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Bioinformatics of the Brain

FIGURE 5.4

Sample simulation of primary Glioma growth.

By applying Taylor’s expansion series such that

∂c

∂x ^{= cx+1,y,z −cx−1,y,z}

2hx

= ^{(cx+1,y,z −cx,y,z) + (cx,y,z −cx−1,y,z)}

2hx

(5.23)

∂²c

∂x^{2 = cx+1,y,z −2cx,y,z + cx−1,y,z}

h²x

= ^{(cx+1,y,z −cx,y,z) −(cx,y,z −cx−1,y,z)}

h²x

(5.24)

∂²c

∂x∂y ^{= cx+1,y+1,z −cx−1,y+1,z −cx+1,y−1,z + cx−1,y−1,z}

4hxhy

(5.25)

where h’s are the displacements between nodes in each direction. Deriving the

first derivative as a central difference will provide less solution error than the

forward or backward difference. The term dtR(c) is set to 1 when the desired

location is considered tumor starting point.

5.6

Comparison between Different Model Combinations

Using a C++ finite element model, spatiotemporal simulation of glioma

growth has been achieved. The RDE equation has been used to simulate pri-

mary glioma growth using the same brain model for the same time duration

of 10000 time units (where one keyframe is saved after every 100 times unites)

and from the same starting point where the WM, GM, and CSF of the brain

model have been segmented for applying inhomogeneous diffusion coefficients

(Figure 5.4). The five diffusion methods (M1 to M5) in addition to the three

reaction equations (R1 to R3) have been used in the RDE equation. As a

guide, the homogenous-isotropic (M0-Iso) scenario is used.