Extension of Murray's law using a non-Newtonian model of blood flow

Abstract

Background: So far, none of the existing methods on Murray's law deal with the non-Newtonian behavior of blood flow although the non-Newtonian approach for blood flow modelling looks more accurate.

A Critical Reassessment of the Hess–Murray Law

“…This applies also to stress situations: when a higher rate of blood (or sap) flow is needed, the existence (or the emergence) of a self-regulatory response which suitably adjusts the vessels radii, as originally proposed by Hess, seems reasonable [15]; (h) The H-M law also allows for an "allometric scaling" between different species [20]: if the relative sizes of the respective blood (or sap) vessels scale according to this law, then the physical dimensions depend on the radius of the main branch, which in turn depends on the metabolic rate of the species [21]; (i) If blood is modelled as a non-Newtonian fluid, the 2´1 /3 law does not apply anymore, because-as expected-the exponent is found to have a non negligible dependence on the viscosity law (from 1 to 2.5 in [22,23]). Miguel [17,18] analyzed the irreversibility in branched flows of a power fluid and showed that (a) the "optimal" radius ratio depends on the "degree of non-newtonianity" of the fluid, ω; (b) that in the limit of Newtonian behavior, ω tends to 1 and the H-M law is recovered; (j) It is also possible to derive the optimal radius ratio in a dendritic structure by reformulating the problem as a constrained multi-variable optimization [11,24,25]: this approach, useful as it may be in engineering applications, loses somewhat of its significance in biological investigations.…”

Mathematical analysis on effect of non-Newtonian behavior of blood on optimal geometry of microvascular bifurcation system

“…In this study, a Newtonian fluid model is considered to describe the rheology of blood and Murray's principle of energy minimization is effectively used. The validity of the classical Murray's law using a Power law model of blood flow in a rigid arterial bifurcation is demonstrated by Revellin et al [8]. The energy expenditure by the wall of blood vessel (the metabolic state of blood vessel wall) has not been taken into account in papers [3][4][5][6][7][8], which will be realistic to be considered in the sense that with smaller vessels, where introduction of such an optimality principle will in general be best met, the proportion of the thickness of the vessel wall to the vessel size is known to become greater.…”

Classification of parametrically constrained bifurcations

“…Such constraints widely exist in electricity [1] , nuclear physics [2] , biology [3][4] , bifurcation control [5][6] , model interaction [7] , non-smooth dynamics [8] , etc. Only using the unconstrained singularity theory, one cannot get all the bifurcation types of a constrained bifurcation, and thus one cannot master bifurcation behaviors on the whole [7][8] .…”