# American Institute of Mathematical Sciences

June  2013, 18(4): 847-863. doi: 10.3934/dcdsb.2013.18.847

## Modeling chemotaxis from $L^2$--closure moments in kinetic theory of active particles

 1 Department of Mathematica Sciences, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy 2 University Cadi Ayyad, Ecole Nationale des Sciences Appliquées, Safi 3 Departamento de Matemática Aplicada, Universidad de Granada, Spain, Spain

Received  March 2012 Revised  May 2012 Published  February 2013

This paper deals with the derivation of macroscopic tissue models from the underlying description delivered by a class of equations modeling binary mixtures of multi-cellular systems by methods of the kinetic theory for active particles. Cellular interactions generate both modification of biological functions and proliferative-destructive events. The analysis refers to a suitable hyperbolic approximation to show how the macroscopic tissue behavior can be described from the underlying cellular description. The approach is specifically focused on the modeling of chemotaxis phenomena by the Keller--Segel approximation.
Citation: Nicola Bellomo, Abdelghani Bellouquid, Juanjo Nieto, Juan Soler. Modeling chemotaxis from $L^2$--closure moments in kinetic theory of active particles. Discrete & Continuous Dynamical Systems - B, 2013, 18 (4) : 847-863. doi: 10.3934/dcdsb.2013.18.847
##### References:
 [1] A. R. A. Anderson and V. Quaranta, Integrative mathematical oncology,, Nature Reviews - Cancer, 8 (2008), 227. Google Scholar [2] N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, Complexity and mathematical tools toward the modelling of multicellular growing systems,, Math. Comput. Modelling., 51 (2010), 441. doi: 10.1016/j.mcm.2009.12.002. Google Scholar [3] N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, Multiscale biological tissue models and flux-limited chemotaxis for multicellular growing systems,, Math. Models Methods Appl. Sci., 20 (2010), 1. doi: 10.1142/S0218202510004568. Google Scholar [4] N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, On the asymptotic theory from microscopic to macroscopic growing tissue models: An overview with perspectives,, Math. Models Methods Appl. Sci., 22 (2012). doi: 10.1142/S0218202512005885. Google Scholar [5] N. Bellomo and B. Carbonaro, Towards a mathematical theory of living systems focusing on developmental biology and evolution: A review and perspectives,, Physics of Life Reviews, 8 (2011), 1. Google Scholar [6] A. Bellouquid, On the asymptotic analysis of kinetic models towards the compressible Euler and acoustic equations,, Math. Models Methods Appl. Sci., 14 (2004), 853. doi: 10.1142/S0218202504003477. Google Scholar [7] A. Bellouquid and E. De Angelis, From kinetic models of multicellular growing systems to macroscopic biological tissue models,, Nonlinear Analysis: Real World. Appl., 12 (2011), 1111. doi: 10.1016/j.nonrwa.2010.09.005. Google Scholar [8] A. Bellouquid, E. De Angelis and D. Knopoff, From the modeling of the immune hallmarks of cancer to a black swan in biology,, Math. Models Methods Appl. Sci., 23 (2013). doi: 10.1142/S0218202512500650. Google Scholar [9] A. Bellouquid and M. Delitala, Kinetic (cellular) models of cell progression and competition with the immune system,, Z. Agnew. Math. Phys., 55 (2004), 295. doi: 10.1007/s00033-003-3057-9. Google Scholar [10] A. Bellouquid and M. Delitala, Mathematical models and tools of kinetic theory towards modelling complex biological systems,, Math. Models