American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Differential impact of sickle cell trait on symptomatic and asymptomatic malaria
  • MBE Home
  • This Issue
  • Next Article
    Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes
2012, 9(4): 843-876. doi: 10.3934/mbe.2012.9.843

Evolution of uncontrolled proliferation and the angiogenic switch in cancer

John D. Nagy 1, and  Dieter Armbruster 2,

1. 

Department of Life Sciences, Scottsdale Community College, 9000 E. Chaparral Rd., Scottsdale, AZ 85256, United States
2. 

School of Mathematical and Statistical Sciences, Arizona State University, PO Box 874501, Tempe AZ, 85287-1804, United States

Received  February 2012 Revised  May 2012 Published  October 2012

The major goal of evolutionary oncology is to explain how malignant traits evolve to become cancer "hallmarks." One such hallmark---the angiogenic switch---is difficult to explain for the same reason altruism is difficult to explain. An angiogenic clone is vulnerable to "cheater" lineages that shunt energy from angiogenesis to proliferation, allowing the cheater to outcompete cooperative phenotypes in the environment built by the cooperators. Here we show that cell- or clone-level selection is sufficient to explain the angiogenic switch, but not because of direct selection on angiogenesis factor secretion---angiogenic potential evolves only as a pleiotropic afterthought. We study a multiscale mathematical model that includes an energy management system in an evolving angiogenic tumor. The energy management model makes the counterintuitive prediction that ATP concentration in resting cells increases with increasing ATP hydrolysis, as seen in other theoretical and empirical studies. As a result, increasing ATP hydrolysis for angiogenesis can increase proliferative potential, which is the trait directly under selection. Intriguingly, this energy dynamic allows an evolutionary stable angiogenesis strategy, but this strategy is an evolutionary repeller, leading to runaway selection for extreme vascular hypo- or hyperplasia. The former case yields a tumor-on-a-tumor, or hypertumor, as predicted in other studies, and the latter case may explain vascular hyperplasia evident in certain tumor types.
Keywords: evolutionary suicide, adaptive dynamics, energy charge, multiscale models, perturbation expansion., Cancer, ATP.
Mathematics Subject Classification: Primary: 92C40, 92D1.
Citation: John D. Nagy, Dieter Armbruster. Evolution of uncontrolled proliferation and the angiogenic switch in cancer. Mathematical Biosciences & Engineering, 2012, 9 (4) : 843-876. doi: 10.3934/mbe.2012.9.843
References:
[1]

B. Alberts, D. Bray, J. Lewis, M. Raff, K. Roberts and J. D. Watson, "Molecular Biology of the Cell," $3^{rd}$ edition, Garland, New York, 1994. Google Scholar

[2]

F. I. Ataullakhanov, S. V. Komarova, M. V. Martynov and V. M. Vitvitsky, A possible role of adenylate metabolism in human erythrocytes: 2. adenylate metabolism is able to improve the erythrocyte volume stabilization, J. Theor. Biol., 183 (1996), 307-316. doi: 10.1006/jtbi.1996.0222.  Google Scholar

[3]

F. I. Ataullakhanov, S. V. Komarova and V. M. Vitvitsky, A possible role of adenylate metabolism in human erythrocytes: simple mathematical model, J. Theor. Biol., 179 (1996), 75-86. doi: 10.1006/jtbi.1996.0050.  Google Scholar

[4]

F. I. Ataullakhanov and V. M. Vitvitsky, What determines the intracellular ATP concentration?, Biosci. Rep., 22 (2002), 501-511. doi: 10.1023/A:1022069718709.  Google Scholar

[5]

F. I. Ataullakhanov, V. M. Vitvitsky, A. M. Zhabotinsky, A. V. Pichugin, O. V. Platonova, B. N. Kholodenko and L. I. Ehrlich, The regulation of glycolysis in human erythrocytes: the dependence of the glycolytic flux on the ATP concentration, Eur. J. Biochem., 115 (1981), 359-365. doi: 10.1111/j.1432-1033.1981.tb05246.x.  Google Scholar

[6]

D. E. Atkinson, "Cellular Energy Metabolism and Its Regulation," Academic Press, New York, 1977. Google Scholar

[7]

L. E. Benjamin, I. Hemo and E. Keshet, A plasticity window for blood vessel remodelling is defined by pericyte coverage of the preformed endothelial network and is regulated by PDGF-B and VEGF, Development, 125 (1998), 1591-1598. Google Scholar

[8]

T. Bønsdorff, M. Gautier, W. Farstad, K. Rønningen, F. Lingaas and I. Olsaker, Mapping of the bovine genes of the de novo AMP synthesis pathway, Anim. Genet., 35 (2004), 438-444. doi: 10.1111/j.1365-2052.2004.01201.x.  Google Scholar

