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On optimal and suboptimal treatment strategies for a mathematical model of leukemia
A multiple timescale computational model of a tumor and its micro environment
1.  University of California, Irvine, Dept. of Statistics, School of Information and Computer Science, 3019 Bren Hall, Irvine, CA 926175100, United States 
2.  Princeton University, Dept. of Computer Science, 35 Olden Street, Princeton, NJ 085405233, United States 
3.  Wesleyan University, Dept. of Mathematics and Computer Science, 265 Church St. Middletown, CT 06459, United States 
4.  Pomona College, Dept. of Mathematics, 610 N. College Ave., Claremont, CA 91711, United States 
References:
[1] 
T. Alarcón, H. M. Byrne and P. K. Maini, A cellular automaton model for tumour growth in inhomogeneous environment,, Journal of Theoretical Biology, 225 (2003), 257. 
[2] 
A. AltHolland, W. Zhang, A. Margulis and J. Garlick, Microenvironmental control of premalignant disease: the role of intercellular adhesion in the progression of squamous cell carcinoma,, Seminars in Cancer Biology, 15 (2005), 84. 
[3] 
A. R. A. Anderson and M. A. J. Chaplain, Continuous and discrete mathematical models of tumorinduced angiogenesis,, Bulletin of Mathematical Biology, 60 (1998), 857. 
[4] 
M. Bernaschi and F. Castiglione, Design and implementation of an immune system simulator,, Computers in Biology and Medicine, 31 (2001), 303. 
[5] 
A. Bertuzzi, A. d'Onofrio, A. Fasano and A. Gandolfi, Regression and regrowth of tumor cords following single dose anticancer treatment,, Bulletin of Mathematical Biology, 65 (2003), 903. 
[6] 
A. Bertuzzi, A. Fasano, A. Gandolfi and C. Sinisgalli, ATP production and necrosis formation in a tumour spheroid model,, Mathematical Modelling of Natural Phenomena, 2 (2007), 30. 
[7] 
A. Bertuzzi, A. Fasano, A. Gandolfi and C. Sinisgalli, Necrotic core in EMT6/Ro tumour spheroids: Is it caused by an ATP deficit?,, Journal of Theoretical Biology, 262 (2010), 142. 
[8] 
B. Blouw, H. Song, T. Tihan, J. Bosze, N. Ferrara, H. Gerber, R. Johnson and G. Bergers, The hypoxic response of tumors is dependent on their microenvironment,, Cancer Cell, 4 (2003), 133. 
[9] 
R. Bristow and R. Hill, Molecular and cellular basis of radiotherapy,, in, (1998), 295. 
[10] 
H. Byrne and M. Chaplain, Modelling the role of cellcell adhesion in the growth and development of carcinomas,, Mathematical and Computational Modelling, 24 (1996), 1. 
[11] 
H. Byrne and M. Chaplain, Free boundary value problems associated with the growth and development of multicellular spheroids,, European Journal of Applied Mathematics, 8 (1997), 639. 
[12] 
R. Cairns and R. Hill, Acute hypoxia enhances spontaneous lymph node metastasis in an orthotopic murine model of human cervical carcinoma,, Cancer Research, 64 (2004), 2054. 
[13] 
J. Casciari and J. Rasey, Determination of the radiobiologically hypoxic fraction in multicellular spheroids from data on the uptake of [3H]fluoromisonidazole,, Radiat Res., 141 (1995), 28. 
[14] 
J. Casciari, S. Sotirchos and R. Sutherland, Variations in tumor cell growth rates and metabolism with oxygen concentration, glucose concentration, and extracellular pH,, Journal of Cellular Physiology, 151 (1992), 386. 
[15] 
M. Chaplain and A. Matzavinos, Mathematical modelling of spatiotemporal phenomena in tumour immunology,, Lect. Notes Math., 1872 (2006), 131. 
[16] 
V. Cristini, J. Lowengrub and Q. Nie, Nonlinear simulation of tumor growth,, J Math Biol., 46 (2003), 191. 
[17] 
L. de Pillis, W. Gu and A. Radunskaya, Mixed immunotherapy and chemotherapy of tumors: Modeling applications and biological interpretations,, Journal of Theoretical Biology, 238 (2005), 841. 
