Including Blood Vasculature into a Game-Theoretic Model of Cancer Dynamics

Li You, Maximilian von Knobloch, Teresa Lopez, Vanessa Peschen, Sidney Radcliffe, Praveen Koshy Sam, Frank Thuijsman, Katerina Stankova, Joel S Brown

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

For cancer, we develop a 2-D agent-based continuous-space game-theoretical model that considers cancer cells’ proximity to a blood vessel. Based on castrate resistant metastatic prostate cancer (mCRPC), the model considers the density and frequency (eco-evolutionary) dynamics of three cancer cell types: those that require exogenous testosterone ( T+ ), those producing testosterone ( TP ), and those independent of testosterone ( T− ). We model proximity to a blood vessel by imagining four zones around the vessel. Zone 0 is the blood vessel. As rings, zones 1–3 are successively farther from the blood vessel and have successively lower carrying capacities. Zone 4 represents the space too far from the blood vessel and too poor in nutrients for cancer cell proliferation. Within the other three zones that are closer to the blood vessel, the cells’ proliferation probabilities are determined by zone-specific payoff matrices. We analyzed how zone width, dispersal, interactions across zone boundaries, and blood vessel dynamics influence the eco-evolutionary dynamics of cell types within zones and across the entire cancer cell population. At equilibrium, zone 3’s composition deviates from its evolutionary stable strategy (ESS) towards that of zone 2. Zone 2 sees deviations from its ESS because of dispersal from zones 1 and 3; however, its composition begins to resemble zone 1’s more so than zone 3’s. Frequency-dependent interactions between cells across zone boundaries have little effect on zone 2’s and zone 3’s composition but have decisive effects on zone 1. The composition of zone 1 diverges dramatically from both its own ESS, but also that of zone 2. That is because T+ cells (highest frequency in zone 1) benefit from interacting with TP cells (highest frequency in zone 2). Zone 1 T+ cells interacting with cells in zone 2 experience a higher likelihood of encountering a TP cell than when restricted to their own zone. As expected, increasing the width of zones decreases these impacts of cross-boundary dispersal and interactions. Increasing zone widths increases the persistence likelihood of the cancer subpopulation in the face of blood vessel dynamics, where the vessel may die or become occluded resulting in the “birth” of another blood vessel elsewhere in the space. With small zone widths, the cancer cell subpopulations cannot persist. With large zone widths, blood vessel dynamics create cancer cell subpopulations that resemble the ESS of zone 3 as the larger area of zone 3 and its contribution to cells within the necrotic zone 4 mean that zones 3 and 4 provide the likeliest colonizers for the new blood vessel. In conclusion, our model provides an alternative modeling approach for considering density-dependent, frequency-dependent, and dispersal dynamics into cancer models with spatial gradients around blood vessels. Additionally, our model can consider the occurrence of circulating tumor cells (cells that disperse into the blood vessel from zone 1) and the presence of live cancer cells within the necrotic regions of a tumor.
Original languageEnglish
Article number13
JournalGames
Volume10
Issue number1
DOIs
Publication statusPublished - 1 Mar 2019

Keywords

  • tumorigenesis
  • angiogenesis
  • metastatic castrate-resistant prostate cancer
  • spatial game theory

