Modeling of Tumor Occurrence and Growth - I
УДК 517:612.6
Abstract
Cancer has recently overtaken heart disease as the world’s biggest killer. Cancer is initiated by gene mutations that result in local proliferation of abnormal cells and their migration to other parts of the human body, a process called metastasis. The metastasized cancer cells then interfere with the normal functions of the body, eventually leading to death. There are 200 types of cancer, classified by their point of origin. Most of them share some common features, but they also have their specific character. In this paper, we consider mathematical models of non-specific solid tumors in a tissue. The models incorporate the constitutive nature of the tissue, and the need for growing tumors to attract blood vessels. We also describe a general multiscale approach that involves cell cycle and incorporates non-specific genes mutation. Also, the trends and general features of modeling tumor growth are discussed. The main goal is set at revealing some trends and challenges on cancer modeling, especially related to the development of multiphase and multiscale models.
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References
Friedman A. Cancer as Multifaceted Disease // Math. Model. Nat. Phenom. 2012. 7 1. DOI: 10.1051/mmnp/20127102.
Astanin S., Tosin A. Mathematical model of tumour cord growth along the source of nutrient // Math. Model. Nat. Phenom. 2007. 2 3. DOI: 10.1051/mmnp:2007007.
Araujo R.P., S. McElwain D.L. A mixture theory for the genesis of residual stresses in growing tissues. I. A general formulation // SIAM J. Appl. Math. 2005. 65. DOI: 10.1137/040607113.
Araujo R.P., S. McElwain D.L. A mixture theory for the genesis of residual stresses in growing tissues. II. Solutions to the biphasic equations for a multicell spheroid // SIAM J. Appl. Math. 2005. 66. DOI: 10.1137/040607125.
Astanin S., Preziosi L. Mathematical modelling of the Warburg effect in tumour cords // J. Theor. Biol. 2009. 258. DOI: 10.1016/j.jtbi.2009.01.034.
Breward C.J.W., Byrne H.M., Lewis C.E. A multiphase model describing vascular tumour growth // Bull. Math. Biol. 2003. 65. DOI: 10.1016/S0092-8240(03)00027-2.
Chaplain M.A.J., Graziano L., Preziosi L. Mathematical modelling of the loss of tissue compression responsiveness and its role in solid tumour development // Math. Med. Biol. 2006. 23. DOI: 10.1093/imammb/dql009.
Franks S.J., King J.R. Interactions between a uniformly proliferating tumour and its surroundings: uniform material properties // Math. Med. Biol. 2003. 20. DOI: 10.1093/imammb/20.1.47.
Preziosi L., Tosin A. Multiphase modelling of tumour growth and extracellular matrix interaction: mathematical tools and applications // J. Math. Biol. 2009. 58. DOI: 10.1007/s00285-008-0218-7.
Cristini V., Li A., Lowengrub J.S., Wise S.M. Nonlinear simulations of solid tumor growth using a mixture model: invasion and branching // J. Math. Biol. 2009. 58. DOI: 10.1007/s00285-008-0215-x.
Wise S.M., Lowengrub J.S., Frieboes H.B., Cristini V. Three-dimensional multispecies nonlinear tumor growth - I. Model and numerical method // J. Theor. Biol. 2008. 253. DOI: 10.1016/j.jtbi.2008.03.027.
Galle J., Loeffler M., Drasdo D. Modelling the effect of deregulated proliferation and apoptosis on the growth dynamics of epithelial cell populations in vitro // Biophys. J. 2005. 88. DOI: 10.1529/biophysj.104.041459.
Hoehme S., Drasdo D. Biomechanical versus nutrient control: what determines the growth dynamics of mammalian cell populations // An International Journal of Mathematical Demography. 2010. 17. DOI: 10.1080/08898480.2010.491032.
Radszuweit M., Block M., Hengstler J.G., Scholl E., Drasdo D. Comparing the growth kinetics of cell populations in two and three dimensions // Phys. Rev. E. 2009. 79. DOI: 10.1007/978-1-4939-6506-9_22.
Ambrosi D., Mollica F. On the mechanics of a growing tumor // Internat. J. Engrg. Sci. 2002. 40. DOI: 10.1016/S0020-7225(02)00014-9.
Ambrosi D., Mollica F. The role of stress in the growth of a multicell spheroid // J. Math. Biol. 2004. 48. DOI: 10.1007/s00285-003-0238-2.
Ambrosi D., Preziosi L. Cell adhesion mechanisms and stress relaxation in the mechanics of tumours // Biomech. Model. Mechanobiol. DOI: 10.1007/s10237-008-0145-y.
