doi: 10.3934/dcdsb.2018071

On a coupled SDE-PDE system modeling acid-mediated tumor invasion

1. 

Technische Universität Kaiserslautern, Felix-Klein-Zentrum für Mathematik, Paul-Ehrlich-Str. 31, 67663 Kaiserslautern, Germany

2. 

Karl-Franzens-Universität Graz, Institut für Mathematik und Wissenschaftliches Rechnen, Heinrichstr. 36, 8010 Graz, Austria

* Corresponding author: Sandesh Athni Hiremath

Received  July 2017 Revised  September 2017 Published  January 2018

Fund Project: This research was supported by the DFG, grant SU807/1-1

We propose and analyze a multiscale model for acid-mediated tumor invasion accounting for stochastic effects on the subcellular level. The setting involves a PDE of reaction-diffusion-taxis type describing the evolution of the tumor cell density, the movement being directed towards pH gradients in the local microenvironment, which is coupled to a PDE-SDE system characterizing the dynamics of extracellular and intracellular proton concentrations, respectively. The global well-posedness of the model is shown and numerical simulations are performed in order to illustrate the solution behavior.

Citation: Sandesh Athni Hiremath, Christina Surulescu, Anna Zhigun, Stefanie Sonner. On a coupled SDE-PDE system modeling acid-mediated tumor invasion. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2018071
References:
[1]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in Function spaces, differential operators and nonlinear analysis (Friedrichroda, 1992), vol. 133 of Teubner-Texte Math., Teubner, Stuttgart, 1993, 9-126. doi: 10.1007/978-3-663-11336-2_1.

[2]

P. Bartel, F. Ludwig, A. Schwab and C. Stock, ph-taxis: directional tumor cell migration along ph-gradients, Acta Physiol. , 204 (2012), p113.

[3]

P. -L. Chow, Stochastic Partial Differential Equations, 2nd edition, Advances in Applied Mathematics, CRC Press, Boca Raton, FL, 2015.

[4]

J. Cresson, B. Puig, S. Sonner, Stochastic models in biology and the invariance problem, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2145-2168. doi: 10.3934/dcdsb.2016041.

[5]

M. Damaghi, J. W. Wojtkowiak and R. J. Gillies, ph sensing and regulation in cancer, Frontiers in Physiology 4 (2013). doi: 10.3389/fphys.2013.00370.

[6]

F. Delarue, G. Guatteri, Weak existence and uniqueness for forward-backward SDEs, Stochastic Process. Appl., 116 (2006), 1712-1742. doi: 10.1016/j.spa.2006.05.002.

[7]

A. Fasano, M.A. Herrero, M.R. Rodrigo, Slow and fast invasion waves in a model of acid-mediated tumour growth, Math. Biosci., 220 (2009), 45-56. doi: 10.1016/j.mbs.2009.04.001.

[8]

R.F. Fox, Y.-n. Lu, Emergent collective behavior in large numbers of globally coupled independently stochastic ion channels, Physical Review E, 49 (1994), 3421-3431. doi: 10.1103/PhysRevE.49.3421.

[9]

R.A. Gatenby, E.T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Research, 56 (1996), 5745-5753.

[10]

R.A. Gatenby, E.T. Gawlinski, The glycolytic phenotype in carcinogenesis and tumor invasion insights through mathematical models, Cancer Research, 63 (2003), 3847-3854.

[11]

I. I. Gikhman and A. V. Skorohod, Stochastic Differential Equations, Springer-Verlag, New York-Heidelberg, 1972, Translated from the Russian by Kenneth Wickwire, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 72.

[12]

A. Giese, L. Kluwe, H. Meissner, E. Michael, M. Westphal, Migration of human glioma cells on myelin., Neurosurgery, 38 (1996), 755-764.

[13]

D. Hanahan, R.A. Weinberg, Hallmarks of cancer: The next generation, Cell, 144 (2011), 646-674. doi: 10.1016/j.cell.2011.02.013.

[14]

S.A. Hiremath, 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.

[15]

S.A. Hiremath, 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.

[16]

L. Jerby, L. Wolf, C. Denkert, G. Stein, M. Hilvo, M. Oresic, T. Geiger, E. Ruppin, Metabolic associations of reduced proliferation and oxidative stress in advanced breast cancer, Cancer Res., 72 (2012), 5712-5720. doi: 10.1158/0008-5472.CAN-12-2215.

