The asymmetry of the reflection coefficients for wave fronts approaching the bifurcation in the forward and backward directions is dramatic and will play an important role in the following analysis.

The goal of this study is to develop a model of the arterial system that can be described my a minal number of parameters which is accurate enough to describe the essential features of arterial hemodynamics and  to predict the effects of various pathologies such as hypertension and ageing. Some of the parameters in the model can be derived from experimental data about the anatomy of arterial trees. The parameters will also be chosen to reflect some of the features of arterial flow that have been measured in experimental studies. It is not meant to model the large arteries accurately, but to provide a functionally realistic model for the arterial tree as a whole, including the myriad medium and small arteries that make up the bulk of the arterial tree.

An appealing way to model the arterial tree is to assume that the model is self-similar so that the diameter and length of any vessel can be related to the diameter and length of the parent vessel by the scaling ratios between generations \(\delta\) and \(\lambda\) and the asymmetry ratios between the major and minor daughter \(\nu\) and \(\mu\). Although we will not assume the tree is uniform, it is instructive to consider the distribution of lengths and diameters in the different generations of a uniform binary tree. Consider first the diameter of the edges at generation \(g\). There will be \(2^g\) edges and iterating from the root we see that their diameters relative to the diameter of the root will have have all of the values \(\nu^k\delta^g,\ k\ =0,1,2,\ ..\ ,g\) where the number of edges with the different diameters are given by the binomial coefficient \(\frac{n!}{k!\left(n-k\right)!}\). These values can be found in Pascal's triangle. Any tree can be embedded in a uniform binary tree of the number of generations equal to the ehight of the tree. The diameters of the edges in the will correspond to the diameters in the uniform tree, but the distribution of the number of edges with any particular diameter will depend on the structure of the tree.

This approach is commonly combined with the assumption that there is a minimum diameter \(D_{\min}\), usually related to the diameter of a capillary which is limited physiologically by the size of the red blood cells whcih must be able to pass through them. Using this threshold diameter, the vessel crossing this threshold is assumed to be a terminal branch and the outlet node is assumed to be a leaf.  We will argue that a tree defined in this way is not a good model for the arterial tree. Assume that the height of the tree \(H\) is specified. This bounds the permissible values of \(\delta\) because the diameter of the major branch of the tree must be greater than \(D_{\min}\) or else the height would be less than \(H\). Similarly, the diameter of the next generation of the major branch must be less than \(D_{\min}\) or else the height would be greater than \(H\). combining these two limits

where \(D_0\) is the diameter of the root vessel. If, for example, we assume \(D_{\min}=\ 20\ \mu m\) and \(D_0=2\ cm\) for a tree with \(H=20\) the scale factor for the diameter would be limited to the range \(0.708\ <\ \delta\ <0.720\), which is a very limited range.

A second consideration is the number of internal nodes (bifurcations) in the model. A uniform tree of height H will have H generations with \(N=2^H-1\) internal nodes. From the calculation for the average height of all trees with \(N\) internal nodes (for the limit \(N\ \rightarrow\infty\), \(N=\frac{H^2}{4\pi}\). As \(H\) increases the difference between these predictions becomes very large. If, for example, we consider a tree with \(H=20\), a uniform tree would have \(N=1,048,575\) internal nodes whereas the 'average' tree would have \(N\ \sim32\) internal nodes. This disparity in predictions must be considered.

These considerations suggest that a single self-similar model will note be adequate to characterise the arterial tree. A straightforward way to proceed is to consider that the arterial tree is composed of two a tree of conduit arteries which can be described by one self-similar model feeding local trees that provide perfusion to the local tissue which can be described by another self-similar model. The division between these two categories of vessels is necessarily blurred but, for the sake of an adequate simplistic model, can be represented by a single parameter. This approach ha the advantage that the seperate functional requirements of the two types of tree, efficient convection of blood to the periphery in the conduit arteries and efficient perfusion of local tissues by the dispersive arteries.

