Lumped parameter model for the cardiovascular system

A lumped parameter cardiovascular model is a mathematical model used to describe the hemodynamic of the cardiovascular system. It allows to study, given a set of parameters that have a physical meaning (e.g. resistances to blood flow), the changes in blood pressures or fluxes throughout the cardiovascular system. Modifying the parameters, it is possible to study the effects of a specific disease, e.g. arterial hypertension is modeled increasing the arterial resistances of the model.

The lumped parameter model studies the hemodynamic of a three-dimensional space (the cardiovascular system) by means of a zero-dimensional space that exploits the analogy between pipes and electrical circuits. The reduction from three to zero dimensions is performed by splitting the cardiovascular system into several compartments each of them representing a specific component of the system, e.g. right atrium or systemic arteries. Each compartment is made up of simple circuit components, like resistances or capacitors and the blood flux behaves like the current flowing through the circuit according to Kirchoff's laws.

The lumped parameter model consists in a system of ordinary differential equations that describes the evolution in time of the volumes of the heart chambers and the blood pressures and fluxes through the blood vessels.

Model description

The lumped parameter model consists in a system of ordinary differential equations that adhere to the principles of conservation of mass and momentum. The model is obtained exploiting the electrical analogy where the current represents the blood flow, the voltage represents the pressure difference, the electric resistance plays the role of the vascular resistance (determined by the section and the length of the blood vessel), the capacitance plays the role of the vascular compliance (the ability of the vessel to distend and increase volume with increasing transmural pressure, that is the difference in pressure between two sides of a wall or equivalent separator) and the inductance represents the blood inertia. Each heart chamber is modeled by means of the elastances that describe the contractility of the cardiac muscle and the unloaded volume, that is the blood volume contained in the chamber at zero-pressure. The valves are modeled as diodes.

The system of ordinary differential equations is solved by means of a numerical method for temporal discretization, e.g., a Runge-Kutta method.

The cardiovascular system is split into different compartments:

• the four heart chambers: left and right atrium and left and right ventricles;
• the four cardiac valves: tricuspid, pulmonary, mitral and aorta valves;
• the systemic circulation that can be split in arteries, veins and, if needed, in other compartments accounting for different blood vessels;
• the pulmonary circulation that can be split in arteries, veins and, if needed, in other compartments accounting for different blood vessels.

The splitting of the pulmonary and systemic circulation is not fixed, for example, if the interest of the study is in systemic capillaries, the compartment accounting for the systemic capillaries can be added to the lumped parameter model.

In what follows the focus will be on a specific lumped parameter model. Each compartment is described by a Windkessel circuit with the number of elements depending on the specific compartment. The ordinary differential equations are derived from the Windkessel circuit and the Kirchoff's laws .

Heart chambers equations

The parameters related to the four heart chambers are the passive and active elastances $EA_{XX}$ and $EB_{XX}$ (where the subscripts vary among $RA,RV,LA$ and $LV$ if the elastances refer to the right atrium or ventricle or the left atrium or ventricle, respectively) and the unloaded volumes $V0_{XX}$. The dynamics of the heart chambers are described by the time-dependent elastance:

$E_{XX}(t)=EB_{XX}+EA_{XX}f_{XX}(t)$

where $f_{XX}(t)$ is a periodic (period of an heartbeat) time dependent function ranging from $0$ to $1$ that accounts for the activation phases of the heart during a heartbeat. From the above equation, the passive elastance represents the minimum elastance of the heart chamber, whereas the sum of $EA_{XX}$ and $EB_{XX}$ the maximum elastance of it. The time-dependent elastance allows the computation of the pressure inside a specific heart chamber as follows:

$p_{XX}(t)=E_{XX}(t)(V_{XX}(t)-V0_{XX})$

where $V_{XX}(t)$ is the volume of blood contained in the heart chamber and the volumes for each chamber are the solutions to the following ordinary differential equations that account for inward and outward blood fluxes associated with the heart chamber:

${\frac {dV_{LA}(t)}{dt}}=Q_{VEN}^{PUL}(t)-Q_{MV}(t)$

${\frac {dV_{LV}(t)}{dt}}=Q_{MV}(t)-Q_{AV}(t)$

${\frac {dV_{RA}(t)}{dt}}=Q_{VEN}^{SYS}(t)-Q_{TV}(t)$

${\frac {dV_{RV}(t)}{dt}}=Q_{TV}(t)-Q_{PV}(t)$

where $Q_{MV}(t),Q_{AV}(t),Q_{TV}(t)$ and $Q_{PV}(t)$ are the fluxes through the mitral, aortic, tricuspid and pulmonary valves respectively and $Q_{VEN}^{PUL}$ and$Q_{VEN}^{SYS}$ are the fluxes through the pulmonary and systemic veins respectively.