[18] 
L. de Pillis, D. Mallet and A. Radunskaya, Spatial tumorimmune modeling,, Journal of Computational and Mathematical Models in Medicine, 7 (2006), 159. 
[19] 
L. de Pillis and A. Radunskaya, A mathematical tumor model with immune resistance and drug therapy: An optimal control approach,, J Theor Med., 3 (2001), 79. 
[20] 
L. de Pillis and A. Radunskaya, The dynamics of an optimally controlled tumor model: A case study,, Math Comput Model. (Special Issues), 37 (2003), 1221. 
[21] 
L. de Pillis and A. Radunskaya, Immune response to tumor invasion,, in, 2 (2003), 1661. 
[22] 
L. de Pillis, A. Radunskaya and C. Wiseman, A validated mathematical model of cellmediated immune response to tumor growth,, Cancer Research, 65 (2005), 7950. 
[23] 
M. Dewhirst, Concepts of oxygen transport at the microcirculatory level,, Semin Radiat Oncol., 8 (1998), 143. 
[24] 
S. Dormann and A. Deutsch, Modeling of selforganized avascular tumor growth with a hybrid cellular automaton,, In Silico Biology, 2 (2002). 
[25] 
A. dos Reis, J. Mombach, M. Walter and de Avila L. F., The interplay between cell adhesion and environment rigidity in the morphology of tumors,, Phyisca AStatistical Mechanics and its Applications, 322 (2003), 546. 
[26] 
R. D'Souza, N. Margolus and M. Smith, Dimensionsplitting for simplifying diffusion in latticegas models,, Journal of Statistical Physics, 107 (2002). 
[27] 
S. C. Ferreira, M. L. Martins and M. J. Vilela, Reactiondiffusion model for the growth of avascular tumor,, Phys Rev E, 65 (2002). 
[28] 
P. Gassmann, J. Haier and G. Nicolson, Cell adhesion and invasion during secondary tumor formation,, Cancer Growth and Progression, 3 (2004). 
[29] 
R. Gatenby and J. Gillies, Why do cancers have high aerobic glycolysis?,, Nature Reviews Cancer, 4 (2004), 891. 
[30] 
I. Georgoudas, G. Sirakoulis and I. Andreadis, An intelligent cellular automaton model for crowd evacuation in fire spreading conditions,, in, 1 (2007), 36. 
[31] 
S. Gobron and N. Chiba, Visual simulation of crack pattern based on 3D surface cellular automaton,, in, (2000), 181. 
[32] 
M. Gryczynskia, J. Kobos and W. Pietruszewska, Intratumoral microvessels density and morphometric study of angiogenesis as prognostic factor in laryngeal cancer,, International Congress Series, 1240 (2003), 1113. 
[33] 
M. Guppy, P. Leedman, X. Zu and V. Russel, Contribution by different fuels and metabolic pathways to the total ATP turnover of proliferating MCF7 breast cancer cells,, Biochem. J., 364 (2002), 309. 
[34] 
J. Haier and G. Nicolson, Role of tumor cell adhesion as an important factor in formation of distant metastases,, Diseases Colon Rect., 44 (2001), 876. 
[35] 
A. Harris, "Hypoxia  A Key Regulatory Factor in Tumour Growth,", 2002., (). 
[36] 
G. Helmlinger, A. Sckell, M. Dellian, N. Forbes and R. Jain, Acid production in glycolysisimpaired tumors provides new insights into tumor metabolism,, Clinical Cancer Research, 8 (2002), 1284. 
[37] 
M. Hockel and P. Vaupel, "Tumor Hypoxia: Definitions and Current Clinical, Biologic, and Molecular Aspects,", 2001., (). 
[38] 
M. Hystad and E. Rofstad, Oxygen consumption rate and mitochondrial density in human melanoma monolayer cultures and multicellular spheroids,, Int J Cancer., 57 (1994), 532. 
