Systems Biology (Methods in Molecular Biology)

The diagnosis of tumor proliferation capacity and invasion

capacity is a very complex issue, because these terms include many

factors such as the tumor aggressiveness, which is related to the

tumor growth rateξ_, and the tumor invasion capacity, which is

associated with the fractal dimensiondf[75] among other factors.

We have thatS_ican be approximately determined (seeEq. ( 8)) from

the rate of mitosis and apoptosis,ψ,ηas

S_iRðÞψηlnψ

η

ð 26 Þ

whereðÞψη ξ_is a tumor growth rate.

The fractal dimensiondfquantifies the tumor malignancy, in

other words, the tumor capacity to invade and infiltrate healthy

tissue [75]. In fact, the fractal dimension, as has been pointed out

by other authors [76], can be considered “... a quantitative shape

descriptors which possess thermodynamics meaning and they could

provide insights into the complexity score of the observed system....”

As shown in previous works [35], the fractal dimensiondfcan

be given as a function of the quotient between mitosisψ and

apoptosisηrates, which quantify the tumor aggressiveness as

df¼

5 ψη

1 þψη

!

ð 27 Þ

Substituting (Eq.27) into (Eq.26) we obtain that the entropy

production rate can be expressed as a function of fractal dimension

dfas

S_i¼R_ξln^5 df

1 þdf



ð 28 Þ

In equation (28) two properties fundamentals of the tumor

growth are exhibited: the rate of growthξ_, which is associated with

their invasiveness capacity and a morphological characteristic as the

fractal dimensiondfof the tumor interface that as we have said,

quantifies the capacity of the tumor to invade and infiltrate healthy

tissue, that is, its complexity. This represents the “degree of malig-

nancy,” which quantifies the capacity of the tumor invasiveness and

infiltration into the healthy tissue.

An increment ofdfis associated with a reduction of the values of

the quotient mitosis/apoptosis and to higher values of entropy

production rate and this is consistent with the findings of Luo [77].

Starting from Eqs. (28) and (13) and substituting in (12) the

following results hold:

Ψ¼TSi¼q_Gl¼TRξ_ln 

1

dfþ 1

ðÞdf 5



ð 29 Þ

Equation (29) shows a relationship between the entropy pro-

duction per unit time, the dissipation function of metabolic rate,

Parameters Estimation in Phase-Space Landscape Reconstruction of Cell Fate... 143