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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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Bifurcation diagram varying γ. The bifurcation diagram of  varying γ, showing the attracting regions. In (a) we show wide range of variation of γ, and in (b), a zoom near origin. For γ <γth (b) the trivial  is attractor for all initial conditions (region Ic). For γ > γth (a), with respect to the coordinates of small equilibrium point , the trivial  is attractor for regions I, Ia and Ib; and we have three possibilities: (1) for , the non-trivial  is attractor for initial conditions in regions II and III; (2) for , stable limit cycle circulating unstable , in regions IIa and IIIa; and (3) for , trivial  is attractor for regions IV and V. In the latter case, the way to reaching the trivial equilibrium is different for initial conditions in Ib and IV or V. 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 10: Bifurcation diagram varying γ. The bifurcation diagram of varying γ, showing the attracting regions. In (a) we show wide range of variation of γ, and in (b), a zoom near origin. For γ <γth (b) the trivial is attractor for all initial conditions (region Ic). For γ > γth (a), with respect to the coordinates of small equilibrium point , the trivial is attractor for regions I, Ia and Ib; and we have three possibilities: (1) for , the non-trivial is attractor for initial conditions in regions II and III; (2) for , stable limit cycle circulating unstable , in regions IIa and IIIa; and (3) for , trivial is attractor for regions IV and V. In the latter case, the way to reaching the trivial equilibrium is different for initial conditions in Ib and IV or V. The scales of vertical and horizontal axes must be multiplied by the factors shown in the legends to obtain the actual values.

Mentions: In Figure 10, we show the bifurcation diagram, considering as a function of γ (curve T in (a) and (b) of Figure 9, which appear at γ = γth). In (b), a zoom near zero is shown. In Figures 10(a) and 10(b), for γ >γth initial conditions set in regions marked with I, Ia and Ib are attracted to the trivial equilibrium point . However, initial conditions set in regions marked with II and III are attracted to ; to a limit cycle circulating in regions IIa and IIIa ; and to in regions IV and V , where E decreases and, then, increases to equilibrium value E0. The Hopf bifurcation occurs at (supercritical) and (subcritical) [17]. The special values are: γth = 5.450 × 10-4 and and , and and . Figure 9(b) showed that at , E is very low. For γ < γth (b), all initial conditions set in region marked with Ic are attracted to the trivial equilibrium point , which is the unique equilibrium, because there is not any positive solution for equation (5). Hence, there is a threshold of the parameter γ, denoted γth, below which all trajectories go to trivial equilibrium.


Mathematical modeling of solid cancer growth with angiogenesis.

Yang HM - Theor Biol Med Model (2012)

Bifurcation diagram varying γ. The bifurcation diagram of  varying γ, showing the attracting regions. In (a) we show wide range of variation of γ, and in (b), a zoom near origin. For γ <γth (b) the trivial  is attractor for all initial conditions (region Ic). For γ > γth (a), with respect to the coordinates of small equilibrium point , the trivial  is attractor for regions I, Ia and Ib; and we have three possibilities: (1) for , the non-trivial  is attractor for initial conditions in regions II and III; (2) for , stable limit cycle circulating unstable , in regions IIa and IIIa; and (3) for , trivial  is attractor for regions IV and V. In the latter case, the way to reaching the trivial equilibrium is different for initial conditions in Ib and IV or V. 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 10: Bifurcation diagram varying γ. The bifurcation diagram of varying γ, showing the attracting regions. In (a) we show wide range of variation of γ, and in (b), a zoom near origin. For γ <γth (b) the trivial is attractor for all initial conditions (region Ic). For γ > γth (a), with respect to the coordinates of small equilibrium point , the trivial is attractor for regions I, Ia and Ib; and we have three possibilities: (1) for , the non-trivial is attractor for initial conditions in regions II and III; (2) for , stable limit cycle circulating unstable , in regions IIa and IIIa; and (3) for , trivial is attractor for regions IV and V. In the latter case, the way to reaching the trivial equilibrium is different for initial conditions in Ib and IV or V. The scales of vertical and horizontal axes must be multiplied by the factors shown in the legends to obtain the actual values.
Mentions: In Figure 10, we show the bifurcation diagram, considering as a function of γ (curve T in (a) and (b) of Figure 9, which appear at γ = γth). In (b), a zoom near zero is shown. In Figures 10(a) and 10(b), for γ >γth initial conditions set in regions marked with I, Ia and Ib are attracted to the trivial equilibrium point . However, initial conditions set in regions marked with II and III are attracted to ; to a limit cycle circulating in regions IIa and IIIa ; and to in regions IV and V , where E decreases and, then, increases to equilibrium value E0. The Hopf bifurcation occurs at (supercritical) and (subcritical) [17]. The special values are: γth = 5.450 × 10-4 and and , and and . Figure 9(b) showed that at , E is very low. For γ < γth (b), all initial conditions set in region marked with Ic are attracted to the trivial equilibrium point , which is the unique equilibrium, because there is not any positive solution for equation (5). Hence, there is a threshold of the parameter γ, denoted γth, below which all trajectories go to trivial equilibrium.

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