Methods Appl. Sci., 15 (2005), 1639. doi: 10.1142/S0218202505000923. Google Scholar [11] A. Bellouquid and M. Delitala, "Mathematical Modeling of Complex Biological Systems. A Kinetic Theory Approach,", Birkäuser, (2006). Google Scholar [12] C. Cattani and A. Ciancio, Separable transition density in the hybrid model for tumor-immune system competition,, Comp. Math. Meth. in Medicine, (2012). Google Scholar [13] F. Cerreti, B. Perthame, C. Schmeiser, M. Tang and V. Vauchelet, Waves for the hyperbolic Keller-Segel model,, Math. Models Methods Appl. Sci., 21 (2011), 825. Google Scholar [14] F. A. Chalub, P. Markovich, B. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits,, Monatsh. Math., 142 (2004), 123. doi: 10.1007/s00605-004-0234-7. Google Scholar [15] F. A. Chalub, Y. Dolak-Struss, P. Markowich, D. Oeltz, C. Schmeiser and A. Soref, Model hierarchies for cell aggregation by chemotaxis,, Math. Models Methods Appl. Sci., 16 (2006), 1173. doi: 10.1142/S0218202506001509. Google Scholar [16] M. A. J. Chaplain, M. Lachowicz, Z. Szyman'ska and D. Wrzosek, Mathematical modelling of cancer invasion: The importance of cell-cell adhesion and cell-matrix adhesion, Math. Models Methods Appl. Sci., 21 (2011), 719. doi: 10.1142/S0218202511005192. Google Scholar [17] K. C. Chen, R. M. Ford and P. T. Cummings, Perturbation expansion of Alt's cell balance equations reduces to Segel's 1d equation for shallow chemoattractant gradients,, SIAM J. Appl. Math., 59 (1999), 35. doi: 10.1137/S0036139996301283. Google Scholar [18] A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux,, Kinetic and Related Models., 5 (2012), 51. doi: 10.3934/krm.2012.5.51. Google Scholar [19] Y. Dolak and T. Hillen, Cattaneo models for chemotaxis, numerical solution and pattern formation,, J. Math. Biol., 2 (2003), 153. doi: 10.1007/s00285-002-0173-7. Google Scholar [20] A. D'Onofrio, P. Cerrai and A. Gandolfi, "New Challenges for Cancer Systems Biomedicine,'', SIMAI-Springer Series, (2012). Google Scholar [21] H. Du, Z. Xu, J. D. Shrout and M. Alber, Multiscale modeling of Pesudomonas Aeruginosa swarming,, Math. Models Methods Appl. Sci., 21 (2011), 939. Google Scholar [22] F. Filbet, P. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement,, J. Math. Biol., 50 (2005), 189. doi: 10.1007/s00285-004-0286-2. Google Scholar [23] J. Folkman, Role of angiogenesis in tumor growth and methastasis,, Seminar Oncology, 29 (2002), 15. Google Scholar [24] H. L. Hartwell, J. J. Hopfield, S. Leibner and A. W. Murray, From molecular to modular cell biology,, Nature, 402 (1999). Google Scholar [25] J. Haskovec and C. Schmeiser, Convergence of a stochastic particle approximation for measure solutions of the 2D Keller-Segel system,, Comm. Part. Diff. Eqs., 36 (2011), 940. doi: 10.1080/03605302.2010.538783. Google Scholar [26] M. A. Herrero, A. Köhn and J. M. Pérez-Pomares, Modelling vascular morphogenesis: Current views on blood vessels development,, Math. Models Methods Appl. Sci., 19 (2009), 1483. doi: 10.1142/S021820250900384X. Google Scholar [27] M. A. Herrero, E. Medina and J. J. L. Velázquez, Finite-time aggregation into a single point in a reaction-diffusion system,, Nonlinearity, 10 (1997), 1739. doi: 10.1088/0951-7715/10/6/016. Google Scholar [28] T. Hillen, On the $L^2$-moment closure of transport equation: The Cattaneo approximation,, Disc. Cont. Dyn. Syst. B, 4 (2004), 961. doi: 10.3934/dcdsb.2004.4.961. Google Scholar [29] T. Hillen and J. K. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183. doi: 10.1007/s00285-008-0201-3. Google Scholar [30] T. Hillen and H. G. Othmer, The diffusion limit of transport equations derived from velocity jump processes,, SIAM J. Appl. Math., 61 (2000), 751. doi: 10.1137/S0036139999358167. Google Scholar [31] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences: Jahresber I,, Deutsch. Math.-Verein, 105 (2003), 103. Google Scholar [32] E. F. Keller and L. A. Segel, Initiation od slide mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. Google Scholar [33] E. F. Keller and L. A. Segel, Model for chemotaxis,, J. Theor. Biol., 30 (1971), 225. Google Scholar [34] E. F. Keller and L. A. Segel, Traveling Bands of chemotactic Bacteria: A Theoretical Analysis,, J. Theor. Biol., 30 (1971), 235. Google Scholar [35] E. F. Keller, Assessing the Keller-Segel model: How has it fared? In biological growth and spread,, Proc. Conf. Math. Biol. Heidelberg, (1980), 379. Google Scholar [36] P. Koumoutsakos, B. Bayati, F. Milde and G. Tauriello, Particle simulations of morphogenesis,, Math. Models Methods Appl. Sci., 21 (2011), 955. Google Scholar [37] R. Kowalczyck, Preventing blow-up in a chemotaxis model,, J. Math. Anal. Appl., 305 (2005), 566. doi: 10.1016/j.jmaa.2004.12.009. Google Scholar [38] M. Lachowicz, Micro and meso scales of description corresponding to a model of tissue invasion by solid tumors,, Math. Models Methods Appl. Sci., 15 (2005), 1667. doi: 10.1142/S0218202505000935. Google Scholar [39] M. Lachowicz, Microscopic, mesoscopic and macroscopic descriptions of complex systems,, Probab. Eng. Mechanics, 26 (2011), 54. Google Scholar [40] M. Lachowicz, Individually-based Markov processes modeling nonlinear systems in mathematical biology,, Nonlinear Analysis: Real World Applications, 12 (2011), 2396. doi: 10.1016/j.nonrwa.2011.02.014. Google Scholar [41] C. D. Levermore, Moment closure hierarchies for kinetic theories,, J. Stat. Phys., 83 (1996), 1021. doi: 10.1007/BF02179552. Google Scholar [42] A. Marciniak-Czochra and M. Ptashnyk, Derivation of a macroscopic receptor-based model using homogenization techniques,, SIAM J. Math. Analysis, 40 (2008), 215. doi: 10.1137/050645269. Google Scholar [43] A. Marciniak-Czochra and M. Ptashnyk, Boundedness of solutions of a haptotaxis model,, Math. Models Methods Appl. Sci., 20 (2010), 440. doi: 10.1142/S0218202510004301. Google Scholar [44] I. Muller and T. Ruggeri, "Rational Extended Thermodynamics,", Springer, (1998). doi: 10.1007/978-1-4612-2210-1. Google Scholar [45] J. D. Murray, "Mathematical Biology,", Springer, (1989). Google Scholar [46] H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems,, J. Math. Biol., 26 (1988), 263. doi: 10.1007/BF00277392. Google Scholar [47] H. G. Othmer and T. Hillen, The diffusion limit of transport equations II: Chemotaxis equations,, SIAM J. Appl. Math., 62 (2002), 1222. doi: 10.1137/S0036139900382772. Google Scholar [48] K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement,, Canadian Appl. Math. Quart., 10 (2002), 501. Google Scholar [49] C. S. Patlak, Random walk with persistant and external bias,, Bull. Math. Biol., 15 (1953), 311. Google Scholar [50] L. E. Payne and B. Straughan, Decay for a Keller-Segel chemotaxis model,, Studies Appl. Math, 123 (2009), 337. doi: 10.1111/j.1467-9590.2009.00457.x. Google Scholar [51] C. Ringhofer, C. Schmeiser and A. Zwirchmayr, Moment methods for the semiconductor Boltzmann equation on bounded position domains,, SIAM J. Num. Anal., 39 (2001), 1078. doi: 10.1137/S0036142998335984. Google Scholar [52] L. A. Segel, A theoretical study of receptor mechanisms in bacterial chemotaxis,, SIAM J. Appl. Math., 32 (1977), 653. Google Scholar [53] M. A. Stolarska, K. I. M. Yangjin and H. G. Othmer, Multi-scale models of cell and tissue dynamics,, Phil. Trans. Royal Society A: Math. Phys. Eng. Sci., 367 (2009), 3525. doi: 10.1098/rsta.2009.0095. Google Scholar [54] Y. Tao, Global existence for a haptotaxis model of cancer invasion with tissue remodeling,, Nonlinear Analysis: RWA, 12 (2011), 418. doi: 10.1016/j.nonrwa.2010.06.027. Google Scholar [55] R. A. Weinberg, "The Biology of Cancer,", Garland Sciences - Taylor and Francis, (2007). Google Scholar

show all references

##### References:
 [1] A. R. A. Anderson and V. Quaranta, Integrative mathematical oncology,, Nature Reviews - Cancer, 8 (2008), 227. Google Scholar [2] N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, Complexity and mathematical tools toward the modelling of multicellular growing systems,, Math. Comput. Modelling., 51 (2010), 441. doi: 10.1016/j.mcm.2009.12.002. Google Scholar [3] N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, Multiscale biological tissue models and flux-limited chemotaxis for multicellular growing systems,, Math. Models Methods Appl. Sci., 20 (2010), 1. doi: 10.1142/S0218202510004568. Google Scholar [4] N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, On the asymptotic theory from microscopic to macroscopic growing tissue models: An overview with perspectives,, Math. Models Methods Appl. Sci., 22 (2012). doi: 10.1142/S0218202512005885. Google Scholar [5] N. Bellomo and B. Carbonaro, Towards a mathematical theory of living systems focusing on developmental biology and evolution: A review and perspectives,, Physics of Life Reviews, 8 (2011), 1. Google Scholar [6] A. Bellouquid, On the asymptotic analysis of kinetic models towards the compressible Euler and acoustic equations,, Math. Models Methods Appl. Sci., 14 (2004), 853. doi: 10.1142/S0218202504003477. Google Scholar [7] A. Bellouquid and E. De Angelis, From kinetic models of multicellular growing systems to macroscopic biological tissue models,, Nonlinear Analysis: Real World. Appl., 12 (2011), 1111. doi: 10.1016/j.nonrwa.2010.09.005. Google Scholar [8] A. Bellouquid, E. De Angelis and D. Knopoff, From the modeling of the immune hallmarks of cancer to a black swan in biology,, Math. Models Methods Appl. Sci., 23 (2013). doi: 10.1142/S0218202512500650. Google Scholar [9] A. Bellouquid and M. Delitala, Kinetic (cellular) models of cell progression and competition with the immune system,, Z. Agnew. Math. Phys., 55 (2004), 295. doi: 10.1007/s00033-003-3057-9. Google Scholar [10] A. Bellouquid and M. Delitala, Mathematical models and tools of kinetic theory towards modelling complex biological systems,, Math. Models Methods Appl. Sci., 15 (2005), 1639. doi: 10.1142/S0218202505000923. Google Scholar [11] A. Bellouquid and M. Delitala, "Mathematical Modeling of Complex Biological Systems. A Kinetic Theory Approach,", Birkäuser, (2006). Google Scholar [12] C. Cattani and A. Ciancio, Separable transition density in the hybrid model for tumor-immune system competition,, Comp. Math. Meth. in Medicine, (2012). Google Scholar [13] F. Cerreti, B. Perthame, C. Schmeiser, M. Tang and V. Vauchelet, Waves for the hyperbolic Keller-Segel model,, Math. Models Methods Appl. Sci., 21 (2011), 825. Google Scholar [14] F. A. Chalub, P. Markovich, B. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits,, Monatsh. Math., 142 (2004), 123. doi: 10.1007/s00605-004-0234-7. Google Scholar [15] F. A. Chalub, Y. Dolak-Struss, P. Markowich, D. Oeltz, C. Schmeiser and A. Soref, Model hierarchies for cell aggregation by chemotaxis,, Math. Models Methods Appl. Sci., 16 (2006), 1173. doi: 10.1142/S0218202506001509. Google Scholar [16] M. A. J. Chaplain, M. Lachowicz, Z. Szyman'ska and D. Wrzosek, Mathematical modelling of cancer invasion: The importance of cell-cell adhesion and cell-matrix adhesion, Math. Models Methods Appl. Sci., 21 (2011), 719. doi: 10.1142/S0218202511005192. Google Scholar [17] K. C. Chen, R. M. Ford and P. T. Cummings, Perturbation expansion of Alt's cell balance equations reduces to Segel's 1d equation for shallow chemoattractant gradients,, SIAM J. Appl. Math., 59 (1999), 35. doi: 10.1137/S0036139996301283. Google Scholar [18] A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux,, Kinetic and Related Models., 5 (2012), 51. doi: 10.3934/krm.2012.5.51. Google Scholar [19] Y. Dolak and T. Hillen, Cattaneo models for chemotaxis, numerical solution and pattern formation,, J. Math. Biol., 2 (2003), 153. doi: 10.1007/s00285-002-0173-7. Google Scholar [20] A. D'Onofrio, P. Cerrai and A. Gandolfi, "New Challenges for Cancer Systems Biomedicine,'', SIMAI-Springer Series, (2012). Google Scholar [21] H. Du, Z. Xu, J. D. Shrout and M. Alber, Multiscale modeling of Pesudomonas Aeruginosa swarming,, Math. Models Methods Appl. Sci., 21 (2011), 939. Google Scholar [22] F. Filbet, P. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement,, J. Math. Biol., 50 (2005), 189. doi: 10.1007/s00285-004-0286-2. Google Scholar [23] J. Folkman, Role of angiogenesis in tumor growth and methastasis,, Seminar Oncology, 29 (2002), 15. Google Scholar [24] H. L. Hartwell, J. J. Hopfield, S. Leibner and A. W. Murray, From molecular to modular cell biology,, Nature, 402 (1999). Google Scholar [25] J. Haskovec and C. Schmeiser, Convergence of a stochastic particle approximation for measure solutions of the 2D Keller-Segel system,, Comm. Part. Diff. Eqs., 36 (2011), 940. doi: 10.1080/03605302.2010.538783. Google Scholar [26] M. A. Herrero, A. Köhn and J. M. Pérez-Pomares, Modelling vascular morphogenesis: Current views on blood vessels development,, Math. Models Methods Appl. Sci., 19 (2009), 1483. doi: 10.1142/S021820250900384X. Google Scholar [27] M. A. Herrero, E. Medina and J. J. L. Velázquez, Finite-time aggregation into a single point in a reaction-diffusion system,, Nonlinearity, 10 (1997), 1739. doi: 10.1088/0951-7715/10/6/016. Google Scholar [28] T. Hillen, On the $L^2$-moment closure of transport equation: The Cattaneo approximation,, Disc. Cont. Dyn. Syst. B, 4 (2004), 961. doi: 10.3934/dcdsb.2004.4.961. Google Scholar [29] T. Hillen and J. K. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183. doi: 10.1007/s00285-008-0201-3. Google Scholar [30] T. Hillen and H. G. Othmer, The diffusion limit of transport equations derived from velocity jump processes,, SIAM J. Appl. Math., 61 (2000), 751. doi: 10.1137/S0036139999358167. Google Scholar [31] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences: Jahresber I,, Deutsch. Math.