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Strong well-posedness and inverse identification problem of a non-local phase field tumour model with degenerate mobilities

Part of: Equations of mathematical physics and other areas of application Parabolic equations and systems Existence theories Physiological, cellular and medical topics

Published online by Cambridge University Press:  22 February 2021

SERGIO FRIGERI ,
KEI FONG LAM  and
ANDREA SIGNORI Open the ORCID record for ANDREA SIGNORI [Opens in a new window]
SERGIO FRIGERI
Affiliation:
Dipartimento di Matematica “Federigo Enriques”, Università degli Studi di Milano via Saldini 50, I-20133Milano, Italy, e-mail: [email protected]
KEI FONG LAM
Affiliation:
Department of Mathematics, Hong Kong Baptist University Kowloon Tong, Hong Kong, e-mail: [email protected]
ANDREA SIGNORI
Affiliation:
Dipartimento di Matematica e Applicazioni, Università di Milano–Bicocca via Cozzi 55, 20125Milano, Italy, e-mail: [email protected]
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Abstract

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We extend previous weak well-posedness results obtained in Frigeri et al. (2017, Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs, Vol. 22, Springer, Cham, pp. 217–254) concerning a non-local variant of a diffuse interface tumour model proposed by Hawkins-Daarud et al. (2012, Int. J. Numer. Method Biomed. Engng.28, 3–24). The model consists of a non-local Cahn–Hilliard equation with degenerate mobility and singular potential for the phase field variable, coupled to a reaction–diffusion equation for the concentration of a nutrient. We prove the existence of strong solutions to the model and establish some high-order continuous dependence estimates, even in the presence of concentration-dependent mobilities for the nutrient variable in two spatial dimensions. Then, we apply the new regularity results to study an inverse problem identifying the initial tumour distribution from measurements at the terminal time. Formulating the Tikhonov regularised inverse problem as a constrained minimisation problem, we establish the existence of minimisers and derive first-order necessary optimality conditions.

Keywords

Tumour growthnon-local Cahn–Hilliard equationdegenerate mobilitysingular potentialsstrong solutionswell-posednessinverse problemGâteaux differentiability

MSC classification

Primary: 35K86: Nonlinear parabolic unilateral problems and nonlinear parabolic variational inequalities 35K61: Nonlinear initial-boundary value problems for nonlinear parabolic equations 49J20: Optimal control problems involving partial differential equations
Secondary: 35Q92: PDEs in connection with biology and other natural sciences 92C50: Medical applications (general)
Type
Papers
Information
European Journal of Applied Mathematics , Volume 33 , Issue 2 , April 2022 , pp. 267 - 308
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

