### #18 The scaling law of the metabolism of tumors

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Malignancy has always been the top killer worldwide. Theoretically, we should be able to describe the metabolism of malignant tumor since we do know something about diffusion, chemical reaction and cell mitosis. However, the complexity and diversity of malignant tumors such as their angiogenesis activities, local necrosis, and diverse inner structures prevent us from formulating a physical model.

To study the problem, the researchers build a simplified model of tumor growth using the kinetics of oxygen, glucose, amino acid, ATP, cell cycles, and lactate metabolism. From this simplified model, they were able to observe the proportion of living cells in the malignant tissues, and they found that the proportion of living cells is related to the distance 's' from that region to the border of tumor by

f(s) = exp(-s/λ), in which λ denotes a characteristic length determined by diffusion and biochemical reactions of neoplastic cells. This observation allows us to derive the metabolic rate of a tumor since we know how much living cells there are. After appropriate normalization with respect to the surface area and difference in metabolic rates among various tumors，the researchers found that the metabolic rate is predictable from the volume of the tumor, which follows a 2/3-power law. For more details, stay tuned to the physics explanation.

/*Physics explanation*/

Assume that there is a tumor with volume V, proportion of living cell F=F(V), proliferation rate of single living cell α, collapsing rate of death cells δ. Define Va(t) = V(t)F(t), which is the volume taken by living cells. Based on the above assumption, we could write down the following ODE:
How could we get the proportion of living cell, F? Assume that the tumor is a sphere. We substitute
V = 4πr^3/3 into the above formula and we get:
We have previously mentioned that the authors derived the proportion of living cells f(s) = exp(-s/λ). Thus from simple calculus and integration by parts, we could derive the volume taken by living cells:
Divide the above equation with V and we will get F:
Perform Taylor expansion upon the above exponential and we could simplify the formula into
However, this formula is not easy to interpret, so we would like to look for its asymptotic behavior. We found that as  λ→∞, F(r)→1；r→0, F(r)→1；r→∞，F ~ 3λ/r；λ→0，F(r)→0. Based on these properties, we could reasonably replace our original formula with
And this allows us to derive something interesting. In our reference, the explicit formula of F could be used to solve the time evolution of tumor size. And even more excitingly, we could discard our original spherical assumption on tumor shape, and allow any shape if only we could define some characteristic length x of the tumor. Based on scaling properties, the volume of tumor could now be written as V=Ax^3, in which A is something related to its shape. Assume that the metabolic rate of single living cell c, volume of single cell v, define η=c/v. Then the metabolic rate of whole tumor, μ, could be expressed as:
We could noticed from this formula that, as the size of tumor becomes very small, μ~ηV. However, μ~3ηλA^(1/3)V^(2/3) when the tumor size is large.

Since different kinds of tumors have different η and A, normalization is necessary to fairly compare different kinds of tumors. With some simple arithmetics we could write

And we use this formula for curve fitting to the real metabolic data of various tumors. The authors found that for different tumor cell lines, the values of  λ were all around 100μm while A varies largely. This indicates the shared physical mechanism of diffusion and intracellular biochemical reaction. Since we have derived the value A for each tumor cell lines, define z = V/A and we could modify our formula into:
Then we could compare the data from various kinds of tumors in the same curve, and the result is shown above. We found that the tumors behave as we predicted: μ'_hat ~ z in small-sized tumors and μ'_hat ~3λz^2/3 in large-sized tumors.

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The figure came from the Figure 2 of the suggested reading.