[17]

B. Jourdain, C. Le Bris and T. Lelièvre, Coupling PDEs and SDEs: the illustrative example of the multiscale simulation of viscoelastic flows, in Multiscale methods in science and engineering, vol. 44 of Lect. Notes Comput. Sci. Eng., Springer, Berlin, 2005,149-168. doi: 10.1007/3-540-26444-2_7.

[18]

P.E. Kloeden, T. Lorenz, Stochastic differential equations with nonlocal sample dependence, Stoch. Anal. Appl., 28 (2010), 937-945. doi: 10.1080/07362994.2010.515194.

[19]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, vol. 23 of Applications of Mathematics (New York), Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.

[20]

P.E. Kloeden, S. Sonner, C. Surulescu, A nonlocal sample dependence SDE-PDE system modeling proton dynamics in a tumor, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2233-2254. doi: 10.3934/dcdsb.2016045.

[21]

O. Ladyzhenskaya, V. Solonnikov and N. Ural'tseva, Linear and Quasi-Linear Equations of Parabolic Type. Translated from the Russian by S. Smith. , Translations of Mathematical Monographs. 23. Providence, RI: American Mathematical Society (AMS). XI, 648 p. (1968)., 1968.

[22]

A.H. Lee, I.F. Tannock, Heterogeneity of intracellular ph and of mechanisms that regulate intracellular ph in populations of cultured cells, Cancer Research, 58 (1998), 1901-1908.

[23]

G.M. Lieberman, Hölder continuity of the gradient of solutions of uniformly parabolic equations with conormal boundary conditions, Ann. Mat. Pura Appl. (4), 148 (1987), 77-99. doi: 10.1007/BF01774284.

[24]

W. Liu, M. Röckner, SPDE in Hilbert space with locally monotone coefficients, J. Funct. Anal., 259 (2010), 2902-2922. doi: 10.1016/j.jfa.2010.05.012.

[25]

J. Ma and J. Yong, Forward-backward Stochastic Differential Equations and Their Applications, vol. 1702 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-540-48831-6.

[26]

X. Mao, Stochastic Differential Equations and Applications, 2nd edition, Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402.

[27]

N.K. Martin, E.A. Gaffney, R.A. Gatenby, P.K. Maini, Tumour-stromal interactions in acid-mediated invasion: A mathematical model, J. Theoret. Biol., 267 (2010), 461-470. doi: 10.1016/j.jtbi.2010.08.028.

[28]

G. Meral, C. Stinner, C. Surulescu, A multiscale model for acid-mediated tumor invasion: Therapy approaches, Journal of Coupled Systems and Multiscale Dynamics, 3 (2015), 135-142. doi: 10.1166/jcsmd.2015.1071.

[29]

G. Meral, C. Stinner, C. Surulescu, On a multiscale model involving cell contractivity and its effects on tumor invasion, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 189-213. doi: 10.3934/dcdsb.2015.20.189.

[30]

G. Meral, C. Surulescu, Mathematical modelling, analysis and numerical simulations for the influence of heat shock proteins on tumour invasion, J. Math. Anal. Appl., 408 (2013), 597-614. doi: 10.1016/j.jmaa.2013.06.017.

[31]

A. Milian, Stochastic viability and a comparison theorem, Colloq. Math., 68 (1995), 297-316. doi: 10.4064/cm-68-2-297-316.

[32]

R.K. Paradise, M.J. Whitfield, D.A. Lauffenburger, K.J. VanVliet, Directional cell migration in an extracellular ph gradient: a model study with an engineered cell line and primary microvascular endothelial cells, Experimental Cell Research, 319 (2013), 487-497. doi: 10.1016/j.yexcr.2012.11.006.

[33]

E. Pardoux, S. Tang, Forward-backward stochastic differential equations and quasilinear parabolic PDEs, Probab. Theory Related Fields, 114 (1999), 123-150. doi: 10.1007/s004409970001.

[34]

S. J. Reshkin, M. R. Greco, R. A. Cardone, Role of pHi, and proton transporters in oncogene-driven neoplastic transformation, Phil. Trans. R. Soc. B, 369 (2014), 20130100. doi: 10.1098/rstb.2013.0100.

[35]

K. Smallbone, D.J. Gavaghan, R.A. Gatenby, P.K. Maini, The role of acidity in solid tumour growth and invasion, J. Theoret. Biol., 235 (2005), 476-484. doi: 10.1016/j.jtbi.2005.02.001.