References

Agmon, S. (1965) Lectures on Elliptic Boundary Value Problems, Vol. 369, AMS Chelsea Publishing, AMS, Providence, R.I.Google Scholar
Armstrong, N. J., Painter, K. J. & Sherratt, J. A. (2006) A continuum approach to modelling cell-cell adhesion. J. Theor. Biol. 243, 98–113.CrossRefGoogle ScholarPubMed
Bedrossian, J., Rodríguez, N. & Bertozzi, A. (2011) Local and global well-posedness for an aggregation equation and Patlak–Keller–Segel models with degenerate diffusion. Nonlinearity 24, 1683–1714.CrossRefGoogle Scholar
Beretta, E., Ratti, L. & Verani, M. (2017) A phase-field approach for the interface reconstruction in a nonlinear elliptic problem arising from cardiac electrophysiology. Preprint:1709.05646 [math.AP].Google Scholar
Boyer, F. (1999) Mathematical study of multi-phase flow under shear through order parameter formulation. Asymptot. Anal. 20, 175–212.Google Scholar
Brézis, H. (1968) Équations et inéquations non-linéaires dans les espaces vectoriel en dualité. Ann. Inst. Fourier 18, 115–176.CrossRefGoogle Scholar
Brezis, H. & Mironescu, P. (2018) Gagliardo–Nirenberg inequalities and non-inequalities: the full story. Ann. I. H. Poincare-AN 35, 1355–1376.Google Scholar
Cavaterra, C., Rocca, E. & Wu, H. (2019) Long-time dynamics and optimal control of a diffuse interface model for tumor growth. Appl. Math. Optim. https://doi.org/10.1007/s00245-019-09562-5.Google Scholar
Chaplain, M. A. J., Lachowicz, M., Szymańska, Z. & Wrzosek, D. (2011) Mathematical modelling of cancer invasion: the important of cell-cell adhesion and cell-matrix adhesion. Math. Models Methods Appl. Sci. 21, 719–743.CrossRefGoogle Scholar
Colli, P., Frigeri, S. & Grasselli, M. (2012) Global existence of weak solutions to a nonlocal Cahn–Hilliard–Navier–Stokes system. J. Math. Anal. Appl. 386, 428–444.CrossRefGoogle Scholar
Colli, P., Gilardi, G. & Hilhorst, D. (2015) On a Cahn–Hilliard type phase field system related to tumor growth. Discrete Contin. Dyn. Syst. 35, 2423–2442.CrossRefGoogle Scholar
Colli, P., Gilardi, G., Rocca, E. & Sprekels, J. (2015) Vanishing viscosities and error estimate for a Cahn–Hilliard type phase field system related to tumor growth. Nonlinear Anal. Real World Appl. 26, 93–108.CrossRefGoogle Scholar
Colli, P., Gilardi, G., Rocca, E. & Sprekels, J. (2017) Optimal distributed control of a diffuse interface model of tumor growth. Nonlinearity 30, 2518–2546.CrossRefGoogle Scholar
Colli, P., Gilardi, G., Rocca, E. & Sprekels, J. (2017) Asymptotic analyses and error estimates for a Cahn–Hilliard type phase field system modeling tumor growth. Discrete Contin. Dyn. Syst. Ser. S 10, 37–54.Google Scholar
Colli, P., Signori, A. & Sprekels, J. (2019) Optimal control of a phase field system modelling tumor growth with chemotaxis and singular potentials. Appl. Math. Optim. https://doi.org/10.1007/s00245-019-09618-6.CrossRefGoogle Scholar
Cristini, V., Li, X., Lowengrub, J. S. & Wise, S. M. (2009) Nonlinear simulations of solid tumor growth using a mixture model: invasion and branching. J. Math. Biol. 58, 723–763.CrossRefGoogle Scholar
Cristini, V. & Lowengrub, J. (2010) Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach, Cambridge University Press, United Kingdom. ISBN 9780521884426.CrossRefGoogle Scholar
Dai, M., Feireisl, E., Rocca, E., Schimperna, G. & Schonbek, M. (2017) Analysis of a diffuse interface model of multispecies tumor growth. Nonlinearity 30, 1639–1658.CrossRefGoogle Scholar
