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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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Related in: MedlinePlus

The impact of phenotypic switching rates on tumour mass.(a) The tumour mass at  for the 2-dimensional model as a function of the phenotypic switching rates  (the rate at which cells become proliferative) and  (the rate at which they become motile). (b) The tumour mass at  for the 3-dimensional as a function of  and . The results in 2 and 3 dimensions are similar, although a larger variability seems to exists in the 3-dimensional case.
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pcbi-1002556-g003: The impact of phenotypic switching rates on tumour mass.(a) The tumour mass at for the 2-dimensional model as a function of the phenotypic switching rates (the rate at which cells become proliferative) and (the rate at which they become motile). (b) The tumour mass at for the 3-dimensional as a function of and . The results in 2 and 3 dimensions are similar, although a larger variability seems to exists in the 3-dimensional case.

Mentions: In order to quantify the dependence on the phenotypic switching rates we measured the tumour mass at in the parameter range . The results are displayed in figure 3a and show a strong dependence on the two parameters. For all cells are in the proliferative state, and as expected the mass is independent of . The other extreme where gives rise to tumours with a zero mass, which occurs since the motile cells cannot multiply and eventually die off due to the small but non-zero apoptosis rate . These results are intuitive, but what is more interesting is that tumour cells with intermediate switching rates are the ones that give rise to the largest tumours. Although migratory behaviour does not directly contribute to an increase in the number of cancer cells it has the secondary effect of freeing up space which accelerates growth compared to the tumours dominated purely by proliferation (). The results suggest that for each there is a which gives a maximal tumour growth rate. These results also hold for the more biologically plausible 3-dimensional case (see figure 3b). Although the maximal tumour mass seems to occur for a smaller , and the region of parameter space giving rise to small tumours is considerably larger (upper left region), the qualitative behaviour is similar. The implications of the observation that influences tumour size in a non-monotone way will be discussed later, and we will now proceed to an analytical treatment of the problem.


The impact of phenotypic switching on glioblastoma growth and invasion.

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

The impact of phenotypic switching rates on tumour mass.(a) The tumour mass at  for the 2-dimensional model as a function of the phenotypic switching rates  (the rate at which cells become proliferative) and  (the rate at which they become motile). (b) The tumour mass at  for the 3-dimensional as a function of  and . The results in 2 and 3 dimensions are similar, although a larger variability seems to exists in the 3-dimensional case.
© Copyright Policy
Related In: Results  -  Collection

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

pcbi-1002556-g003: The impact of phenotypic switching rates on tumour mass.(a) The tumour mass at for the 2-dimensional model as a function of the phenotypic switching rates (the rate at which cells become proliferative) and (the rate at which they become motile). (b) The tumour mass at for the 3-dimensional as a function of and . The results in 2 and 3 dimensions are similar, although a larger variability seems to exists in the 3-dimensional case.
Mentions: In order to quantify the dependence on the phenotypic switching rates we measured the tumour mass at in the parameter range . The results are displayed in figure 3a and show a strong dependence on the two parameters. For all cells are in the proliferative state, and as expected the mass is independent of . The other extreme where gives rise to tumours with a zero mass, which occurs since the motile cells cannot multiply and eventually die off due to the small but non-zero apoptosis rate . These results are intuitive, but what is more interesting is that tumour cells with intermediate switching rates are the ones that give rise to the largest tumours. Although migratory behaviour does not directly contribute to an increase in the number of cancer cells it has the secondary effect of freeing up space which accelerates growth compared to the tumours dominated purely by proliferation (). The results suggest that for each there is a which gives a maximal tumour growth rate. These results also hold for the more biologically plausible 3-dimensional case (see figure 3b). Although the maximal tumour mass seems to occur for a smaller , and the region of parameter space giving rise to small tumours is considerably larger (upper left region), the qualitative behaviour is similar. The implications of the observation that influences tumour size in a non-monotone way will be discussed later, and we will now proceed to an analytical treatment of the problem.

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