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Preface: Special issue on cancer modeling, analysis and control
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 |
References:
[1] |
A. R. A. Anderson and V. Quaranta, Integrative mathematical oncology, Nature Reviews - Cancer, 8 (2008), 227-234. |
[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), 441-451.
doi: 10.1016/j.mcm.2009.12.002. |
[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), 1-29.
doi: 10.1142/S0218202510004568. |
[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), 1130001 (37 pages).
doi: 10.1142/S0218202512005885. |
[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), 1-18. |
[6] |
A. Bellouquid, On the asymptotic analysis of kinetic models towards the compressible Euler and acoustic equations, Math. Models Methods Appl. Sci., 14 (2004), 853-882.
doi: 10.1142/S0218202504003477. |
[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), 1111-1122.
doi: 10.1016/j.nonrwa.2010.09.005. |
[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. |
[9] |
A. Bellouquid and M. Delitala, Kinetic (cellular) models of cell progression and competition with the immune system, Z. Agnew. Math. Phys., 55 (2004), 295-317.
doi: 10.1007/s00033-003-3057-9. |
[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), 1639-1666.
doi: 10.1142/S0218202505000923. |
[11] |
A. Bellouquid and M. Delitala, "Mathematical Modeling of Complex Biological Systems. A Kinetic Theory Approach," Birkäuser, Boston, 2006. |
[12] |
C. Cattani and A. Ciancio, Separable transition density in the hybrid model for tumor-immune system competition, Comp. Math. Meth. in Medicine, Article ID 610126, (2012). |
[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), 825-842. |
[14] |
F. A. Chalub, P. Markovich, B. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits, Monatsh. Math., 142 (2004), 123-141.
doi: 10.1007/s00605-004-0234-7. |
[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), 1173-1198.
doi: 10.1142/S0218202506001509. |
[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), 719-743.
doi: 10.1142/S0218202511005192. |
[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), 35-57.
doi: 10.1137/S0036139996301283. |
[18] |
A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux, Kinetic and Related Models., 5 (2012), 51-95.
doi: 10.3934/krm.2012.5.51. |
[19] |
Y. Dolak and T. Hillen, Cattaneo models for chemotaxis, numerical solution and pattern formation, J. Math. Biol., 2 (2003), 153-170.
doi: 10.1007/s00285-002-0173-7. |
[20] |
A. D'Onofrio, P. Cerrai and A. Gandolfi, "New Challenges for Cancer Systems Biomedicine,'' SIMAI-Springer Series, Springer-Italia, 2012. |
[21] |
H. Du, Z. Xu, J. D. Shrout and M. Alber, Multiscale modeling of Pesudomonas Aeruginosa swarming, Math. Models Methods Appl. Sci., 21 (2011), 939-954. |
[22] |
F. Filbet, P. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement, J. Math. Biol., 50 (2005), 189-207.
doi: 10.1007/s00285-004-0286-2. |
[23] |
J. Folkman, Role of angiogenesis in tumor growth and methastasis, Seminar Oncology, 29 (2002), 15-18. |
[24] |
H. L. Hartwell, J. J. Hopfield, S. Leibner and A. W. Murray, From molecular to modular cell biology, Nature, 402 (1999), c47-c52. |
[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), 940-960.
doi: 10.1080/03605302.2010.538783. |
[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), 1483-1538.
doi: 10.1142/S021820250900384X. |
[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), 1739-1754.
doi: 10.1088/0951-7715/10/6/016. |
[28] |
T. Hillen, On the $L^2$-moment closure of transport equation: The Cattaneo approximation, Disc. Cont. Dyn. Syst. B, 4 (2004), 961-982.
doi: 10.3934/dcdsb.2004.4.961. |
[29] |
T. Hillen and J. K. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[30] |
T. Hillen and H. G. Othmer, The diffusion limit of transport equations derived from velocity jump processes, SIAM J. Appl. Math., 61 (2000), 751-775.
