Cardiovascular models
Cardiovascular models describe how the heart generates pressure and flow, how blood moves through vessels, and how vessel properties shape the changing pressure waveform.
Pressure difference drives flow
A basic lumped relation is
\[\boxed{Q=\frac{P_1-P_2}{R}=\frac{\Delta P}{R}.}\]Here \(Q\) is volumetric flow, \(P_1-P_2\) is the pressure difference, and \(R\) is hydraulic resistance.
Only a pressure difference drives flow in this relation. If \(P_1=P_2\), then \(Q=0\).
How resistance changes flow
For fixed pressure difference,
\[Q(\Delta P)=\frac{\Delta P}{R}.\]Increasing resistance reduces the flow produced by the same pressure difference.
Where vascular resistance comes from
For ideal steady laminar flow of a Newtonian fluid through a rigid cylindrical tube, Poiseuille's law gives
\[\boxed{R=\frac{8\mu L}{\pi r^4}.}\]Here \(\mu\) is viscosity, \(L\) is tube length and \(r\) is radius.
The fourth power is important: under these assumptions, relatively small changes in radius can cause large changes in resistance.
Radius has a strong effect
Since
\[R\propto\frac1{r^4},\]doubling radius would reduce resistance by a factor of \(2^4=16\) in the ideal Poiseuille model.
Compliance
Blood vessels can expand when pressure rises. Compliance measures the change in volume associated with a change in pressure:
\[\boxed{C=\frac{dV}{dP}.}\]If compliance is approximately constant over the range considered,
\[\Delta V\approx C\,\Delta P.\]A highly compliant compartment can accept more volume for a given increase in pressure.
Conservation of blood volume
For a vascular compartment,
\[\boxed{\frac{dV}{dt}=Q_{in}-Q_{out}.}\]This is simply conservation: volume increases when inflow exceeds outflow and decreases when outflow exceeds inflow.
From volume balance to pressure dynamics
If \(dV=C\,dP\), then
\[C\frac{dP}{dt}=Q_{in}-Q_{out}.\]If outflow passes through a resistance \(R\) toward a downstream pressure \(P_v\),
\[Q_{out}=\frac{P-P_v}{R}.\]Therefore
\[\boxed{C\frac{dP}{dt}=Q_{in}(t)-\frac{P-P_v}{R}.}\]This single equation is a fundamental lumped cardiovascular model.
The RC time constant
When inflow stops,
\[C\frac{dP}{dt}=-\frac{P-P_v}{R}.\]The solution is
\[\boxed{P(t)=P_v+(P_0-P_v)e^{-t/(RC)}.}\]The product
\[\boxed{\tau=RC}\]is the time constant. It controls how quickly pressure decays toward downstream pressure.
Pressure decay from the model
The Windkessel idea
Large arteries do more than resist flow. Their elasticity stores some of the blood ejected during systole and releases it during diastole.
This smooths the strongly pulsatile output of the heart into a less intermittent peripheral flow.
A two-element Windkessel model
The simplest Windkessel contains arterial compliance \(C\) and peripheral resistance \(R\):
\[\boxed{C\frac{dP}{dt}=Q_h(t)-\frac{P-P_v}{R}.}\]The heart-flow function \(Q_h(t)\) drives the system, and the equation determines arterial pressure.
A mathematically defined pulsatile input
For illustration, a non-negative periodic ejection function can be defined by
\[Q_h(t)=Q_0\max\{0,\sin(2\pi t/T)\}.\]This is not hand-drawn: at every time, cardiac inflow is calculated from the stated function.
Why arterial pressure does not instantly follow cardiac flow
Compliance stores volume while the heart is ejecting and releases stored volume after ejection falls. Pressure therefore has its own dynamics and does not simply equal a scaled copy of instantaneous cardiac flow.
Systolic and diastolic pressure
Systolic pressure is associated with ventricular ejection and arterial loading. During diastole, arterial pressure is maintained partly by elastic recoil while blood continues to leave through the peripheral resistance.
The Windkessel model captures this mechanism in simplified mathematical form.
Pulse pressure
Pulse pressure is
\[\boxed{P_{pulse}=P_{systolic}-P_{diastolic}.}\]In lumped models, pulse pressure depends on the interaction of stroke input, arterial compliance, peripheral resistance and timing.
Mean arterial pressure
A common coarse relationship is
\[\boxed{MAP- P_v\approx CO\times SVR,}\]where \(CO\) is cardiac output and \(SVR\) is systemic vascular resistance.
This is the same pressure–flow–resistance principle applied to the systemic circulation.
