Analytical solution to a minimal cardiovascular model

Lumped-parameter models of the cardiovascular system have been used extensively in the past, especially when coupled to time-varying elastance (or compliance) representations of ventricular contraction. In this paper, we use a physiologically motivated approximation to the time- varying compliance waveform to derive analytical solutions to the short-term (intra-beat) and long-term (inter- beat) evolution of a minimal cardiovascular model. We show that this approximation turns a non-linear, periodically varying model into one that is periodically-varying and linear for physiologically reasonable initial conditions. The solution to the equations of motion takes the form of an affine transformation of the initial conditions. The methodology outlined in this paper is applicable to larger versions of the type of models considered here.