doi: 10.1137/S0036139999358167. |
[31] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences: Jahresber I, Deutsch. Math.-Verein, 105 (2003) 103-165. |
[32] |
E. F. Keller and L. A. Segel, Initiation od slide mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. |
[33] |
E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234. |
[34] |
E. F. Keller and L. A. Segel, Traveling Bands of chemotactic Bacteria: A Theoretical Analysis, J. Theor. Biol., 30 (1971), 235-248. |
[35] |
E. F. Keller, Assessing the Keller-Segel model: How has it fared? In biological growth and spread, Proc. Conf. Math. Biol. Heidelberg, Springer, Berlin, (1980), 379-387. |
[36] |
P. Koumoutsakos, B. Bayati, F. Milde and G. Tauriello, Particle simulations of morphogenesis, Math. Models Methods Appl. Sci., 21 (2011), 955-1006. |
[37] |
R. Kowalczyck, Preventing blow-up in a chemotaxis model, J. Math. Anal. Appl., 305 (2005), 566-588.
doi: 10.1016/j.jmaa.2004.12.009. |
[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), 1667-1683.
doi: 10.1142/S0218202505000935. |
[39] |
M. Lachowicz, Microscopic, mesoscopic and macroscopic descriptions of complex systems, Probab. Eng. Mechanics, 26 (2011), 54-60. |
[40] |
M. Lachowicz, Individually-based Markov processes modeling nonlinear systems in mathematical biology, Nonlinear Analysis: Real World Applications, 12 (2011), 2396-2407.
doi: 10.1016/j.nonrwa.2011.02.014. |
[41] |
C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065.
doi: 10.1007/BF02179552. |
[42] |
A. Marciniak-Czochra and M. Ptashnyk, Derivation of a macroscopic receptor-based model using homogenization techniques, SIAM J. Math. Analysis, 40 (2008), 215-237
doi: 10.1137/050645269. |
[43] |
A. Marciniak-Czochra and M. Ptashnyk, Boundedness of solutions of a haptotaxis model, Math. Models Methods Appl. Sci., 20 (2010), 440-476.
doi: 10.1142/S0218202510004301. |
[44] |
I. Muller and T. Ruggeri, "Rational Extended Thermodynamics," Springer, New, York, 2nd edition, 1998.
doi: 10.1007/978-1-4612-2210-1. |
[45] |
J. D. Murray, "Mathematical Biology," Springer, Berlin, New York, 1989. |
[46] |
H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.
doi: 10.1007/BF00277392. |
[47] |
H. G. Othmer and T. Hillen, The diffusion limit of transport equations II: Chemotaxis equations, SIAM J. Appl. Math., 62 (2002), 1222-1250.
doi: 10.1137/S0036139900382772. |
[48] |
K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Canadian Appl. Math. Quart., 10 (2002), 501-543. |
[49] |
C. S. Patlak, Random walk with persistant and external bias, Bull. Math. Biol., 15 (1953), 311-338. |
[50] |
L. E. Payne and B. Straughan, Decay for a Keller-Segel chemotaxis model, Studies Appl. Math, 123 (2009), 337-360.
doi: 10.1111/j.1467-9590.2009.00457.x. |
[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), 1078-1095.
doi: 10.1137/S0036142998335984. |
[52] |
L. A. Segel, A theoretical study of receptor mechanisms in bacterial chemotaxis, SIAM J. Appl. Math., 32 (1977), 653-665. |
[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), 3525-3553
doi: 10.1098/rsta.2009.0095. |
[54] |
Y. Tao, Global existence for a haptotaxis model of cancer invasion with tissue remodeling, Nonlinear Analysis: RWA, 12 (2011), 418-435.
