Advertisement

Parameter Identification via Optimal Control for a Cahn–Hilliard-Chemotaxis System with a Variable Mobility

  • Christian Kahle
  • Kei Fong Lam
Article

Abstract

We consider the inverse problem of identifying parameters in a variant of the diffuse interface model for tumour growth proposed by Garcke et al. (Math Models Methods Appl Sci 26(6):1095–1148, 2016). The model contains three constant parameters; namely the tumour growth rate, the chemotaxis parameter and the nutrient consumption rate. We study the inverse problem from the viewpoint of PDE-constrained optimal control theory and establish first order optimality conditions. A chief difficulty in the theoretical analysis lies in proving high order continuous dependence of the strong solutions on the parameters, in order to show the solution map is continuously Fréchet differentiable when the model has a variable mobility. Due to technical restrictions, our results hold only in two dimensions for sufficiently smooth domains. Analogous results for polygonal domains are also shown for the case of constant mobilities. Finally, we propose a discrete scheme for the numerical simulation of the tumour model and solve the inverse problem using a trust-region Gauss–Newton approach.

Keywords

Cahn–Hilliard equation Chemotaxis Parameter identification Optimal control Variable mobility 

Mathematics Subject Classification

35Q92 35R30 49J20 49J50 65M32 92B05 92C17 

References

  1. 1.
    Allmaras, M., Bangerth, W., Linhart, J., Polanco, J., Wang, F., Wang, K., Webster, J., Zedler, S.: Estimating parameters in physical models through Bayesian inversion: a complete example. SIAM Rev. 55(1), 149–167 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Becker, R., Braack, M., Vexler, B.: Numerical parameter estimation for chemical models in multidimensional reactive flows. Combust. Theory Model. 8(4), 661–682 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Blank, L., Butz, M., Garcke, H.: Solving the Cahn–Hilliard variational inequality with a semi-smooth Newton method. ESAIM: Control Optim. Calc. Var. 17(4), 931–954 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Blowey, J., Elliott, C.: The Cahn–Hilliard gradient theory for phase separation with non-smooth free energy. Part I: mathematical analysis. Eur. J. Appl. Math. 2(3), 233–280 (1991)CrossRefzbMATHGoogle Scholar
  5. 5.
    Brézis, H., Gallouet, T.: Nonlinear Schrödinger evolution equations. Nonlinear Anal. 4(4), 677–681 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Campillo-Funollet, E., Venkataraman, C., Madzvamuse, A.: A Bayesian approach to parameter identification with an application to Turing systems (2016). Preprint. arXiv:1605.04718
  7. 7.
    Colli, P., Gilardi, G., Rocca, E., Sprekels, J.: Optimal distributed control of a diffuse interface model of tumor growth. Nonlinearity 30, 2518–2546 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Collis, J., Connor, A., Paczkowski, M., Kannan, P., Pitt-Francis, J., Byrne, H., Hubbard, M.: Bayesian calibration, validation and uncertainty quantification for predictive modelling of tumour growth: a tutorial. Bull. Math. Biol. 79(4), 939–974 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cristini, V., Lowengrub, J., Nie, Q.: Nonlinear simulations of tumor growth. J. Math. Biol. 46(3), 191–224 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cristini, V., Li, X., Lowengrub, J.S., Wise, S.M.: Nonlinear simulations of solid tumor growth using a mixture model: invasion and branching. J. Math. Biol. 58(4–5), 723–763 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33(3), 1106–1124 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Engler, H.: An alternative proof of the Brezis–Wainger inequality. Commun. Partial Differ. Equ. 14(4), 541–544 (1989)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Garcke, H., Lam, K.: Analysis of a Cahn–Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis. Discret. Contin. Dyn. Syst. 37(8), 4277–4308 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Garcke, H., Lam, K.: Well-posedness of a Cahn–Hilliard system modelling tumour growth with chemotaxis and active transport. Eur. J. Appl. Math. 28(2), 284–316 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Garcke, H., Hinze, M., Kahle, C.: A stable and linear time discretization for a thermodynamically consistent model for two-phase incompressible flow. Appl. Numer. Math. 99, 151–171 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Garcke, H., Lam, K., Sitka, E., Styles, V.: A Cahn–Hilliard–Darcy model for tumour growth with chemotaxis and active transport. Math. Models Methods Appl. Sci. 26(6), 1095–1148 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Garcke, H., Lam, K., Nürnberg, R., Sitka, E.: A multiphase Cahn-Hilliard-Darcy model for tumour growth with necrosis (2018). Math. Models Methods Appl. Sci. (To appear).  https://doi.org/10.1142/S0218202518500148
