Tumor Growth Simulation

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5.2

Reaction Diffusion Equations (RDE) for Modeling

Brain Tumors

In the literature, spatiotemporal models of tumor growth have been exten-

sively employed to simulate the growth of brain gliomas [5–24]. Based on the

Reaction–Diffusion Equation (RDE), these models explain how the pathology

progresses through the tumor cells’ time of proliferation and the space where

they infiltrate into the surrounding tissue. Equation 5.1 and the second Fick’s

law of Kolmogorov-Petrovsky-Piskounov can be used to create the most basic

RDE. Equation 5.2 represents the isotropic version of the formula.

Anisotropic:

dc

dt ^{= ∇. (D(x)∇c) + R(c)}

(5.1)

Isotropic:

dc

dt ^{= D(x)∇2c + R (c)}

(5.2)

Where D(x) is the spatially resolved diffusion tensor that describes cell

diffusion rate at certain point x in time t, c is the Glioma cell concentration

of the same point x and at the same time t. The function R(c) represents

the proliferation component where it is the temporal evolution pattern of the

growth. Some research [8, 9, 14, 16, 17] tends to include treatment therapy

function T(c) to the equation which makes Equation 5.1 takes the form:

dc

dt ^{= ∇. (D(x)∇c) + R (c) −T(c)}

(5.3)

RDE is bounded by some condition where for each point x:

x ∈B

(5.4)

where B is the brain tissue domain

t ≥0

(5.5)

c (x, 0) = c0

(5.6)

where, c0 is initial distribution of tumor cells

n.∇c = 0 on∂B

(5.7)

5.3

Reaction Models

There are numerous ways to express the reaction part, some of them are as

follows [8, 10, 13–18, 20, 21]: