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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 IB-model and analytical result.The wave speed of the propagating tumour margin determined from both the individual-based model (dashed line) and phase space analysis of the continuum approximation (solid line). In (a) the switch rate to proliferation is fixed at , while in (b) we have fixed .
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pcbi-1002556-g006: Comparison between IB-model and analytical result.The wave speed of the propagating tumour margin determined from both the individual-based model (dashed line) and phase space analysis of the continuum approximation (solid line). In (a) the switch rate to proliferation is fixed at , while in (b) we have fixed .

Mentions: When it comes to the IB model, we have to take into account the stochastic nature of the model, and therefore need to estimate the average margin velocity from a large number of simulations (100 independent realisations). Each simulation was started with a single P-cell at the center of the lattice and the model was simulated for 100 time steps (cell cycles). In each time step the location of the cells was recorded and from this we calculated the occupation probability of finding any cell at location at time . The wave speed was then approximated by taking the average propagation speed of in the and direction (as in the PDE case). In comparing with the two-dimensional simulations we need to rescale the diffusion coefficient , since cell movement occurring tangential to the two-dimensional front does not contribute to its propagation. The result can be seen in figure 6, which shows that the analytical result is in good agreement with the discrete individual-based model. The disparity between the IB-model and the analytic answer is largest for small , when the dynamics are dominated by proliferation. This is to be expected since for larger the movement of the cells decorrelates the sites, and hence our assumption about site independence is closer to truth. The analytical results recapitulates the non-monotone dependence on and using this method we found that the largest tumours occur when , i.e. when the ratio between the switching rates is 1∶2.


The impact of phenotypic switching on glioblastoma growth and invasion.

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

Comparison between IB-model and analytical result.The wave speed of the propagating tumour margin determined from both the individual-based model (dashed line) and phase space analysis of the continuum approximation (solid 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-g006: Comparison between IB-model and analytical result.The wave speed of the propagating tumour margin determined from both the individual-based model (dashed line) and phase space analysis of the continuum approximation (solid line). In (a) the switch rate to proliferation is fixed at , while in (b) we have fixed .
Mentions: When it comes to the IB model, we have to take into account the stochastic nature of the model, and therefore need to estimate the average margin velocity from a large number of simulations (100 independent realisations). Each simulation was started with a single P-cell at the center of the lattice and the model was simulated for 100 time steps (cell cycles). In each time step the location of the cells was recorded and from this we calculated the occupation probability of finding any cell at location at time . The wave speed was then approximated by taking the average propagation speed of in the and direction (as in the PDE case). In comparing with the two-dimensional simulations we need to rescale the diffusion coefficient , since cell movement occurring tangential to the two-dimensional front does not contribute to its propagation. The result can be seen in figure 6, which shows that the analytical result is in good agreement with the discrete individual-based model. The disparity between the IB-model and the analytic answer is largest for small , when the dynamics are dominated by proliferation. This is to be expected since for larger the movement of the cells decorrelates the sites, and hence our assumption about site independence is closer to truth. The analytical results recapitulates the non-monotone dependence on and using this method we found that the largest tumours occur when , i.e. when the ratio between the switching rates is 1∶2.

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