[9]

J. J. Boza, D. Moënnoz, C. E. Bournot, S. Blum, I. Zbinden, P. A. Finot and O. Ballèvre, Role of glutamine on the de novo purine nucleotide synthesis in Caco-2 cells, Eur. J. Nutr., 39 (2000), 38-46. Google Scholar

[10]

D. J. Brat and E. G. Van Meir, Vaso-occlusive and prothrombotic mechanisms associated with tumor hypoxia, necrosis, and accelerated growth in glioblastoma, Lab. Invest., 84 (2004), 397-405. doi: 10.1038/labinvest.3700070.  Google Scholar

[11]

J. P. Collins, "Evolutionary ecology" and the use of natural selection in ecological theory, J. Hist. Biol., 19 (1986), 257-288. doi: 10.1007/BF00138879.  Google Scholar

[12]

J. de Grouchy and C. de Nava, A chromosomal theory of carcinogenesis, Ann. Intern. Med., 69 (1968), 381-391. Google Scholar

[13]

F. Du, X.-H. Zhu, Y. Zhang, M. Friedman, N. Zhang adn K. Uqurbil and W. Chen, Tightly coupled brain activity and cerebral ATP metabolic rate, Proc. Nat. Acad. Sci. USA, 105 (2008), 6409-6414. doi: 10.1073/pnas.0710766105.  Google Scholar

[14]

I. F. Dunn, O. Heese and P. McL. Black, Growth factors in glioma angiogenesis: FGFs, PDGF, EGF, and TGFs, J. Neuro-Onco., 50 (2000), 121-137. doi: 10.1023/A:1006436624862.  Google Scholar

[15]

D. Gammack, H. M. Byrne and C. E. Lewis, Estimating the selective advantage of mutant p53 tumour cells to repeated rounds of hypoxia, Bull. Math. Biol., 63 (2001), 135-166. doi: 10.1006/bulm.2000.0210.  Google Scholar

[16]

S. A. H. Geritz, É. Kisdi, G. Meszéna and J. A. J. Metz, Evolutionarily singular stategies and the adaptive growth and branching of the evolutionary tree, Evol. Ecol., 12 (1998), 35-57. doi: 10.1023/A:1006554906681.  Google Scholar

[17]

A. C. Giese, "Cell Physiology," $5^{th}$ edition, Saunders, Philadelphia, 1973. Google Scholar

[18]

M. Greaves, Darwinian medicine: A case for cancer, Nature Rev. Cancer, 7 (2007), 213-221. Google Scholar

[19]

M. Greaves and C. C. Maley, Clonal evolution in cancer, Nature, 481 (2012), 306-313. doi: 10.1038/nature10762.  Google Scholar

[20]

D. Hanahan and J. Folkman, Patterns and emerging mechanisms of the angiogenic switch during tumorigenesis, Cell, 86 (1996), 353-364. doi: 10.1016/S0092-8674(00)80108-7.  Google Scholar

[21]

D. Hanahan and R. A. Weinberg, The hallmarks of cancer, Cell, 100 (2000), 57-70. doi: 10.1016/S0092-8674(00)81683-9.  Google Scholar

[22]

D. Hanahan and R. A. Weinberg, Hallmarks of cancer: The next generation, Cell, 144 (2011), 646-674. doi: 10.1016/j.cell.2011.02.013.  Google Scholar

[23]

D. G. Hardie, D. Carling and M. Carlson, The AMP-activated/SNF1 protein kinase subfamily: Metabolic sensors of the eukaryotic cell?, Ann. Rev. Biochem., 67 (1998), 821-855. doi: 10.1146/annurev.biochem.67.1.821.  Google Scholar

[24]

T. S. Hauschka, The chromosomes in ontogeny and oncogeny, Cancer Res., 21 (1961), 957-974. Google Scholar

[25]

J. Holash, P. C. Maisonpierre, D. Compton, P. Boland, C. R. Alexander, D. Zagzag, G. D. Yancopolous and S. J. Weigand, Vessel cooperation, regression and growth in tumors mediated by angiopoietins and VEGF, Science, 221 (1998), 1994-1998. Google Scholar

[26]

J. Maynard Smith, "Evolution and the Theory of Games," Cambridge University Press, Cambridge, 1982. Google Scholar

[27]

J. Maynard Smith and G. R. Price, The logic of animal conflict, Nature, 246 (1973), 15-18. doi: 10.1038/246015a0.  Google Scholar

[28]

