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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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Impact of model parameters.The wave speed of the propagating tumour margin as a function of (a) , (b)  and (c) . The phenotypic switching rates were fixed at . The dashed line in the inset of (c) has slope 1/2 and shows that .
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pcbi-1002556-g007: Impact of model parameters.The wave speed of the propagating tumour margin as a function of (a) , (b) and (c) . The phenotypic switching rates were fixed at . The dashed line in the inset of (c) has slope 1/2 and shows that .

Mentions: Naturally the other model parameters also affect the rate of tumour invasion (see figure 7). Increasing the proliferation rate leads initially (for small ) to an increase in velocity according to , while for there is a cross-over to a linear dependence with , with . The motility rate also influences the wave speed in a non-linear way according to the relation , which holds for all . Finally, increasing the rate of apoptosis , as expected, decreases the wave speed, and does so in a non-linear way. Actually the dependency on looks very much like that of a second-order phase transition, where the derivative diverges at a critical point , and we have for that (see inset of figure 7c). We observed that the critical apoptosis rate , above which no travelling wave solutions exists and hence the tumour disappears, depends on the other parameters of the model, but that the critical exponent is independent of the other parameters.


The impact of phenotypic switching on glioblastoma growth and invasion.

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

Impact of model parameters.The wave speed of the propagating tumour margin as a function of (a) , (b)  and (c) . The phenotypic switching rates were fixed at . The dashed line in the inset of (c) has slope 1/2 and shows that .
© Copyright Policy
Related In: Results  -  Collection

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

pcbi-1002556-g007: Impact of model parameters.The wave speed of the propagating tumour margin as a function of (a) , (b) and (c) . The phenotypic switching rates were fixed at . The dashed line in the inset of (c) has slope 1/2 and shows that .
Mentions: Naturally the other model parameters also affect the rate of tumour invasion (see figure 7). Increasing the proliferation rate leads initially (for small ) to an increase in velocity according to , while for there is a cross-over to a linear dependence with , with . The motility rate also influences the wave speed in a non-linear way according to the relation , which holds for all . Finally, increasing the rate of apoptosis , as expected, decreases the wave speed, and does so in a non-linear way. Actually the dependency on looks very much like that of a second-order phase transition, where the derivative diverges at a critical point , and we have for that (see inset of figure 7c). We observed that the critical apoptosis rate , above which no travelling wave solutions exists and hence the tumour disappears, depends on the other parameters of the model, but that the critical exponent is independent of the other parameters.

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