Systems Biology (Methods in Molecular Biology)

∂c

∂t

¼FcðÞþD

∂^2 c

∂x^2

þ

∂^2 c

∂y^2

þ

∂^2 c

∂z^2



whereDis the (effective) homogeneous and isotropic diffusion

coefficient resulting from a suitable smoothing of the heteroge-

neous character of the tissue. On the other handF(c) represents the

reaction term which accounts for the generation and disappearance

of cancer cells. The diffusive part of the equation by including

partial derivatives couples each point of tissue to the closer ones

(a communication term which is welcome in TOFT). The reaction

term instead does not contain derivatives so that it deals with what

happens to the point itself only (and internal cancer cell dynamics

which is fine for SMT). Both terms in the equation interact to have

a space-time dynamics for cancer growth. The proposed form for

the functionF(c) is the simplest one, i.e. a cubic one:

FcðÞ¼kcðÞc 1 ðÞcc 2 ðÞc 3 c

wherec 1 ,c 2 ,c 3 andkare constants with 0<c 1 <c 2 <c 3 .In

dynamical systems theory, this is a basic way to obtain bistability.

In fact the value of speciescwill settle down asymptotically in time

whether toc 1 (in our case this is zero corresponding to no cell

Fig. 1Finite elements simulation at a given time of a brain cancer cells concentration concentric isosurfaces
(indifferent colors) in a realistic NMR imported geometry

208 Christian Cherubini et al.