Cardiac output
Cardiac output is the volume pumped per unit time:
\[\boxed{CO=HR\times SV,}\]where \(HR\) is heart rate and \(SV\) is stroke volume.
This algebraic relation is useful, but models of stroke volume require ventricular filling, contraction and afterload to be considered.
The heart as a time-varying pump
A common lumped approach models ventricular pressure using time-varying elastance:
\[\boxed{P_v(t)=E(t)\,[V_v(t)-V_0].}\]The function \(E(t)\) is low during relaxation and high during contraction.
This allows ventricular pressure to change because both ventricular volume and myocardial stiffness change during the cardiac cycle.
Valves as directional flow elements
An idealised valve can be represented by
\[\boxed{Q=\max\left(0,\frac{P_{up}-P_{down}}{R_v}\right).}\]Flow occurs when upstream pressure exceeds downstream pressure and is zero in the reverse direction.
This simple rule creates switching between filling and ejection phases.
Pressure–volume loops
Plotting ventricular pressure against ventricular volume over one cardiac cycle produces a pressure–volume loop.
Its shape reflects filling, isovolumetric contraction, ejection and relaxation. Mechanistic heart models generate this loop from ventricular, valve and vascular equations rather than drawing it independently.
Multiple cardiovascular compartments
A whole-circulation model may divide blood volume among compartments such as systemic arteries, systemic veins, pulmonary arteries, pulmonary veins and heart chambers.
For each compartment \(i\),
\[\frac{dV_i}{dt}=\sum Q_{in}-\sum Q_{out}.\]Pressures are then related to volumes through compliance or elastance relations, while flows depend on pressure differences and resistances.
Series resistances
For resistive elements in series carrying the same flow,
\[\boxed{R_{total}=R_1+R_2+\cdots.}\]The total pressure drop is the sum of the pressure drops across the individual elements.
Parallel resistances
For parallel pathways exposed to the same pressure difference,
\[\boxed{\frac1{R_{total}}=\frac1{R_1}+\frac1{R_2}+\cdots.}\]This structure is relevant to organ circulations arranged in parallel.
Autoregulation
Vascular resistance is not necessarily constant. Vessels can constrict or dilate in response to pressure, metabolites, neural signals and hormones.
A dynamic resistance model might therefore include
\[\frac{dR}{dt}=F(P,Q,\text{signals})\]instead of treating \(R\) as a fixed parameter.
The baroreflex
Blood-pressure regulation involves negative feedback. Changes in arterial pressure alter baroreceptor signalling, which can affect heart rate, contractility and vascular resistance.
Mathematical baroreflex models can therefore couple cardiovascular mechanics with control equations.
Spatial blood-flow models
Lumped models ignore position within a vessel. One-dimensional models introduce cross-sectional area \(A(x,t)\) and flow \(Q(x,t)\).
Mass conservation has the form
\[\boxed{\frac{\partial A}{\partial t}+\frac{\partial Q}{\partial x}=0.}\]A momentum equation is then added to describe acceleration, pressure gradients and friction.
Pulse waves
Because arteries are elastic, pressure disturbances travel as waves. Wave speed depends on vessel geometry and wall properties.
Spatial models can therefore represent propagation and reflection phenomena that a single Windkessel compartment cannot resolve.
Zero-dimensional, one-dimensional and three-dimensional models
| Model | Main idea |
|---|---|
| 0D lumped | pressure, volume and flow vary with time but not spatial position |
| 1D | flow and vessel area vary along vessel length and through time |
| 3D | full spatial fluid velocity and pressure fields are resolved |
Increasing dimension adds spatial detail but also increases data and computational requirements.
Parameter estimation
Cardiovascular parameters can be estimated from measurements such as pressure, flow, ventricular volume and vessel geometry.
Resistance and compliance are effective model parameters whose values depend on the model scale and assumptions.
Identifiability
Different combinations of resistance, compliance and cardiac input can sometimes produce similar pressure observations.
Using multiple measured quantities can help distinguish parameters, but identifiability should be checked before assigning strong physiological interpretations.
Model validation
A model fitted to one pressure waveform should not automatically be assumed predictive in other conditions. Validation should test whether it reproduces relevant observations outside the data used for fitting.
A practical modelling workflow
Define the cardiovascular region and variables of interest. Apply conservation of volume. Relate flows to pressure differences and resistances. Relate pressure to volume through compliance or elastance. Specify cardiac input or a heart model. Solve the resulting equations. Compare calculated pressure and flow with data. Add regulation, spatial wave propagation or additional compartments only when required by the biological question.