Systems Biology (Methods in Molecular Biology)

cancer density) or toc 3 (in our case this is non-zero corresponding

to appreciable cell cancer density). The quantityc 2 plays a role of a

switch in this dynamics. If cancer cells concentrationcgets higher

thanc 2 , then the fate of the point of tissue is in principle deter-

mined, and a stably dense cancer scenario there arises if diffusion

does not carry away cancer cells fast enough. On the other hand, if

the value ofcis lower, the organism can avoid this pathological

situation. It is clear that in this model thenc 2 (in union with

diffusion which—by construction—lowers cellular population)

characterizes the role of the immune system against cancer

progression.

3 Results

The model previously described can be cast in a non-dimensional

form, a standard procedure in any numerical situation and studied

in different dimensions.

In one dimension (1D), the resulting equation is

∂C

∂T

¼

∂^2 C

∂X^2

þaCðÞ 1 CðÞCα

where Cis non-dimensional concentration, and X and Tare

non-dimensional space and time respectively. Parametersa> 0

and 0<α<1 describe the entire dynamics then. The 1D equation

above can be both studied analytically and numerically. In the

former case, one can find a traveling wave exact solution for an

infinite spatial domain given by

C¼

1

2

1 þtanh

ffiffiffi

a

p

ðÞXþVT

2

ffiffiffi

2

p



with constant non-dimensional velocityV¼

ffiffia

2

p

ðÞ 1  2 α. This is a

traveling wave that transfers high concentrations of cancer cells in

regions that had before almost zero values for this species.

These equations have been numerically investigated in 1D and

2D numerical simulations, showing how a nontrivial initial data for

cancer cells leads to a tumor cells cancellation in some regions but

finally to a growth in the space of tumor mass.

Similar studies can also be performed in 3D, even in the pres-

ence of more complicated situations in which the diffusion coeffi-

cient is not a constant but a function of space and time

(inhomogeneous diffusion) instead, i.e.,D¼Dt;x

⇀

.

This is what can be done for instance in studying tumor diffu-

sion processes in the brain, a standard application of mathematical

modeling extremely useful in Neurosurgery [6, 33]. We point out

that a more realistic (but complicated) scenario would be the one in

which the diffusion coefficient is not a scalar but a matrix (diffusion

Systems Biology Modelling 209