DiBenedetto, E. (1993) Degenerate Parabolic Equations. Universitext. Springer–Verlag, New York.CrossRefGoogle Scholar
DiBenedetto, E. (2002) Real Analysis, Advanced Text Series, Birkhäuser, Boston.Google Scholar
Ebenbeck, M. & Garcke, H. (2019) Analysis of a Cahn–Hilliard–Brinkman model for tumour growth with chemotaxis. J. Differ. Equ. 266, 5998–6036.CrossRefGoogle Scholar
Ebenbeck, M. & Garcke, H. (2019) On a Cahn–Hilliard–Brinkman model for tumor growth and its singular limits. SIAM J. Math. Anal. 51, 1868–1912.CrossRefGoogle Scholar
Ebenbeck, M., Garcke, H. & Nürnberg, R. (2020) Cahn–Hilliard–Brinkman systems for tumour growth. Preprint arXiv:2003.08314 [math.AP].CrossRefGoogle Scholar
Ebenbeck, M. & Knopf, P. (2020) Optimal control theory and advanced optimality conditions for a diffuse interface model of tumor growth. ESAIM COCV 26, Article number 71, 38 pp.CrossRefGoogle Scholar
Ebenbeck, M. & Knopf, P. (2019) Optimal medication for tumors modeled by a Cahn–Hilliard–Brinkman equation. Calc. Var. 58, 131.CrossRefGoogle Scholar
Ebenbeck, M. & Lam, K. F. (2020) Weak and stationary solutions to a Cahn–Hilliard–Brinkman model with singular potentials and source terms. Adv. Nonlinear Anal. 10, 24–65.CrossRefGoogle Scholar
Elliott, C. M. & Garcke, H. (1996) On the Cahn–Hilliard equation with degenerate mobility. SIAM J. Math. Anal. 27, 404–423.CrossRefGoogle Scholar
Engl, H. W., Hanke, M. & Neubauer, A. (1996) Regularization of Inverse Problems, Mathematics and its Applications, Springer, Netherlands.Google Scholar
Evans, L. C. (1998) Partial differential equations, Graduate Studies in Mathematics, Vol. 19, Americal Mathematical Society, Providence, RI.Google Scholar
Frigeri, S. (2016) Global existence of weak solutions for a nonlocal model for two-phase flows of incompressible fluids with unmatched densities. Math. Models Methods Appl. Sci. 26, 1957–1993.CrossRefGoogle Scholar
Frigeri, S. (2020) On a nonlocal Cahn–Hilliard/Navier–Stokes system with degenerate mobility and singular potential for incompressible fluids with different densities. Ann. Inst. H. Poincaré Anal. Non Linéaire. https://doi.org/10.1016/j.anihpc.2020.08.005.Google Scholar
Frigeri, S., Gal, C. G. & Grasselli, M. (2016) On nonlocal Cahn–Hilliard–Navier–Stokes systems in two dimensions. J. Nonlinear Sci. 26, 847–893.CrossRefGoogle Scholar
Frigeri, S., Gal, C. G., Grasselli, M. & Sprekels, J. (2019) Two-dimensional nonlocal Cahn–Hilliard–Navier–Stokes systems with variable viscosity, degenerate mobility and singular potential. Nonlinearity 32, 678–727.CrossRefGoogle Scholar
Frigeri, S. & Grasselli, M. (2012) Global and trajectory attractors for a nonlocal Cahn–Hilliard–Navier–Stokes system. J. Dyn. Differ. Equ. 24, 827–856.CrossRefGoogle Scholar
Frigeri, S. & Grasselli, M. (2012) Nonlocal Cahn–Hilliard–Navier–Stokes systems with singular potentials. Dyn. Partial Differ. Equ. 9, 273–304.CrossRefGoogle Scholar
Frigeri, S., Grasselli, M. & Krejčí, P. (2013) Strong solutions for two-dimensional nonlocal Cahn–Hilliard–Navier–Stokes systems. J. Differ. Equ. 255, 2587–2614.CrossRefGoogle Scholar
Frigeri, S., Grasselli, M. & Rocca, E. (2015) On a diffuse interface model of tumor growth. Eur. J. Appl. Math. 26, 215–243.CrossRefGoogle Scholar
