Systems Biology (Methods in Molecular Biology)

Steps 1, 2 are related to the process of mitosis and apoptosis,

wherexrepresents the population of proliferating tumor cells in an

avascular phase, and the reactions are associated with the mitosiskm

and apoptosiskapconstants. TheNrepresents the population of

normal cells,Hthe population of the host cells [38]. The numbers

of members ofN,Hare considered constants. Finally, ncp repre-

sent non-cancerous products.

The dynamic behavior of the chemical network model for

avascular growth is given by the following ordinary differential

equation [21]:

dx

dt

¼γxx^2 , ð 19 Þ

where γ¼NkmH. The exact solution of Eq. (19)is

xtðÞ¼ 1 þγeγt. The stability analysis of Eq. (19) shows that at

γ¼0 a transcritical bifurcation [47] takes place and the system

evolves (avascular growth) toward a stable steady state, known as

the dormant state [48]. Hence, as we postulated in previous works

[21, 49], this process resembles a “second-order” phase transition,

whose biological implication is clear: the difficulty of early detection

of cancer.

Such formalism discussed above [21] can be extended to the

study of the dynamics of prostate tumor cell lines, LNCaP and PC3

[49]. The system in question is a 2D region with characteristic length

Lin which initially there are very few tumor cells. The cell density

increases with time due to the proliferation of these cells (seeFig. 1).

The morphology observed in this region has fractal nature as a

result of the stochastic nature of the mitosis and apoptosis processes

that occur at the level of single cells [28]. Thus we have

x¼Ldf, ð 20 Þ

L

T = 1

T = 5 T = 6 T = 7 T = 8

T = 2 T = 3 T = 4

Fig. 1In vitro growth of prostate tumor cell line LNCaP.Tis time in days. Reprinted from [49]

Parameters Estimation in Phase-Space Landscape Reconstruction of Cell Fate... 135