A nonlocal sample dependence SDE-PDE system modeling proton dynamics in a tumor

Previous Article
Neurotransmitter concentrations in the presence of neural switching in one dimension
DCDS-B Home
This Issue
Next Article
Ghosts of bump attractors in stochastic neural fields: Bottlenecks and extinction

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

A nonlocal sample dependence SDE-PDE system modeling proton dynamics in a tumor

Peter E. Kloeden ^1, , Stefanie Sonner ^2, and Christina Surulescu ^3,

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

Abstract
Related Papers

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.

Keywords: partial differential equations, nonlocal sample dependence, intra- and extracellular proton dynamics, tumor acidity, stochastic differential equations, mean-field model., Multiscale model.

Mathematics Subject Classification: Primary: 60H10, 34F05, 35R60; Secondary: 92C5.

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.

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.

[1]	Yinggu Chen, Said HamadÈne, Tingshu Mu. Mean-field doubly reflected backward stochastic differential equations. Numerical Algebra, Control and Optimization, 2022 doi: 10.3934/naco.2022012
[2]	Jianhui Huang, Xun Li, Jiongmin Yong. A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Mathematical Control and Related Fields, 2015, 5 (1) : 97-139. doi: 10.3934/mcrf.2015.5.97
[3]	Haiyan Zhang. A necessary condition for mean-field type stochastic differential equations with correlated state and observation noises. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1287-1301. doi: 10.3934/jimo.2016.12.1287
[4]	Juan Li, Wenqiang Li. Controlled reflected mean-field backward stochastic differential equations coupled with value function and related PDEs. Mathematical Control and Related Fields, 2015, 5 (3) : 501-516. doi: 10.3934/mcrf.2015.5.501
[5]	Michael Herty, Torsten Trimborn, Giuseppe Visconti. Mean-field and kinetic descriptions of neural differential equations. Foundations of Data Science, 2022, 4 (2) : 271-298. doi: 10.3934/fods.2022007
[6]	Yves Achdou, Mathieu Laurière. On the system of partial differential equations arising in mean field type control. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 3879-3900. doi: 10.3934/dcds.2015.35.3879
[7]	Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Mean-field backward stochastic Volterra integral equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (7) : 1929-1967. doi: 10.3934/dcdsb.2013.18.1929
[8]	Jun Moon. Linear-quadratic mean-field type stackelberg differential games for stochastic jump-diffusion systems. Mathematical Control and Related Fields, 2022, 12 (2) : 371-404. doi: 10.3934/mcrf.2021026
[9]	Patrick Gerard, Christophe Pallard. A mean-field toy model for resonant transport. Kinetic and Related Models, 2010, 3 (2) : 299-309. doi: 10.3934/krm.2010.3.299
[10]	Seung-Yeal Ha, Jinwook Jung, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. A mean-field limit of the particle swarmalator model. Kinetic and Related Models, 2021, 14 (3) : 429-468. doi: 10.3934/krm.2021011
[11]	Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264
[12]	Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515
[13]	Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[14]	Chang Zhang, Fang Li, Jinqiao Duan. Long-time behavior of a class of nonlocal partial differential equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 749-763. doi: 10.3934/dcdsb.2018041
[15]	Jiaohui Xu, Tomás Caraballo, José Valero. Asymptotic behavior of nonlocal partial differential equations with long time memory. Discrete and Continuous Dynamical Systems - S, 2022, 15 (10) : 3059-3078. doi: 10.3934/dcdss.2021140
[16]	Franco Flandoli, Enrico Priola, Giovanni Zanco. A mean-field model with discontinuous coefficients for neurons with spatial interaction. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3037-3067. doi: 10.3934/dcds.2019126
[17]	Seung-Yeal Ha, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. Uniform stability and mean-field limit for the augmented Kuramoto model. Networks and Heterogeneous Media, 2018, 13 (2) : 297-322. doi: 10.3934/nhm.2018013
[18]	Matthew Rosenzweig. The mean-field limit of the Lieb-Liniger model. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 3005-3037. doi: 10.3934/dcds.2022006
[19]	Min Yang, Guanggan Chen. Finite dimensional reducing and smooth approximating for a class of stochastic partial differential equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1565-1581. doi: 10.3934/dcdsb.2019240
[20]	Qi Lü, Xu Zhang. A concise introduction to control theory for stochastic partial differential equations. Mathematical Control and Related Fields, 2021 doi: 10.3934/mcrf.2021020

2021 Impact Factor: 1.497

Tools

Metrics

Other articles
by authors

on AIMS
Peter E. Kloeden Stefanie Sonner Christina Surulescu
on Google Scholar
Peter E. Kloeden Stefanie Sonner Christina Surulescu

[Back to Top]