Anderson A.R.A., Chaplain M.A.J., Rejniak K.A. Single-cell-based models in biology and medicine // Mathematical Medicine and Biology. 2007. DOI: 10.1093/imammb/dqn008.
Anderson A.R.A., Rejniak K.A., Gerlee P., Quaranta V. Modelling of cancer growth, evolution and invasion: bridging scales and models // Math. Model. Nat. Phenom. 2007. 2. DOI: 10.1051/mmnp:2007001.
Gerlee P., Anderson A.R.A. A hybrid cellular automaton model of clonal evolution in cancer: The emergence of the glycolytic phenotype // J. Theor. Biol. 2007. 250. DOI: 10.1016/j.jtbi.2007.10.038.
Gerlee P., Anderson A.R.A. An evolutionary hybrid cellular automaton model of solid tumour growth // J. Theor. Biol. 2007. 246. DOI: 10.1016/j.jtbi.2007.01.027.
Rejniak K.A. A single-cell approach in modeling the dynamics of tumor microregions // Math. Biosci. Eng. 2005. 2. DOI: 10.3934/mbe.2005.2.643.
Ramis-Conde I., Drasdo D., Anderson A.R.A., Chaplain M.A.J. Modeling the infuence of the E-cadherin-b-catenin pathway in cancer cell invasion: A multiscale approach // Biophys. J. 2008. 95. DOI: 10.1529/biophysj.107.114678.
Rejniak K.A., Dillon R.H. A single cell-based model of the ductal tumour microarchitecture // Comp. Math. Methods Med. 2007. 8. DOI: 10.1080/17486700701303143.
Jiang Y., Pjesivac-Grbovic J., Cantrell C., Freyer J.P. A multiscale model for avascular tumor growth // Biophys. J. 2005. 89. DOI: 10.1529/biophysj.105.060640.
Doumic M. Analysis of a population model structured by the cells molecular content // Math. Model. Nat. Phenom. 2007. 2. DOI: 10.1051/mmnp:2007006.
Ribba B., Colin T., Schnell S. A multiscale mathematical model of cancer, and its use in analyzing irradiation therapies // Theor. Biol. Medical Model. 2006. 3 7. DOI: 10.1186/1742-46823-7.
Ribba B., Saut O., Colin T., Bresch D., Grenier E., Boissel J.P. A multiscale mathematical model of avascular tumor growth to investigate the therapeutic benefit of antiinvasive agents // J. Theor. Biol. 2006. 243. DOI: 10.1016/j.jtbi.2006.07.013.
Bertuzzi A., Fasano A., Gandolfi A., Sinisgalli C. ATP production and necrosis formation in a tumour spheroid model // Math. Model. Nat. Phenom. 2007. 2. DOI: 10.1051/mmnp:2007002.
Venkatasubramanian R., Henson M.A., Forbes N.S. Incorporating energy metabolism into a growth model of multicellular tumor spheroids // J. Theor. Biol. 2006. 242. DOI: 10.1016/j.jtbi.2006.03.011.
Preziosi L., Ambrosi D., Verdier C. A elasto-visco-plastic model of cell aggregates // Journal of Theoretical Biology. 2010. 262 1. DOI: 10.1016/j.jtbi.2009.08.023.
Chapman S.J., Shipley R.J., Jawad R. Multiscale modeling of fluid transport in tumors // Bull. Math. Biol. 2008. 70. DOI: 10.1007/s11538-008-9349-7.
Chauviere A., Hillen T., Preziosi L. Modeling cell movement in anisotropic and heterogeneous network tissues // Netw. Heterog. Media. 2007. 2. DOI: 10.3934/nhm.2007.2.333.
Hillen T. M5 mesoscopic and macroscopic models for mesenchymal motion // J. Math. Biol. 2006. 53. DOI: 10.1007/s00285-006-0017-y.
Lachowicz M. From microscopic to macroscopic description for generalized kinetic models // Math. Models Methods Appl. Sci. 2002. 12. DOI: 10.1142/S0218202502001994.
Secomb T.W., El-Kareh A.W. A theoretical model for the elastic properties of very soft tissues // Biorheology. 2001. 38.
Shipley R.J., Jones G.W., Dyson R.J., Sengers B.G., Bailey C.L., Catt C.J., Please C.P., Malda J. Design criteria for a printed tissue engineering construct: A mathematical homogenization approach // J. Theor. Biol. DOI: 10.1016/j.jtbi.2009.03.037.
