Lumped parameter model for the cardiovascular system
[1][2] Modifying the parameters, it is possible to study the effects of a specific disease.
[1] 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.
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 Kirchoff's laws, under the action of the blood pressure (voltage drop).
[2] 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.
[3] 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 parameter of the model are the resistances, the capacitances, the inductances and the elastances.
[3] The cardiovascular system is split into different compartments: 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 ordinary differential equations of the model are derived from the Windkessel circuits and the Kirchoff's laws.
The compartments considered are the four heart chambers, the systemic and pulmonary arteries and veins.
[5] The parameters related to the four heart chambers are the passive and active elastances
The time-dependent elastance allows the computation of the pressure inside a specific heart chamber as follows:[5] where
are the fluxes through the mitral, aortic, tricuspid and pulmonary valves respectively and
[5] The valves are modeled as diodes and the blood fluxes across the valves depend on the pressure jumps between the upstream and downstream compartment:[5] where the pressure inside each heart chamber is defined in the previous section,
are the time-dependent pressures inside the systemic and pulmonary artery compartment and
[5] Each compartment of the blood vessels is characterized by a combination of resistances, capacitances and inductances.
that represent the arterial systemic resistance, capacitance and inductance.
The ordinary differential equations that describes the systemic arterial circulation are:[5] where
[5] Analogous equations with similar notation hold for the other compartments describing the blood circulation.
The first two equations are related to the volumes in the left atrium and ventricles respectively.
[5] From a mathematical point of view, the well-posedness of the problem is a consequence of the Cauchy–Lipschitz theorem, so its solution exists and it is unique.
The solution of the system is approximated by means of a numerical method.
depends on the number of heartbeats and the heart rate) to approach the limit cycle of the dynamical system, so that the solution behaves in a similar way to a periodic function emulating the periodicity of the cardiac cycle.
The equations that govern the new or the modified compartments are the Kirchoff's laws as before.
Complex models can describe different dynamics, but the increase in complexity entails a larger computational cost to solve the system of differential equations.
) 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.