A. Joshi and B. O. Palsson, Metabolic dynamics in the human red cell. Parts 1-2, J. Theor. Biol., 141 (1989), 515-545. doi: 10.1016/S0022-5193(89)80233-4.  Google Scholar

[29]

A. Joshi and B. O. Palsson, Metabolic dynamics in the human red cell. Parts 3-4, J. Theor. Biol., 142 (1990), 41-85. doi: 10.1016/S0022-5193(05)80012-8.  Google Scholar

[30]

W. G. Kaelin and P. J. Ratcliffe, Oxygen sensiing by metazoans: The central role of the HIF hydroxylase pathway, Mol. Cell, 30 (2008), 393-402. doi: 10.1016/j.molcel.2008.04.009.  Google Scholar

[31]

G. Karoubi, D. J. Stewart and D. W. Courtman, A population analysis of VEGF transgene expression and secretion, Biotech. Bioeng., 101 (2008), 1083-1093. doi: 10.1002/bit.21993.  Google Scholar

[32]

B. Kaur, C. Tan, D. J. Brat, D. E. Post and E. G. Van Meir, Gene and hypoxic regulation of angiogenesis in gliomas, J. Neuro-Oncol., 70 (2004), 229-243. doi: 10.1007/s11060-004-2752-5.  Google Scholar

[33]

D. G. Kilburn, M. D. Lilly and F. C. Webb, The energetics of mammalian cell growth, J. Cell Sci., 4 (1969), 645-654. Google Scholar

[34]

L. A. Lai, R. Kostadivov, M. T. Barrett, D. A. Peiffer, D. Pokholok, R. Odze, C. A. Sanchez, C. C. Maley, B. J. Reid, K. L. Gunderson and P. S. Rabinovitch, Deletion at fragile sites is a common and early event in Barrett's esophagus, Mol. Cancer Res., 8 (2010), 1084-1094. Google Scholar

[35]

L. W. Law, Origin of the resistance of leukaemic cells to folic acid antagonists, Nature, 169 (1952), 628-629. doi: 10.1038/169628a0.  Google Scholar

[36]

A. M. Leroi, V. Koufopanou and A. Burt, Cancer selection, Nature Rev. Cancer, 3 (2003), 226-231. Google Scholar

[37]

A. Levan and J. J. Biesele, Role of chromosomes in cancerogenesis, as studied in serial tissue culture of mammalian cells, Ann. N. Y. Acad. Sci., 71 (1958), 1022-1053. doi: 10.1111/j.1749-6632.1958.tb46820.x.  Google Scholar

[38]

M. V. Martinov, A. G. Plotnikov, V. M. Vitvitsky and F. I. Ataullakhanov, Deficiencies of glycolytic enzymes as a possible cause of hemolytic anemia, Biochim. Biophys. Acta, 1474 (2000), 75-87. doi: 10.1016/S0304-4165(99)00218-4.  Google Scholar

[39]

L. M. Merlo, J. W. Pepper, B. J. Reid and C. C. Maley, Cancer as an evolutionary and ecological process, Nature Rev. Cancer, 6 (2006), 924-935. Google Scholar

[40]

L. M. Merlo, N. A. Shah, X. Li, P. L. Blount, T. L. Vaughan, B. J. Reid and C. C. Maley, A comprehensive survey of clonal diversity measures in Barrett's esophagus as biomarkers of progression to esophageal adenocarcinoma, Cancer Prev. Res., 3 (2010), 1388-1397. Google Scholar

[41]

J. A. J. Metz, R. Nesbit and S. A. H. Geritz, How should we define 'fitness' for general ecological scenarios?, Trends Ecol. Evol., 7 (1992), 198-202. Google Scholar

[42]

J. D. Nagy, Competition and natural selection in a mathematical model of cancer, Bull. Math. Biol., 66 (2004), 663-687. doi: 10.1016/j.bulm.2003.10.001.  Google Scholar

[43]

J. D. Nagy, The ecology and evolutionary biology of cancer: A review of mathematical models of necrosis and tumor cell diversity, Math. Biosci. Eng., 2 (2005), 381-418.  Google Scholar

[44]

J. D. Nagy, E. M. Victor and J. H. Cropper, Why don't all whales have cancer? A novel hypothesis resolving Peto's paradox, Int. Comp. Biol., 47 (2007), 317-328. doi: 10.1093/icb/icm062.  Google Scholar

[45]

N. Navin, J. Kendall, J. Troge, P. Andrews, L. Rodgers, J. McIndoo, K. Cook, A. Stapansky, D. Levy, D. Esposito, L. Muthuswamy, A. Krasnitz, W. R. McCombie, J. Hicks and M. Wiglerm, Tumour evolution inferred by single-cell sequencing, Nature, 472 (2011), 90-94. doi: 10.1038/nature09807.  Google Scholar

