Impact of parallel heterogeneity on a continuum model of the pulmonary arterial tree

Model arterial trees were constructed following rules consistent with morphometric data, Nj = (Dj/Da)-beta 1 and Lj = La(Dj/Da)beta 2, where Nj, Dj, and Lj are number, diameter, and length, respectively, of vessels in the jth level; Da and La are diameter and length, respectively, of the inlet artery, and -beta 1 and beta 2 are power law slopes relating vessel number and length, respectively, to vessel diameter. Simulated heterogeneous trees approximating these rules were constructed by assigning vessel diameters Dm = Da[2/(m + 1)]1/beta 1, such that m-1 vessels were larger than Dm (vessel length proportional to diameter). Vessels were connected, forming random bifurcating trees. Longitudinal intravascular pressure [P(Qcum)] with respect to cumulative vascular volume [Qcum] was computed for Poiseuille flow. Strahler-ordered tree morphometry yielded estimates of La, Da, beta 1, beta 2, and mean number ratio (B); B is defined by Nk + 1 = Bk, where k is total number of Strahler orders minus Strahler order number. The parameters were used in P(Qcum) = Pa [formula: see text] and the resulting P(Qcum) relationship was compared with that of the simulated tree, where Pa is total arterial pressure drop, Q is flow rate, Ra = (128 microLa)/(pi D4a (where mu is blood viscosity), and Qa (volume of inlet artery) = 1/4D2a pi La. Results indicate that the equation, originally developed for homogeneous trees (J. Appl. Physiol. 72: 2225-2237, 1992), provides a good approximation to the heterogeneous tree P(Qcum).