Lejeune E., Linder C. Modeling tumor growth with peridynamics. // Biomechanics and Modeling in Mechanobiology. 2017. 16 4. DOI: 10.1007/s10237-017-0876-8.
Deutsch A., Dormann S. Tumor Growth and Invasion // Springer Science+Business Media New York. 2017. DOI: 10.1007/978-1-4899-7980-3.
Drasdo D., Hoehme S. Individual-based approaches to birth and death in avascular tumors // Math. Comput. Mod. 2003. 37. DOI: 10.1016/S0895-7177(03)00128-6.
Drasdo D., Hoehme S. A single-cell-based model of tumor growth in vitro: monolayers and spheroids // Phys. Biol. 2005. 2. DOI: 10.1088/14783975/2/3/001.
Anderson A.R.A., Chaplain M.A.J. Continuous and discrete mathematical models of tumor-induced angiogenesis // Bull. Math. Biol. 1998. 60. DOI: 10.1006/bulm.1998.0042.
McDougall S.R., Anderson A.R.A., Chaplain M.A.J. Mathematical modelling of dynamic adaptive tumour-induced angiogenesis: clinical implications and therapeutic targeting strategies // J. Theor. Biol. 2006. 241. DOI: 10.1016/j.jtbi.2005.12.022.
McDougall S.R., Anderson A.R.A., Chaplain M.A.J., Sherratt J.A. Mathematical modelling of flow through vascular networks: implications for tumour-induced angiogenesis and chemotherapy strategies // Bull. Math. Biol. 2002. 64. DOI: 10.1006/bulm.2002.0293.
Stephanou A., McDougall S.R., Anderson A.R.A., Chaplain M.A.J. Mathematical modelling of flow in 2D and 3D vascular networks: applications to anti-angiogenic and chemotherapeutic drug strategies // Math. Comput. Modelling. 2005. 41. DOI: 10.1016/j.mcm.2005.05.008.
Macklin P., McDougall S., Anderson A.R.A., Chaplain M.A.J., Cristini V., Lowengrub J. Multiscale modelling and nonlinear simulation of vascular tumour growth // J. Math. Biol. 2009. 58. DOI: 10.1007/s00285-008-0216-9.
Zheng X., Wise S.M., Cristini V. Nonlinear simulation of tumor necrosis, neovascularization and tissue invasion via an adaptive finite-element/level-set method // Bull. Math. Biol. 2005. 67. DOI: 10.1016/j.bulm.2004.08.001.
Ambrosi D., Preziosi L. On the closure of mass balance models for tumor growth // Math. Models Methods Appl. Sci. 2002. 12. DOI: 10.1142/S0218202502001878.
Pettet G.J., Please C.P., Tindall M.J.,S. McElwain D.L. The migration of cells in multicell tumor spheroids // Bull. Math. Biol. 2001. 63. DOI: 10.1006/bulm.2000.0217.
Bazaliy B.V., Friedman A. A free boundary problem for an elliptic-parabolic system: Application to a model of tumor growth // Comm. in PDE. 2003. 28. DOI: 10.1081/PDE-120020486.
Chen X., Friedman A. A free boundary problem for elliptic-hyperbolic system: An application to tumor growth // SIAM J. Math. Anal. 2003. 35. DOI: 10.1137/S0036141002418388.
Cui S., Friedman A. A hyperbolic free boundary problem modeling tumor growth // Interfaces & Free Boundaries. 2003. 5. DOI: 10.4171/IFB/76.
Chen X., Cui S., Friedman A. A hyperbolic free boundary problem modeling tumor growth: Asymptotic behavior // Trans. Amer. Math. Soc. 2005. 357. DOI: 0.1090/S0002-9947-05-03784-0.
Cui S., Friedman A. A free boundary problem for a singular system of differential equations: An application to a model of tumor growth // Trans. Amer. Math. Soc. 2003. 355. DOI: 10.1090/S0002-9947-03-03137-4.
Friedman A., Reitich F. Analysis of a mathematical model for growth of tumors // J. Math. Biol. 1999. 38. DOI: 10.1007/s002850050149.
Bazaliy B., Friedman A. Global existence and stability for an elliptic-parabolic free boundary problem: An application to a model with tumor growth // Indiana Univ. Math. J. 2003. 52. DOI: 10.1512/iumj.2003.52.2317.
Friedman A., Hu B. Asymptotic stability for a free boundary problem arising in a tumor model // J. Diff. Eqs. 2006. 227. DOI: 10.1016/j.jde.2005.09.008.