[46]

G. Neufeld, T. Cohen, S. Gengrinovitch and Z. Poltorak, Vascular endothelial growth factor and its receptors, FASEB J., 13 (1999), 9-22. Google Scholar

[47]

P. C. Nowell, The clonal evolution of tumor cell populations, Science, 194 (1976), 23-28. doi: 10.1126/science.959840.  Google Scholar

[48]

K. Parvinen, Evolutionary suicide, Acta Biotheor., 53 (2005), 241-264. doi: 10.1007/s10441-005-2531-5.  Google Scholar

[49]

K. Pavlov and C. C. Maley, New models of neoplastic progression in Barrett's esophagus, Biochem. Soc. Trans., 38 (2010), 331-336. doi: 10.1042/BST0380331.  Google Scholar

[50]

C. M. Perrins, Survival of young swifts in relation to brood size, Nature, 201 (1964), 1147-1148. doi: 10.1038/2011147b0.  Google Scholar

[51]

K. H. Plate, G. Breier, H. A. Weich and W. Risau, Vascular endothelial growth factor is a potent tumour angiogenesis factor in human gliomas in vivo, Nature, 359 (1992), 845-848. doi: 10.1038/359845a0.  Google Scholar

[52]

C. M. Robbins, W. A. Tembe, A. Baker, S. Sinari, T. Y. Moses, S. Beckstrom-Sternberg, J. Beckstrom-Sternberg, M. Barrett, J. Long, A. Chinnaiyan, J. Lowey, E. Suh, J. V. Pearson, D. W. Craig, D. B. Angus, K. J. Pienta and J. D. Carpten, Copy number and targeted mutational analysis reveals novel somatic events in metastatic prostate tumors, Genome Res., 21 (2011), 47-55. doi: 10.1101/gr.107961.110.  Google Scholar

[53]

Y. Rong, D. L. Durden, E. G. Van Meir and D. J. Brat, 'Pseudopalisading' necrosis in glioblastoma: A familiar morphologic feature that links vascular pathology, hypoxia and angiogenesis, J. Neuropathol. Exp. Neurol., 65 (2006), 529-539. doi: 10.1097/00005072-200606000-00001.  Google Scholar

[54]

M. Tehrani, T. M. Friedman, J. J. Olson and D. J. Brat, Intravascular thrombosis in central nervous system malignancies: a potential role in astrocytoma progression to glioma, Brain Pathol., 18 (2008), 164-171. doi: 10.1111/j.1750-3639.2007.00108.x.  Google Scholar

[55]

P. Vajkoczy, M. Farhadi, A. Gaumann, R. Heidenreich, R. Erber, A. Wunder, J. C. Tonn, M. D. Menger and G. Breier, Microtumor growth initiates angiogenic sprouting with simultaneous expression of VEGF VEGF receptor-2 and angopietin-2, J. Clin. Invest., 109 (2002), 777-785. doi: 10.1172/JCI200214105.  Google Scholar

[56]

G. C. Williams, "Adaptation and Natural Selection: A Critique of Some Current Evolutionary Thought," Princeton U Press, Princeton, 1966. Google Scholar

[57]

V. C. Wynn-Edwards, Intergroup selection in the evolution of social systems, Nature, 200 (1963), 623-626. doi: 10.1038/200623a0.  Google Scholar

show all references

References:
[1]

B. Alberts, D. Bray, J. Lewis, M. Raff, K. Roberts and J. D. Watson, "Molecular Biology of the Cell," $3^{rd}$ edition, Garland, New York, 1994. Google Scholar

[2]

F. I. Ataullakhanov, S. V. Komarova, M. V. Martynov and V. M. Vitvitsky, A possible role of adenylate metabolism in human erythrocytes: 2. adenylate metabolism is able to improve the erythrocyte volume stabilization, J. Theor. Biol., 183 (1996), 307-316. doi: 10.1006/jtbi.1996.0222.  Google Scholar

[3]

F. I. Ataullakhanov, S. V. Komarova and V. M. Vitvitsky, A possible role of adenylate metabolism in human erythrocytes: simple mathematical model, J. Theor. Biol., 179 (1996), 75-86. doi: 10.1006/jtbi.1996.0050.  Google Scholar

[4]

F. I. Ataullakhanov and V. M. Vitvitsky, What determines the intracellular ATP concentration?, Biosci. Rep., 22 (2002), 501-511. doi: 10.1023/A:1022069718709.  Google Scholar

[5]

