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Ghosts of bump attractors in stochastic neural fields: Bottlenecks and extinction
A nonlocal sample dependence SDE-PDE system modeling proton dynamics in a tumor
1. | School of Mathematics and Statistics, Huazhong University of Science & Technology, Wuhan 430074 |
2. | Felix-Klein-Zentrum für Mathematik, TU Kaiserslautern, Paul-Ehrlich-Str. 31, 67663 Kaiserslautern, Germany |
3. | Technische Universität Kaiserslautern, Felix-Klein-Zentrum für Mathematik, Paul-Ehrlich-Str. 31, 67663 Kaiserslautern |
References:
[1] |
M. J. Boyer and I. F. Tannock, Regulation of intracellular pH in tumor cell lines: Influence of microenvironmental conditions, Cancer Res., 52 (1992), 4441-4447. |
[2] |
P. L. Chow, Stochastic Partial Differential Equations, Chapman & Hall /CRC, Boca Raton, 2015. |
[3] |
J. Cresson, B. Puig and S. Sonner, Stochastic models in biology and the invariance problem, Discrete Contin. Dyn. Syst. Ser. B, (2016), to appear. |
[4] |
A. De Milito and S. Fais, Tumor acidity, chemoresistance and proton pump inhibitors, Future Oncol., 1 (2005), 779-786.
doi: 10.2217/14796694.1.6.779. |
[5] |
M. L. Freeman and E. Sierra, An acidic extracellular environment reduces the fixation of radiation damage, Radiat. Resist., 97 (1984), 154-161.
doi: 10.2307/3576196. |
[6] |
R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Res., 56 (1996), 5745-5753. |
[7] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, New York, 1981. |
[8] |
S. Hiremath and C. Surulescu, A stochastic multiscale model for acid mediated cancer invasion, Nonlinear Anal. Real World Appl., 22 (2015), 176-205.
doi: 10.1016/j.nonrwa.2014.08.008. |
[9] |
S. Hiremath and C. Surulescu, A stochastic model featuring acid induced gaps during tumor progression, Nonlinearity, 29 (2016), 851-914.
doi: 10.1088/0951-7715/29/3/851. |
[10] |
E. Jakobsson and S. W. Chiu, Stochastic theory of ion movement in channels with single-ion occupancy. Application to sodium permeation of gramicidin channel, Biophys. J., 52 (1987), 33-45.
doi: 10.1016/S0006-3495(87)83186-7. |
[11] |
A. Jentzen and P. E. Kloeden, Taylor Approximations for Stochastic Partial Differential Equations, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 2011.
doi: 10.1137/1.9781611972016. |
[12] |
Y. Jiongmin, Linear-quadratic optimal control problems for mean-field stochastic differential equations, SIAM J. Control Optim., 51 (2013), 2809-2838.
doi: 10.1137/120892477. |
[13] |
P. E. Kloeden and T. Lorenz, Stochastic differential equations with nonlocal sample dependence, Stoch. Anal. Appl., 28 (2010), 937-945.
doi: 10.1080/07362994.2010.515194. |
[14] |
P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin-Heidelberg, 1992.
doi: 10.1007/978-3-662-12616-5. |
[15] |
A. H. Lee and I. F. Tannock, Heterogeneity of intracellular pH and of mechanisms that regulate intracellular pH in populations of cultured cells, Cancer Res., 58 (1998), 1901-1908. |
[16] |
X. Mao, Stochastic Differential Equations and Applications, $2^{nd}$ edition, Woodhead Publishing, Cambridge, 2008.
doi: 10.1533/9780857099402. |
[17] |
N. K. Martin, R. A. Gatenby, E. T. Gawlinski and P. K. Maini, Tumor-stromal interactions in acid-mediated invasion: A mathematical model, J. Theor. Biol., 267 (2010), 461-470.
