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Games 2019, 10(1), 13; https://doi.org/10.3390/g10010013

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

1
Department of Molecular Genetics, Erasmus Medical Center, P.O. Box 2040, NL-3000 CA Rotterdam, The Netherlands
2
Department of Data Science and Knowledge Engineering, Maastricht University, Bouillonstraat 8-10, 6211 LH Maastricht, The Netherlands
3
Department of Integrated Mathematical Oncology, Moffitt Cancer Center, Tampa, FL 33612, USA
4
Delft Institute of Applied Mathematics, Delft University of Technology, Mourik Broekmanweg 6, 2628 XE Delft, The Netherlands
*
Author to whom correspondence should be addressed.
Received: 23 December 2018 / Revised: 10 February 2019 / Accepted: 25 February 2019 / Published: 11 March 2019
(This article belongs to the Special Issue Mathematical Biology and Game Theory)
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PDF [22267 KB, uploaded 12 March 2019]
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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 ( T P ), 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 T P cells (highest frequency in zone 2). Zone 1 T + cells interacting with cells in zone 2 experience a higher likelihood of encountering a T P 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.
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Keywords: tumorigenesis; angiogenesis; metastatic castrate-resistant prostate cancer; spatial game theory tumorigenesis; angiogenesis; metastatic castrate-resistant prostate cancer; spatial game theory
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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You, L.; von Knobloch, M.; Lopez, T.; Peschen, V.; Radcliffe, S.; Koshy Sam, P.; Thuijsman, F.; Staňková, K.; Brown, J.S. Including Blood Vasculature into a Game-Theoretic Model of Cancer Dynamics. Games 2019, 10, 13.

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