F. I. Ataullakhanov, V. M. Vitvitsky, A. M. Zhabotinsky, A. V. Pichugin, O. V. Platonova, B. N. Kholodenko and L. I. Ehrlich, The regulation of glycolysis in human erythrocytes: the dependence of the glycolytic flux on the ATP concentration, Eur. J. Biochem., 115 (1981), 359-365. doi: 10.1111/j.1432-1033.1981.tb05246.x.  Google Scholar

[6]

D. E. Atkinson, "Cellular Energy Metabolism and Its Regulation," Academic Press, New York, 1977. Google Scholar

[7]

L. E. Benjamin, I. Hemo and E. Keshet, A plasticity window for blood vessel remodelling is defined by pericyte coverage of the preformed endothelial network and is regulated by PDGF-B and VEGF, Development, 125 (1998), 1591-1598. Google Scholar

[8]

T. Bønsdorff, M. Gautier, W. Farstad, K. Rønningen, F. Lingaas and I. Olsaker, Mapping of the bovine genes of the de novo AMP synthesis pathway, Anim. Genet., 35 (2004), 438-444. doi: 10.1111/j.1365-2052.2004.01201.x.  Google Scholar

[9]

J. J. Boza, D. Moënnoz, C. E. Bournot, S. Blum, I. Zbinden, P. A. Finot and O. Ballèvre, Role of glutamine on the de novo purine nucleotide synthesis in Caco-2 cells, Eur. J. Nutr., 39 (2000), 38-46. Google Scholar

[10]

D. J. Brat and E. G. Van Meir, Vaso-occlusive and prothrombotic mechanisms associated with tumor hypoxia, necrosis, and accelerated growth in glioblastoma, Lab. Invest., 84 (2004), 397-405. doi: 10.1038/labinvest.3700070.  Google Scholar

[11]

J. P. Collins, "Evolutionary ecology" and the use of natural selection in ecological theory, J. Hist. Biol., 19 (1986), 257-288. doi: 10.1007/BF00138879.  Google Scholar

[12]

J. de Grouchy and C. de Nava, A chromosomal theory of carcinogenesis, Ann. Intern. Med., 69 (1968), 381-391. Google Scholar

[13]

F. Du, X.-H. Zhu, Y. Zhang, M. Friedman, N. Zhang adn K. Uqurbil and W. Chen, Tightly coupled brain activity and cerebral ATP metabolic rate, Proc. Nat. Acad. Sci. USA, 105 (2008), 6409-6414. doi: 10.1073/pnas.0710766105.  Google Scholar

[14]

I. F. Dunn, O. Heese and P. McL. Black, Growth factors in glioma angiogenesis: FGFs, PDGF, EGF, and TGFs, J. Neuro-Onco., 50 (2000), 121-137. doi: 10.1023/A:1006436624862.  Google Scholar

[15]

D. Gammack, H. M. Byrne and C. E. Lewis, Estimating the selective advantage of mutant p53 tumour cells to repeated rounds of hypoxia, Bull. Math. Biol., 63 (2001), 135-166. doi: 10.1006/bulm.2000.0210.  Google Scholar

[16]

S. A. H. Geritz, É. Kisdi, G. Meszéna and J. A. J. Metz, Evolutionarily singular stategies and the adaptive growth and branching of the evolutionary tree, Evol. Ecol., 12 (1998), 35-57. doi: 10.1023/A:1006554906681.  Google Scholar

[17]

A. C. Giese, "Cell Physiology," $5^{th}$ edition, Saunders, Philadelphia, 1973. Google Scholar

[18]

M. Greaves, Darwinian medicine: A case for cancer, Nature Rev. Cancer, 7 (2007), 213-221. Google Scholar

[19]

M. Greaves and C. C. Maley, Clonal evolution in cancer, Nature, 481 (2012), 306-313. doi: 10.1038/nature10762.  Google Scholar

[20]

D. Hanahan and J. Folkman, Patterns and emerging mechanisms of the angiogenic switch during tumorigenesis, Cell, 86 (1996), 353-364. doi: 10.1016/S0092-8674(00)80108-7.  Google Scholar

[21]

D. Hanahan and R. A. Weinberg, The hallmarks of cancer, Cell, 100 (2000), 57-70. doi: 10.1016/S0092-8674(00)81683-9.  Google Scholar

[22]

D. Hanahan and R. A. Weinberg, Hallmarks of cancer: The next generation, Cell, 144 (2011), 646-674. doi: 10.1016/j.cell.2011.02.013.  Google Scholar

[23]

D. G. Hardie, D. Carling and M. Carlson, The AMP-activated/SNF1 protein kinase subfamily: Metabolic sensors of the eukaryotic cell?, Ann. Rev. Biochem., 67 (1998), 821-855. doi: 10.1146/annurev.biochem.67.1.821.  Google Scholar

