Vessel distensibility and flow distribution in vascular trees
Journal of Mathematical Biology
In a class of model vascular trees having distensible blood vessels, we prove that flow partitioning throughout the tree remains constant, independent of the nonzero driving flow (or nonzero inlet to terminal outlet pressure difference). Underlying assumptions are: (1) every vessel in the tree exhibits the same distensibility relationship given by DD0=f(P) where D is the diameter which results from distending pressure P and D0 is the diameter of the individual vessel at zero pressure (each vessel may have its own individual D0). The choice of f(P) includes distensibilities often used in vessel biomechanics modeling, e.g., f(P)=1+P or f(P)=b+(1−b)exp(−cP), as well as f(P) which exhibit autoregulatory behavior. (2) Every terminal vessel in the tree is subjected to the same terminal outlet pressure. (3) Bernoulli effects are ignored. (4) Flow is nonpulsatile. (5) Blood viscosity within any individual vessel is constant. The results imply that for a vascular tree consistent with assumptions 2–5, the flow distribution calculations based on a rigid geometry, e.g., D=D0, also gives the flow distribution when assuming the common distensibility relationships.