Systems Biology (Methods in Molecular Biology)

Lyapunov function forpif for certain neighborhoodNofpthe

following conditions hold:

1.V(x)>0 for everyx 6 ¼pinNandV(p)¼ 0

2. The Eulerian derivative,dV xdtðÞ<0 for everyxinN.

In this way, it can be stated for alltt 0 thatpis globally

asymptotically stable, i.e., the system evolves to a minimum of the

functionV(x).

Thus, we have that the entropy production per unit time meets

the necessary and sufficient conditions for Lyapunov function

[26]. Indeed:

S_i¼fðÞΩ > 0 ð 17 Þ

whereΩis the vector of control parameters. The Eulerian derivative

(Eq.17) must meet

dS_i

dt

¼

∂S_i

∂Ω

dΩ

dt

< 0 ð 18 Þ

3 What Can Be Learned from a Phase Transition in Tumor Growth?

The thermodynamics formalism of irreversible processes [17], sys-

tems biology [3], and complex systems theory [27] offers a theo-

retical framework appropriate for the characterization of the

emergence and evolution of cancer. In such a sense, in a previous

work [28], we presented a conceptual definition of the cancer,

which is expressed as follows: “Cancer is a generic name given to a

complex network of interactions of malignant cells, which have lost

their specialization and control over normal growth. This network of

malignant cells could be considered a nonlinear dynamical system,

self-organized in time and space, far from thermodynamic equilib-

rium, exhibiting high complexity[29],robustness[30],and adapt-

ability[31]”.

Tumor cells show two aspects of robustness: functional redun-

dancy, due to cellular heterogeneity, and those arising because of

feedback control systems [28]. The robustness of tumor cells allows

a system to maintain its functionality against various external and

internal perturbations [28]. The control of the robustness of the

tumor cells constitutes a potential strategy for the development of

drugs and therapies.

The evolution of a tumor runs through three basic stages:

avascular, vascular, and metastasis [32]. In the avascular stage, the

tumor grows to a state known as “dormant” state [33], with a

microscopic nature (1 mm diameter). This dormant stage can

remain silent for a long time and is not macroscopically perceptible.

For reasons that are still unknown [34], the tumors that are in

the dormant state leave this state and begin a process of

Parameters Estimation in Phase-Space Landscape Reconstruction of Cell Fate... 133