[36]

C. Stinner, C. Surulescu, 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.

[37]

C. Stinner, C. Surulescu, M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007. doi: 10.1137/13094058X.

[38]

C. Stock, A. Schwab, Protons make tumor cells move like clockwork, Pflügers Archiv-European Journal of Physiology, 458 (2009), 981-992. doi: 10.1007/s00424-009-0677-8.

[39]

M. Stubbs, P.M. McSheehy, J.R. Griffiths, C.L. Bashford, Causes and consequences of tumour acidity and implications for treatment, Molecular Medicine Today, 6 (2000), 15-19. doi: 10.1016/S1357-4310(99)01615-9.

[40]

B.A. Webb, M. Chimenti, M.P. Jacobson, D.L. Barber, Dysregulated ph: A perfect storm for cancer progression, Nature Reviews Cancer, 11 (2011), 671-677. doi: 10.1038/nrc3110.

[41]

D. Widmer, Hypoxia contributes to melanoma heterogeneity by triggering hif1α-dependent phenotype switching., J. Invest. Dermat., 133 (2013), 2436-2443. doi: 10.1038/jid.2013.115.

[42]

L. Zhang, K. Radtke, L. Zheng, A. Q. Cai, T. F. Schilling and Q. Nie, Noise drives sharpening of gene expression boundaries in the zebrafish hindbrain, Molecular Systems Biology, 8 (2012), p613. doi: 10.1038/msb.2012.45.

[43]

A. Zhigun, The Malliavin derivative and compactness: application to a degenerate PDE-SDE coupling, Preprint, arXiv: 1609.01495, submitted, 2016.

[44]

A. Zhigun, C. Surulescu and A. Hunt, Global existence for a degenerate haptotaxis model of tumor invasion under the go-or-grow dichotomy hypothesis, Preprint, arXiv: 1605.09226, submitted, 2016.

[45]

A. Zhigun, C. Surulescu and A. Uatay, Global existence for a degenerate haptotaxis model of cancer invasion, Z. Angew. Math. Phys. , 67 (2016), Art. 146, 29pp. doi: 10.1007/s00033-016-0741-0.

show all references

References:
[1]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in Function spaces, differential operators and nonlinear analysis (Friedrichroda, 1992), vol. 133 of Teubner-Texte Math., Teubner, Stuttgart, 1993, 9-126. doi: 10.1007/978-3-663-11336-2_1.

[2]

P. Bartel, F. Ludwig, A. Schwab and C. Stock, ph-taxis: directional tumor cell migration along ph-gradients, Acta Physiol. , 204 (2012), p113.

[3]

P. -L. Chow, Stochastic Partial Differential Equations, 2nd edition, Advances in Applied Mathematics, CRC Press, Boca Raton, FL, 2015.

[4]

J. Cresson, B. Puig, S. Sonner, Stochastic models in biology and the invariance problem, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2145-2168. doi: 10.3934/dcdsb.2016041.

[5]

M. Damaghi, J. W. Wojtkowiak and R. J. Gillies, ph sensing and regulation in cancer, Frontiers in Physiology 4 (2013). doi: 10.3389/fphys.2013.00370.

[6]

F. Delarue, G. Guatteri, Weak existence and uniqueness for forward-backward SDEs, Stochastic Process. Appl., 116 (2006), 1712-1742. doi: 10.1016/j.spa.2006.05.002.

[7]

A. Fasano, M.A. Herrero, M.R. Rodrigo, Slow and fast invasion waves in a model of acid-mediated tumour growth, Math. Biosci., 220 (2009), 45-56. doi: 10.1016/j.mbs.2009.04.001.

[8]

R.F. Fox, Y.-n. Lu, Emergent collective behavior in large numbers of globally coupled independently stochastic ion channels, Physical Review E, 49 (1994), 3421-3431. doi: 10.1103/PhysRevE.49.3421.

[9]

R.A. Gatenby, E.T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Research, 56 (1996), 5745-5753.

[10]

R.A. Gatenby, E.T. Gawlinski, The glycolytic phenotype in carcinogenesis and tumor invasion insights through mathematical models, Cancer Research, 63 (2003), 3847-3854.

[11]

I. I. Gikhman and A. V. Skorohod, Stochastic Differential Equations, Springer-Verlag, New York-Heidelberg, 1972, Translated from the Russian by Kenneth Wickwire, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 72.