The division of the arterial tree into conduit and dispersive arteries follows from the 1-D momentum equation written in non-dimensional form using the mean velocity into the tree \(\hat{U}\) as the charcteristic velocity and the water-hammer pressure \(\rho c\hat U\)as the characteristic pressure

where \(Re\) is the Reynolds number based on the characteristic velocity and diameter of the vessel. When the local \(Re\ \ll1\)the only way to balance the RHS of this equation is to require that \(P'_x\sim\frac{-\kappa m}{Re}\frac{U'}{A'}\) with \(\frac{m}{Re}=O\left(1\right)\). This can be written succinctly by observing that \(\frac{Re}{m}=\frac{\rho cD}{\mu}\equiv Re_c\), the Reynolds number based on the wave speed and the diameter of the vessel. The wave Reynolds number can be rearranged to show that it is the ratio of the rate at which momentum is transported by the wave \(\frac{c}{D}\)to the rate at which momentum diffuses due to the kinematic viscosity \(\frac{\mu}{\rho D^2}\), Thus \(Re_c=1\) will be taken as the transition from conduit to dispersive vessels.

In the conduit arteries we argue that convective transport is enhanced when the reflection of a forward travelling wave is small at each bifurcation, which is realised when \(y_1\approx y_2+y_3\), where \(y_1\)is the admittance of the parent vessel and \(y_2\) and \(y_3\) are the admittances of the major and minor daughters. Using the scaling parameters defined above, \(D_2=\delta D_1\) and \(D_3=\nu D_2\) and the definition of the characteristic admittance \(y\ =\ \frac{A}{\rho c}\) we impose the constraint

where we have used \(A\ \sim D^2\) and \(c\ \sim D^{-b}\), where \(b\) is an empirical constant describing how wave speed varies with the diameter of the vessel.

These relationships are not enough to determine \(\delta\) and \(\nu\). Do do this we will have to use information about the sparseness of the conduit arterial tree; in particular its relatively large height relative to the number of internal nodes (bifurcations). As a concrete example of this we turn to the heavily used 55-artery model of the large systemic arteries introduced by Westerhof, extended by Stergiopolus, studied experimentally by Segers and used as the basis for 1-D computational modelling of the systemic arteries by many researchers, including Alustruey. This model is a binary tree with \(E=55\) edges (vessels) with \(N_{int}=27\) internal nodes and \(N_{ext}=29\) external nodes (the root and 28 leaves (terminal nodes). The height of the model \(H=14\) which corresponds to the number of internal nodes between the heart and the terminal arteries in the lower legs. If this tree was a uniform bifurcating tree its height would be the height of a full uniform binary tree with \(31=2^H-1\) internal nodes; i.e. \(H=5\), approximately 1/3 of its actual height. Interestingly the average height of all binary trees with 27 internal notes \(\bar H=2\sqrt{\pi N_{int}}\approx18.4\) is a much better estimate of the height of the tree than the uniform binary tree model although it is not clear that a formula valid in the limit of the number of internal nodes approaches infinity should be expected to be valid for 27 internal nodes.

The capillary density in a full uniform binary tree of meso-arteries is defined as the number of capillaries perfused by the tree divided by the volume occupied by the tree. If there are \(G\) generations ending in terminal arteries (arterioles??) there will be \(2^G\) arterioles and so the number of capillaries perfused by the tree will be \(N_{cap}=2^GC_a\) where \(C_a\) is the number of capillaries served by each arteriole. This number will depend on the topology of the microcirculation and will most likely vary in different organs and tissues. We will assume that it is a constant that can be determined anatomically or, alternatively, treated as a parameter of the model.

The volume occupied by the tree \(V_{tree}\) requires a bit of analysis. We assume that the branches of the tree lie in random directions in space. Any particular branch of the tree can be thought of as a random walk with the step size decreasing by the scale factor \(\lambda\) from generation to generation. Thus, the spatial location of the node after the \(n^{th}\) step will be \(X_n=X_{n-1}+L_n\cos\ \theta_n\) where \(L_n=\lambda^nL_0\) is the length of the step and \(\theta_n\) is the angle between the direction of the step and the \(X\)-axis, taken to be a random variable.  Squaring this equation and taking the average over all of the branches steps

\(\langle X_n^2\rangle=\langle X_{n-1}^2+2X_{n-1}\lambda^nL_0\cos\theta_n+\lambda^{2n}L_0^2\cos^2\theta_n\rangle\) \(=\langle X_{n-1}^2\rangle+\frac{1}{2}\lambda^{2n}L_0^2\)

Here we have used \(\left\langle\cos\theta_n\right\rangle=0\) and \(\langle\cos^2\theta_n\rangle=\frac{1}{2}\). Taking \(X_0=0\), we find by iteration that

Thus, the average distance of the terminal vessels from the origin of the zero generation vessel is the square root of this expression and so the average volume occupied by the tree is