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Equilibrium solutions for microscopic stochastic systems in population dynamics
Spatial stochastic models of cancer: Fitness, migration, invasion
1.  Department of Mathematics, University of California Irvine, Irvine CA 92697, United States 
References:
[1] 
A. Anderson, M. Chaplain, K. Rejniak and J. Fozard, Singlecell based models in biology and medicine,, Math. Med. Biol., 25 (2008), 185. 
[2] 
A. Anderson and V. Quaranta, Integrative mathematical oncology,, Nature Reviews Cancer, 8 (2008), 227. 
[3] 
R. M. Anderson and R. M. May, Coevolution of hosts and parasites,, Parasitology, 85 (1982), 411. 
[4] 
M. Boots, P. J. Hudson and A. Sasaki, Large shifts in pathogen virulence relate to host population structure,, Science, 303 (2004), 842. 
[5] 
J. Breivik and G. Gaudernack, Carcinogenesis and natural selection: A new perspective to the genetics and epigenetics of colorectal cancer,, Adv. Cancer Res., 76 (1999), 187. 
[6] 
H. Byrne, T. Alarcón, M. Owen, S. Webb and P. Maini, Modeling aspects of cancer dynamics: A review,, Phi. Trans. R. Soc. A, 364 (2006), 1563. doi: 10.1098/rsta.2006.1786. 
[7] 
A. Chauvière, L. Preziosi and C. Verdier, "Cell Mechanics: From Single ScaleBased Models to Multiscale Modeling,", CRC Press, 32 (2009). doi: 10.1201/9781420094558. 
[8] 
B. Chopard, R. Ouared, A. Deutsch, H. Hatzikirou and D. WolfGladrow, Latticegas cellular automaton models for biology: from fluids to cells,, Acta Biotheoretica, 58 (2010), 329. 
[9] 
T. Deisboeck and G. Stamatakos, "Multiscale Cancer Modeling,", CRC Press, 34 (2010). 
[10] 
T. Deisboeck, L. Zhang, J. Yoon and J. Costa, In silico cancer modeling: is it ready for prime time?,, in press., (). 
[11] 
A. Deutsch and S. Dormann, "Cellular Automaton Modeling of Biological Pattern Formation,", Birkhauser, (2005). 
[12] 
D. Drasdo and S. Höhme, On the role of physics in the growth and pattern of multicellular systems: What we learn from individualcell based models?,, J. Stat. Phys., 128 (2007), 287. doi: 10.1007/s109550079289x. 
[13] 
D. Ebert and E. A. Herre, The evolution of parasitic diseases,, Parasitol Today, 12 (1996), 96. 
[14] 
D. Ebert and K. L. Mangin, The influence of host demography on the evolution of virulence of a microsporidian gut parasite,, Evolution, 51 (1997), 1828. 
[15] 
A. Fasano, A. Bertuzzi and A. Gandolfi, Complex systems in biomedicine chapter mathematical modelling of tumour growth and treatment,, Milan: Springer, (2006), 71. doi: 10.1007/8847003962_3. 
[16] 
S. A. Frank, Models of parasite virulence,, Q. Rev. Biol., 71 (1996), 37. 
[17] 
J. Galle, G. Aust, G. Schaller, T. Beyer and D. Drasdo, Individual cellbased models of the spatial temporal organization of multicellular systems achievements and limitations,, Cytometry, 69A (2006), 704. 
[18] 
R. Gatenby and P. Maini, Mathematical oncology: Cancer summed up,, Nature, 421 (2003). 
[19] 
D. Hanahan and R. Weinberg, The hallmarks of cancer,, CELL, 100 (2000), 57. 
[20] 
P. Hinow,, P. Gerlee, L. McCawley, V. Quaranta, M. Ciobanu, S. Wang,, J. Graham, B. Ayati, J. Claridge, K. Swanson, et al., A spatial model of tumorhost interaction: Application of chemotherapy,, Mathematical Biosciences and Engineering: MBE, 6 (2009). doi: 10.3934/mbe.2009.6.521. 
[21] 
Y. Iwasa, F. Michor and M. A. Nowak, Stochastic tunnels in evolutionary dynamics,, Genetics, 166 (2004), 1571. 
[22] 
Y. Jiao and S. Torquato, A cellular automaton model for tumor growth in heterogeneous environment,, Bulletin of the American Physical Society, 56 (2011). 