[12]

A. Giese, L. Kluwe, H. Meissner, E. Michael, M. Westphal, Migration of human glioma cells on myelin., Neurosurgery, 38 (1996), 755-764.

[13]

D. Hanahan, R.A. Weinberg, Hallmarks of cancer: The next generation, Cell, 144 (2011), 646-674. doi: 10.1016/j.cell.2011.02.013.

[14]

S.A. Hiremath, 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.

[15]

S.A. Hiremath, 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.

[16]

L. Jerby, L. Wolf, C. Denkert, G. Stein, M. Hilvo, M. Oresic, T. Geiger, E. Ruppin, Metabolic associations of reduced proliferation and oxidative stress in advanced breast cancer, Cancer Res., 72 (2012), 5712-5720. doi: 10.1158/0008-5472.CAN-12-2215.

[17]

B. Jourdain, C. Le Bris and T. Lelièvre, Coupling PDEs and SDEs: the illustrative example of the multiscale simulation of viscoelastic flows, in Multiscale methods in science and engineering, vol. 44 of Lect. Notes Comput. Sci. Eng., Springer, Berlin, 2005,149-168. doi: 10.1007/3-540-26444-2_7.

[18]

P.E. Kloeden, T. Lorenz, Stochastic differential equations with nonlocal sample dependence, Stoch. Anal. Appl., 28 (2010), 937-945. doi: 10.1080/07362994.2010.515194.

[19]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, vol. 23 of Applications of Mathematics (New York), Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.

[20]

P.E. Kloeden, S. Sonner, C. Surulescu, A nonlocal sample dependence SDE-PDE system modeling proton dynamics in a tumor, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2233-2254. doi: 10.3934/dcdsb.2016045.

[21]

O. Ladyzhenskaya, V. Solonnikov and N. Ural'tseva, Linear and Quasi-Linear Equations of Parabolic Type. Translated from the Russian by S. Smith. , Translations of Mathematical Monographs. 23. Providence, RI: American Mathematical Society (AMS). XI, 648 p. (1968)., 1968.

[22]

A.H. Lee, I.F. Tannock, Heterogeneity of intracellular ph and of mechanisms that regulate intracellular ph in populations of cultured cells, Cancer Research, 58 (1998), 1901-1908.

[23]

G.M. Lieberman, Hölder continuity of the gradient of solutions of uniformly parabolic equations with conormal boundary conditions, Ann. Mat. Pura Appl. (4), 148 (1987), 77-99. doi: 10.1007/BF01774284.

[24]

W. Liu, M. Röckner, SPDE in Hilbert space with locally monotone coefficients, J. Funct. Anal., 259 (2010), 2902-2922. doi: 10.1016/j.jfa.2010.05.012.

[25]

J. Ma and J. Yong, Forward-backward Stochastic Differential Equations and Their Applications, vol. 1702 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-540-48831-6.

[26]

X. Mao, Stochastic Differential Equations and Applications, 2nd edition, Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402.

[27]

N.K. Martin, E.A. Gaffney, R.A. Gatenby, P.K. Maini, Tumour-stromal interactions in acid-mediated invasion: A mathematical model, J. Theoret. Biol., 267 (2010), 461-470. doi: 10.1016/j.jtbi.2010.08.028.

[28]

G. Meral, C. Stinner, C. Surulescu, A multiscale model for acid-mediated tumor invasion: Therapy approaches, Journal of Coupled Systems and Multiscale Dynamics, 3 (2015), 135-142. doi: 10.1166/jcsmd.2015.1071.

[29]

G. Meral, C. Stinner, C. Surulescu, On a multiscale model involving cell contractivity and its effects on tumor invasion, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 189-213. doi: 10.3934/dcdsb.2015.20.189.

[30]

G. Meral, C. Surulescu, Mathematical modelling, analysis and numerical simulations for the influence of heat shock proteins on tumour invasion, J. Math. Anal. Appl., 408 (2013), 597-614. doi: 10.1016/j.jmaa.2013.06.017.

[31]

A. Milian, Stochastic viability and a comparison theorem, Colloq. Math., 68 (1995), 297-316. doi: 10.4064/cm-68-2-297-316.