-Verein, 105 (2003), 103. Google Scholar [32] E. F. Keller and L. A. Segel, Initiation od slide mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. Google Scholar [33] E. F. Keller and L. A. Segel, Model for chemotaxis,, J. Theor. Biol., 30 (1971), 225. Google Scholar [34] E. F. Keller and L. A. Segel, Traveling Bands of chemotactic Bacteria: A Theoretical Analysis,, J. Theor. Biol., 30 (1971), 235. Google Scholar [35] E. F. Keller, Assessing the Keller-Segel model: How has it fared? In biological growth and spread,, Proc. Conf. Math. Biol. Heidelberg, (1980), 379. Google Scholar [36] P. Koumoutsakos, B. Bayati, F. Milde and G. Tauriello, Particle simulations of morphogenesis,, Math. Models Methods Appl. Sci., 21 (2011), 955. Google Scholar [37] R. Kowalczyck, Preventing blow-up in a chemotaxis model,, J. Math. Anal. Appl., 305 (2005), 566. doi: 10.1016/j.jmaa.2004.12.009. Google Scholar [38] M. Lachowicz, Micro and meso scales of description corresponding to a model of tissue invasion by solid tumors,, Math. Models Methods Appl. Sci., 15 (2005), 1667. doi: 10.1142/S0218202505000935. Google Scholar [39] M. Lachowicz, Microscopic, mesoscopic and macroscopic descriptions of complex systems,, Probab. Eng. Mechanics, 26 (2011), 54. Google Scholar [40] M. Lachowicz, Individually-based Markov processes modeling nonlinear systems in mathematical biology,, Nonlinear Analysis: Real World Applications, 12 (2011), 2396. doi: 10.1016/j.nonrwa.2011.02.014. Google Scholar [41] C. D. Levermore, Moment closure hierarchies for kinetic theories,, J. Stat. Phys., 83 (1996), 1021. doi: 10.1007/BF02179552. Google Scholar [42] A. Marciniak-Czochra and M. Ptashnyk, Derivation of a macroscopic receptor-based model using homogenization techniques,, SIAM J. Math. Analysis, 40 (2008), 215. doi: 10.1137/050645269. Google Scholar [43] A. Marciniak-Czochra and M. Ptashnyk, Boundedness of solutions of a haptotaxis model,, Math. Models Methods Appl. Sci., 20 (2010), 440. doi: 10.1142/S0218202510004301. Google Scholar [44] I. Muller and T. Ruggeri, "Rational Extended Thermodynamics,", Springer, (1998). doi: 10.1007/978-1-4612-2210-1. Google Scholar [45] J. D. Murray, "Mathematical Biology,", Springer, (1989). Google Scholar [46] H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems,, J. Math. Biol., 26 (1988), 263. doi: 10.1007/BF00277392. Google Scholar [47] H. G. Othmer and T. Hillen, The diffusion limit of transport equations II: Chemotaxis equations,, SIAM J. Appl. Math., 62 (2002), 1222. doi: 10.1137/S0036139900382772. Google Scholar [48] K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement,, Canadian Appl. Math. Quart., 10 (2002), 501. Google Scholar [49] C. S. Patlak, Random walk with persistant and external bias,, Bull. Math. Biol., 15 (1953), 311. Google Scholar [50] L. E. Payne and B. Straughan, Decay for a Keller-Segel chemotaxis model,, Studies Appl. Math, 123 (2009), 337. doi: 10.1111/j.1467-9590.2009.00457.x. Google Scholar [51] C. Ringhofer, C. Schmeiser and A. Zwirchmayr, Moment methods for the semiconductor Boltzmann equation on bounded position domains,, SIAM J. Num. Anal., 39 (2001), 1078. doi: 10.1137/S0036142998335984. Google Scholar [52] L. A. Segel, A theoretical study of receptor mechanisms in bacterial chemotaxis,, SIAM J. Appl. Math., 32 (1977), 653. Google Scholar [53] M. A. Stolarska, K. I. M. Yangjin and H. G. Othmer, Multi-scale models of cell and tissue dynamics,, Phil. Trans. Royal Society A: Math. Phys. Eng. Sci., 367 (2009), 3525. doi: 10.1098/rsta.2009.0095. Google Scholar [54] Y. Tao, Global existence for a haptotaxis model of cancer invasion with tissue remodeling,, Nonlinear Analysis: RWA, 12 (2011), 418. doi: 10.1016/j.nonrwa.2010.06.027. Google Scholar [55] R. A. Weinberg, "The Biology of Cancer,", Garland Sciences - Taylor and Francis, (2007). Google Scholar
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