[23] 
N. Komarova, Loss and gainoffunction mutations in cancer: Massaction, spatial and hierarchical models,, Jour. Stat. Phys., 128 (2007), 413. doi: 10.1007/s1095500692380. 
[24] 
N. L. Komarova, Spatial stochastic models for cancer initiation and progression,, Bull. Math. Biol., 68 (2006), 1573. doi: 10.1007/s1153800590468. 
[25] 
N. L. Komarova, A. Sengupta and M. A. Nowak, Mutationselection networks of cancer initiation: Tumor suppressor genes and chromosomal instability,, J. Theor. Biol., 223 (2003), 433. doi: 10.1016/S00225193(03)001206. 
[26] 
B. R. Levin, The evolution and maintenance of virulence in microparasites,, Emerg. Infect. Dis., 2 (1996), 93. 
[27] 
L. Merlo, J. Pepper, B. Reid and C. Maley, Cancer as an evolutionary and ecological process,, Nat. Rev. Cancer, 6 (2006), 924. 
[28] 
F. Michor, Y. Iwasa, H. Rajagopalan, C. Lengauer and M. A. Nowak, Linear model of colon cancer initiation,, Cell Cycle, 3 (2004), 358. 
[29] 
P. Moran, "The Statistical Processes of Evolutionary Theory,", Clarendon, (1962). 
[30] 
M. Nowak and K. Sigmund, Evolutionary dynamics of biological games,, Science, 303 (2004), 793. 
[31] 
M. A. Nowak, N. L. Komarova, A. Sengupta, P. V. Jallepalli, I.M. Shih, B. Vogelstein and C. Lengauer, The role of chromosomal instability in tumor initiation,, Proc. Natl. Acad. Sci. U S A, 99 (2002), 16226. 
[32] 
M. A. Nowak and R. M. May, Superinfection and the evolution of parasite virulence,, Proc. Biol. Sci., 255 (1994), 81. 
[33] 
M. A. Nowak, F. Michor, N. L. Komarova and Y. Iwasa, Evolutionary dynamics of tumor suppressor gene inactivation,, Proc. Natl. Acad. Sci. U S A, 101 (2004), 10635. 
[34] 
P. Nowell, The clonal evolution of tumor cell populations,, Science, 194 (1976), 23. 
[35] 
V. Quaranta, K. Rejniak, P. Gerlee and A. Anderson, Invasion emerges from cancer cell adaptation to competitive microenvironments: Quantitative predictions from multiscale mathematical models,, Sem. Cancer Biol., (2008). 
[36] 
K. Rejniak and A. Anderson, Hybrid models of tumor growth,, Wiley Interdisciplinary Reviews: Systems Biology and Medicine, 3 (2011), 115. 
[37] 
C. W. RinkerSchaeffer, J. P. O'Keefe, D. R. Welch and D. Theodorescu, Metastasis suppressor proteins: Discovery, molecular mechanisms, and clinical application,, CLINICAL CANCER RESEARCH, 12 (2006), 3882. 
[38] 
J. Sagotsky and T. Deisboeck, Simulating cancer growth with agentbased models,, Multiscale Cancer Modeling, 34 (2010). 
[39] 
A. Sasaki and M. Boots, Parasite evolution and extinctions,, Ecology Letters, 6 (2003). 
[40] 
J. M. Smith, "Evolution and the Theory of Games,", Cambridge University Press, (1982). 
[41] 
C. Thalhauser, J. Lowengrub, D. Stupack and N. Komarova, Research selection in spatial stochastic models of cancer: Migration as a key modulator of fitness,, Biology Direct, 5 (2010). 
[42] 
T. L. Vincent and J. S. Brown, "Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics,", Cambridge University Press, (2005). 
[43] 
P. Vineis and M. Berwick, The population dynamics of cancer: A Darwinian perspective,, Int. J. Epidemiol, 35 (2006), 1151. 
[44] 
D. Wodarz and N. Komarova, "Computational Biology of Cancer: Lecture Notes and Mathematical Modeling,", World Scientific, (2005). 
