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The Energy Minimization Component
of the West Model of Biological Distribution Systems

The Model

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 network consists of K levels of elements. At the k-th level the elements are tubes of radius rk and length ;k. The k-th level elements branch into nk smaller elements. The total number of elements at level k is then

Nk = n0· n1· · ·nk-1
with N0=1.

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

Z = Σ(Rk/Nk)

where Rk is the resistance of a single k-th level element and Nk 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 rk and length lk for laminar flow is given by the Poiseuille equation:

Rk = 8νlk/(πrk4Nk)

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

Nkπrk²uk = Q
for all k

where uk is the cross section average fluid velocity in the k-th level elements.

The article instead has constraints on Nkl³. 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 rk<<lk 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)³Nk.

This make no sense. It does make sense for the volumes to depend upon lk³ if rk is proportional to lk at all levels; i.e., rklk for all k. Then the volume of a k-th level element, which is πrk²lk, becomes πδ²lk³. But in the energy minimization process the rk and lk 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.

The Space Volume-filling Constraint

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,

π[(rK+ρ)² − rK²] = π[2ρrK + ρ²].

The volume served by one terminal element of length lK is then π[2rKρ + ρ²]llK. Altogether the NK terminal elements must fill the space S; i.e.,

NKπ(2rKρ+ρ)²lK = S

Each of the (K-1) level elements serves nK-1 elements which altogether cover an area of nK-1π(rK+ρ)². Thus the service volume for each (K-1) level element is nK-1π(rK+ρ)²lK−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.,

NK-1nK-1π(rK+ρ)²lK−1 = S + NKrKlK
but NK-1nK-1=NK so the constraint reduces to
NKπ(rK+ρ)²lK−1 = S + NKrKlK

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, VTotal. Thus the constraint for the (K-1) elements is

NKπ(rK+ρ)²lK−1 = S + VTotal

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

nK-2nK-1π(rK+ρ)²lK−2
and hence the constraint for the (K-2)
NK-2nK-2nK-1π(rK+ρ)²lK−2 = S + VTotal
which is the same as
NKπ(rK+ρ)²lK−2 = S + VTotal

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

NKπ(rK+ρ)²lk = S + VTotal

Note that the number of elements in the above constraint is for all k the number of terminal level elements NK.

The Constrained Minimization Problem

The optimization problem for the network is now

Minimize Q²Z
with respect to rk, lk, uk and nk
subject to the constraints
Nkπrk²uk = Q
and
NKπ[(rK+ρ)²−rK²]lK = S
and
NKπ(rK+ρ)²lk = S + VTotal
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., rk≥rmin.

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

L = Q²Z − Σλk(Nkπrk²uk − Q)
− Σμk(NKπ(rK+ρ)²lk−(S+VTotal)) − μK(NKπ[(rK+ρ)²−rK²]lK−S)
where
Z = Σ(8πνlk)/(rk4Nk)
and where the summations
for the λk's and for Z are over k from 0 to K
but the summation for the μk is from k=0 to k=(K-1).

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.

The First Order Conditions

(To be continued.)

Appendix I: Fundamentals of Flow in a Network

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

Q = Δp/R

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 R1 and R2 connected sequentially (in series) the resistance of the combination is (R1+R1).


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