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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 β2. The bifurcation diagram of  varying β2, showing the attracting regions. With respect to the coordinates of small equilibrium point  for , the trivial  is attractor for initial conditions in region I, and for regions II and III, trajectories are attracted to . For , all initial conditions (region Ia) are attracted to . 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 7: Bifurcation diagram varying β2. The bifurcation diagram of varying β2, showing the attracting regions. With respect to the coordinates of small equilibrium point for , the trivial is attractor for initial conditions in region I, and for regions II and III, trajectories are attracted to . For , all initial conditions (region Ia) are attracted to . 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 7, we show the bifurcation diagram, considering as a function of β2 (curve T in (a) and (b) of Figure 6). For , initial conditions set in region marked with I are attracted to the trivial equilibrium point , while for those set in regions II and III are attracted to big non-trivial equilibrium point . However, for all initial conditions are attracted to , which is the unique equilibrium, because there is not any positive solution for equation (5). Hence, there is a threshold of the parameter β2, denoted by , above which all trajectories go to trivial equilibrium. At the threshold value both roots assume same value, that is, .


Mathematical modeling of solid cancer growth with angiogenesis.

Yang HM - Theor Biol Med Model (2012)

Bifurcation diagram varying β2. The bifurcation diagram of  varying β2, showing the attracting regions. With respect to the coordinates of small equilibrium point  for , the trivial  is attractor for initial conditions in region I, and for regions II and III, trajectories are attracted to . For , all initial conditions (region Ia) are attracted to . 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 7: Bifurcation diagram varying β2. The bifurcation diagram of varying β2, showing the attracting regions. With respect to the coordinates of small equilibrium point for , the trivial is attractor for initial conditions in region I, and for regions II and III, trajectories are attracted to . For , all initial conditions (region Ia) are attracted to . 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 7, we show the bifurcation diagram, considering as a function of β2 (curve T in (a) and (b) of Figure 6). For , initial conditions set in region marked with I are attracted to the trivial equilibrium point , while for those set in regions II and III are attracted to big non-trivial equilibrium point . However, for all initial conditions are attracted to , which is the unique equilibrium, because there is not any positive solution for equation (5). Hence, there is a threshold of the parameter β2, denoted by , above which all trajectories go to trivial equilibrium. At the threshold value both roots assume same value, that is, .

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