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The West Model of Allometric Scaling
in 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. This material is an explanation of the model. 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 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:

In the course of the analysis a number of other assumptions were made. Some of those assumption made were rather artifical and could be replaced with equivalent more plausible assumptions. The implicit assumptions will be pointed out in the explanation of the model and a summary presented after the model has been covered.

Distribution systems the model applies to include the bronchial tubes in the lungs of animals and plant vascular systems as well as the blood circulatory systems of animals. They have a tree-like structure that terminates in some minimal size element. This is called fractal-like because a true fractal would have no terminal elements. Blood circulation system of animals branches out until the capillaries are reached then the collection system goes from capillaries to veins. The circulatory system is thus two fractal-like systems back-to-back. The exposition of the West model is concerned just with the distribution system but the model applies equally well to the collection system.

Let the branching network have K levels starting with level zero. In the West article the number of levels was denoted as N but N was also used in reference to the number of elements in a level. It is less confusing to let the number of levels be denoted by K (uppercase). Lower case k is used as the index of the levels. Let the length and radius of the tubes in the k-th level be denoted as lk and rk, respectively. The pressure drop over the length of the level k tubes will be denoted as Δpk, and the cross-section average flow rate as uk. At each level there is a branching number for the number of smaller tubes each tube branches into. This is denoted as nk.

The total number of tubes in the k-th level, Nk, is then given by the product of the branching numbers of the previous levels; i.e.,

Nk = n0·n1···nk-1

The total flow of fluid through the k-th level, Qk, is Nkπr<k²uk. Since the fluid is conserved

Qk = Nkπr<k²uk is the same for all levels

The volume of fluid in a k-th level element, Vk, is given by

Vk = πr<k²lk

But West implicitly assumes that the ratio of radius to length is the same in all levels. This is not explicitly stated in the 1997 paper but is equivalent to what is stated. Let d be this common ratio. Then

NkVk = Nkpdl.

At this point in the analysis West limits consideration to networks in which the branching number is the same for all levels; i.e., nk=n for all k. West makes two additional assumptions. One, that the volume of fluid in one level is the same as in the previous level. Two, that the cross-sectional area of the branches of a tube is the same as that of the tube.

Thus, it is assumed that

NkVk = Nk+1Vk+1
for all k, which implies that
Nklk = Nk+1lk+1
and hence
(lk+1/lk) = Nk/Nk+1 = 1/n
and therefore
γ = lk+1/lk = n-1/3

From the assumption that the cross-section area of the n branches of a tube is equal to the area of that tube it follows that

nπr<k+1 = πr<k
and therefore
rk+1/rk = n-1/2

This common ratio of successive radii West calls .

These two assumptions are reasonable if one assumes the average fluid velocity is constant from one level to the next. West makes these assumptions and then derives the constancy of fluid velocity. It is more sensible to make the constancy of the fluid velocity the assumption and derive the area-preserving and volume-preserving assumptions from it.

The total volume of fluid in the distribution system is the sum of the volumes in each level; i.e.,

Vtotal = Σnkπrklk
= πr0l0Σnk(߲)kγk
= πr0l0Σ(n߲γ)k
= πr0l0[1-(nγ߲)K+1]/[1-(nγ߲)]

Note that πr0l0=V0 so the above can be expressed as

Vtotal = V0[1-(nγ߲)(K+1)]/[(nγ߲)-1]
which for nγ߲<1 and K large reduces to
Vtotal = V0/[(nγ߲)-1]

West presumes that nγ߲<1 but without supporting arguments. The parameters γ and are less than unity but n is greater than unity. In fact, West's assumption of area preservation between levels implies that n߲=1 and hence nγ߲ is really just γ and hence is necessarily less than unity.

Note that the volume of the terminal elements is given by

VK = πr0l0(γ߲)K = V0(γ߲)K
so
Vtotal = VK(γ߲)-K/(1-nγ߲)
and, taking into account that n߲=1
Vtotal = VK(γ߲)-K/(1-γ)

West presumes that the total fluid volume needs to be proportional to body mass. This is plausible in that each unit of body mass needs a definite quantity of nutrients and oxygen. However changes in fluid velocity could compensate for reduced fluid volume. Under West's assumption and the above equation it follows that (γ߲)-K is proportional to body mass.

The number of terminal network elements NK is given by

NK = nK

The metabolism rate corresponds to the total fluid flow rate, which under the previous assumptions is constant for all levels in the distribution network. The geometric characteristics of the terminal elements of the distribution network are assumed to be the same at all creature scales. This means that the total fluid flow in the terminal elements would be proportional to the number of terminal elements. Thus, if B represents the metabolic rate

B = c0nK
M = c1(γ߲)-K
where c0 and c1 are constants.

These relationship provide a link between B and M through the variable K; i.e.,

ln(B) = ln(c0) + K*ln(n)
ln(M) = ln(c1) -K*ln(γ߲)

It was previously found that γ is proportional to n-1/3
and to n-1/2 so (γ߲) is proportional to n-1/3-2(1/2)=n-4/3. This means that

ln(M) = ln(c1) + (4/3)K*ln(n)

Solving this last for K*ln(n) in terms of M and substituting the result into the equation for ln(B) gives B proportional to M to the (3/4) power.

Deductions

The size characteristics of the terminal elements are scale invariant so the dependencies of the sizes of the other elements on scale are easily determined; i.e.,

r0 = -KrK
Since = n-1/2,
r0 = (nK)1/2rK
r0 = (NK)1/2rK

The total number of terminal elements NK was previously found to be proportional to the metabolic rate B, which in turn was found proportional to M3/4. Therefore r0 is proportional to (M3/4)1/2=M3/8.

Likewise

l0 = γ-KlK
and γ=n1/3 so
l0 = NK1/3lK
and thus l0 is proportional to
B1/3=(M3/4)1/3 = M1/4

(To be continued.)

The energy minimization component of the model is dealt with elsewhere. That part of the model is not a success but its failure does not detract from the quality of the model.


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