[45] 
S. Wright, The roles of mutation, inbreeding, crossbreeding, and selection in evolution,, in, (1932), 355. 
[46] 
L. Zhang, Z. Wang, J. Sagotsky and T. Deisboeck, Multiscale agentbased cancer modeling,, Journal of Mathematical Biology, 58 (2009), 545. doi: 10.1007/s0028500802111. 
show all references
References:
[1] 
A. Anderson, M. Chaplain, K. Rejniak and J. Fozard, Singlecell based models in biology and medicine,, Math. Med. Biol., 25 (2008), 185. 
[2] 
A. Anderson and V. Quaranta, Integrative mathematical oncology,, Nature Reviews Cancer, 8 (2008), 227. 
[3] 
R. M. Anderson and R. M. May, Coevolution of hosts and parasites,, Parasitology, 85 (1982), 411. 
[4] 
M. Boots, P. J. Hudson and A. Sasaki, Large shifts in pathogen virulence relate to host population structure,, Science, 303 (2004), 842. 
[5] 
J. Breivik and G. Gaudernack, Carcinogenesis and natural selection: A new perspective to the genetics and epigenetics of colorectal cancer,, Adv. Cancer Res., 76 (1999), 187. 
[6] 
H. Byrne, T. Alarcón, M. Owen, S. Webb and P. Maini, Modeling aspects of cancer dynamics: A review,, Phi. Trans. R. Soc. A, 364 (2006), 1563. doi: 10.1098/rsta.2006.1786. 
[7] 
A. Chauvière, L. Preziosi and C. Verdier, "Cell Mechanics: From Single ScaleBased Models to Multiscale Modeling,", CRC Press, 32 (2009). doi: 10.1201/9781420094558. 
[8] 
B. Chopard, R. Ouared, A. Deutsch, H. Hatzikirou and D. WolfGladrow, Latticegas cellular automaton models for biology: from fluids to cells,, Acta Biotheoretica, 58 (2010), 329. 
[9] 
T. Deisboeck and G. Stamatakos, "Multiscale Cancer Modeling,", CRC Press, 34 (2010). 
[10] 
T. Deisboeck, L. Zhang, J. Yoon and J. Costa, In silico cancer modeling: is it ready for prime time?,, in press., (). 
[11] 
A. Deutsch and S. Dormann, "Cellular Automaton Modeling of Biological Pattern Formation,", Birkhauser, (2005). 
[12] 
D. Drasdo and S. Höhme, On the role of physics in the growth and pattern of multicellular systems: What we learn from individualcell based models?,, J. Stat. Phys., 128 (2007), 287. doi: 10.1007/s109550079289x. 
[13] 
D. Ebert and E. A. Herre, The evolution of parasitic diseases,, Parasitol Today, 12 (1996), 96. 
[14] 
D. Ebert and K. L. Mangin, The influence of host demography on the evolution of virulence of a microsporidian gut parasite,, Evolution, 51 (1997), 1828. 
[15] 
A. Fasano, A. Bertuzzi and A. Gandolfi, Complex systems in biomedicine chapter mathematical modelling of tumour growth and treatment,, Milan: Springer, (2006), 71. doi: 10.1007/8847003962_3. 
[16] 
S. A. Frank, Models of parasite virulence,, Q. Rev. Biol., 71 (1996), 37. 
[17] 
J. Galle, G. Aust, G. Schaller, T. Beyer and D. Drasdo, Individual cellbased models of the spatial temporal organization of multicellular systems achievements and limitations,, Cytometry, 69A (2006), 704. 
[18] 
R. Gatenby and P. Maini, Mathematical oncology: Cancer summed up,, Nature, 421 (2003). 
[19] 
D. Hanahan and R. Weinberg, The hallmarks of cancer,, CELL, 100 (2000), 57. 
[20] 
P. Hinow,, P. Gerlee, L. McCawley, V. Quaranta, M. Ciobanu, S. Wang,, J. Graham, B. Ayati, J. Claridge, K. Swanson, et al., A spatial model of tumorhost interaction: Application of chemotherapy,, Mathematical Biosciences and Engineering: MBE, 6 (2009). doi: 10.3934/mbe.2009.6.521. 
[21] 
Y. Iwasa, F. Michor and M. A. Nowak, Stochastic tunnels in evolutionary dynamics,, Genetics, 166 (2004), 1571. 
[22] 
Y. Jiao and S. Torquato, A cellular automaton model for tumor growth in heterogeneous environment,, Bulletin of the American Physical Society, 56 (2011). 
