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of the West Model of Biological Distribution Systems |

Geoffrey B. West along with his collaborators James H. Brown and Brian J. Enquist in
the Biology Department of the University of New Mexico at Albuquerque developed a model of the distribution systems of plants and animals that
explains allometric scaling laws. A distribution system such as the circulatory system that
carries blood to the body cells in animals consists of a branching network of tubes starting
with the aorta at the heart which feeds into successively smaller tubes until the smallest
capillaries are reached. The model was published in *Science* Magazine with
the title, "A General Model for the Origin of Allometric Scaling Laws in Biology," Although it
was published in 1997 it is still not as well known as it should be. Material on an
explanation of the model is given elsewhere. The original article is a bit hard to follow in crucial spots. However
even if there are some flaws of expostion in the article the model is sound and clearly the
right overall approach to modeling such biological systems.
To make reference easier the
model will be referred to as the West Model.

The West model treats a system such as the cariovascular circulatory system of animals as a branching network of smaller and smaller tubes that has some of the characteristics of a fractal but is not actually a fractal. The acknowledges three explicit assumptions but there are a number of implicit assumptions utilized in the development. The explicit assumptions are:

- The branching network is fractal-like and volume-filling.
- The minimal elements in the network are the same size for all creatures. This means that not only the radii the same but also length and other characteristics.
- The structure of the network minimizes the energy required to supply the body mass with its necessary nutrients. This reduces to minimizing fluid resistance. of the network

The network consists of K levels of elements. At the k-th level the elements are
tubes of radius r_{k} and length *;*_{k}. The k-th level elements
branch into n_{k} smaller elements. The total number of elements at level k is then

with N

The material here focuses on this third aspect of the model. The analysis of energy minimization in the article is simply technically defective. Its being a failed effort does not detract from the value of the conceptual model.

What follows here is an explanation and critique of the energy minimization component of the West model.

The rate of energy expenditure W required to pump fluid through the system is given by QΔp where Q is the rate of fluid flow and Δp is the pressure difference through the network. Because Q=Δp/Z, where Z is the resistance/impediance of the network, this means that Δp=QZ and hence W=(QZ)Q=Q²Z. Q is given as a requirement based upon a creature's mass. Therefore minimizing energy use is equivalent to minimizing resistance Z.

The resistance of the network can be expressed as

where R_{k} is the resistance of a single k-th level element and N_{k}
is the number of k-th level elements.

The k-th level elements are in parallel and the different levels are in series. Appendix I
gives some fundamentals of flows in networks. The resistance of a tube of radius r_{k}
and length *l*_{k} for laminar flow is given by the Poiseuille equation:

where ν is the fluid viscosity.

This far there is no major problem with the analysis although it is a minor mistake to literally replace the objective function of energy use W=Q²Z with Z because some information will be lost in the later analysis.

The problems in the West article are with the constraints. One constraint used in the analysis is that the total volume of fluid in the system should be a specified value. If the network size and other characteristics were specified first then it would make some sense that there would be a requirement that volume of fluid be sufficient to keep the network inflated. But that is not how the analysis is working in the article. The analysis is saying that given a specific volume of fluid the network is constrained to accomodate exactly that volume of fluid. This is saying that a creature is going to have five pints of blood and its circulatory system is constrained to accomodate that volume of blood.

The proper constraint is that the creature on the basis of its size requires a certain level of fluid flow to deliver the nutrients and oxygen it needs and to take away waste products and heat. It is in the terminal elements of the network that the fluid is in contact with the body cells. In the case of animal circulatory systems these are the capillaries. So the constraint is in terms of the fluid flow rate in the terminal elements, but the fluid flow rates in all of the prior elements of the network have to have the same fluid flow rate to deliver the required amount to the terminal elements. Therefore the network must satisfy the constraints that

for all k

where u_{k} is the cross section average fluid velocity in the k-th level elements.

The article instead has constraints on N_{k}*l*³. This is a botched
version of a constraint in terms of the volume of the k-th level elements of the network.
Early in the article it is asserted that:

Because r_{k}<<l_{k}and the total number of branchings N is large, the volume supplied by the total network can be approximated by the sum of spheres whose diameters are that of a typical kth level vessel, namely (4/3)π(l/2)³N_{k}.