[39] 
T. L. Jackson, Vascular tumor growth and treatment: Consequenes of polyclonality, competition and dynamic vascular support,, J Math Biol., 44 (2002), 201. 
[40] 
S. Kooijman, "Dynamic Energy and Mass Budgets in Biological Systems,", Cambridge University Press, (2000). 
[41] 
M. I. Koukourakis, M. Pitiakoudis, A. Giatromanolaki, A. Tsarouha, A. Polychronidis, E. Sivridis and C. Simopoulos, Oxygen and glucose consumption in gastrointestinal adenocarcinomas: Correlation with markers of hypoxia, acidity and anaerobic glycolysis,, Cancer Science, 97 (2006), 1056. 
[42] 
M. Kunz, S. Moeller, D. Koczan, P. Lorenz, R. Wenger, M. Glocker, H. Thiesen, G. Gross and S. Ibrahim, Mechanisms of hypoxic gene regulation of angiogenesis factor Cyr61 in melanoma cells,, Journal of Biological Chemistry, 278 (2003), 45651. 
[43] 
M. Li, Z. Ru and J. He, Cellular automata to simulate rock failure,, in, (2006), 110. 
[44] 
P. Macklin, S. McDougall, A. Anderson, M. Chaplain, V. Cristini and J. Lowengrub, Multiscale modelling and nonlinear simulation of vascular tumour growth,, Journal of Mathematical Biology, 58 (2009), 765. doi: 10.1007/s0028500802169. 
[45] 
D. Mallet and L. de Pillis, A cellular automata model of tumorimmune system interactions,, Journal of Theoretical Biology, 239 (2006), 334. 
[46] 
C. Menon, G. Polin, I. Prabakaran, A. Hsi, C. Cheung, J. Culver, J. Pingpank, C. Sehgal, A. Yodh, D. Buerk and D. Fraker, An integrated approach to measuring tumor oxygen status using human melanoma xenografts as a model,, Cancer Research, 63 (2003), 7232. 
[47] 
B. Mueller, R. Reisfeld, T. Edgington and W. Ruf, Expression of tissue factor by melanoma cells promotes efficient hematogenous metastasis,, Proc. Natl. Acad. Sci. USA, 89 (1992), 11832. 
[48] 
D. Nelson and M. Cox, "Lehninger Principles of Biochemistry,", W. H. Freeman and Co., (2004). 
[49] 
N. Oriuchi, T. Higuchi, T. Ishikita, M. Miyakubo, H. Hanaoka, Y. Iida and K. Endo, Present role and future prospects of positron emission tomography in clinical oncology,, Cancer Science, (2006). 
[50] 
M. Owen, H. Byrne and C. Lewis, Mathematical modelling of the use of macrophages as vehicles for drug delivery to hypoxic tumour sites,, Journal of Theoretical Biology, 226 (2004), 377. 
[51] 
A. Patel, E. Gawlinski, S. Lemieux and R. Gatenby, A cellular automaton model of early tumor growth and invasion: The effects of native tissue vascularity and increased anaerobic tumor metabolism,, Journal of Theoretical Biology, 213 (2001), 315. 
[52] 
L. Preziosi, "Cancer Modelling and Simulation,", Mathematical Biology and Medicine Series. Chapman & Hall/CRC, (2003). 
[53] 
R. Puzone, B. Kohler, P. Seiden and F. Celada, IMMSIM, a flexible model for in machina experiments on immune system responses,, Future Generation Computer Systems, 18 (2002), 961. 
[54] 
A. Radunskaya and M. Villasana, A delay differential equation model for tumor growth,, J. Math.Biol., 47 (2003), 270. 
[55] 
K. A. Rejniak, An immersed boundary framework for modelling the growth of individual cells: An application to the early tumour development,, Journal of Theoretical Biology, 247 (2007), 186. 
[56] 
K. A. Rejniak and A. R. A. Anderson, Hybrid models of tumor growth,, Wiley Interdisciplinary Reviews  Systems Biology and Medicine, 3 (2011), 115. 
[57] 
K. A. Rejniak and L. J. McCawley, Current trends in mathematical modeling of tumormicroenvironment interactions: a survey of tools and applications,, Experimental Biology and Medicine, 235 (2010), 411. 
