On indicator dilution and perfusion heterogeneity: A stochastic model

Unidirectional extraction of a substrate S in the capillaries following the arterial injection of a bolus containing S and a reference tracer R is assumed to follow first-order kinetics. If C_R and C_S denote normalized venous effluent concentrations of R and S, respectively, let L(t)=ln[C_R(t){plus 45 degree rule}C_S(t)]. We derive a formula which expresses the experimental L(t) data in terms of the mean μ(t) and variance of the transit times of those capillaries which are contributing indicators at each sample time t. We examine the information thus contained in the L data about capillary and noncapillary transit times under several kinematic assumptions. We show that if the capillary and noncapillary transit times are stochastically independent with frequency functions h_c(t) and h_av(t), respectively, then the shapes of the graphs of L(t) and μ(t) depend on the variances and skewnesses of h_c(t) and h_av(t). Specifically, let r₂ be the ratio of the variance of h_c(t) to the variance of h_av(t), and let r₃ be the ratio of skewnesses in the same order. Then the graph of μ(t) is concave downward if r₂{plus 45 degree rule}r₃ > 1, concave upward if r₂{plus 45 degree rule}r₃< 1, and linear if r₂{plus 45 degree rule}r₃ = 1. If the fraction of S extracted is not too large, L(t) has nearly the same shape as μ(t), and therefore, L(t) contains information about h_c(t) and h_av(t).