Analytical solution to Windkessel models using piecewise linear aortic flow waveform

Abstract

Abstract Objective. Deriving time-domain analytical solutions to two- three- and four-element Windkessel models, which are commonly used in teaching and research to analyse the behaviour of the arterial pressure-flow relationship. Approach. The governing (first-order, non-homogeneous, linear) differential equations are solved analytically, based on a piecewise linear function that can accurately approximate typical aortic flow waveforms. Main results. Closed-form expressions for arterial pressure are obtained both in transient conditions and in steady-state periodic regime. Significance. In most cases Windkessel models are studied in the frequency domain and when studied in the time domain, numerical methods are used. The main advantage of the proposed expressions is that they are an explicit, exact, and easily understood mathematical description of the model behaviour. Moreover, they avoid the use of Fourier analysis or numerical solvers to integrate the differential equations.