Frigeri, S., Grasselli, M. & Rocca, E. (2015) A diffuse interface model for two-phase incompressible flows with non-local interactions and non-constant mobility. Nonlinearity 28, 1257–1293.CrossRefGoogle Scholar
Frigeri, S., Grasselli, M. & Sprekels, J. (2018) Optimal distributed control of two-dimensional nonlocal Cahn–Hilliard–Navier–Stokes systems with degenerate mobility and singular potential. Appl. Math. Optim. 81, 899–931.CrossRefGoogle Scholar
Frigeri, S., Lam, K. F., Rocca, E. & Schimperna, G. (2018) On a multi-species Cahn–Hilliard–Darcy tumor growth model with singular potentials. Commun. Math. Sci. 16, 821–856.CrossRefGoogle Scholar
Frigeri, S., Lam, K. F. & Rocca, E. (2017) On a diffuse interface model for tumour growth with non-local interactions and degenerate mobilities. In: Colli, P., Favini, A., Rocca, E., Schimperna, G. and Sprekels, J. (editors), Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs, Springer INdAM Series, Vol. 22, Springer, Cham, pp. 217–254.CrossRefGoogle Scholar
Fritz, M., Lima, E. A. F., Nikolić, V., Oden, J. T. & Wohlmuth, B. (2019) Local and nonlocal phase-field models of tumor growth and invasion due to ECM degradation. Math. Models Methods Appl. Sci. 29, 2433–2468.CrossRefGoogle Scholar
Fritz, M., Lima, E. A. F., Oden, J. T. & Wohlmuth, B. (2019) On the unsteady Darcy–Forcheimer–Brinkman equation in local and nonlocal tumor growth models. Math. Models Methods Appl. Sci. 29, 1691–1731.CrossRefGoogle Scholar
Gajewski, H. & Zacharias, K. (2003) On a nonlocal phase separation model. J. Math. Anal. Appl. 286, 11–31.CrossRefGoogle Scholar
Gal, C. G., Giorgini, A. & Grasselli, M. (2017) The nonlocal Cahn–Hilliard equation with singular potential: Well-posedness, regularity and strict separation property. J. Differ. Equ. 263, 5253–5297.CrossRefGoogle Scholar
Garcke, H., Lam, K. F., Nürnberg, R. & Sitka, E. (2018) A multiphase Cahn–Hilliard–Darcy model for tumour growth with necrosis. Math. Models Methods Appl. Sci. 28, 525–577.CrossRefGoogle Scholar
Garcke, H., Lam, K. F. & Signori, A. (2021) On a phase field model of Cahn–Hilliard type for tumour growth with mechanical effects. Nonlinear Anal. Real World Appl. 57, 103192.CrossRefGoogle Scholar
Garcke, H., Lam, K. F., Stika, E. & Styles, V. (2016) A Cahn–Hilliard–Darcy model for tumour growth with chemotaxis and active transport. Math. Models Methods Appl. Sci. 26, 1095–1148.CrossRefGoogle Scholar
Gerisch, A. & Chaplain, M. A. J. (2008) Mathematical modelling of cancer cell invasion of tissue: Local and non-local models and the effect of adhesion. J. Theor. Biol. 25, 684–704.CrossRefGoogle Scholar
Giacomin, G. & Lebowitz, J. L. (1996) Exact macroscopic description of phase segregation in model alloys with long range interactions. Phys. Rev. Lett. 76, 1094–1097.CrossRefGoogle ScholarPubMed
Giacomin, G. & Lebowitz, J. L. (1997) Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits. J. Statist. Phys. 87, 37–61.CrossRefGoogle Scholar
Giacomin, G. & Lebowitz, J. L. (1998) Phase segregation dynamics in particle systems with long range interactions. II. Phase motion. SIAM J. Appl. Math. 58, 1707–1729.CrossRefGoogle Scholar
Hawkins-Daarud, A., Prudhomme, S., van der Zee, K. G. & Oden, J. T. (2013) Bayesian calibration, validation, and uncertainty quantification of diffuse interface models of tumor growth. J. Math. Biol. 67, 1457–1485.CrossRefGoogle ScholarPubMed
