An analytic method to investigate hemodynamics of the cardiovascular system – single ventricular system

Based on the lumped parameter model (LPM) of the cardiovascular system, an analytic method is developed to derive its hemodynamics theoretically. As soon as the LPM (a series of differential equations) is solved, the hemodynamics would be obtained immediately. However, because of time-varying ventricular elastance and high order, it is difficult to solve analytically. Through simplifying the LPM, the original biventricular system with continuously varying elastance becomes a single ventricular system with discrete elastance which keeps constant during the systolic or diastolic phase. As a consequence, the original time-varying and high-order system becomes a time-invariant and first-order system during each phase. From the analytic solutions of the simplified system, a set of algebraic equations is carried out. Then the hemodynamics are obtained from the solutions of the algebraic equations. The nature of the algebraic equations is an integral form of the differential equations. A connection between the equations and PV loop is established. All of these equations are deduced based on the idealization of replacing the continuous elastance with the discrete elastance. However, there exist algebraic equations, that can be derived directly from volume conservation, still hold for the case of continuous elastance. As a preliminary application, the method is utilized to deduce the hemodynamics of left heart failure (LHF). The results show that the theoretical hemodynamics of LHF are coincident with simulated results. The analytic method can be generalized to investigate biventricular system. A program for developing a more general framework is presented in the last part.