[58] 
E. Rofstad and K. Maseide, Radiobiological and immunohistochemical assessment of hypoxia in human melanoma xenografts: acute and chronic hypoxia in individual tumours,, Int J Radiat Biol., 75 (1999), 1377. 
[59] 
S. Sanga, H. B. Frieboes, X. Zheng, R. Gatenby, E. L. Bearer and V. Cristini, Predictive oncology: A review of multidisciplinary, multiscale in silico modeling linking phenotype, morphology and growth,, NeuroImage, 37 (2007). 
[60] 
P. Schornack and R. Gillies, Contributions of cell metabolism and $H^+$ diffusion to the acidic pH of tumors,, Neoplasia, 5 (2003), 135. 
[61] 
T. J. Schulz, R. Thierbach, A. Voigt, G. Drewes, B. Mietzner, P. Steinberg, A. F. H. Pfeiffer and M. Ristow, Induction of oxidative metabolism by mitochondrial frataxin inhibits cancer growth: Otto warburg revisited,, J. Biol. Chem., 281 (2006), 977. 
[62] 
R. Skoyum, K. Eide, K. Berg and E. Rofstad, Energy metabolism in human melanoma cells under hypoxic and acidic conditions in vitro,, Br J Cancer, 76 (1997), 421. 
[63] 
K. Smallbone, R. A. Gatenby, R. Gillies, P. K. Maini and D. Gavaghan, Metabolic changes during carcinogenesis: Potential impact on invasiveness,, Journal of Theoretical Biology, 244 (2006), 703. 
[64] 
K. Smallbone, D. J. Gavaghan, R. A. Gatenby and P. K. Maini, The role of acidity in solid tumour growth and invasion,, Journal of Theoretical Biology, 234 (2005), 476. 
[65] 
J. Smolle, Cellular automaton simulation of tumour growth  equivocal relationships between simulation parameters and morphologic pattern features,, Anal Cell Pathol., 17 (1998), 71. 
[66] 
J. Smolle, R. HofmannWellenhof and H. Kerl, Pattern interpretation by cellular automata (pica)evaluation of tumour cell adhesion in human melanomas,, Anal Cell Pathol., 7 (1994), 91. 
[67] 
P. Subarsky and R. Hill, The hypoxic tumour microenvironment and metastatic progression,, Clinical & Experimental Metastasis, 20 (2003), 237. 
[68] 
I. Tufto, H. Lyng and E. K. Forstad, Vascular density in human melanoma xenografts: Relationship to angiogenesis, perfusion and necrosis,, Cancer Letters, 123 (1998), 159. 
[69] 
S. Turner, Using cell potential energy to model the dynamics of adhesive biological cells,, Physical Review E, 71 (2005). 
[70] 
S. Turner, J. Sherratt, K. Painter and N. Savill, From a discrete to a continuous model of biological cell movement,, Physical Review E, 69 (2004). 
[71] 
I. van Leeuwen, C. Zonneveld and S. Kooijman, The embedded tumour: Host physiology is important for the evaluation of tumour growth,, British Journal of Cancer, 89 (2003), 2254. 
[72] 
P. Vaupel, O. Thews, D. Kelleher and M. Hoeckel, Current status of knowledge and critical issues in tumor oxygenation. results from 25 years research in tumor pathophysiology,, Adv Exp Med Biol., 454 (1998), 591. 
[73] 
R. Venkatasubramanian, M. A. Henson and N. S. Forbes, Incorporating energy metabolism into a growth model of multicellular tumor spheroids,, Journal of Theoretical Biology, 242 (2006), 440. 
[74] 
J. von Neumann, "Theory of SelfReproducing Automata,", University of Illinois Press, (1966). 
[75] 
S. Wise, J. Lowengrubb and V. Cristini, An adaptive multigrid algorithm for simulating solid tumor growth using mixture models,, Mathematical and Computer Modelling, 53 (2011), 1. doi: 10.1016/j.mcm.2010.07.007. 