Cite this

You, L., von Knobloch, M., Lopez, T., Peschen, V., Radcliffe, S., Koshy Sam, P., ... Brown, J. S. (2019). Including Blood Vasculature into a Game-Theoretic Model of Cancer Dynamics. Games, 10(1), [13]. https://doi.org/10.3390/g10010013
You, Li ; von Knobloch, Maximilian ; Lopez, Teresa ; Peschen, Vanessa ; Radcliffe, Sidney ; Koshy Sam, Praveen ; Thuijsman, Frank ; Stankova, Katerina ; Brown, Joel S. / Including Blood Vasculature into a Game-Theoretic Model of Cancer Dynamics. In: Games. 2019 ; Vol. 10, No. 1.
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title = "Including Blood Vasculature into a Game-Theoretic Model of Cancer Dynamics",
abstract = "For cancer, we develop a 2-D agent-based continuous-space game-theoretical model that considers cancer cells’ proximity to a blood vessel. Based on castrate resistant metastatic prostate cancer (mCRPC), the model considers the density and frequency (eco-evolutionary) dynamics of three cancer cell types: those that require exogenous testosterone ( T+ ), those producing testosterone ( TP ), and those independent of testosterone ( T− ). We model proximity to a blood vessel by imagining four zones around the vessel. Zone 0 is the blood vessel. As rings, zones 1–3 are successively farther from the blood vessel and have successively lower carrying capacities. Zone 4 represents the space too far from the blood vessel and too poor in nutrients for cancer cell proliferation. Within the other three zones that are closer to the blood vessel, the cells’ proliferation probabilities are determined by zone-specific payoff matrices. We analyzed how zone width, dispersal, interactions across zone boundaries, and blood vessel dynamics influence the eco-evolutionary dynamics of cell types within zones and across the entire cancer cell population. At equilibrium, zone 3’s composition deviates from its evolutionary stable strategy (ESS) towards that of zone 2. Zone 2 sees deviations from its ESS because of dispersal from zones 1 and 3; however, its composition begins to resemble zone 1’s more so than zone 3’s. Frequency-dependent interactions between cells across zone boundaries have little effect on zone 2’s and zone 3’s composition but have decisive effects on zone 1. The composition of zone 1 diverges dramatically from both its own ESS, but also that of zone 2. That is because T+ cells (highest frequency in zone 1) benefit from interacting with TP cells (highest frequency in zone 2). Zone 1 T+ cells interacting with cells in zone 2 experience a higher likelihood of encountering a TP cell than when restricted to their own zone. As expected, increasing the width of zones decreases these impacts of cross-boundary dispersal and interactions. Increasing zone widths increases the persistence likelihood of the cancer subpopulation in the face of blood vessel dynamics, where the vessel may die or become occluded resulting in the “birth” of another blood vessel elsewhere in the space. With small zone widths, the cancer cell subpopulations cannot persist. With large zone widths, blood vessel dynamics create cancer cell subpopulations that resemble the ESS of zone 3 as the larger area of zone 3 and its contribution to cells within the necrotic zone 4 mean that zones 3 and 4 provide the likeliest colonizers for the new blood vessel. In conclusion, our model provides an alternative modeling approach for considering density-dependent, frequency-dependent, and dispersal dynamics into cancer models with spatial gradients around blood vessels. Additionally, our model can consider the occurrence of circulating tumor cells (cells that disperse into the blood vessel from zone 1) and the presence of live cancer cells within the necrotic regions of a tumor.",
keywords = "tumorigenesis, angiogenesis, metastatic castrate-resistant prostate cancer, spatial game theory",
author = "Li You and {von Knobloch}, Maximilian and Teresa Lopez and Vanessa Peschen and Sidney Radcliffe and {Koshy Sam}, Praveen and Frank Thuijsman and Katerina Stankova and Brown, {Joel S}",
year = "2019",
month = "3",
day = "1",
doi = "10.3390/g10010013",
language = "English",
volume = "10",
journal = "Games",
issn = "2073-4336",
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You, L, von Knobloch, M, Lopez, T, Peschen, V, Radcliffe, S, Koshy Sam, P, Thuijsman, F, Stankova, K & Brown, JS 2019, 'Including Blood Vasculature into a Game-Theoretic Model of Cancer Dynamics', Games, vol. 10, no. 1, 13. https://doi.org/10.3390/g10010013

Including Blood Vasculature into a Game-Theoretic Model of Cancer Dynamics. / You, Li; von Knobloch, Maximilian ; Lopez, Teresa ; Peschen, Vanessa ; Radcliffe, Sidney ; Koshy Sam, Praveen ; Thuijsman, Frank; Stankova, Katerina; Brown, Joel S.

In: Games, Vol. 10, No. 1, 13, 01.03.2019.

