A lumped parameter cardiovascular model is a zero-dimensional mathematical model used to describe the hemodynamics of the cardiovascular system. Given a set of parameters that have a physical meaning (e.g. resistances to blood flow), it allows to study the changes in blood pressures or flow rates throughout the cardiovascular system. Modifying the parameters, it is possible to study the effects of a specific disease. For example, arterial hypertension is modeled increasing the arterial resistances of the model. The lumped parameter model is used to study the hemodynamics 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 different 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 circuital components, like resistances or capacitors, while the blood flux behaves like the current flowing through the circuit according to Kirchhoff's laws, under the action of the blood pressure (voltage drop). 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 balance. 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 vessel wall) 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 parameter of the model are the resistances, the capacitances, the inductances and the elastances. The unknowns of the system are the blood volumes inside each heart chamber, the blood pressures and fluxes inside each compartment of the circulation. 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 systemic circulation that can be split into arteries, veins and, if needed, in other compartments accounting for different blood vessels; the pulmonary circulation that can be split into arteries, veins and, if needed, in other compartments accounting for different blood vessels.
Downstream of the left atrium and ventricle and right atrium and ventricle there are the four cardiac valves: mitral, aortic, tricuspid and pulmonary valves, respectively. 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. Each compartment is described by a Windkessel circuit with the number of elements depending on the specific compartment. The ordinary differential equations of the model are derived from the Windkessel circuits and the Kirchhoff's laws. In what follows the focus will be on a specific lumped parameter model. The compartments considered are the four heart chambers, the systemic and pulmonary arteries and veins.
Heart chambers equations The parameters related to the four heart chambers are the passive and active elastances E A X X {\displaystyle {EA}_{\mathrm {XX} }} and E B X X {\displaystyle EB_{\mathrm {XX} }} (where the subscript X X {\displaystyle \mathrm {XX} } varies among R A , R V , L A {\displaystyle \mathrm {RA} ,\mathrm {RV} ,\mathrm {LA} } and L V {\displaystyle \mathrm {LV} } if the elastances refer to the right atrium or ventricle or the left atrium or ventricle, respectively) and the unloaded volumes V 0 X X {\displaystyle V0_{\mathrm {XX} }} . The dynamics of the heart chambers are described by the time-dependent elastance:
E X X ( t ) = E B X X + E A X X f X X ( t ) {\displaystyle E_{\mathrm {XX} }(t)=EB_{\mathrm {XX} }+EA_{\mathrm {XX} }f_{\mathrm {XX} }(t)}
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