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

Model arterial trees were constructed following rules consistent with morphometric data, N(j) = (D(j)/D(a))(-β₁) and L(j) = L(a)(D(j)/D(a))(β₂), where N(j), D(j), and L(j) are number, diameter, and length, respectively, of vessels in the jth level; D(a) and L(a) are diameter and length, respectively, of the inlet artery, and -β₁ and β₂ 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 D(m) = D(a)[2/(m + 1)](1/β₁), such that m - 1 vessels were larger than D(m) (vessel length proportional to diameter). Vessels were connected, forming random bifurcating trees. Longitudinal intravascular pressure [P(Q(cum))] with respect to cumulative vascular volume [Q(cum)] was computed for Poiseuille flow. Strahler-ordered tree morphometry yielded estimates of L(a), D(a), β₁, β₂, and mean number ratio (B); B is defined by N(k + 1) = B(k), where k is total number of Strahler orders minus Strahler order number. The parameters and the resulting P(Q(cum)) relationship was compared with that of the simulated tree, where Pa is total arterial pressure drop, Q̇ is flow rate, R(a) = (128μL(a))/(πD(a)/⁴) (where μ is blood viscosity), and Q(a) (volume of inlet artery) = 1/4 D(a)/²πL(a). 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(Q(cum)).