Research output: Contribution to journalArticleAcademicpeer-review

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AU - You, Li

AU - von Knobloch, Maximilian

AU - Lopez, Teresa

AU - Peschen, Vanessa

AU - Radcliffe, Sidney

AU - Koshy Sam, Praveen

AU - Thuijsman, Frank

AU - Stankova, Katerina

AU - Brown, Joel S

PY - 2019/3/1

Y1 - 2019/3/1

N2 - For cancer, we develop a 2-D agent-based continuous-space game-theoretical model that considers cancer cells’ proximity to a blood vessel. Based on castrate resistant metastatic prostate cancer (mCRPC), the model considers the density and frequency (eco-evolutionary) dynamics of three cancer cell types: those that require exogenous testosterone ( T+ ), those producing testosterone ( TP ), and those independent of testosterone ( T− ). We model proximity to a blood vessel by imagining four zones around the vessel. Zone 0 is the blood vessel. As rings, zones 1–3 are successively farther from the blood vessel and have successively lower carrying capacities. Zone 4 represents the space too far from the blood vessel and too poor in nutrients for cancer cell proliferation. Within the other three zones that are closer to the blood vessel, the cells’ proliferation probabilities are determined by zone-specific payoff matrices. We analyzed how zone width, dispersal, interactions across zone boundaries, and blood vessel dynamics influence the eco-evolutionary dynamics of cell types within zones and across the entire cancer cell population. At equilibrium, zone 3’s composition deviates from its evolutionary stable strategy (ESS) towards that of zone 2. Zone 2 sees deviations from its ESS because of dispersal from zones 1 and 3; however, its composition begins to resemble zone 1’s more so than zone 3’s. Frequency-dependent interactions between cells across zone boundaries have little effect on zone 2’s and zone 3’s composition but have decisive effects on zone 1. The composition of zone 1 diverges dramatically from both its own ESS, but also that of zone 2. That is because T+ cells (highest frequency in zone 1) benefit from interacting with TP cells (highest frequency in zone 2). Zone 1 T+ cells interacting with cells in zone 2 experience a higher likelihood of encountering a TP cell than when restricted to their own zone. As expected, increasing the width of zones decreases these impacts of cross-boundary dispersal and interactions. Increasing zone widths increases the persistence likelihood of the cancer subpopulation in the face of blood vessel dynamics, where the vessel may die or become occluded resulting in the “birth” of another blood vessel elsewhere in the space. With small zone widths, the cancer cell subpopulations cannot persist. With large zone widths, blood vessel dynamics create cancer cell subpopulations that resemble the ESS of zone 3 as the larger area of zone 3 and its contribution to cells within the necrotic zone 4 mean that zones 3 and 4 provide the likeliest colonizers for the new blood vessel. In conclusion, our model provides an alternative modeling approach for considering density-dependent, frequency-dependent, and dispersal dynamics into cancer models with spatial gradients around blood vessels. Additionally, our model can consider the occurrence of circulating tumor cells (cells that disperse into the blood vessel from zone 1) and the presence of live cancer cells within the necrotic regions of a tumor.

AB - For cancer, we develop a 2-D agent-based continuous-space game-theoretical model that considers cancer cells’ proximity to a blood vessel. Based on castrate resistant metastatic prostate cancer (mCRPC), the model considers the density and frequency (eco-evolutionary) dynamics of three cancer cell types: those that require exogenous testosterone ( T+ ), those producing testosterone ( TP ), and those independent of testosterone ( T− ). We model proximity to a blood vessel by imagining four zones around the vessel. Zone 0 is the blood vessel. As rings, zones 1–3 are successively farther from the blood vessel and have successively lower carrying capacities. Zone 4 represents the space too far from the blood vessel and too poor in nutrients for cancer cell proliferation. Within the other three zones that are closer to the blood vessel, the cells’ proliferation probabilities are determined by zone-specific payoff matrices. We analyzed how zone width, dispersal, interactions across zone boundaries, and blood vessel dynamics influence the eco-evolutionary dynamics of cell types within zones and across the entire cancer cell population. At equilibrium, zone 3’s composition deviates from its evolutionary stable strategy (ESS) towards that of zone 2. Zone 2 sees deviations from its ESS because of dispersal from zones 1 and 3; however, its composition begins to resemble zone 1’s more so than zone 3’s. Frequency-dependent interactions between cells across zone boundaries have little effect on zone 2’s and zone 3’s composition but have decisive effects on zone 1. The composition of zone 1 diverges dramatically from both its own ESS, but also that of zone 2. That is because T+ cells (highest frequency in zone 1) benefit from interacting with TP cells (highest frequency in zone 2). Zone 1 T+ cells interacting with cells in zone 2 experience a higher likelihood of encountering a TP cell than when restricted to their own zone. As expected, increasing the width of zones decreases these impacts of cross-boundary dispersal and interactions. Increasing zone widths increases the persistence likelihood of the cancer subpopulation in the face of blood vessel dynamics, where the vessel may die or become occluded resulting in the “birth” of another blood vessel elsewhere in the space. With small zone widths, the cancer cell subpopulations cannot persist. With large zone widths, blood vessel dynamics create cancer cell subpopulations that resemble the ESS of zone 3 as the larger area of zone 3 and its contribution to cells within the necrotic zone 4 mean that zones 3 and 4 provide the likeliest colonizers for the new blood vessel. In conclusion, our model provides an alternative modeling approach for considering density-dependent, frequency-dependent, and dispersal dynamics into cancer models with spatial gradients around blood vessels. Additionally, our model can consider the occurrence of circulating tumor cells (cells that disperse into the blood vessel from zone 1) and the presence of live cancer cells within the necrotic regions of a tumor.

KW - tumorigenesis

KW - angiogenesis

KW - metastatic castrate-resistant prostate cancer

KW - spatial game theory

U2 - 10.3390/g10010013

DO - 10.3390/g10010013

M3 - Article

VL - 10

JO - Games

JF - Games

SN - 2073-4336

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You L, von Knobloch M, Lopez T, Peschen V, Radcliffe S, Koshy Sam P et al. Including Blood Vasculature into a Game-Theoretic Model of Cancer Dynamics. Games. 2019 Mar 1;10(1). 13. https://doi.org/10.3390/g10010013