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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

Dynamical trajectories using δ = 0.001 and δ = 10.0. Dynamical trajectories of the system (1) considering the fixed values of parameters given in Table 2, except those in the column marked with **. The initial conditions determine the region of attraction. For δ = 0.001 non-trivial  is attractor when T(0) = 2.493 × 10-1 (a), while for δ = 10.0, when T(0) = 6.609 × 10-3 (b). 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 15: Dynamical trajectories using δ = 0.001 and δ = 10.0. Dynamical trajectories of the system (1) considering the fixed values of parameters given in Table 2, except those in the column marked with **. The initial conditions determine the region of attraction. For δ = 0.001 non-trivial is attractor when T(0) = 2.493 × 10-1 (a), while for δ = 10.0, when T(0) = 6.609 × 10-3 (b). The scales of vertical and horizontal axes must be multiplied by the factors shown in the legends to obtain the actual values.

Mentions: We introduced in the model an intermediate phase between epithelial cells and angiogenesis cells. The purpose was to consider a delay in new blood vessel formation (angiogenesis) by the period of time δ-1. This can be suppressed by letting δ → ∞. In Figure 15 we show dynamical trajectories using the initial conditions given in equation (2), and values of the parameters given in the column marked with ** of Table 2, and other parameters are those given in fixed values, except δ. Dynamical trajectories are attracted: (1) for δ = 0.001, to (not shown) when T(0) = 2.492 × 10-1, while for T(0) = 2.493 × 10-1, to (a); and (2) for δ = 10.0, to (not shown) when T(0) = 6.608 × 10-3, while for T(0) = 6.609 × 10-3, to (b). The cancer is triggered at around 900 and 360 days, respectively for δ = 0.001 and 10.0. Including Figure 14(b), the cancer trigger is delayed and initial cancer formation due to mutation must be increased as δ decreases.


Mathematical modeling of solid cancer growth with angiogenesis.

Yang HM - Theor Biol Med Model (2012)

Dynamical trajectories using δ = 0.001 and δ = 10.0. Dynamical trajectories of the system (1) considering the fixed values of parameters given in Table 2, except those in the column marked with **. The initial conditions determine the region of attraction. For δ = 0.001 non-trivial  is attractor when T(0) = 2.493 × 10-1 (a), while for δ = 10.0, when T(0) = 6.609 × 10-3 (b). 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 15: Dynamical trajectories using δ = 0.001 and δ = 10.0. Dynamical trajectories of the system (1) considering the fixed values of parameters given in Table 2, except those in the column marked with **. The initial conditions determine the region of attraction. For δ = 0.001 non-trivial is attractor when T(0) = 2.493 × 10-1 (a), while for δ = 10.0, when T(0) = 6.609 × 10-3 (b). The scales of vertical and horizontal axes must be multiplied by the factors shown in the legends to obtain the actual values.
Mentions: We introduced in the model an intermediate phase between epithelial cells and angiogenesis cells. The purpose was to consider a delay in new blood vessel formation (angiogenesis) by the period of time δ-1. This can be suppressed by letting δ → ∞. In Figure 15 we show dynamical trajectories using the initial conditions given in equation (2), and values of the parameters given in the column marked with ** of Table 2, and other parameters are those given in fixed values, except δ. Dynamical trajectories are attracted: (1) for δ = 0.001, to (not shown) when T(0) = 2.492 × 10-1, while for T(0) = 2.493 × 10-1, to (a); and (2) for δ = 10.0, to (not shown) when T(0) = 6.608 × 10-3, while for T(0) = 6.609 × 10-3, to (b). The cancer is triggered at around 900 and 360 days, respectively for δ = 0.001 and 10.0. Including Figure 14(b), the cancer trigger is delayed and initial cancer formation due to mutation must be increased as δ decreases.

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