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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 γ near but greater than . Dynamical trajectories of the system (1) considering the values of parameters given in Table 2, except γ near but greater than . When T(0) = 0.42 ( is attracting for T(0) = 0.41), limit cycle disappears for γ = 2.520 × 10-2. Being  unstable, the dynamical trajectories go to trivial  after one oscillation (a). However, the number of oscillations increases if γ is very close to . For γ = 2.519675 × 10-2 and T(0) = 0.42,  is attained after three oscillations: C and E (b), T (c), P and A (d). 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 21: Dynamical trajectories using γ near but greater than . Dynamical trajectories of the system (1) considering the values of parameters given in Table 2, except γ near but greater than . When T(0) = 0.42 ( is attracting for T(0) = 0.41), limit cycle disappears for γ = 2.520 × 10-2. Being unstable, the dynamical trajectories go to trivial after one oscillation (a). However, the number of oscillations increases if γ is very close to . For γ = 2.519675 × 10-2 and T(0) = 0.42, is attained after three oscillations: C and E (b), T (c), P and A (d). The scales of vertical and horizontal axes must be multiplied by the factors shown in the legends to obtain the actual values.

Mentions: In Figures 19, 20 and 21 we illustrate the Hopf bifurcation (see Figure 10 in the main text), using values of parameters given in Table 2, except γ assuming higher values. The following figures are mathematical results, not cancer in an organ.


Mathematical modeling of solid cancer growth with angiogenesis.

Yang HM - Theor Biol Med Model (2012)

Dynamical trajectories using γ near but greater than . Dynamical trajectories of the system (1) considering the values of parameters given in Table 2, except γ near but greater than . When T(0) = 0.42 ( is attracting for T(0) = 0.41), limit cycle disappears for γ = 2.520 × 10-2. Being  unstable, the dynamical trajectories go to trivial  after one oscillation (a). However, the number of oscillations increases if γ is very close to . For γ = 2.519675 × 10-2 and T(0) = 0.42,  is attained after three oscillations: C and E (b), T (c), P and A (d). 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 21: Dynamical trajectories using γ near but greater than . Dynamical trajectories of the system (1) considering the values of parameters given in Table 2, except γ near but greater than . When T(0) = 0.42 ( is attracting for T(0) = 0.41), limit cycle disappears for γ = 2.520 × 10-2. Being unstable, the dynamical trajectories go to trivial after one oscillation (a). However, the number of oscillations increases if γ is very close to . For γ = 2.519675 × 10-2 and T(0) = 0.42, is attained after three oscillations: C and E (b), T (c), P and A (d). The scales of vertical and horizontal axes must be multiplied by the factors shown in the legends to obtain the actual values.
Mentions: In Figures 19, 20 and 21 we illustrate the Hopf bifurcation (see Figure 10 in the main text), using values of parameters given in Table 2, except γ assuming higher values. The following figures are mathematical results, not cancer in an organ.

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