Wu J., Cui S. Asymptotic stability of stationary solutions of a free boundary problem modeling the growth of tumors with fluid tissues // SIAM J. Math. Anal. 2010. 41.
Fontelos M.A., Friedman A. Symmetry-breaking bifurcations of free boundary problems in three dimensions // Asymptotic Anal. 2003. 35.
Friedman A., Hu B. Bifurcation from stability to instability for a free boundary problem arising in tumor model // Arch. Rat. Mech. Anal. 2006. 180. DOI: 10.1016/j.jmaa.2006.04.034
Friedman A., Hu B. Stability and instability of Liapounov-Schmidt and Hopf bifurcations for a free boundary problem arising in a tumor model // Trans. Amer. Math. Soc. 2008. 360. DOI: 10.1090/S0002-9947-08-04468-1.
Morozov A., Ptashnyk M., Volpert V. Preface. Bifurcations and pattern formation in biological applications // Mathematical Modelling of Natural Phenomena. 2016. 11 5. DOI: 10.1051/mmnp/201611501.
Friedman A. A free boundary problem for a coupled system of elliptic, hyperbolic, and Stokes equations modeling tumor growth // Interfaces and Free Boundaries 2006. 8. DOI: 10.4171/IFB/142.
Friedman A., Hu B. Bifurcation from stability to instability for a free boundary problem modeling tumor growth by Stokes equation // Math. Anal & Appl. 2007. 327. DOI: 10.1016/j.jmaa.2006.04.034.
Friedman A., Hu B. Bifurcation for a free boundary problem modeling tumor growth by Stokes equation // SIAM J. Math. Anal. 2007. 39. DOI: 10.1016/j.jmaa.2006.04.034.
Kim Y., Stolarska M., Othmer H. A hybrid model for tumor spheroid growth in vitro I: theoretical development and early results // Math. Mod. Meth. Appl. Sci. 2007. 17. DOI: 10.1142/S0218202507002479.
Levine H.A., Nilsen-Hamilton M. Angiogenesis-A biochemical/mathematical perspective // Lecture Notes Math. 2006. 1872. DOI: 10.1007/11561606_2.
Levine H.A., Pamuk S.L., Sleeman B.D., Nilsen-Hamilton M. Mathematical modeling of capillary formation and development in tumor angiogenesis: penetration into the stroma // Bull. Math. Biol. 2001. 63. DOI: 10.1006/bulm.2001.0240.
Mantzaris N., Webb S., Othmer H.G. Mathematical Modeling of Tumor-Induced Angiogenesis // J. Math. Biol. 2004. 49. DOI: 10.1007/s00285-003-0262-2.
Xu J., Vilanova G., Gomez H. A mathematical model coupling tumor growth and angiogenesis // PLOS One. 2016. DOI: 10.1371/journal.pone.0149422.
Byrne H.M., Chaplain M.A.J. Growth of necrotic tumors in the presence and absence of inhibitors // Math. Biosci. 1996. 135. DOI: 10.1016/0025-5564(94)00117-3.
Cui S., Friedman A. Analysis of a mathematical model of the growth of necrotic tumors // J. Math. Anal. & Appl. 2001. 255. DOI: 10.1006/jmaa.2000.7306.
Friedman A. A multiscale tumor model // Interfaces & Free Boundaries. 2008. 10.
Friedman A. Free boundary value problems associated with multiscale tumor models // Mathematical Modeling of Natural Phenomena. 2009. 4. DOI: 10.1051/mmnp/20094306.
Friedman A., Hu B. The role of oxygen in tissue maintenance: Mathematical modeling and qualitative analysis // Math. Mod. Meth. Appl. Sci. 2008. 18. DOI: 10.1142/S021820250800308X.
Ayati B.P., Webb G.F., Anderson A.R.A. Computational methods and results for structured multiscale methods of tumor invasion // Multiscale Model. Simul. 2006. 5. DOI: 10.1137/050629215.
Jiang Y., Pjesivac-Grbovic J., Cantrell C., Freyer J.P. A multiscale model for avascular tumor growth // Biophy. J. 2005. 89. DOI: 10.1529/biophysj.105.060640.
Ribba R., Colin T., Schnell S. A multiscale model of cancer, and its use in analyzing irradiation therapies // Theor. Biol. & Med. Mod. 2006. 3. No. 7. DOI: 10.1186/1742-4682-3-7.
Ribba B., Sant O., Colin T., Bresch D., Grenien E., Boissel J.P. A multiscale model of avascular tumor growth to investigate agents // J. Theor. Biol. 2006. 243. DOI: 10.1016/j.jtbi.2006.07.013.
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