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

  • * Corresponding author: Sandesh Athni Hiremath

    * Corresponding author: Sandesh Athni Hiremath 
This research was supported by the DFG, grant SU807/1-1.
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  • 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.

    Mathematics Subject Classification: Primary: 34F05, 35R60, 92C17; Secondary: 35Q92, 60H10, 92C50.


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  • 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 -
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    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
     | Show Table
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  • [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. CressonB. Puig and 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 and 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. FasanoM.A. Herrero and 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 and 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 and E.T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Research, 56 (1996), 5745-5753. 
    [10] R.A. Gatenby and 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. GieseL. KluweH. MeissnerE. Michael and M. Westphal, Migration of human glioma cells on myelin., Neurosurgery, 38 (1996), 755-764. 
    [13] D. Hanahan and 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 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.
    [15] S.A. 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.
    [16] L. JerbyL. WolfC. DenkertG. SteinM. HilvoM. OresicT. Geiger and 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 and 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. KloedenS. Sonner and 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 and 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 and 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. MartinE.A. GaffneyR.A. Gatenby and 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. MeralC. Stinner and 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. MeralC. Stinner and 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 and 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. ParadiseM.J. WhitfieldD.A. Lauffenburger and 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 and 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. ReshkinM. R. Greco and 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. SmallboneD.J. GavaghanR.A. Gatenby and 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. StinnerC. 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.
    [37] C. StinnerC. Surulescu and 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 and 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. StubbsP.M. McSheehyJ.R. Griffiths and 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. WebbM. ChimentiM.P. Jacobson and 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, et al., 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.
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