[23] 
N. Komarova, Loss and gainoffunction mutations in cancer: Massaction, spatial and hierarchical models,, Jour. Stat. Phys., 128 (2007), 413. doi: 10.1007/s1095500692380. 
[24] 
N. L. Komarova, Spatial stochastic models for cancer initiation and progression,, Bull. Math. Biol., 68 (2006), 1573. doi: 10.1007/s1153800590468. 
[25] 
N. L. Komarova, A. Sengupta and M. A. Nowak, Mutationselection networks of cancer initiation: Tumor suppressor genes and chromosomal instability,, J. Theor. Biol., 223 (2003), 433. doi: 10.1016/S00225193(03)001206. 
[26] 
B. R. Levin, The evolution and maintenance of virulence in microparasites,, Emerg. Infect. Dis., 2 (1996), 93. 
[27] 
L. Merlo, J. Pepper, B. Reid and C. Maley, Cancer as an evolutionary and ecological process,, Nat. Rev. Cancer, 6 (2006), 924. 
[28] 
F. Michor, Y. Iwasa, H. Rajagopalan, C. Lengauer and M. A. Nowak, Linear model of colon cancer initiation,, Cell Cycle, 3 (2004), 358. 
[29] 
P. Moran, "The Statistical Processes of Evolutionary Theory,", Clarendon, (1962). 
[30] 
M. Nowak and K. Sigmund, Evolutionary dynamics of biological games,, Science, 303 (2004), 793. 
[31] 
M. A. Nowak, N. L. Komarova, A. Sengupta, P. V. Jallepalli, I.M. Shih, B. Vogelstein and C. Lengauer, The role of chromosomal instability in tumor initiation,, Proc. Natl. Acad. Sci. U S A, 99 (2002), 16226. 
[32] 
M. A. Nowak and R. M. May, Superinfection and the evolution of parasite virulence,, Proc. Biol. Sci., 255 (1994), 81. 
[33] 
M. A. Nowak, F. Michor, N. L. Komarova and Y. Iwasa, Evolutionary dynamics of tumor suppressor gene inactivation,, Proc. Natl. Acad. Sci. U S A, 101 (2004), 10635. 
[34] 
P. Nowell, The clonal evolution of tumor cell populations,, Science, 194 (1976), 23. 
[35] 
V. Quaranta, K. Rejniak, P. Gerlee and A. Anderson, Invasion emerges from cancer cell adaptation to competitive microenvironments: Quantitative predictions from multiscale mathematical models,, Sem. Cancer Biol., (2008). 
[36] 
K. Rejniak and A. Anderson, Hybrid models of tumor growth,, Wiley Interdisciplinary Reviews: Systems Biology and Medicine, 3 (2011), 115. 
[37] 
C. W. RinkerSchaeffer, J. P. O'Keefe, D. R. Welch and D. Theodorescu, Metastasis suppressor proteins: Discovery, molecular mechanisms, and clinical application,, CLINICAL CANCER RESEARCH, 12 (2006), 3882. 
[38] 
J. Sagotsky and T. Deisboeck, Simulating cancer growth with agentbased models,, Multiscale Cancer Modeling, 34 (2010). 
[39] 
A. Sasaki and M. Boots, Parasite evolution and extinctions,, Ecology Letters, 6 (2003). 
[40] 
J. M. Smith, "Evolution and the Theory of Games,", Cambridge University Press, (1982). 
[41] 
C. Thalhauser, J. Lowengrub, D. Stupack and N. Komarova, Research selection in spatial stochastic models of cancer: Migration as a key modulator of fitness,, Biology Direct, 5 (2010). 
[42] 
T. L. Vincent and J. S. Brown, "Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics,", Cambridge University Press, (2005). 
[43] 
P. Vineis and M. Berwick, The population dynamics of cancer: A Darwinian perspective,, Int. J. Epidemiol, 35 (2006), 1151. 
[44] 
D. Wodarz and N. Komarova, "Computational Biology of Cancer: Lecture Notes and Mathematical Modeling,", World Scientific, (2005). 
[45] 
S. Wright, The roles of mutation, inbreeding, crossbreeding, and selection in evolution,, in, (1932), 355. 
[46] 
L. Zhang, Z. Wang, J. Sagotsky and T. Deisboeck, Multiscale agentbased cancer modeling,, Journal of Mathematical Biology, 58 (2009), 545. doi: 10.1007/s0028500802111. 
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