Accuracy of simple difference-differential equations for blood flow in arteries

This paper presents an analysis of the numerical accuracy of the representation of an arterial segment by difference-differential equations (dde) of low order (N). The analysis is done by comparing, in the linear autonomous case, the two-port representations of the exact solution (Womersley) and of the lumped approximation (Rideout) of the flow by N annuli of constant flows. For low N (2, 3) it is shown that the dde's overestimate greatly the longitudinal impedance of the artery. For higher N, we give a closed-form expression for this longitudinal impedance approximation and show that it converges toward the Womersley solution for N → ∞. However for N ⩽ 5 (which are the only practical cases) the error in longitudinal impedance remains large. Finally, the practical influence of these errors, is shown on a model of the pulmonary vascular bed of the dog for which the use of Rideout's method completely destroys the input impedance pattern shown by the Womersley solution.