[76] 
X. Zu and M. Guppy, Cancer metabolism: Facts, fantasy, and fiction,, Biochemical and Biophysical Research Communications, 313 (2004), 459. doi: 10.1016/j.bbrc.2003.11.136. 
show all references
References:
[1] 
T. Alarcón, H. M. Byrne and P. K. Maini, A cellular automaton model for tumour growth in inhomogeneous environment,, Journal of Theoretical Biology, 225 (2003), 257. 
[2] 
A. AltHolland, W. Zhang, A. Margulis and J. Garlick, Microenvironmental control of premalignant disease: the role of intercellular adhesion in the progression of squamous cell carcinoma,, Seminars in Cancer Biology, 15 (2005), 84. 
[3] 
A. R. A. Anderson and M. A. J. Chaplain, Continuous and discrete mathematical models of tumorinduced angiogenesis,, Bulletin of Mathematical Biology, 60 (1998), 857. 
[4] 
M. Bernaschi and F. Castiglione, Design and implementation of an immune system simulator,, Computers in Biology and Medicine, 31 (2001), 303. 
[5] 
A. Bertuzzi, A. d'Onofrio, A. Fasano and A. Gandolfi, Regression and regrowth of tumor cords following single dose anticancer treatment,, Bulletin of Mathematical Biology, 65 (2003), 903. 
[6] 
A. Bertuzzi, A. Fasano, A. Gandolfi and C. Sinisgalli, ATP production and necrosis formation in a tumour spheroid model,, Mathematical Modelling of Natural Phenomena, 2 (2007), 30. 
[7] 
A. Bertuzzi, A. Fasano, A. Gandolfi and C. Sinisgalli, Necrotic core in EMT6/Ro tumour spheroids: Is it caused by an ATP deficit?,, Journal of Theoretical Biology, 262 (2010), 142. 
[8] 
B. Blouw, H. Song, T. Tihan, J. Bosze, N. Ferrara, H. Gerber, R. Johnson and G. Bergers, The hypoxic response of tumors is dependent on their microenvironment,, Cancer Cell, 4 (2003), 133. 
[9] 
R. Bristow and R. Hill, Molecular and cellular basis of radiotherapy,, in, (1998), 295. 
[10] 
H. Byrne and M. Chaplain, Modelling the role of cellcell adhesion in the growth and development of carcinomas,, Mathematical and Computational Modelling, 24 (1996), 1. 
[11] 
H. Byrne and M. Chaplain, Free boundary value problems associated with the growth and development of multicellular spheroids,, European Journal of Applied Mathematics, 8 (1997), 639. 
[12] 
R. Cairns and R. Hill, Acute hypoxia enhances spontaneous lymph node metastasis in an orthotopic murine model of human cervical carcinoma,, Cancer Research, 64 (2004), 2054. 
[13] 
J. Casciari and J. Rasey, Determination of the radiobiologically hypoxic fraction in multicellular spheroids from data on the uptake of [3H]fluoromisonidazole,, Radiat Res., 141 (1995), 28. 
[14] 
J. Casciari, S. Sotirchos and R. Sutherland, Variations in tumor cell growth rates and metabolism with oxygen concentration, glucose concentration, and extracellular pH,, Journal of Cellular Physiology, 151 (1992), 386. 
[15] 
M. Chaplain and A. Matzavinos, Mathematical modelling of spatiotemporal phenomena in tumour immunology,, Lect. Notes Math., 1872 (2006), 131. 
[16] 
V. Cristini, J. Lowengrub and Q. Nie, Nonlinear simulation of tumor growth,, J Math Biol., 46 (2003), 191. 
[17] 
L. de Pillis, W. Gu and A. Radunskaya, Mixed immunotherapy and chemotherapy of tumors: Modeling applications and biological interpretations,, Journal of Theoretical Biology, 238 (2005), 841. 
[18] 
L. de Pillis, D. Mallet and A. Radunskaya, Spatial tumorimmune modeling,, Journal of Computational and Mathematical Models in Medicine, 7 (2006), 159. 
[19] 
L. de Pillis and A. Radunskaya, A mathematical tumor model with immune resistance and drug therapy: An optimal control approach,, J Theor Med., 3 (2001), 79. 
