Systems Biology (Methods in Molecular Biology)

tensor) taking into account not only heterogeneity but also anisot-

ropy [34] associated with the biological fibered structures of

tissues.

Specifically in this case the symmetric diffusion tensor results

into

Db¼

Dxx t;x

⇀

Dxy t;x

⇀

Dxz t;x

⇀

Dyx t;x

⇀



Dyy t;x

⇀



Dyz t;x

⇀



Dzx t;x

⇀



Dzy t;x

⇀



Dzz t;x

⇀



0

B

B

B

@

1

C

C

C

A

so that the matter flux vectorJ

⇀

¼Dbn∇

⇀

cis not directed as the

concentration gradient anymore. Stated more technically, such a

vector is not orthogonal to the surfaces of constant cancer cells

density.

In ref.32 however, a further relevant improvement of the

formulation just discussed was presented. Taking advantage of

modeling works on animal dispersal, in fact, the authors hypothe-

sized that cancer cells would act as a predator against prey. In

ecological models, this effect can be mathematically described in

several ways although a significant contribution can be given by

changing the diffusive dynamics as follows. If there are many ani-

mals in a given region, they tend to eat the food supplies rapidly. In

this terminology, we could think about several animal species which

eat the same food as grass for instance. Alternatively, we could talk

about lions or others eating smaller mammals. In any case, these

“eaters,” after realizing that food is becoming scarce tend to reach

zones where there are fewer competitors and almost settle down

there. This process should be a sort of random walk type. When the

random walkers are many, one can substitute the dynamics with a

diffusive one. Here, however, one must take into account that this

is not a conventional diffusive process but an anomalous one,

because when eaters are locally many, they want to diffuse away

rapidly, while when their population locally reduces, their dynamics

is of “walkabout” type, i.e., a little diffusion. In other words, we are

saying that the diffusion coefficient is a function of the concentra-

tion of the diffusive species itself, i.e.,Dct;x

⇀



.

This mathematical modification has dramatic consequences

because it leads to a porous media type nonlinear partial differential

equation [6], which for the sake of simplicity in dimensionless form

results schematically in

∂C

∂T

¼σm∇

⇀

n Cm∇

⇀

C



þFCðÞ:

In the limit of vanishingm(hereσis a constant), this process

gives the previously discussed reaction-diffusion equation. On the

210 Christian Cherubini et al.