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Mathematical modeling of solid cancer growth with angiogenesis.

Yang HM - Theor Biol Med Model (2012)

Bottom Line: Thresholds of interacting parameters were obtained from the steady states analysis.The existence of two equilibrium points determine the strong dependency of dynamical trajectories on the initial conditions.Cancer can be settled in an organ if the following combination matches: better fitness of cancer cells, decrease in the efficiency of the repairing systems, increase in the capacity of sprouting from existing vascularization, and higher capacity of mounting up new vascularization.

View Article: PubMed Central - HTML - PubMed

Affiliation: UNICAMP - IMECC - DMA, Praça Sérgio Buarque de Holanda, Campinas, SP, Brazil. hyunyang@ime.unicamp.br

ABSTRACT

Background: Cancer arises when within a single cell multiple malfunctions of control systems occur, which are, broadly, the system that promote cell growth and the system that protect against erratic growth. Additional systems within the cell must be corrupted so that a cancer cell, to form a mass of any real size, produces substances that promote the growth of new blood vessels. Multiple mutations are required before a normal cell can become a cancer cell by corruption of multiple growth-promoting systems.

Methods: We develop a simple mathematical model to describe the solid cancer growth dynamics inducing angiogenesis in the absence of cancer controlling mechanisms.

Results: The initial conditions supplied to the dynamical system consist of a perturbation in form of pulse: The origin of cancer cells from normal cells of an organ of human body. Thresholds of interacting parameters were obtained from the steady states analysis. The existence of two equilibrium points determine the strong dependency of dynamical trajectories on the initial conditions. The thresholds can be used to control cancer.

Conclusions: Cancer can be settled in an organ if the following combination matches: better fitness of cancer cells, decrease in the efficiency of the repairing systems, increase in the capacity of sprouting from existing vascularization, and higher capacity of mounting up new vascularization. However, we show that cancer is rarely induced in organs (or tissues) displaying an efficient (numerically and functionally) reparative or regenerative mechanism.

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Related in: MedlinePlus

Positive equilibrium values varying β1. The coordinates of the positive equilibrium points varying β1. In (a) we show the coordinates of the small equilibrium point , and in (b), of the big equilibrium point . The coordinates of  is quite insensitive with variation in β1, and in (b) we also show the curves of TC ( at ) and Tg (Ā = 0 at ). The small root T<decreases very slowly, while the big one T> increases up to T> = Tg, and then, decreases. The scales of vertical and horizontal axes must be multiplied by the factors shown in the legends to obtain the actual values.
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Figure 3: Positive equilibrium values varying β1. The coordinates of the positive equilibrium points varying β1. In (a) we show the coordinates of the small equilibrium point , and in (b), of the big equilibrium point . The coordinates of is quite insensitive with variation in β1, and in (b) we also show the curves of TC ( at ) and Tg (Ā = 0 at ). The small root T<decreases very slowly, while the big one T> increases up to T> = Tg, and then, decreases. The scales of vertical and horizontal axes must be multiplied by the factors shown in the legends to obtain the actual values.

Mentions: Let us vary β1 and compute the corresponding equilibrium value using equation (5). As shown in Figure 2, we have two positive solutions: small, T<, and big, T>. Substituting this solution into equation (4), we obtain the coordinates of the non-trivial equilibrium point . In Figure 3 we show the coordinates of the equilibrium points (a) and (b). In (b) we also show the curves of TC and Tg, which intercept the curve of big solution T>. When T>= TC, which occurs at , we have , at which the big non-trivial disappears and arises an another equilibrium , with and other coordinates given by equation (9), which has fixed value for . Coordinates of are same for all . Mathematically, there is another value of β1, which does not change the existing equilibrium point , at which we have T>= Tg, that is, . At this value we have Ā = 0, with .


Mathematical modeling of solid cancer growth with angiogenesis.

Yang HM - Theor Biol Med Model (2012)

Positive equilibrium values varying β1. The coordinates of the positive equilibrium points varying β1. In (a) we show the coordinates of the small equilibrium point , and in (b), of the big equilibrium point . The coordinates of  is quite insensitive with variation in β1, and in (b) we also show the curves of TC ( at ) and Tg (Ā = 0 at ). The small root T<decreases very slowly, while the big one T> increases up to T> = Tg, and then, decreases. The scales of vertical and horizontal axes must be multiplied by the factors shown in the legends to obtain the actual values.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3344686&req=5

Figure 3: Positive equilibrium values varying β1. The coordinates of the positive equilibrium points varying β1. In (a) we show the coordinates of the small equilibrium point , and in (b), of the big equilibrium point . The coordinates of is quite insensitive with variation in β1, and in (b) we also show the curves of TC ( at ) and Tg (Ā = 0 at ). The small root T<decreases very slowly, while the big one T> increases up to T> = Tg, and then, decreases. The scales of vertical and horizontal axes must be multiplied by the factors shown in the legends to obtain the actual values.
Mentions: Let us vary β1 and compute the corresponding equilibrium value using equation (5). As shown in Figure 2, we have two positive solutions: small, T<, and big, T>. Substituting this solution into equation (4), we obtain the coordinates of the non-trivial equilibrium point . In Figure 3 we show the coordinates of the equilibrium points (a) and (b). In (b) we also show the curves of TC and Tg, which intercept the curve of big solution T>. When T>= TC, which occurs at , we have , at which the big non-trivial disappears and arises an another equilibrium , with and other coordinates given by equation (9), which has fixed value for . Coordinates of are same for all . Mathematically, there is another value of β1, which does not change the existing equilibrium point , at which we have T>= Tg, that is, . At this value we have Ā = 0, with .

Bottom Line: Thresholds of interacting parameters were obtained from the steady states analysis.The existence of two equilibrium points determine the strong dependency of dynamical trajectories on the initial conditions.Cancer can be settled in an organ if the following combination matches: better fitness of cancer cells, decrease in the efficiency of the repairing systems, increase in the capacity of sprouting from existing vascularization, and higher capacity of mounting up new vascularization.

View Article: PubMed Central - HTML - PubMed

Affiliation: UNICAMP - IMECC - DMA, Praça Sérgio Buarque de Holanda, Campinas, SP, Brazil. hyunyang@ime.unicamp.br

ABSTRACT

Background: Cancer arises when within a single cell multiple malfunctions of control systems occur, which are, broadly, the system that promote cell growth and the system that protect against erratic growth. Additional systems within the cell must be corrupted so that a cancer cell, to form a mass of any real size, produces substances that promote the growth of new blood vessels. Multiple mutations are required before a normal cell can become a cancer cell by corruption of multiple growth-promoting systems.

Methods: We develop a simple mathematical model to describe the solid cancer growth dynamics inducing angiogenesis in the absence of cancer controlling mechanisms.

Results: The initial conditions supplied to the dynamical system consist of a perturbation in form of pulse: The origin of cancer cells from normal cells of an organ of human body. Thresholds of interacting parameters were obtained from the steady states analysis. The existence of two equilibrium points determine the strong dependency of dynamical trajectories on the initial conditions. The thresholds can be used to control cancer.

Conclusions: Cancer can be settled in an organ if the following combination matches: better fitness of cancer cells, decrease in the efficiency of the repairing systems, increase in the capacity of sprouting from existing vascularization, and higher capacity of mounting up new vascularization. However, we show that cancer is rarely induced in organs (or tissues) displaying an efficient (numerically and functionally) reparative or regenerative mechanism.

Show MeSH
Related in: MedlinePlus