[24]

T. S. Hauschka, The chromosomes in ontogeny and oncogeny, Cancer Res., 21 (1961), 957-974. Google Scholar

[25]

J. Holash, P. C. Maisonpierre, D. Compton, P. Boland, C. R. Alexander, D. Zagzag, G. D. Yancopolous and S. J. Weigand, Vessel cooperation, regression and growth in tumors mediated by angiopoietins and VEGF, Science, 221 (1998), 1994-1998. Google Scholar

[26]

J. Maynard Smith, "Evolution and the Theory of Games," Cambridge University Press, Cambridge, 1982. Google Scholar

[27]

J. Maynard Smith and G. R. Price, The logic of animal conflict, Nature, 246 (1973), 15-18. doi: 10.1038/246015a0.  Google Scholar

[28]

A. Joshi and B. O. Palsson, Metabolic dynamics in the human red cell. Parts 1-2, J. Theor. Biol., 141 (1989), 515-545. doi: 10.1016/S0022-5193(89)80233-4.  Google Scholar

[29]

A. Joshi and B. O. Palsson, Metabolic dynamics in the human red cell. Parts 3-4, J. Theor. Biol., 142 (1990), 41-85. doi: 10.1016/S0022-5193(05)80012-8.  Google Scholar

[30]

W. G. Kaelin and P. J. Ratcliffe, Oxygen sensiing by metazoans: The central role of the HIF hydroxylase pathway, Mol. Cell, 30 (2008), 393-402. doi: 10.1016/j.molcel.2008.04.009.  Google Scholar

[31]

G. Karoubi, D. J. Stewart and D. W. Courtman, A population analysis of VEGF transgene expression and secretion, Biotech. Bioeng., 101 (2008), 1083-1093. doi: 10.1002/bit.21993.  Google Scholar

[32]

B. Kaur, C. Tan, D. J. Brat, D. E. Post and E. G. Van Meir, Gene and hypoxic regulation of angiogenesis in gliomas, J. Neuro-Oncol., 70 (2004), 229-243. doi: 10.1007/s11060-004-2752-5.  Google Scholar

[33]

D. G. Kilburn, M. D. Lilly and F. C. Webb, The energetics of mammalian cell growth, J. Cell Sci., 4 (1969), 645-654. Google Scholar

[34]

L. A. Lai, R. Kostadivov, M. T. Barrett, D. A. Peiffer, D. Pokholok, R. Odze, C. A. Sanchez, C. C. Maley, B. J. Reid, K. L. Gunderson and P. S. Rabinovitch, Deletion at fragile sites is a common and early event in Barrett's esophagus, Mol. Cancer Res., 8 (2010), 1084-1094. Google Scholar

[35]

L. W. Law, Origin of the resistance of leukaemic cells to folic acid antagonists, Nature, 169 (1952), 628-629. doi: 10.1038/169628a0.  Google Scholar

[36]

A. M. Leroi, V. Koufopanou and A. Burt, Cancer selection, Nature Rev. Cancer, 3 (2003), 226-231. Google Scholar

[37]

A. Levan and J. J. Biesele, Role of chromosomes in cancerogenesis, as studied in serial tissue culture of mammalian cells, Ann. N. Y. Acad. Sci., 71 (1958), 1022-1053. doi: 10.1111/j.1749-6632.1958.tb46820.x.  Google Scholar

[38]

M. V. Martinov, A. G. Plotnikov, V. M. Vitvitsky and F. I. Ataullakhanov, Deficiencies of glycolytic enzymes as a possible cause of hemolytic anemia, Biochim. Biophys. Acta, 1474 (2000), 75-87. doi: 10.1016/S0304-4165(99)00218-4.  Google Scholar

[39]

L. M. Merlo, J. W. Pepper, B. J. Reid and C. C. Maley, Cancer as an evolutionary and ecological process, Nature Rev. Cancer, 6 (2006), 924-935. Google Scholar

[40]

L. M. Merlo, N. A. Shah, X. Li, P. L. Blount, T. L. Vaughan, B. J. Reid and C. C. Maley, A comprehensive survey of clonal diversity measures in Barrett's esophagus as biomarkers of progression to esophageal adenocarcinoma, Cancer Prev. Res., 3 (2010), 1388-1397. Google Scholar

[41]

J. A. J. Metz, R. Nesbit and S. A. H. Geritz, How should we define 'fitness' for general ecological scenarios?, Trends Ecol. Evol., 7 (1992), 198-202. Google Scholar

[42]

