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The impact of phenotypic switching on glioblastoma growth and invasion.

Gerlee P, Nelander S - PLoS Comput. Biol. (2012)

Bottom Line: At the microscopic level, however, proliferation and migration appear to be mutually exclusive phenotypes, as indicated by recent in vivo imaging data.Here, we develop a mathematical model to analyse how the phenotypic switching between proliferative and migratory states of individual cells affects the macroscopic growth of the tumour.From the model we derive a continuum approximation in the form of two coupled reaction-diffusion equations, which exhibit travelling wave solutions whose speed of invasion depends on the model parameters.

View Article: PubMed Central - PubMed

Affiliation: Sahlgrenska Cancer Center, Institute of Medicine, Göteborg, Sweden. philip.gerlee@gu.se

ABSTRACT
The brain tumour glioblastoma is characterised by diffuse and infiltrative growth into surrounding brain tissue. At the macroscopic level, the progression speed of a glioblastoma tumour is determined by two key factors: the cell proliferation rate and the cell migration speed. At the microscopic level, however, proliferation and migration appear to be mutually exclusive phenotypes, as indicated by recent in vivo imaging data. Here, we develop a mathematical model to analyse how the phenotypic switching between proliferative and migratory states of individual cells affects the macroscopic growth of the tumour. For this, we propose an individual-based stochastic model in which glioblastoma cells are either in a proliferative state, where they are stationary and divide, or in motile state in which they are subject to random motion. From the model we derive a continuum approximation in the form of two coupled reaction-diffusion equations, which exhibit travelling wave solutions whose speed of invasion depends on the model parameters. We propose a simple analytical method to predict progression rate from the cell-specific parameters and demonstrate that optimal glioblastoma growth depends on a non-trivial trade-off between the phenotypic switching rates. By linking cellular properties to an in vivo outcome, the model should be applicable to designing relevant cell screens for glioblastoma and cytometry-based patient prognostics.

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Comparison between continuum model and analytical result.The wave speed of the propagating tumour margin determined from both phase space analysis (solid line) and numerical simulation (dashed line). In (a) the switch rate to proliferation is fixed at , while in (b) we have fixed .
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pcbi-1002556-g005: Comparison between continuum model and analytical result.The wave speed of the propagating tumour margin determined from both phase space analysis (solid line) and numerical simulation (dashed line). In (a) the switch rate to proliferation is fixed at , while in (b) we have fixed .

Mentions: In order to test the validity of the wave speed analysis we compared the wave speeds obtained in the continuum and IB models with those from the phase space analysis. For the continuum model an estimate of the wave speed was obtained by, from the initial condition (for proliferating cells), and (for migrating cells), integrating the equations (3)–(4) for 200 time steps (cell cycles). From these solutions we estimated the velocity of the front by measuring the position of a reference point , defined as the point where , as a function of time. The comparison between the speed of propagation in the numerical solution and the wave speed obtained from the phase space analysis is shown in figure 5. The agreement is fairly good and the discrepancies are probably due to error in integration and the deviation in the numerical solution from a perfect travelling wave, which from a given set of initial conditions, is only attained in the limit . However, since we are interested in biologically relevant scenarios the time frame considered is reasonable.


The impact of phenotypic switching on glioblastoma growth and invasion.

Gerlee P, Nelander S - PLoS Comput. Biol. (2012)

Comparison between continuum model and analytical result.The wave speed of the propagating tumour margin determined from both phase space analysis (solid line) and numerical simulation (dashed line). In (a) the switch rate to proliferation is fixed at , while in (b) we have fixed .
© Copyright Policy
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC3375261&req=5

pcbi-1002556-g005: Comparison between continuum model and analytical result.The wave speed of the propagating tumour margin determined from both phase space analysis (solid line) and numerical simulation (dashed line). In (a) the switch rate to proliferation is fixed at , while in (b) we have fixed .
Mentions: In order to test the validity of the wave speed analysis we compared the wave speeds obtained in the continuum and IB models with those from the phase space analysis. For the continuum model an estimate of the wave speed was obtained by, from the initial condition (for proliferating cells), and (for migrating cells), integrating the equations (3)–(4) for 200 time steps (cell cycles). From these solutions we estimated the velocity of the front by measuring the position of a reference point , defined as the point where , as a function of time. The comparison between the speed of propagation in the numerical solution and the wave speed obtained from the phase space analysis is shown in figure 5. The agreement is fairly good and the discrepancies are probably due to error in integration and the deviation in the numerical solution from a perfect travelling wave, which from a given set of initial conditions, is only attained in the limit . However, since we are interested in biologically relevant scenarios the time frame considered is reasonable.

Bottom Line: At the microscopic level, however, proliferation and migration appear to be mutually exclusive phenotypes, as indicated by recent in vivo imaging data.Here, we develop a mathematical model to analyse how the phenotypic switching between proliferative and migratory states of individual cells affects the macroscopic growth of the tumour.From the model we derive a continuum approximation in the form of two coupled reaction-diffusion equations, which exhibit travelling wave solutions whose speed of invasion depends on the model parameters.

View Article: PubMed Central - PubMed

Affiliation: Sahlgrenska Cancer Center, Institute of Medicine, Göteborg, Sweden. philip.gerlee@gu.se

ABSTRACT
The brain tumour glioblastoma is characterised by diffuse and infiltrative growth into surrounding brain tissue. At the macroscopic level, the progression speed of a glioblastoma tumour is determined by two key factors: the cell proliferation rate and the cell migration speed. At the microscopic level, however, proliferation and migration appear to be mutually exclusive phenotypes, as indicated by recent in vivo imaging data. Here, we develop a mathematical model to analyse how the phenotypic switching between proliferative and migratory states of individual cells affects the macroscopic growth of the tumour. For this, we propose an individual-based stochastic model in which glioblastoma cells are either in a proliferative state, where they are stationary and divide, or in motile state in which they are subject to random motion. From the model we derive a continuum approximation in the form of two coupled reaction-diffusion equations, which exhibit travelling wave solutions whose speed of invasion depends on the model parameters. We propose a simple analytical method to predict progression rate from the cell-specific parameters and demonstrate that optimal glioblastoma growth depends on a non-trivial trade-off between the phenotypic switching rates. By linking cellular properties to an in vivo outcome, the model should be applicable to designing relevant cell screens for glioblastoma and cytometry-based patient prognostics.

Show MeSH
Related in: MedlinePlus