On deriving lumped models for blood flow and pressure in the systemic arteries

Windkessel and similar lumped models are often used to represent blood flow and pressure in systemic arteries. The windkessel model was originally developed by Stephen Hales (1733) and Otto Frank (1899) who used it to describe blood flow in the heart. In this paper we start with the one-dimensional axisymmetric Navier-Stokes equations for time-dependent blood flow in a rigid vessel to derive lumped models relating flow and pressure. This is done through Laplace transform and its inversion via residue theory. Upon keeping contributions from one, two, or more residues, we derive lumped models of successively higher order. We focus on zeroth, first and second order models and relate them to electrical circuit analogs, in which current is equivalent to flow and voltage to pressure. By incorporating effects of compliance through addition of capacitors, windkessel and related lumped models are obtained. Our results show that given the radius of a blood vessel, it is possible to determine the order of the model that would be appropriate for analyzing the flow and pressure in that vessel. For instance, in small rigid vessels ($R <$ 0.2 cm) it is adequate to use Poiseuille's law to express the relation between flow and pressure, whereas for large vessels it might be necessary to incorporate spatial dependence by using a one-dimensional model accounting for axial variations.