J. D. Nagy, Competition and natural selection in a mathematical model of cancer, Bull. Math. Biol., 66 (2004), 663-687. doi: 10.1016/j.bulm.2003.10.001.  Google Scholar

[43]

J. D. Nagy, The ecology and evolutionary biology of cancer: A review of mathematical models of necrosis and tumor cell diversity, Math. Biosci. Eng., 2 (2005), 381-418.  Google Scholar

[44]

J. D. Nagy, E. M. Victor and J. H. Cropper, Why don't all whales have cancer? A novel hypothesis resolving Peto's paradox, Int. Comp. Biol., 47 (2007), 317-328. doi: 10.1093/icb/icm062.  Google Scholar

[45]

N. Navin, J. Kendall, J. Troge, P. Andrews, L. Rodgers, J. McIndoo, K. Cook, A. Stapansky, D. Levy, D. Esposito, L. Muthuswamy, A. Krasnitz, W. R. McCombie, J. Hicks and M. Wiglerm, Tumour evolution inferred by single-cell sequencing, Nature, 472 (2011), 90-94. doi: 10.1038/nature09807.  Google Scholar

[46]

G. Neufeld, T. Cohen, S. Gengrinovitch and Z. Poltorak, Vascular endothelial growth factor and its receptors, FASEB J., 13 (1999), 9-22. Google Scholar

[47]

P. C. Nowell, The clonal evolution of tumor cell populations, Science, 194 (1976), 23-28. doi: 10.1126/science.959840.  Google Scholar

[48]

K. Parvinen, Evolutionary suicide, Acta Biotheor., 53 (2005), 241-264. doi: 10.1007/s10441-005-2531-5.  Google Scholar

[49]

K. Pavlov and C. C. Maley, New models of neoplastic progression in Barrett's esophagus, Biochem. Soc. Trans., 38 (2010), 331-336. doi: 10.1042/BST0380331.  Google Scholar

[50]

C. M. Perrins, Survival of young swifts in relation to brood size, Nature, 201 (1964), 1147-1148. doi: 10.1038/2011147b0.  Google Scholar

[51]

K. H. Plate, G. Breier, H. A. Weich and W. Risau, Vascular endothelial growth factor is a potent tumour angiogenesis factor in human gliomas in vivo, Nature, 359 (1992), 845-848. doi: 10.1038/359845a0.  Google Scholar

[52]

C. M. Robbins, W. A. Tembe, A. Baker, S. Sinari, T. Y. Moses, S. Beckstrom-Sternberg, J. Beckstrom-Sternberg, M. Barrett, J. Long, A. Chinnaiyan, J. Lowey, E. Suh, J. V. Pearson, D. W. Craig, D. B. Angus, K. J. Pienta and J. D. Carpten, Copy number and targeted mutational analysis reveals novel somatic events in metastatic prostate tumors, Genome Res., 21 (2011), 47-55. doi: 10.1101/gr.107961.110.  Google Scholar

[53]

Y. Rong, D. L. Durden, E. G. Van Meir and D. J. Brat, 'Pseudopalisading' necrosis in glioblastoma: A familiar morphologic feature that links vascular pathology, hypoxia and angiogenesis, J. Neuropathol. Exp. Neurol., 65 (2006), 529-539. doi: 10.1097/00005072-200606000-00001.  Google Scholar

[54]

M. Tehrani, T. M. Friedman, J. J. Olson and D. J. Brat, Intravascular thrombosis in central nervous system malignancies: a potential role in astrocytoma progression to glioma, Brain Pathol., 18 (2008), 164-171. doi: 10.1111/j.1750-3639.2007.00108.x.  Google Scholar

[55]

P. Vajkoczy, M. Farhadi, A. Gaumann, R. Heidenreich, R. Erber, A. Wunder, J. C. Tonn, M. D. Menger and G. Breier, Microtumor growth initiates angiogenic sprouting with simultaneous expression of VEGF VEGF receptor-2 and angopietin-2, J. Clin. Invest., 109 (2002), 777-785. doi: 10.1172/JCI200214105.  Google Scholar

[56]

G. C. Williams, "Adaptation and Natural Selection: A Critique of Some Current Evolutionary Thought," Princeton U Press, Princeton, 1966. Google Scholar

[57]

V. C. Wynn-Edwards, Intergroup selection in the evolution of social systems, Nature, 200 (1963), 623-626. doi: 10.1038/200623a0.  Google Scholar

[1]

Ebraheem O. Alzahrani, Muhammad Altaf Khan. Androgen driven evolutionary population dynamics in prostate cancer growth. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3419-3440. doi: 10.3934/dcdss.2020426
[2]

