# 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 & 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. Google Scholar [2] P. L. Chow, Stochastic Partial Differential Equations,, Chapman & Hall /CRC, (2015). Google Scholar [3] J. Cresson, B. Puig and S. Sonner, Stochastic models in biology and the invariance problem,, Discrete Contin. Dyn. Syst. Ser. B, (2016). Google Scholar [4] A. De Milito and S. Fais, Tumor acidity, chemoresistance and proton pump inhibitors,, Future Oncol., 1 (2005), 779. doi: 10.2217/14796694.1.6.779. Google Scholar [5] M. L. Freeman and E. Sierra, An acidic extracellular environment reduces the fixation of radiation damage,, Radiat. Resist., 97 (1984), 154. doi: 10.2307/3576196. Google Scholar [6] R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion,, Cancer Res., 56 (1996), 5745. Google Scholar [7] D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981). Google Scholar [8] S. Hiremath and C. Surulescu, A stochastic multiscale model for acid mediated cancer invasion,, Nonlinear Anal. Real World Appl., 22 (2015), 176. doi: 10.1016/j.nonrwa.2014.08.008. Google Scholar [9] S. Hiremath and C. Surulescu, A stochastic model featuring acid induced gaps during tumor progression,, Nonlinearity, 29 (2016), 851. doi: 10.1088/0951-7715/29/3/851. Google Scholar [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. doi: 10.1016/S0006-3495(87)83186-7. Google Scholar [11] A. Jentzen and P. E. Kloeden, Taylor Approximations for Stochastic Partial Differential Equations,, CBMS-NSF Regional Conference Series in Applied Mathematics, (2011). doi: 10.1137/1.9781611972016. Google Scholar [12] Y. Jiongmin, Linear-quadratic optimal control problems for mean-field stochastic differential equations,, SIAM J. Control Optim., 51 (2013), 2809. doi: 10.1137/120892477. Google Scholar [13] P. E. Kloeden and T. Lorenz, Stochastic differential equations with nonlocal sample dependence,, Stoch. Anal. Appl., 28 (2010), 937. doi: 10.1080/07362994.2010.515194. Google Scholar [14] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations,, Springer-Verlag, (1992). doi: 10.1007/978-3-662-12616-5. Google Scholar [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. Google Scholar [16] X. Mao, Stochastic Differential Equations and Applications,, $2^{nd}$ edition, (2008). doi: 10.1533/9780857099402. Google Scholar [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. doi: 10.1016/j.jtbi.2010.08.028. Google Scholar [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. Google Scholar [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. doi: 10.1166/jcsmd.2015.1071. Google Scholar [20] A. Milian, Stochastic viability and comparison theorem,, Colloq. Math., 68 (1995), 297. Google Scholar [21] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar [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. doi: 10.1016/j.jtbi.2005.02.001. Google Scholar [23] C. W. Song, R. Griffin and H. J. Park, Influence of Tumor pH on Therapeutic Response,, in Cancer Drug Resistance, (2006), 21. doi: 10.1007/978-1-59745-035-5_2. Google Scholar [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. doi: 10.1093/imamat/hxu055. Google Scholar [25] C. Stock and A. Schwab, Protons make tumor cells move like clockwork,, Pflugers Arch. - European J. Physiology, 458 (2009), 981. doi: 10.1007/s00424-009-0677-8. Google Scholar [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. Google Scholar [27] J. Touboul, G. Herrmann and O. Faugeras, Noise-induced behaviors in neural mean field dynamics,, SIAM J. Appl. Dyn. Syst., 11 (2012), 49. doi: 10.1137/110832392. Google Scholar [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. Google Scholar

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. Google Scholar [2] P. L. Chow, Stochastic Partial Differential Equations,, Chapman & Hall /CRC, (2015). Google Scholar [3] J. Cresson, B. Puig and S. Sonner, Stochastic models in biology and the invariance problem,, Discrete Contin. Dyn. Syst. Ser. B, (2016). Google Scholar [4] A. De Milito and S. Fais, Tumor acidity, chemoresistance and proton pump inhibitors,, Future Oncol., 1 (2005), 779. doi: 10.2217/14796694.1.6.779. Google Scholar [5] M. L. Freeman and E. Sierra, An acidic extracellular environment reduces the fixation of radiation damage,, Radiat. Resist., 97 (1984), 154. doi: 10.2307/3576196. Google Scholar [6] R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion,, Cancer Res., 56 (1996), 5745. Google Scholar [7] D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981). Google Scholar [8] S. Hiremath and C. Surulescu, A stochastic multiscale model for acid mediated cancer invasion,, Nonlinear Anal. Real World Appl., 22 (2015), 176. doi: 10.1016/j.nonrwa.2014.08.008. Google Scholar [9] S. Hiremath and C. Surulescu, A stochastic model featuring acid induced gaps during tumor progression,, Nonlinearity, 29 (2016), 851. doi: 10.1088/0951-7715/29/3/851. Google Scholar [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. doi: 10.1016/S0006-3495(87)83186-7. Google Scholar [11] A. Jentzen and P. E. Kloeden, Taylor Approximations for Stochastic Partial Differential Equations,, CBMS-NSF Regional Conference Series in Applied Mathematics, (2011). doi: 10.1137/1.9781611972016. Google Scholar [12] Y. Jiongmin, Linear-quadratic optimal control problems for mean-field stochastic differential equations,, SIAM J. Control Optim., 51 (2013), 2809. doi: 10.1137/120892477. Google Scholar [13] P. E. Kloeden and T. Lorenz, Stochastic differential equations with nonlocal sample dependence,, Stoch. Anal. Appl., 28 (2010), 937. doi: 10.1080/07362994.2010.515194. Google Scholar [14] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations,, Springer-Verlag, (1992). doi: 10.1007/978-3-662-12616-5. Google Scholar [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. Google Scholar [16] X. Mao, Stochastic Differential Equations and Applications,, $2^{nd}$ edition, (2008). doi: 10.1533/9780857099402. Google Scholar [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. doi: 10.1016/j.jtbi.2010.08.028. Google Scholar [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. Google Scholar [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. doi: 10.1166/jcsmd.2015.1071. Google Scholar [20] A. Milian, Stochastic viability and comparison theorem,, Colloq. Math., 68 (1995), 297. Google Scholar [21] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar [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. doi: 10.1016/j.jtbi.2005.02.001. Google Scholar [23] C. W. Song, R. Griffin and H. J. Park, Influence of Tumor pH on Therapeutic Response,, in Cancer Drug Resistance, (2006), 21. doi: 10.1007/978-1-59745-035-5_2. Google Scholar [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. doi: 10.1093/imamat/hxu055. Google Scholar [25] C. Stock and A. Schwab, Protons make tumor cells move like clockwork,, Pflugers Arch. - European J. Physiology, 458 (2009), 981. doi: 10.1007/s00424-009-0677-8. Google Scholar [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. Google Scholar [27] J. Touboul, G. Herrmann and O. Faugeras, Noise-induced behaviors in neural mean field dynamics,, SIAM J. Appl. Dyn. Syst., 11 (2012), 49. doi: 10.1137/110832392. Google Scholar [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. Google Scholar
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