[20] 
L. de Pillis and A. Radunskaya, The dynamics of an optimally controlled tumor model: A case study,, Math Comput Model. (Special Issues), 37 (2003), 1221. 
[21] 
L. de Pillis and A. Radunskaya, Immune response to tumor invasion,, in, 2 (2003), 1661. 
[22] 
L. de Pillis, A. Radunskaya and C. Wiseman, A validated mathematical model of cellmediated immune response to tumor growth,, Cancer Research, 65 (2005), 7950. 
[23] 
M. Dewhirst, Concepts of oxygen transport at the microcirculatory level,, Semin Radiat Oncol., 8 (1998), 143. 
[24] 
S. Dormann and A. Deutsch, Modeling of selforganized avascular tumor growth with a hybrid cellular automaton,, In Silico Biology, 2 (2002). 
[25] 
A. dos Reis, J. Mombach, M. Walter and de Avila L. F., The interplay between cell adhesion and environment rigidity in the morphology of tumors,, Phyisca AStatistical Mechanics and its Applications, 322 (2003), 546. 
[26] 
R. D'Souza, N. Margolus and M. Smith, Dimensionsplitting for simplifying diffusion in latticegas models,, Journal of Statistical Physics, 107 (2002). 
[27] 
S. C. Ferreira, M. L. Martins and M. J. Vilela, Reactiondiffusion model for the growth of avascular tumor,, Phys Rev E, 65 (2002). 
[28] 
P. Gassmann, J. Haier and G. Nicolson, Cell adhesion and invasion during secondary tumor formation,, Cancer Growth and Progression, 3 (2004). 
[29] 
R. Gatenby and J. Gillies, Why do cancers have high aerobic glycolysis?,, Nature Reviews Cancer, 4 (2004), 891. 
[30] 
I. Georgoudas, G. Sirakoulis and I. Andreadis, An intelligent cellular automaton model for crowd evacuation in fire spreading conditions,, in, 1 (2007), 36. 
[31] 
S. Gobron and N. Chiba, Visual simulation of crack pattern based on 3D surface cellular automaton,, in, (2000), 181. 
[32] 
M. Gryczynskia, J. Kobos and W. Pietruszewska, Intratumoral microvessels density and morphometric study of angiogenesis as prognostic factor in laryngeal cancer,, International Congress Series, 1240 (2003), 1113. 
[33] 
M. Guppy, P. Leedman, X. Zu and V. Russel, Contribution by different fuels and metabolic pathways to the total ATP turnover of proliferating MCF7 breast cancer cells,, Biochem. J., 364 (2002), 309. 
[34] 
J. Haier and G. Nicolson, Role of tumor cell adhesion as an important factor in formation of distant metastases,, Diseases Colon Rect., 44 (2001), 876. 
[35] 
A. Harris, "Hypoxia  A Key Regulatory Factor in Tumour Growth,", 2002., (). 
[36] 
G. Helmlinger, A. Sckell, M. Dellian, N. Forbes and R. Jain, Acid production in glycolysisimpaired tumors provides new insights into tumor metabolism,, Clinical Cancer Research, 8 (2002), 1284. 
[37] 
M. Hockel and P. Vaupel, "Tumor Hypoxia: Definitions and Current Clinical, Biologic, and Molecular Aspects,", 2001., (). 
[38] 
M. Hystad and E. Rofstad, Oxygen consumption rate and mitochondrial density in human melanoma monolayer cultures and multicellular spheroids,, Int J Cancer., 57 (1994), 532. 
[39] 
T. L. Jackson, Vascular tumor growth and treatment: Consequenes of polyclonality, competition and dynamic vascular support,, J Math Biol., 44 (2002), 201. 
[40] 
S. Kooijman, "Dynamic Energy and Mass Budgets in Biological Systems,", Cambridge University Press, (2000). 
[41] 
M. I. Koukourakis, M. Pitiakoudis, A. Giatromanolaki, A. Tsarouha, A. Polychronidis, E. Sivridis and C. Simopoulos, Oxygen and glucose consumption in gastrointestinal adenocarcinomas: Correlation with markers of hypoxia, acidity and anaerobic glycolysis,, Cancer Science, 97 (2006), 1056. 