This make no sense. It does make sense for the volumes to depend upon
*l*_{k}³
if r_{k} is proportional to *l*_{k} at all levels; i.e.,
r_{k}=δ*l*_{k} for all k. Then the volume of a k-th level
element, which is πr_{k}²*l*_{k}, becomes πδ²*l*_{k}³.
But in the energy minimization process the r_{k} and *l*_{k} should be
independently determined.

Another flaw in the analysis is that body M mass is included as a constraint *per se*.
M is a parameter involved in the constraints rather than a constraint itself. The body
mass determines the required fluid flow rate. This confusion would be innocuous except in the
analysis the derivative of the Lagrangian with respect to M is taken and set equal to zero.
This makes no sense whatsoever. Clearly the value of M that minimizes energy use is zero.

The additional constraints on the network characteristics have to do with the requirement that the network fill the volume occupied by the creature's mass.

Let S be the volume of body tissue and ρ the service distance beyong the radius of the terminal elements. The service distance is the distance beyond the terminal element for which the element provides nutrients. The diagram below shows a segment of an idealized distribution system for the terminal elements:

In the diagram the cross sections of the capillary-like elements are shown in red. The service area is necessarily a hexagon or a triangle for a uniform pattern. The yellow outline does not represent any structure but is used just to depict the tissue service areas of the the capillaries.

The service area is then, ignoring the minor areal difference between a hexagon and a circle,

The
volume served by one terminal element of length *l*_{K} is then
π[2r_{K}ρ + ρ²]l*l*_{K}.
Altogether the N_{K} terminal elements must fill the space S; i.e.,

Each of the (K-1) level elements serves n_{K-1} elements which altogether cover
an area of n_{K-1}π(r_{K}+ρ)². Thus the service volume for each
(K-1) level
element is n_{K-1}π(r_{K}+ρ)²*l*_{K−1} and this times the
number of (K-1) elements must also fill a space related to S. However in this case the
volume is not only that of the tissues but also the volume of the terminal elements as well;
i.e.,

but N

N

But if one works through the logic of the model the term that needs to be added to the
tissue volume is S is not just the volume in the terminal elements but the volume in the entire
network system, V_{Total}. Thus the constraint for the (K-1) elements is

The service volume of each of the (K-2) level elements is likewise

and hence the constraint for the (K-2)

N

which is the same as

N

The pattern that emerges is that for all k the volume filling constraint is

Note that the number of elements in the above constraint is for all k the number of
terminal level elements N_{K}.

The optimization problem for the network is now

with respect to r

subject to the constraints

N

and

N

and

N

for all k from 1 to (K-1)

where Q and S are functions of M.

There is another type of constraint that is left out of the model at this point; i.e.,
r_{k}≥r_{min}.

This problem may be analysed using the Lagrangian multiplier method.
Let {λ_{k}} and {μ_{k}} be the sets of multipliers. The Lagrangian
for the problem is then

− Σμ

where

Z = Σ(8πν

and where the summations

for the λ

but the summation for the μ

The terms involving the Lagrangian multipliers are in the nature of penalty functions for the non-satisfaction of the constraints.

The diagram below shows an idealized distribution system for three hierarachical levels of distribution elements:

In the diagram the blue aorta-like element feeds the violet arteries which, in turn, feed the capillaries. The capillaries then provide nutrients to a hexagonal service area of pink tissue around them. For the diagram to be valid there have to be capillary elements along side of each artery and the aorta.

For a four-level hierarchy the picture would be of this sort:

In the following diagram there is a five-level hierarchy in which the service areas of most of the terminal elements are outlined in yellow.

(To be continued.)

The rate of fluid flow Q through a network is determined by the expression

where Δp is the pressure difference between the beginning and end of the network. R stands for resistance or, alternatively, impedance.

For N equal tubes each of resistance R operating in parallel the resistance is R/N.
For tubes of resistance of R_{1} and R_{2} connected sequentially (in series)
the resistance of the combination is (R_{1}+R_{1}).

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