[32]

R.K. Paradise, M.J. Whitfield, D.A. Lauffenburger, K.J. VanVliet, Directional cell migration in an extracellular ph gradient: a model study with an engineered cell line and primary microvascular endothelial cells, Experimental Cell Research, 319 (2013), 487-497. doi: 10.1016/j.yexcr.2012.11.006.

[33]

E. Pardoux, S. Tang, Forward-backward stochastic differential equations and quasilinear parabolic PDEs, Probab. Theory Related Fields, 114 (1999), 123-150. doi: 10.1007/s004409970001.

[34]

S. J. Reshkin, M. R. Greco, R. A. Cardone, Role of pHi, and proton transporters in oncogene-driven neoplastic transformation, Phil. Trans. R. Soc. B, 369 (2014), 20130100. doi: 10.1098/rstb.2013.0100.

[35]

K. Smallbone, D.J. Gavaghan, R.A. Gatenby, P.K. Maini, The role of acidity in solid tumour growth and invasion, J. Theoret. Biol., 235 (2005), 476-484. doi: 10.1016/j.jtbi.2005.02.001.

[36]

C. Stinner, C. Surulescu, 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.

[37]

C. Stinner, C. Surulescu, M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007. doi: 10.1137/13094058X.

[38]

C. Stock, A. Schwab, Protons make tumor cells move like clockwork, Pflügers Archiv-European Journal of Physiology, 458 (2009), 981-992. doi: 10.1007/s00424-009-0677-8.

[39]

M. Stubbs, P.M. McSheehy, J.R. Griffiths, C.L. Bashford, Causes and consequences of tumour acidity and implications for treatment, Molecular Medicine Today, 6 (2000), 15-19. doi: 10.1016/S1357-4310(99)01615-9.

[40]

B.A. Webb, M. Chimenti, M.P. Jacobson, D.L. Barber, Dysregulated ph: A perfect storm for cancer progression, Nature Reviews Cancer, 11 (2011), 671-677. doi: 10.1038/nrc3110.

[41]

D. Widmer, Hypoxia contributes to melanoma heterogeneity by triggering hif1α-dependent phenotype switching., J. Invest. Dermat., 133 (2013), 2436-2443. doi: 10.1038/jid.2013.115.

[42]

L. Zhang, K. Radtke, L. Zheng, A. Q. Cai, T. F. Schilling and Q. Nie, Noise drives sharpening of gene expression boundaries in the zebrafish hindbrain, Molecular Systems Biology, 8 (2012), p613. doi: 10.1038/msb.2012.45.

[43]

A. Zhigun, The Malliavin derivative and compactness: application to a degenerate PDE-SDE coupling, Preprint, arXiv: 1609.01495, submitted, 2016.

[44]

A. Zhigun, C. Surulescu and A. Hunt, Global existence for a degenerate haptotaxis model of tumor invasion under the go-or-grow dichotomy hypothesis, Preprint, arXiv: 1605.09226, submitted, 2016.

[45]

A. Zhigun, C. Surulescu and A. Uatay, Global existence for a degenerate haptotaxis model of cancer invasion, Z. Angew. Math. Phys. , 67 (2016), Art. 146, 29pp. doi: 10.1007/s00033-016-0741-0.

