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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

Bifurcation diagram varying ε. The bifurcation diagram of  varying ε, showing the attracting regions. With respect to the coordinates of small equilibrium point , the trivial  is attractor for initial conditions in a very small region I, and for initial conditions in regions II and III  is the attractor (a). There is not a critical value due to influx γP in equation for A. However, for γ = 0, we have bifurcation diagram similar to the Figure 10.b: for ε <εth, trivial  is attractor in I (and for all initial conditions in Ia), and  is attractor in II and III (b) for ε > εth. 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 12: Bifurcation diagram varying ε. The bifurcation diagram of varying ε, showing the attracting regions. With respect to the coordinates of small equilibrium point , the trivial is attractor for initial conditions in a very small region I, and for initial conditions in regions II and III is the attractor (a). There is not a critical value due to influx γP in equation for A. However, for γ = 0, we have bifurcation diagram similar to the Figure 10.b: for ε <εth, trivial is attractor in I (and for all initial conditions in Ia), and is attractor in II and III (b) for ε > εth. 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 12, we show the bifurcation diagram, considering as a function of ε (curve T in (a) and (b) of Figure 11). When γ >0, there are not neither special nor threshold values: when initial conditions are set in a small region I, trajectories are attracted to the trivial equilibrium point , and for initial conditions set in regions II and III, trajectories are attracted to (a). This behavior results by the existence of influx in equation for A, given by the term γP. Hence, if we let γ = 0, a different bifurcation arises (b).


Mathematical modeling of solid cancer growth with angiogenesis.

Yang HM - Theor Biol Med Model (2012)

Bifurcation diagram varying ε. The bifurcation diagram of  varying ε, showing the attracting regions. With respect to the coordinates of small equilibrium point , the trivial  is attractor for initial conditions in a very small region I, and for initial conditions in regions II and III  is the attractor (a). There is not a critical value due to influx γP in equation for A. However, for γ = 0, we have bifurcation diagram similar to the Figure 10.b: for ε <εth, trivial  is attractor in I (and for all initial conditions in Ia), and  is attractor in II and III (b) for ε > εth. 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 12: Bifurcation diagram varying ε. The bifurcation diagram of varying ε, showing the attracting regions. With respect to the coordinates of small equilibrium point , the trivial is attractor for initial conditions in a very small region I, and for initial conditions in regions II and III is the attractor (a). There is not a critical value due to influx γP in equation for A. However, for γ = 0, we have bifurcation diagram similar to the Figure 10.b: for ε <εth, trivial is attractor in I (and for all initial conditions in Ia), and is attractor in II and III (b) for ε > εth. 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 12, we show the bifurcation diagram, considering as a function of ε (curve T in (a) and (b) of Figure 11). When γ >0, there are not neither special nor threshold values: when initial conditions are set in a small region I, trajectories are attracted to the trivial equilibrium point , and for initial conditions set in regions II and III, trajectories are attracted to (a). This behavior results by the existence of influx in equation for A, given by the term γP. Hence, if we let γ = 0, a different bifurcation arises (b).

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