Numerical integration

Abstract

Compartmentalized models of the circulatory system are used to investigate the dynamics of the movement of blood from one region of the circulation to another. When these models are studied using numerical methods—viz., numerical integration—solution stability can often be achieved only at integration step sizes that are far less than the time intervals of interest. Hence, obtaining solution stability at much larger integration step sizes is desirable. Instability results from a (one-iteration) delay between calculation of the derivative and calculation of the integral. The method described here, applied locally within a model, replaces the usual method of estimating flow at the beginning of the interval only, with an estimated average value for the whole interval. The “flow predictor” method is not general; it requires a knowledge of local resistances and capacitances and their effect on flow. But, only one subroutine call is required when a flow is calculated; the expense in computing time is minimal. Using an example four-compartment model, the maximum stable integration interval with rectangular integration (0.0005 min) was improved by a factor of 10⁶ (to 1000 min) with this method.