  18. 18.
    Garcke, H., Lam, K., Rocca, E.: Optimal control of treatment time in a diffuse interface model for tumour growth (2018). Appl. Math. Optim. (To appear).  https://doi.org/10.1007/s00245-017-9414-4
  19. 19.
    Grisvard, P.: Elliptic Problems in Nonsmooth Domains, vol. 69. Classics in Applied Mathematics. SIAM, Philadelphia (2011)Google Scholar
  20. 20.
    Hawkins-Daarud, A., Prudhomme, S., van der Zee, K., Oden, J.: Bayesian calibration, validation, and uncertainty quantification of diffuse interface models of tumour growth. J. Math. Biol. 67(6–7), 1457–1485 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hintermüller, M., Hinze, M., Tber, M.: An adaptive finite element Moreau–Yosida-based solver for a non-smooth Cahn–Hilliard problem. Optim. Methods Softw. 25(4–5), 777–811 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hintermüller, M., Kopacka, I.: A smooth penalty approach and a nonlinear multigrid algorithm for elliptic MPECs. Comput. Optim. Appl. 50(1), 111–145 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hintermüller, M., Hinze, M., Kahle, C.: An adaptive finite element Moreau–Yosida-based solver for a coupled Cahn–Hilliard/Navier–Stokes system. J. Comput. Phys. 235, 810–827 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Mathematical Modelling: Theory and Applications. Springer, Dordrecht (2009)zbMATHGoogle Scholar
  25. 25.
    Lam, K., Wu, H.: Thermodynamically consistent Navier–Stokes–Cahn–Hilliard models with mass transfer and chemotaxis (2018). Eur. J. Appl. Math. (to appear).  https://doi.org/10.1017/S0956792517000298
  26. 26.
    Lima, E., Oden, J., Hormuth II, D., Yankeelov, T., Almeida, R.: Selection, calibration, and validation of models of tumor growth. Math. Models Methods Appl. Sci. 26, 2341–2368 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Lima, E., Oden, J., Wohlmuth, B., Shamoradi, A., Hormuth II, D., Yankeelov, T., Scarabosio, L., Horger, T.: Selection and validation of predictive models of radiation effects on tumor growth based on noninvasive imaging data. Comput. Method Appl. Mech. Eng. 327, 277–305 (2017)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Logg, A., Mardal, K.A., Wells, G.: Automated Solution of Differential Equations by the Finite Element Method—The FEniCS Book, vol. 84. Lecture Notes in Computational Science and Engineering. Springer, Berlin (2012)Google Scholar
  29. 29.
    Nocedal, J., Wright, S.: Numerical Optimization. Springer Series in Operation Research and Financial Engineering. Springer, New York (2006)zbMATHGoogle Scholar
  30. 30.
    Pennacchietti, S., Michieli, P., Galluzzo, M., Mazzone, M., Giordano, S., Comoglio, P.M.: Hypoxia promotes invasive growth by transcriptional activation of the met protooncogene. Cancer Cell 3(4), 347–361 (2003)CrossRefGoogle Scholar
  31. 31.
    Schmidt, A., Siebert, K.: Design of Adaptive Finite Element Software: The Finite Element Toolbox ALBERTA, vol. 42. Lecture Notes in Computational Science and Engineering. Springer, Berlin (2005)Google Scholar
  32. 32.
    Stuart, A.: Inverse problems: a Bayesian perspective. Acta Numer. 19, 451–559 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods, and Applications. Graduate Studies in Mathematics. AMS, Providence (2010)Google Scholar
  34. 34.
    Verfürth, R.: A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner Series: Advances in Numerical Mathematics. Wiley-Teubner, New York (1996)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Zentrum MathematikTechnische Universität MünchenGarching bei MünchenGermany
  2. 2.Department of MathematicsThe Chinese University of Hong KongShatinHong Kong

Personalised recommendations