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

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.

[1]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[2]  Thierry Gallouët,et al.  Nonlinear Schrödinger evolution equations , 1980 .

[3]  H. Garcke,et al.  Optimal Control of Treatment Time in a Diffuse Interface Model of Tumor Growth , 2016, 1608.00488.

[4]  Kunibert G. Siebert,et al.  Design of Adaptive Finite Element Software - The Finite Element Toolbox ALBERTA , 2005, Lecture Notes in Computational Science and Engineering.

[5]  Joe Pitt-Francis,et al.  Bayesian Calibration, Validation and Uncertainty Quantification for Predictive Modelling of Tumour Growth: A Tutorial , 2017, Bulletin of Mathematical Biology.

[6]  K. Lam,et al.  Thermodynamically consistent Navier–Stokes–Cahn–Hilliard models with mass transfer and chemotaxis , 2017, European Journal of Applied Mathematics.

[7]  Stefan Ulbrich,et al.  Optimization with PDE Constraints , 2008, Mathematical modelling.

[8]  Charles M. Elliott,et al.  The Cahn–Hilliard gradient theory for phase separation with non-smooth free energy Part II: Numerical analysis , 1991, European Journal of Applied Mathematics.

[9]  Harald Garcke,et al.  Solving the Cahn-Hilliard variational inequality with a semi-smooth Newton method , 2011 .

[10]  J. Oden,et al.  Selection and Validation of Predictive Models of Radiation Effects on Tumor Growth Based on Noninvasive Imaging Data. , 2017, Computer methods in applied mechanics and engineering.

[11]  Harald Garcke,et al.  Well-posedness of a Cahn–Hilliard system modelling tumour growth with chemotaxis and active transport , 2015, European Journal of Applied Mathematics.

[12]  R. Becker,et al.  Numerical parameter estimation for chemical models in multidimensional reactive flows , 2004 .

[13]  Moulay Hicham Tber,et al.  An adaptive finite-element Moreau–Yosida-based solver for a non-smooth Cahn–Hilliard problem , 2011, Optim. Methods Softw..

[14]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[15]  Stephen J. Wright,et al.  Numerical Optimization (Springer Series in Operations Research and Financial Engineering) , 2000 .

[16]  F. Tröltzsch.  Optimal Control of Partial Differential Equations: Theory, Methods and Applications , 2010 .

[17]  J. Sprekels,et al.  Optimal distributed control of a diffuse interface model of tumor growth , 2016, 1601.04567.

[18]  Rüdiger Verfürth,et al.  A posteriori error estimation and adaptive mesh-refinement techniques , 1994 .

[19]  Christian Kahle,et al.  An adaptive finite element Moreau-Yosida-based solver for a coupled Cahn-Hilliard/Navier-Stokes system , 2013, J. Comput. Phys..

[20]  Anotida Madzvamuse,et al.  A Bayesian approach to parameter identification with an application to Turing systems , 2016, 1605.04718.

[21]  V. Cristini,et al.  Nonlinear simulation of tumor growth , 2003, Journal of mathematical biology.

[22]  T E Yankeelov,et al.  Selection, calibration, and validation of models of tumor growth. , 2016, Mathematical models & methods in applied sciences : M3AS.

[23]  Harald Garcke,et al.  A stable and linear time discretization for a thermodynamically consistent model for two-phase incompressible flow , 2014, 1402.6524.

[24]  Michael Hintermüller,et al.  A smooth penalty approach and a nonlinear multigrid algorithm for elliptic MPECs , 2011, Comput. Optim. Appl..

[25]  Andrew M. Stuart,et al.  Inverse problems: A Bayesian perspective , 2010, Acta Numerica.

[26]  P. Comoglio,et al.  Hypoxia promotes invasive growth by transcriptional activation of the met protooncogene. , 2003, Cancer cell.

[27]  Jean Marie Linhart,et al.  Estimating Parameters in Physical Models through Bayesian Inversion: A Complete Example , 2013, SIAM Rev..

[28]  J. Tinsley Oden,et al.  Bayesian calibration, validation, and uncertainty quantification of diffuse interface models of tumor growth , 2012, Journal of Mathematical Biology.

[29]  W. Dörfler.  A convergent adaptive algorithm for Poisson's equation , 1996 .

[30]  Harald Garcke,et al.  A coupled surface-Cahn--Hilliard bulk-diffusion system modeling lipid raft formation in cell membranes , 2015, 1509.03655.

[31]  Harald Garcke,et al.  Analysis of a Cahn--Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis , 2016, 1604.00287.

[32]  Xiangrong Li,et al.  Nonlinear simulations of solid tumor growth using a mixture model: invasion and branching , 2009, Journal of mathematical biology.

[33]  Harald Garcke,et al.  A multiphase Cahn--Hilliard--Darcy model for tumour growth with necrosis , 2017, 1701.06656.

[34]  Anders Logg,et al.  Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book , 2012 .

[35]  Harald Garcke,et al.  A Cahn–Hilliard–Darcy model for tumour growth with chemotaxis and active transport , 2015, 1508.00437.