Systems Biology (Methods in Molecular Biology)

wherexas we have said, represents the population of proliferating

tumor cells in the region of characteristic sizeLanddfis the fractal

dimension of the contour. The temporal variation of Eq. (20)is

given by

dlnx

dt

¼

∂df

∂t

lnLþdf

∂lnL

∂t

: ð 21 Þ

We show that the temporal variation of the total number of cells

Ncan be described equivalently through a logistic or Gompertz

dynamical equation [21], such that

dlndf

dt

¼lndfþαlnK, ð 22 Þ

whereK¼df0ðÞe

bA

αis the carrying capacity andAbandαare empirical

constants in the Gompertz equation [50]. Equation (22) has the

analytical solution

dfðÞ¼t k exp 

Ab

α

! !etα

ð 23 Þ

In Fig.2, the time dependence of the fractal dimensiondfis

shown for the dynamical behavior of the prostate tumor cell lines,

LNCaP and PC3 respectively.

Unlike other cells lines showing fractal behavior [38], the

prostate tumor cell lines, LNCaP and PC3 exhibit a multifractal

behavior.

The evolution of a tumor, through the three basic stages avas-

cular, vascular, and metastasis [32], could be generalized through-

out the following heuristic mechanism [21] based on a chemical

network model (see Fig.3).

The present model (Fig.3) is a qualitative representation of the

dynamics of the development of cancer, based on the above experi-

mental evidence, without claim to suggest any type of therapy. The

x,y,zrepresents the population of proliferating tumor cells in an

avascular, vascular, and metastases phases respectively;Nrepresents

the population of normal cells,Hthe population of the host cells

[38] is considered constant andIthe population of immune cells

(T lymphocytes, CTL and natural killer, NK cells [39]) is consid-

ered the control parameter. Finally, ncp represents a non-cancerous

product.

In the Fig.3, the reactions 1, 3 and 2, 4, 6 are related to the

process of mitosis and apoptosis of the proliferating tumor cells

respectively; 5 and 7 correspond to the action of the host

H[38]. Finally, reactions 8 and 9 show the action of the population

immune cells. It is assumed that the cells of typezare generated

during cell apoptosisy(reaction 4) representing the occurrence of

micrometastases [40], that is to say, the spread of cancer cells from

136 Sheyla Montero et al.