Valves equations

The valves are modeled as diodes and the blood fluxes across the valves depend on the pressure jumps between the upstream and downstream compartment:

$Q_{MV}(t)=Q_{valve}(p_{LA}(t)-p_{LV}(t))\qquad Q_{AV}(t)=Q_{valve}(p_{LV}(t)-p_{AR}^{SYS}(t))$

$Q_{TV}(t)=Q_{valve}(p_{RA}(t)-p_{RV}(t))\qquad Q_{TV}(t)=Q_{valve}(p_{RV}(t)-p_{AR}^{PUL}(t))$

where the pressure inside each heart chamber is defined in the previous section, $p_{AR}^{SYS}(t)$ and $p_{AR}^{PUL}(t)$ are the time-dependent pressures inside the systemic and pulmonary artery compartment and $Q_{valve}(\Delta p)$ is the flux across the valve depending on the pressure jump:

$Q_{valve}(\Delta p)={\begin{cases}{\frac {\Delta p}{R_{min}}}\qquad &{\text{if }}\Delta p<0\\{\frac {\Delta p}{R_{max}}}\qquad &{\text{if }}\Delta p\geq 0\end{cases}}$

where $R_{min}$ and $R_{max}$ are the resistances of the valves when they are open and closed respectively.

Circulation compartments equations

Each compartment of blood vessels is characterized by a combination of resistances, capacitances and inductances. For example, the arterial systemic circulation can be described by three parameters $R_{AR}^{SYS},C_{AR}^{SYS}$ and $L_{AR}^{SYS}$ that represent the arterial systemic resistance, capacitance and inductance. The ordinary differential equations that describes the systemic arterial circulation are:

$C_{AR}^{SYS}{\frac {dp_{AR}^{SYS}}{dt}}=Q_{AV}(t)-Q_{AR}^{SYS}(t)$

$L_{AR}^{SYS}{\frac {dQ_{AR}^{SYS}(t)}{dt}}=-R_{AR}^{SYS}Q_{AR}^{SYS}(t)+p_{AR}^{SYS}(t)-p_{VEN}^{SYS}(t)$

where $Q_{AR}^{SYS}(t)$ is the blood flux across the systemic arterial compartment and $p_{VEN}^{SYS}(t)$ is the pressure inside the veins compartment.

Analogous equations hold for the other compartments describing the blood circulation.

Further developments

The model described above is a specific lumped parameter model. It can be easily modified adding or removing compartments or circuit components inside any compartment as needed. The equations that govern the new or the modified compartments are the Kirchoff's laws as before.

The cardiovascular lumped parameter models can be enhanced adding a lumped parameter model for the respiratory system. As for the cardiovascular system, the respiratory system is split into different compartments modeling, for example, the larynx, the pharinx or the trachea. Moreover, it can be added a model for blood oxygenation to the cardiopulmonary model to study the levels of blood saturation.

The are several lumped parameter models and the choice of the model depends on the purpose of the work or the research. Complex models can describe different dynamics, but the increase in complexity entails a larger computational cost for the system of differential equations to be solved.

Finally, some of the 0-D compartments of the lumped parameter model could be substituted by $d$-dimensional components ($d=1,2,3$) to describe geometrically a specific component of the cardiovascular system (e.g., the 0-D compartment of the left ventricle can be substituted by a 3-D representation of it). As a consequence, the system of equations will include also partial differential equations to describe the dimensional components and it will entail a larger computational cost to be numerically solved.

See also

• Heart
• Blood vessel
• Lumped-element model
• Discretization
• Finite element method
• Model order reduction

Further readings

• Bazilevs, Yuri; Takizawa, Kenji; Tezduyar, Tayfun E. (2013). Computational fluid-structure interaction: methods and applications. Wiley series in computational mechanics. Chichester: Wiley. ISBN 978-0-470-97877-1.
• Quarteroni, Alfio; Dede', Luca; Manzoni, Andrea; Vergara, Christian (2019). Mathematical modelling of the human cardiovascular system: data, numerical approximation, clinical applications. Cambridge monographs on applied and computational mathematics. New York, NY: Cambridge University Press. ISBN 978-1-108-61609-6.