Figure 1.  Initial conditions in 1D and 2D.
Figure 2.  Time snapshots of a sample solution in the case of a 1D domain. Blue line: cancer cell density $c$; green line: extracellular proton concentration $p$, red line: intracellular proton concentration $h$. Choice of functions and coefficients as in (47).
Figure 3.  Time snapshots of the numerical mean in the case of a 1D domain. Blue line: cancer cell density $c$; green line: extracellular proton concentration $p$, red line: intracellular proton concentration $h$. Choice of functions and coefficients as in (47).
Figure 4.  Qualitative behavior of $J$ as a function of $c$.
Figure 5.  Time snapshots of three different sample solutions (out of 1000 simulations) in a 1D domain. Blue: cancer cell density, green: extracellular proton concentration, red: intracellular proton concentration. Choice of functions and coefficients as in (48).
Figure 6.  Time snapshots of the numerical mean in a 1D domain. Choice of functions and coefficients as in (48).
Figure 7.  Time snapshots of three different sample solutions to (3)-(4) in a 2D domain. Functions and coefficients as in (48).
Figure 8.  Time snapshots of the numerical mean in a 2D domain. Functions and coefficients as in (48).
Figure 9.  Time snapshots of the sample solution 335 in the case of nonlocal coupling and for a 1D domain. Functions $f_3$ and $g$ as in (48), $f_1$ and $f_2$ as in (49).
Figure 10.  Time snapshots of the numerical mean in the case of nonlocal coupling and for a 1D domain. Functions $f_3$ and $g$ as in (48), $f_1$ and $f_2$ as in (49).
Figure 11.  Time snapshots of the numerical mean in the case of nonlocal coupling and for a 2D domain. Functions $f_3$ and $g$ as in (48), $f_1$ and $f_2$ as in (49).
Table 1.  Numerical parameters
Numerical parameters (48), (49) (47)
Parameter 1D 2D 1D
N (# time steps) 8000 1500 5000
M (# Monte Carlo simulations) 1000 1000 1000
$\tau$ (temporal step size) 0.1 0.1 0.1
$\delta_x$ (spatial step size along $x$) 0.01 0.01 0.01
$M_x$ (grid resolution along $x$) 301 41 301
$\delta_y$ (spatial step size along $y$) - 0.01 -
$M_y$ (grid resolution along $y$) - 41 -
Numerical parameters (48), (49) (47)
Parameter 1D 2D 1D
N (# time steps) 8000 1500 5000
M (# Monte Carlo simulations) 1000 1000 1000
$\tau$ (temporal step size) 0.1 0.1 0.1
$\delta_x$ (spatial step size along $x$) 0.01 0.01 0.01
$M_x$ (grid resolution along $x$) 301 41 301
$\delta_y$ (spatial step size along $y$) - 0.01 -
$M_y$ (grid resolution along $y$) - 41 -
Table 2.  Simulation parameters (1D and 2D)
Growth and decay parameters
phenomenological relevance value in (48), (49) value in (47)
$\gamma_{_{f_1}}$ rate const. for cancer proliferation 0.009 0.09
$\gamma_{_{f_2}}$ rate const. for extracellular protons 0.4 36.8
$\rho$ const. within the logistic term of $p$ - $\frac{1}{36.8}$
$\gamma_{_{f_3}}$ rate const. for intracell. protons 1 0.08
$ \gamma_{_g}$ noise intensity intracell. proton dyn. 3 0.03
Migration parameters
phenomenological relevance value in (48), (49) value in (47)
$\gamma_{_{_D}}$ diffusion coefficient for protons 0.0001 0.0001
$\gamma_{_{\Phi}}$ diffusion coefficient for cancer cells 0.00005 0.00005
$\gamma_{_{\Psi}}$ pH-taxis coefficient 0.02 0.002
$k_1$ conversion rate from $h$ to $p$ 0.07 0.06
$k_2$ conversion rate from $p$ to $h$ 0.01 0.07
$k_3$ decay rate $h$ due to $c$ - 0.06
$k_4$ decay rate $c$ due to interaction with $p$ - 0.01
$\alpha_1$ const. in diffusion coefficient $\Phi$ (47) 1 1
$\alpha_2$ const. in diffusion coefficient $\Phi$ (47) 4 4
Growth and decay parameters
phenomenological relevance value in (48), (49) value in (47)
$\gamma_{_{f_1}}$ rate const. for cancer proliferation 0.009 0.09
$\gamma_{_{f_2}}$ rate const. for extracellular protons 0.4 36.8
$\rho$ const. within the logistic term of $p$ - $\frac{1}{36.8}$
$\gamma_{_{f_3}}$ rate const. for intracell. protons 1 0.08
$ \gamma_{_g}$ noise intensity intracell. proton dyn. 3 0.03
Migration parameters
phenomenological relevance value in (48), (49) value in (47)
$\gamma_{_{_D}}$ diffusion coefficient for protons 0.0001 0.0001
$\gamma_{_{\Phi}}$ diffusion coefficient for cancer cells 0.00005 0.00005
$\gamma_{_{\Psi}}$ pH-taxis coefficient 0.02 0.002
$k_1$ conversion rate from $h$ to $p$ 0.07 0.06
$k_2$ conversion rate from $p$ to $h$ 0.01 0.07
$k_3$ decay rate $h$ due to $c$ - 0.06
$k_4$ decay rate $c$ due to interaction with $p$ - 0.01
$\alpha_1$ const. in diffusion coefficient $\Phi$ (47) 1 1
$\alpha_2$ const. in diffusion coefficient $\Phi$ (47) 4 4
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