doi: 10.1016/j.jtbi.2010.08.028. |
[18] |
R. Martínez-Zaguilán, E. A. Seftor, R. E. B. Seftor, Y. W. Chu, R. J. Gillies and M. J. C. Hendrix, Acidic pH enhances the invasive behavior of human melanoma cells, Clin. Exp. Metastasis, 14 (1996), 176-186. |
[19] |
G. Meral, C. Stinner and C. Surulescu, A multiscale model for acid-mediated tumor invasion: Therapy approaches, J Coupled Syst Multiscale Dyn, 3 (2015), 135-142.
doi: 10.1166/jcsmd.2015.1071. |
[20] |
A. Milian, Stochastic viability and comparison theorem, Colloq. Math., 68 (1995), 297-316. |
[21] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[22] |
K. Smallbone, D. J. Gavaghan, R. A. Gatenby and P. K. Maini, The role of acidity in solid tumor growth and invasion, J. Theor. Biol., 235 (2005), 476-484.
doi: 10.1016/j.jtbi.2005.02.001. |
[23] |
C. W. Song, R. Griffin and H. J. Park, Influence of Tumor pH on Therapeutic Response, in Cancer Drug Resistance, Edited B. Teicher, Humana Press Inc., Totowa, NJ, (2006), 21-42.
doi: 10.1007/978-1-59745-035-5_2. |
[24] |
C. Stinner, C. Surulescu and G. Meral, A multiscale model for pH-tactic invasion with time-varying carrying capacities, IMA J. Appl. Math., 80 (2015), 1300-1321.
doi: 10.1093/imamat/hxu055. |
[25] |
C. Stock and A. Schwab, Protons make tumor cells move like clockwork, Pflugers Arch. - European J. Physiology, 458 (2009), 981-992.
doi: 10.1007/s00424-009-0677-8. |
[26] |
C. L. Stokes, D. A. Lauffenburger and S. K. Williams, Migration of individual microvessel endothelial cells: Stochastic model and parameter measurement, J. Cell Science, 99 (1991), 419-430. |
[27] |
J. Touboul, G. Herrmann and O. Faugeras, Noise-induced behaviors in neural mean field dynamics, SIAM J. Appl. Dyn. Syst., 11 (2012), 49-81.
doi: 10.1137/110832392. |
[28] |
S. D. Webb, J. A. Sherratt and R. G. Fish, Mathematical modelling of tumour acidity: Regulation of intracellular pH, J. Theor. Biol., 196 (1999), 237-250. |
show all references
References:
[1] |
M. J. Boyer and I. F. Tannock, Regulation of intracellular pH in tumor cell lines: Influence of microenvironmental conditions, Cancer Res., 52 (1992), 4441-4447. |
[2] |
P. L. Chow, Stochastic Partial Differential Equations, Chapman & Hall /CRC, Boca Raton, 2015. |
[3] |
J. Cresson, B. Puig and S. Sonner, Stochastic models in biology and the invariance problem, Discrete Contin. Dyn. Syst. Ser. B, (2016), to appear. |
[4] |
A. De Milito and S. Fais, Tumor acidity, chemoresistance and proton pump inhibitors, Future Oncol., 1 (2005), 779-786.
doi: 10.2217/14796694.1.6.779. |
[5] |
M. L. Freeman and E. Sierra, An acidic extracellular environment reduces the fixation of radiation damage, Radiat. Resist., 97 (1984), 154-161.
doi: 10.2307/3576196. |
[6] |
R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Res., 56 (1996), 5745-5753. |
[7] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, New York, 1981. |
[8] |
S. Hiremath and C. Surulescu, A stochastic multiscale model for acid mediated cancer invasion, Nonlinear Anal. Real World Appl., 22 (2015), 176-205.
doi: 10.1016/j.nonrwa.2014.08.008. |
[9] |
S. Hiremath and C. Surulescu, A stochastic model featuring acid induced gaps during tumor progression, Nonlinearity, 29 (2016), 851-914.
doi: 10.1088/0951-7715/29/3/851. |
[10] |
E. Jakobsson and S. W. Chiu, Stochastic theory of ion movement in channels with single-ion occupancy. Application to sodium permeation of gramicidin channel, Biophys. J., 52 (1987), 33-45.