Qingshan Yang, Xuerong Mao. Stochastic dynamics of SIRS epidemic models with random perturbation. Mathematical Biosciences & Engineering, 2014, 11 (4) : 1003-1025. doi: 10.3934/mbe.2014.11.1003
[3]

John D. Nagy. The Ecology and Evolutionary Biology of Cancer: A Review of Mathematical Models of Necrosis and Tumor Cell Diversity. Mathematical Biosciences & Engineering, 2005, 2 (2) : 381-418. doi: 10.3934/mbe.2005.2.381
[4]

Gong Chen. Strichartz estimates for charge transfer models. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1201-1226. doi: 10.3934/dcds.2017050
[5]

Walter Allegretto, Liqun Cao, Yanping Lin. Multiscale asymptotic expansion for second order parabolic equations with rapidly oscillating coefficients. Discrete & Continuous Dynamical Systems, 2008, 20 (3) : 543-576. doi: 10.3934/dcds.2008.20.543
[6]

YunKyong Hyon, James E. Fonseca, Bob Eisenberg, Chun Liu. Energy variational approach to study charge inversion (layering) near charged walls. Discrete & Continuous Dynamical Systems - B, 2012, 17 (8) : 2725-2743. doi: 10.3934/dcdsb.2012.17.2725
[7]

Avner Friedman. A hierarchy of cancer models and their mathematical challenges. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 147-159. doi: 10.3934/dcdsb.2004.4.147
[8]

Per Danzl, Ali Nabi, Jeff Moehlis. Charge-balanced spike timing control for phase models of spiking neurons. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1413-1435. doi: 10.3934/dcds.2010.28.1413
[9]

Timothy Blass, Rafael de la Llave. Perturbation and numerical methods for computing the minimal average energy. Networks & Heterogeneous Media, 2011, 6 (2) : 241-255. doi: 10.3934/nhm.2011.6.241
[10]

Jim M. Cushing. The evolutionary dynamics of a population model with a strong Allee effect. Mathematical Biosciences & Engineering, 2015, 12 (4) : 643-660. doi: 10.3934/mbe.2015.12.643
[11]

Minette Herrera, Aaron Miller, Joel Nishimura. Altruistic aging: The evolutionary dynamics balancing longevity and evolvability. Mathematical Biosciences & Engineering, 2017, 14 (2) : 455-465. doi: 10.3934/mbe.2017028
[12]

Amy Veprauskas, J. M. Cushing. Evolutionary dynamics of a multi-trait semelparous model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 655-676. doi: 10.3934/dcdsb.2016.21.655
[13]

Natalia L. Komarova. Spatial stochastic models of cancer: Fitness, migration, invasion. Mathematical Biosciences & Engineering, 2013, 10 (3) : 761-775. doi: 10.3934/mbe.2013.10.761
[14]

Urszula Ledzewicz, Heinz Schättler. On the role of pharmacometrics in mathematical models for cancer treatments. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 483-499. doi: 10.3934/dcdsb.2020213
[15]

Vincent Giovangigli, Wen-An Yong. Volume viscosity and internal energy relaxation: Symmetrization and Chapman-Enskog expansion. Kinetic & Related Models, 2015, 8 (1) : 79-116. doi: 10.3934/krm.2015.8.79
[16]

Urszula Ledzewicz, Heinz Schättler. Controlling a model for bone marrow dynamics in cancer chemotherapy. Mathematical Biosciences & Engineering, 2004, 1 (1) : 95-110. doi: 10.3934/mbe.2004.1.95
[17]

Shan Jiang, Li Liang, Meiling Sun, Fang Su. Uniform high-order convergence of multiscale finite element computation on a graded recursion for singular perturbation. Electronic Research Archive, 2020, 28 (2) : 935-949. doi: 10.3934/era.2020049
[18]

Carsten Hartmann, Juan C. Latorre, Wei Zhang, Grigorios A. Pavliotis. Addendum to "Optimal control of multiscale systems using reduced-order models". Journal of Computational Dynamics, 2017, 4 (1&2) : 167-167. doi: 10.3934/jcd.2017006
[19]

Carsten Hartmann, Juan C. Latorre, Wei Zhang, Grigorios A. Pavliotis. Optimal control of multiscale systems using reduced-order models. Journal of Computational Dynamics, 2014, 1 (2) : 279-306. doi: 10.3934/jcd.2014.1.279
[20]

Elvio Accinelli, Bruno Bazzano, Franco Robledo, Pablo Romero. Nash Equilibrium in evolutionary competitive models of firms and workers under external regulation. Journal of Dynamics & Games, 2015, 2 (1) : 1-32. doi: 10.3934/jdg.2015.2.1

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (33)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy