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Bioinformatics of the Brain

Reaction Method 1 (R1) Exponential Proliferation:

R (c) =ρc

(5.8)

Reaction Method 2 (R2) Verhulst or logistic function:

R (c) =ρc ^{cm −c}

cm

(5.9)

Reaction Method 3 (R3) Gompertz equation:

R (c) =ρc ln

cm

c



(5.10)

Where ρ denotes the proliferation rate of cells (also called geometrical

rate) and it represent the relative increase of cell concentration per time unit,

cm is the maximum tumor cell concentration parameter. In normalized scale,

cm = 1 which makes Verhulst or logistic function becomes Fisher’s equation

[11, 13, 16, 18, 22]:

R (c) =ρc (1 −c)

(5.11)

and Gompertz equation will take the form:

R (c) =ρc ln

1

c



(5.12)

For the HGG, the regularly used value for the proliferation rate is ρ = 0.012

per day, however, the LGG employs much lower value such as ρ = 0.0012 per

day.

5.4

Diffusion Models

The diffusion tensor D(x) has been represented in the literature with different

forms and parameters values. The general form can be represented by the

following equation:

D (x) = DW GC (x) W (x)

(5.13)

where DW GC(x) is the inhomogeneous diffusion coefficient, and W(x) is the

diffusion tensor. Inhomogeneous diffusion coefficient DW GC(x) can be deter-

mined using:

DW GC (x) =







DW M

if x ∈WM

DGM

if x ∈GM

DCSF

if x ∈CFS

(5.14)