doi: 10.1016/S0006-3495(87)83186-7. |
[11] |
A. Jentzen and P. E. Kloeden, Taylor Approximations for Stochastic Partial Differential Equations, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 2011.
doi: 10.1137/1.9781611972016. |
[12] |
Y. Jiongmin, Linear-quadratic optimal control problems for mean-field stochastic differential equations, SIAM J. Control Optim., 51 (2013), 2809-2838.
doi: 10.1137/120892477. |
[13] |
P. E. Kloeden and T. Lorenz, Stochastic differential equations with nonlocal sample dependence, Stoch. Anal. Appl., 28 (2010), 937-945.
doi: 10.1080/07362994.2010.515194. |
[14] |
P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin-Heidelberg, 1992.
doi: 10.1007/978-3-662-12616-5. |
[15] |
A. H. Lee and I. F. Tannock, Heterogeneity of intracellular pH and of mechanisms that regulate intracellular pH in populations of cultured cells, Cancer Res., 58 (1998), 1901-1908. |
[16] |
X. Mao, Stochastic Differential Equations and Applications, $2^{nd}$ edition, Woodhead Publishing, Cambridge, 2008.
doi: 10.1533/9780857099402. |
[17] |
N. K. Martin, R. A. Gatenby, E. T. Gawlinski and P. K. Maini, Tumor-stromal interactions in acid-mediated invasion: A mathematical model, J. Theor. Biol., 267 (2010), 461-470.
doi: 10.1016/j.jtbi.2010.08.028. |
[18] |
R. Martínez-Zaguilán, E. A. Seftor, R. E. B. Seftor, Y. W. Chu, R. J. Gillies and M. J. C. Hendrix, Acidic pH enhances the invasive behavior of human melanoma cells, Clin. Exp. Metastasis, 14 (1996), 176-186. |
[19] |
G. Meral, C. Stinner and C. Surulescu, A multiscale model for acid-mediated tumor invasion: Therapy approaches, J Coupled Syst Multiscale Dyn, 3 (2015), 135-142.
doi: 10.1166/jcsmd.2015.1071. |
[20] |
A. Milian, Stochastic viability and comparison theorem, Colloq. Math., 68 (1995), 297-316. |
[21] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[22] |
K. Smallbone, D. J. Gavaghan, R. A. Gatenby and P. K. Maini, The role of acidity in solid tumor growth and invasion, J. Theor. Biol., 235 (2005), 476-484.
doi: 10.1016/j.jtbi.2005.02.001. |
[23] |
C. W. Song, R. Griffin and H. J. Park, Influence of Tumor pH on Therapeutic Response, in Cancer Drug Resistance, Edited B. Teicher, Humana Press Inc., Totowa, NJ, (2006), 21-42.
doi: 10.1007/978-1-59745-035-5_2. |
[24] |
C. Stinner, C. Surulescu and G. Meral, A multiscale model for pH-tactic invasion with time-varying carrying capacities, IMA J. Appl. Math., 80 (2015), 1300-1321.
doi: 10.1093/imamat/hxu055. |
[25] |
C. Stock and A. Schwab, Protons make tumor cells move like clockwork, Pflugers Arch. - European J. Physiology, 458 (2009), 981-992.
doi: 10.1007/s00424-009-0677-8. |
[26] |
C. L. Stokes, D. A. Lauffenburger and S. K. Williams, Migration of individual microvessel endothelial cells: Stochastic model and parameter measurement, J. Cell Science, 99 (1991), 419-430. |
[27] |
J. Touboul, G. Herrmann and O. Faugeras, Noise-induced behaviors in neural mean field dynamics, SIAM J. Appl. Dyn. Syst., 11 (2012), 49-81.
doi: 10.1137/110832392. |
[28] |
S. D. Webb, J. A. Sherratt and R. G. Fish, Mathematical modelling of tumour acidity: Regulation of intracellular pH, J. Theor. Biol., 196 (1999), 237-250. |
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