a complete description of cardiovascular physiology.

Real circulation is affected by:

• Pulsatile flow
• Vascular compliance
• Ventricular contractility
• Diastolic function
• Autoregulation
• Blood viscosity
• Hematocrit
• Vessel recruitment and distension
• Turbulence
• Stenotic lesions
• Nonlinear pressure-flow relationships
• Respiratory mechanics
• Ventricular–vascular coupling

In stenotic lesions, pressure drop may increase disproportionately with flow. In such settings, the relationship may resemble Bernoulli physiology more than simple Ohmic resistance.

Despite these limitations, ΔP = Q × R remains one of the most useful organizing principles in clinical hemodynamics.

Key Takeaway

Blood flow is governed by a pressure gradient and opposed by resistance.

Q = ΔP / R

This principle explains systemic perfusion, pulmonary blood flow, SVR, PVR, transpulmonary gradient, venous return, Fontan physiology, and shunt circulation.

In congenital heart surgery, understanding the interaction between pressure gradient, resistance, and anatomy is essential for interpreting physiology, anticipating complications, and guiding perioperative management.

References

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[2] Jacobsohn E, Chorn R, O’Connor MF. The role of the vasculature in regulating venous return and cardiac output: historical and graphical approach. Can J Anaesth. 1997;44(8):849-867.

[3] Venkateshvaran A, Tossavainen E, Borneteg C, Tureli HO, Vanoli D, Lund L, Flachskampf F, Lindqvist P. A novel echocardiographic estimate of pulmonary vascular resistance employing the hydraulic analogy to Ohm’s law. Int J Cardiol Heart Vasc. 2022;42:101121.

[4] Gewillig M. The Fontan circulation. Heart. 2005;91(6):839-846.

[5] Sundareswaran K, Pekkan K, Dasi L, Whitehead K, Sharma S, Kanter K, Fogel M, Yoganathan A. The total cavopulmonary connection resistance: a significant impact on single ventricle hemodynamics at rest and exercise. Am J Physiol Heart Circ Physiol. 2008;295(6):H2427-H2435.

[6] Liang F, Senzaki H, Kurishima C, Sughimoto K, Inuzuka R, Liu H. Hemodynamic performance of the Fontan circulation compared with a normal biventricular circulation: a computational model study. Am J Physiol Heart Circ Physiol. 2014;307(7):H1056-H1072.

[7] Mazza G, Gribaudo E, Agnoletti G. The pathophysiology and complications of Fontan circulation. Acta Biomed. 2021;92(5):e2021260.

[8] Mitchell M, Campbell D, Ivy D, Boucek M, Sondheimer H, Pietra B, Das B, Coll J. Evidence of pulmonary vascular disease after heart transplantation for Fontan circulation failure. J Thorac Cardiovasc Surg. 2004;128(5):693-702.

[9] Kawasaki Y, Sasaki T, Kobayashi D. Hemodynamic profiling using a cardiac index–systemic vascular resistance plot in patients with Fontan circulation. Congenit Heart Dis. 2023;18(4):405-415.