# American Institute of Mathematical Sciences

September  2016, 21(7): 2233-2254. doi: 10.3934/dcdsb.2016045

## 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

Received  October 2015 Revised  December 2015 Published  August 2016

A nonlocal stochastic model for intra- and extracellular proton dynamics in a tumor is proposed. The intracellular dynamics is governed by an SDE coupled to a reaction-diffusion equation for the extracellular proton concentration on the macroscale. In a more general context the existence and uniqueness of solutions for local and nonlocal SDE-PDE systems are established allowing, in particular, to analyze the proton dynamics model, both in its local version and in the case with nonlocal path dependence. Numerical simulations are performed to illustrate the behavior of solutions, providing some insights into the effects of randomness on tumor acidity.
Citation: Peter E. Kloeden, Stefanie Sonner, Christina Surulescu. A nonlocal sample dependence SDE-PDE system modeling proton dynamics in a tumor. Discrete and Continuous Dynamical Systems - B, 2016, 21 (7) : 2233-2254. doi: 10.3934/dcdsb.2016045
##### 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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##### 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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