[42] 
M. Kunz, S. Moeller, D. Koczan, P. Lorenz, R. Wenger, M. Glocker, H. Thiesen, G. Gross and S. Ibrahim, Mechanisms of hypoxic gene regulation of angiogenesis factor Cyr61 in melanoma cells,, Journal of Biological Chemistry, 278 (2003), 45651. 
[43] 
M. Li, Z. Ru and J. He, Cellular automata to simulate rock failure,, in, (2006), 110. 
[44] 
P. Macklin, S. McDougall, A. Anderson, M. Chaplain, V. Cristini and J. Lowengrub, Multiscale modelling and nonlinear simulation of vascular tumour growth,, Journal of Mathematical Biology, 58 (2009), 765. doi: 10.1007/s0028500802169. 
[45] 
D. Mallet and L. de Pillis, A cellular automata model of tumorimmune system interactions,, Journal of Theoretical Biology, 239 (2006), 334. 
[46] 
C. Menon, G. Polin, I. Prabakaran, A. Hsi, C. Cheung, J. Culver, J. Pingpank, C. Sehgal, A. Yodh, D. Buerk and D. Fraker, An integrated approach to measuring tumor oxygen status using human melanoma xenografts as a model,, Cancer Research, 63 (2003), 7232. 
[47] 
B. Mueller, R. Reisfeld, T. Edgington and W. Ruf, Expression of tissue factor by melanoma cells promotes efficient hematogenous metastasis,, Proc. Natl. Acad. Sci. USA, 89 (1992), 11832. 
[48] 
D. Nelson and M. Cox, "Lehninger Principles of Biochemistry,", W. H. Freeman and Co., (2004). 
[49] 
N. Oriuchi, T. Higuchi, T. Ishikita, M. Miyakubo, H. Hanaoka, Y. Iida and K. Endo, Present role and future prospects of positron emission tomography in clinical oncology,, Cancer Science, (2006). 
[50] 
M. Owen, H. Byrne and C. Lewis, Mathematical modelling of the use of macrophages as vehicles for drug delivery to hypoxic tumour sites,, Journal of Theoretical Biology, 226 (2004), 377. 
[51] 
A. Patel, E. Gawlinski, S. Lemieux and R. Gatenby, A cellular automaton model of early tumor growth and invasion: The effects of native tissue vascularity and increased anaerobic tumor metabolism,, Journal of Theoretical Biology, 213 (2001), 315. 
[52] 
L. Preziosi, "Cancer Modelling and Simulation,", Mathematical Biology and Medicine Series. Chapman & Hall/CRC, (2003). 
[53] 
R. Puzone, B. Kohler, P. Seiden and F. Celada, IMMSIM, a flexible model for in machina experiments on immune system responses,, Future Generation Computer Systems, 18 (2002), 961. 
[54] 
A. Radunskaya and M. Villasana, A delay differential equation model for tumor growth,, J. Math.Biol., 47 (2003), 270. 
[55] 
K. A. Rejniak, An immersed boundary framework for modelling the growth of individual cells: An application to the early tumour development,, Journal of Theoretical Biology, 247 (2007), 186. 
[56] 
K. A. Rejniak and A. R. A. Anderson, Hybrid models of tumor growth,, Wiley Interdisciplinary Reviews  Systems Biology and Medicine, 3 (2011), 115. 
[57] 
K. A. Rejniak and L. J. McCawley, Current trends in mathematical modeling of tumormicroenvironment interactions: a survey of tools and applications,, Experimental Biology and Medicine, 235 (2010), 411. 
[58] 
E. Rofstad and K. Maseide, Radiobiological and immunohistochemical assessment of hypoxia in human melanoma xenografts: acute and chronic hypoxia in individual tumours,, Int J Radiat Biol., 75 (1999), 1377. 
