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Tumor Microenvironment pp 53–76Cite as

Oxygen in the Tumor Microenvironment: Mathematical and Numerical Modeling

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Part of the book series: Advances in Experimental Medicine and Biology ((AEMB,volume 1259))

Abstract

There are many reasons to try to achieve a good grasp of the distribution of oxygen in the tumor microenvironment. The lack of oxygen – hypoxia – is a main actor in the evolution of tumors and in their growth and appears to be just as important in tumor invasion and metastasis. Mathematical models of the distribution of oxygen in tumors which are based on reaction-diffusion equations provide partial but qualitatively significant descriptions of the measured oxygen concentrations in the tumor microenvironment, especially when they incorporate important elements of the blood vessel network such as the blood vessel size and spatial distribution and the pulsation of local pressure due to blood circulation. Here, we review our mathematical and numerical approaches to the distribution of oxygen that yield insights both on the role of the distribution of blood vessel density and size and on the fluctuations of blood pressure.

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Notes

  1. 1.

    Formally, the diffusion equation for heat is the combined result of Fick’s law applied to thermal current J and temperature, J = −K∇T, where K is the thermal conductivity, of the conservation of energy applied to the thermal current and internal (thermal) energy U,

    $$\displaystyle \begin{aligned}-\frac{\partial U}{\partial t} = \nabla\cdot \boldsymbol{J},\end{aligned}$$

    and of the relation between internal energy and temperature, ΔU = C ΔT, where C is the constant-volume thermal capacity, so that one finds the multidimensional diffusion equation for temperature

    $$\displaystyle \begin{aligned} \frac{\partial T}{\partial t} = D \nabla^2 T, \end{aligned}$$

    with D = K∕C.

  2. 2.

    A “superposition” is just a linear combination as in the text, and whenever the principle holds, then any superposition of solutions is also a solution. Much of the value of the principle comes from experiment rather than theory: if one finds experimentally that the principle holds, then one knows that the underlying equations must be linear, just as the diffusion equation.

  3. 3.

    We recall that the enzymatic activity – and therefore also the individual steps of the metabolic pathways – is often described by the Michaelis-Menten (MM) equation

    $$\displaystyle \begin{aligned} v = v_{\mathrm{max}} \frac{[S]}{K_m+[S]} \end{aligned}$$

    where v is the reaction rate and [S] is the concentration of the substrate (in our case, oxygen). The reaction rate depends on two parameters v max and K m which characterize the specific enzymatic process. The MM equation is unable to fit some of the observed reaction rates, which are sigmoid functions of the substrate concentration, and in this case, it is common to turn to the Hill equation, a phenomenological modification of the MM equation

    $$\displaystyle \begin{aligned} v = v_{\mathrm{max}} \frac{[S]^n}{K_m^n+[S]^n} \end{aligned}$$

    with one more parameter, the exponent n. For more details, see, e.g., [12].

  4. 4.

    In the jargon of High Performance Computing, this is the experimenter’s actual waiting time for the completion of the simulation.

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Author information

Authors and Affiliations

  1. Department of Physics, University of Trieste, Trieste, Italy

    Edoardo Milotti

  2. Center for Biophysics & FB Theoretical Physics, Saarland University, Saarbrücken, Germany

    Thierry Fredrich & Heiko Rieger

  3. Department of Biotechnology, University of Verona, Verona, Italy

    Roberto Chignola

Authors
  1. Edoardo Milotti
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  2. Thierry Fredrich
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  3. Roberto Chignola
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  4. Heiko Rieger
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Correspondence to Edoardo Milotti .

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Editors and Affiliations

  1. Department of Radiology, Columbia University Medical Center, New York, NY, USA

    Alexander Birbrair

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Milotti, E., Fredrich, T., Chignola, R., Rieger, H. (2020). Oxygen in the Tumor Microenvironment: Mathematical and Numerical Modeling. In: Birbrair, A. (eds) Tumor Microenvironment. Advances in Experimental Medicine and Biology, vol 1259. Springer, Cham. https://doi.org/10.1007/978-3-030-43093-1_4

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