doi: 10.1016/j.nonrwa.2010.06.027. |
[55] |
R. A. Weinberg, "The Biology of Cancer," Garland Sciences - Taylor and Francis, New York, 2007. |
show all references
References:
[1] |
A. R. A. Anderson and V. Quaranta, Integrative mathematical oncology, Nature Reviews - Cancer, 8 (2008), 227-234. |
[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), 441-451.
doi: 10.1016/j.mcm.2009.12.002. |
[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), 1-29.
doi: 10.1142/S0218202510004568. |
[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), 1130001 (37 pages).
doi: 10.1142/S0218202512005885. |
[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), 1-18. |
[6] |
A. Bellouquid, On the asymptotic analysis of kinetic models towards the compressible Euler and acoustic equations, Math. Models Methods Appl. Sci., 14 (2004), 853-882.
doi: 10.1142/S0218202504003477. |
[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), 1111-1122.
doi: 10.1016/j.nonrwa.2010.09.005. |
[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. |
[9] |
A. Bellouquid and M. Delitala, Kinetic (cellular) models of cell progression and competition with the immune system, Z. Agnew. Math. Phys., 55 (2004), 295-317.
doi: 10.1007/s00033-003-3057-9. |
[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), 1639-1666.
doi: 10.1142/S0218202505000923. |
[11] |
A. Bellouquid and M. Delitala, "Mathematical Modeling of Complex Biological Systems. A Kinetic Theory Approach," Birkäuser, Boston, 2006. |
[12] |
C. Cattani and A. Ciancio, Separable transition density in the hybrid model for tumor-immune system competition, Comp. Math. Meth. in Medicine, Article ID 610126, (2012). |
[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), 825-842. |
[14] |
F. A. Chalub, P. Markovich, B. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits, Monatsh. Math., 142 (2004), 123-141.
doi: 10.1007/s00605-004-0234-7. |
[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), 1173-1198.
doi: 10.1142/S0218202506001509. |
[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), 719-743.
doi: 10.1142/S0218202511005192. |
[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), 35-57.
doi: 10.1137/S0036139996301283. |
[18] |
A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux, Kinetic and Related Models., 5 (2012), 51-95.
doi: 10.3934/krm.2012.5.51. |
[19] |
Y. Dolak and T. Hillen, Cattaneo models for chemotaxis, numerical solution and pattern formation, J. Math. Biol., 2 (2003), 153-170.
doi: 10.1007/s00285-002-0173-7. |
[20] |
A. D'Onofrio, P. Cerrai and A. Gandolfi, "New Challenges for Cancer Systems Biomedicine,'' SIMAI-Springer Series, Springer-Italia, 2012. |
[21] |
H. Du, Z. Xu, J. D. Shrout and M. Alber, Multiscale modeling of Pesudomonas Aeruginosa swarming, Math. Models Methods Appl. Sci., 21 (2011), 939-954. |
[22] |
F. Filbet, P. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement, J. Math. Biol., 50 (2005), 189-207.
doi: 10.1007/s00285-004-0286-2. |
[23] |
J. Folkman, Role of angiogenesis in tumor growth and methastasis, Seminar Oncology, 29 (2002), 15-18. |
[24] |
H. L. Hartwell, J. J. Hopfield, S. Leibner and A. W. Murray, From molecular to modular cell biology, Nature, 402 (1999), c47-c52. |
[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), 940-960.
doi: 10.1080/03605302.2010.538783. |
[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), 1483-1538.
doi: 10.1142/S021820250900384X. |
[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), 1739-1754.
doi: 10.1088/0951-7715/10/6/016. |
[28] |
T. Hillen, On the $L^2$-moment closure of transport equation: The Cattaneo approximation, Disc. Cont. Dyn. Syst. B, 4 (2004), 961-982.
doi: 10.3934/dcdsb.2004.4.961. |
[29] |
T. Hillen and J. K. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[30] |
T. Hillen and H. G. Othmer, The diffusion limit of transport equations derived from velocity jump processes, SIAM J. Appl. Math., 61 (2000), 751-775.
doi: 10.1137/S0036139999358167. |
[31] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences: Jahresber I, Deutsch. Math.-Verein, 105 (2003) 103-165. |
[32] |
E. F. Keller and L. A. Segel, Initiation od slide mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. |
[33] |
E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234. |
[34] |
E. F. Keller and L. A. Segel, Traveling Bands of chemotactic Bacteria: A Theoretical Analysis, J. Theor. Biol., 30 (1971), 235-248. |
[35] |
E. F. Keller, Assessing the Keller-Segel model: How has it fared? In biological growth and spread, Proc. Conf. Math. Biol. Heidelberg, Springer, Berlin, (1980), 379-387. |
[36] |
P. Koumoutsakos, B. Bayati, F. Milde and G. Tauriello, Particle simulations of morphogenesis, Math. Models Methods Appl. Sci., 21 (2011), 955-1006. |
[37] |
R. Kowalczyck, Preventing blow-up in a chemotaxis model, J. Math. Anal. Appl., 305 (2005), 566-588.
doi: 10.1016/j.jmaa.2004.12.009. |
[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), 1667-1683.
doi: 10.1142/S0218202505000935. |
[39] |
M. Lachowicz, Microscopic, mesoscopic and macroscopic descriptions of complex systems, Probab. Eng. Mechanics, 26 (2011), 54-60. |
[40] |
M. Lachowicz, Individually-based Markov processes modeling nonlinear systems in mathematical biology, Nonlinear Analysis: Real World Applications, 12 (2011), 2396-2407.
doi: 10.1016/j.nonrwa.2011.02.014. |
[41] |
C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065.
doi: 10.1007/BF02179552. |
[42] |
A. Marciniak-Czochra and M. Ptashnyk, Derivation of a macroscopic receptor-based model using homogenization techniques, SIAM J. Math. Analysis, 40 (2008), 215-237
doi: 10.1137/050645269. |
[43] |
A. Marciniak-Czochra and M. Ptashnyk, Boundedness of solutions of a haptotaxis model, Math. Models Methods Appl. Sci., 20 (2010), 440-476.
doi: 10.1142/S0218202510004301. |
[44] |
I. Muller and T. Ruggeri, "Rational Extended Thermodynamics," Springer, New, York, 2nd edition, 1998.
doi: 10.1007/978-1-4612-2210-1. |
[45] |
J. D. Murray, "Mathematical Biology," Springer, Berlin, New York, 1989. |
[46] |
H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.
doi: 10.1007/BF00277392. |
[47] |
H. G. Othmer and T. Hillen, The diffusion limit of transport equations II: Chemotaxis equations, SIAM J. Appl. Math., 62 (2002), 1222-1250.
doi: 10.1137/S0036139900382772. |
[48] |
K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Canadian Appl. Math. Quart., 10 (2002), 501-543. |
[49] |
C. S. Patlak, Random walk with persistant and external bias, Bull. Math. Biol., 15 (1953), 311-338. |
[50] |
L. E. Payne and B. Straughan, Decay for a Keller-Segel chemotaxis model, Studies Appl. Math, 123 (2009), 337-360.
doi: 10.1111/j.1467-9590.2009.00457.x. |
[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), 1078-1095.
doi: 10.1137/S0036142998335984. |
[52] |
L. A. Segel, A theoretical study of receptor mechanisms in bacterial chemotaxis, SIAM J. Appl. Math., 32 (1977), 653-665. |
[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), 3525-3553
doi: 10.1098/rsta.2009.0095. |
[54] |
Y. Tao, Global existence for a haptotaxis model of cancer invasion with tissue remodeling, Nonlinear Analysis: RWA, 12 (2011), 418-435.
doi: 10.1016/j.nonrwa.2010.06.027. |
[55] |
R. A. Weinberg, "The Biology of Cancer," Garland Sciences - Taylor and Francis, New York, 2007. |
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