[59] 
S. Sanga, H. B. Frieboes, X. Zheng, R. Gatenby, E. L. Bearer and V. Cristini, Predictive oncology: A review of multidisciplinary, multiscale in silico modeling linking phenotype, morphology and growth,, NeuroImage, 37 (2007). 
[60] 
P. Schornack and R. Gillies, Contributions of cell metabolism and $H^+$ diffusion to the acidic pH of tumors,, Neoplasia, 5 (2003), 135. 
[61] 
T. J. Schulz, R. Thierbach, A. Voigt, G. Drewes, B. Mietzner, P. Steinberg, A. F. H. Pfeiffer and M. Ristow, Induction of oxidative metabolism by mitochondrial frataxin inhibits cancer growth: Otto warburg revisited,, J. Biol. Chem., 281 (2006), 977. 
[62] 
R. Skoyum, K. Eide, K. Berg and E. Rofstad, Energy metabolism in human melanoma cells under hypoxic and acidic conditions in vitro,, Br J Cancer, 76 (1997), 421. 
[63] 
K. Smallbone, R. A. Gatenby, R. Gillies, P. K. Maini and D. Gavaghan, Metabolic changes during carcinogenesis: Potential impact on invasiveness,, Journal of Theoretical Biology, 244 (2006), 703. 
[64] 
K. Smallbone, D. J. Gavaghan, R. A. Gatenby and P. K. Maini, The role of acidity in solid tumour growth and invasion,, Journal of Theoretical Biology, 234 (2005), 476. 
[65] 
J. Smolle, Cellular automaton simulation of tumour growth  equivocal relationships between simulation parameters and morphologic pattern features,, Anal Cell Pathol., 17 (1998), 71. 
[66] 
J. Smolle, R. HofmannWellenhof and H. Kerl, Pattern interpretation by cellular automata (pica)evaluation of tumour cell adhesion in human melanomas,, Anal Cell Pathol., 7 (1994), 91. 
[67] 
P. Subarsky and R. Hill, The hypoxic tumour microenvironment and metastatic progression,, Clinical & Experimental Metastasis, 20 (2003), 237. 
[68] 
I. Tufto, H. Lyng and E. K. Forstad, Vascular density in human melanoma xenografts: Relationship to angiogenesis, perfusion and necrosis,, Cancer Letters, 123 (1998), 159. 
[69] 
S. Turner, Using cell potential energy to model the dynamics of adhesive biological cells,, Physical Review E, 71 (2005). 
[70] 
S. Turner, J. Sherratt, K. Painter and N. Savill, From a discrete to a continuous model of biological cell movement,, Physical Review E, 69 (2004). 
[71] 
I. van Leeuwen, C. Zonneveld and S. Kooijman, The embedded tumour: Host physiology is important for the evaluation of tumour growth,, British Journal of Cancer, 89 (2003), 2254. 
[72] 
P. Vaupel, O. Thews, D. Kelleher and M. Hoeckel, Current status of knowledge and critical issues in tumor oxygenation. results from 25 years research in tumor pathophysiology,, Adv Exp Med Biol., 454 (1998), 591. 
[73] 
R. Venkatasubramanian, M. A. Henson and N. S. Forbes, Incorporating energy metabolism into a growth model of multicellular tumor spheroids,, Journal of Theoretical Biology, 242 (2006), 440. 
[74] 
J. von Neumann, "Theory of SelfReproducing Automata,", University of Illinois Press, (1966). 
[75] 
S. Wise, J. Lowengrubb and V. Cristini, An adaptive multigrid algorithm for simulating solid tumor growth using mixture models,, Mathematical and Computer Modelling, 53 (2011), 1. doi: 10.1016/j.mcm.2010.07.007. 
[76] 
X. Zu and M. Guppy, Cancer metabolism: Facts, fantasy, and fiction,, Biochemical and Biophysical Research Communications, 313 (2004), 459. doi: 10.1016/j.bbrc.2003.11.136. 
[1] 
Hyun Geun Lee, Yangjin Kim, Junseok Kim. Mathematical model and its fast numerical method for the tumor growth. Mathematical Biosciences & Engineering, 2015, 12 (6) : 11731187. doi: 10.3934/mbe.2015.12.1173 
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