Hawkins-Daarud, A., van der Zee, K. G. & Oden, J. T. (2012) Numerical simulation of a thermodynamically consistent four-species tumor growth model. Int. J. Numer. Method Biomed. Engng. 28, 3–24.CrossRefGoogle ScholarPubMed
Hilhorst, D., Kampmann, J., Nguyen, T. N. & van der Zee, K. G. (2015) Formal asymptotic limit of a diffuse-interface tumor-growth model. Math. Models Methods Appl. Sci. 25, 1011–1043.CrossRefGoogle Scholar
Jiang, J., Wu, H. & Zheng, S. (2015) Well-posedness and long-time behavior of a non-autonomous Cahn–Hilliard–Darcy system with mass source modeling tumor growth. J. Differ. Equ. 259, 3032–3077.CrossRefGoogle Scholar
Lima, E. A. B. F., Oden, J. T., Shahmoradi, A., Hormuth, D. A. II, Yankeelov, T. E. & Almeida, R. C. (2016) Selection, calibration, and validation of models of tumor growth. Math. Models Methods Appl. Sci. 26, 2341–2368.CrossRefGoogle ScholarPubMed
Lima, E. A. B. F., Oden, J. T., Wohlmuth, B., Shahmoradi, A., Hormuth, D. A. II, Yankeelov, T. E., Scarabosio, L. & Horger, T. (2017) Selection and validation of predictive models of radiation effects on tumor growth based on noninvasive imaging data. Comput. Methods Appl. Mech. Engrg. 327, 277–305.CrossRefGoogle ScholarPubMed
Miranville, A., Rocca, E. & Schimperna, G. (2019) On the long time behavior of a tumor growth model. J. Differ. Equ. 267, 2616–2642.CrossRefGoogle Scholar
Roubček, T. (2005) Nonlinear Partial Differential Equations with Applications, Birkhäuser Verlag, Basel, ISBN 3-7643-7293-1.Google Scholar
Scarpa, L. & Signori, A. (2020) On a class of non-local phase-field models for tumor growth with possibly singular potentials, chemotaxis, and active transport. Preprint arXiv:2002.12702 [math.AP].CrossRefGoogle Scholar
Signori, A. (2020) Vanishing parameter for an optimal control problem modeling tumor growth. Asymptot. Anal. 117, 43–66.Google Scholar
Signori, A. (2020) Optimal treatment for a phase field system of Cahn–Hilliard type modeling tumor growth by asymptotic scheme. Math. Control Relat. Fields 10, 305–331.Google Scholar
Signori, A. (2020) Optimality conditions for an extended tumor growth model with double obstacle potential via deep quench approach. Evol. Equ. Control Theory 9, 193–217.CrossRefGoogle Scholar
Signori, A. (2020) Optimal distributed control of an extended model of tumor growth with logarithmic potential. Appl. Math. Optim. 82, 517–549.CrossRefGoogle Scholar
Signori, A. (2020) Penalisation of long treatment time and optimal control of a tumour growth model of Cahn–Hilliard type with singular potential. Discrete Contin. Dyn. Syst. Ser. A. https://doi.org/10.3934/dcds.2020373.Google Scholar
Sprekels, J. & Wu, H. (2019) Optimal distributed control of a Cahn–Hilliard–Darcy system with mass sources. Appl. Math. Optim. https://doi.org/10.1007/s00245-019-09555-4.Google Scholar
Tröltzsch, F. (2010) Optimal Control of Partial Differential Equations. Theory, Methods and Applications , Graduate Studies in Mathematics, Vol. 112, Americal Mathematical Society, Providence, RI.Google Scholar
Wise, S. M., Lowengrub, J. S., Frieboes, H. B. & Cristini, V. (2008) Three-dimensional multispecies nonlinear tumor growth I: Model and numerical method. J. Theor. Biol. 253(3), 524–543.CrossRefGoogle ScholarPubMed
Wu, X., van Zwieten, G. J. & van der Zee, K. G. (2014) Stabilized second-order splitting schemes for Cahn–Hilliard models with applications to diffuse-interface tumor-growth models. Int. J. Numer. Meth. Biomed. Engng. 30, 180–203.CrossRefGoogle Scholar