Skip to main content

Optimal Control of Treatment Time in a Diffuse Interface Model of Tumor Growth

Abstract

We consider an optimal control problem for a diffuse interface model of tumor growth. The state equations couples a Cahn–Hilliard equation and a reaction-diffusion equation, which models the growth of a tumor in the presence of a nutrient and surrounded by host tissue. The introduction of cytotoxic drugs into the system serves to eliminate the tumor cells and in this setting the concentration of the cytotoxic drugs will act as the control variable. Furthermore, we allow the objective functional to depend on a free time variable, which represents the unknown treatment time to be optimized. As a result, we obtain first order necessary optimality conditions for both the cytotoxic concentration and the treatment time.

This is a preview of subscription content, access via your institution.

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Pure and Applied Mathematics, 2nd edn. Elsevier, New York (2003)

  2. Arada, N., Raymond, J.P.: Time optimal problems with Dirichlet boundary controls. Discret. Contin. Dyn. Syst. 9, 1549–1570 (2003)

    MathSciNet  Article  Google Scholar 

  3. Bosia, S., Conti, M., Grasselli, M.: On the Cahn–Hilliard–Brinkman system. Commun. Math. Sci. 13(6), 1541–1567 (2015)

    MathSciNet  Article  Google Scholar 

  4. Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. International Series in Pure and Applied Mathematics. Tata McGraw-Hill, New York (1955)

  5. Colli, P., Farshbaf-Shaker, M.H., Gilardi, G., Sprekels, J.: Optimal boundary control of a viscous Cahn–Hilliard system with dynamic boundary condition and double obstacle potentials. SIAM J. Control Optim. 53(4), 2696–2721 (2015)

    MathSciNet  Article  Google Scholar 

  6. Colli, P., Farshbaf-Shaker, M.H., Gilardi, G., Sprekels, J.: Second-order analysis of a boundary control problem for the viscous Cahn–Hilliard equation with dynamic boundary conditions. Ann. Acad. Rom. Sci. Math. Appl. 7, 41–66 (2015)

    MathSciNet  MATH  Google Scholar 

  7. Colli, P., Gilardi, G., Hilhorst, D.: On a Cahn–Hilliard type phase field model related to tumor growth. Discret. Contin. Dyn. Syst. 35(6), 2423–2442 (2015)

    Article  Google Scholar 

  8. Colli, P., Gilardi, G., Rocca, E., Sprekels, J.: 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 (2015)

    MathSciNet  Article  Google Scholar 

  9. Colli, P., Gilardi, G., Rocca, E., Sprekels, J.: Asymptotic analyses and error estimates for a Cahn–Hilliard type phase field system modelling tumor growth. Discret. Contin. Dyn. Syst. S 10(1), 37–54 (2016)

    MathSciNet  Article  Google Scholar 

  10. Colli, P., Gilardi, G., Rocca, E., Sprekels, J.: Optimal distributed control of a diffuse interface model of tumor growth. Preprint. arXiv:1601.04567 (2016)

  11. Colli, P., Gilardi, G., Sprekels, J.: A boundary control problem for the pure Cahn–Hilliard equation with dynamic boundary conditions. Adv. Nonlinear Anal. 4, 311–325 (2015)

    MathSciNet  MATH  Google Scholar 

  12. Colli, P., Gilardi, G., Sprekels, J.: A boundary control problem for the viscous Cahn–Hilliard equation with dynamic boundary conditions. Appl. Math. Optim. 73(2), 195–225 (2016)

    MathSciNet  Article  Google Scholar 

  13. 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, 723–763 (2009)

    MathSciNet  Article  Google Scholar 

  14. Cristini, V., Lowengrub, J.: Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach. Cambridge University Press, Leiden (2010)

    Book  Google Scholar 

  15. Dai, M., Feireisl, E., Rocca, E., Schimperna, G., Schonbek, M.: Analysis of a diffuse interface model for multispecies tumor growth. Nonlinearity 30, 1639–1658 (2017)

    MathSciNet  Article  Google Scholar 

  16. Friedman, A.: Partial Differential Equations. Holt, Rinehart and Winston, New York (1969)

    MATH  Google Scholar 

  17. Frigeri, S., Grasselli, M., Rocca, E.: On a diffuse interface model of tumor growth. Eur. J. Appl. Math. 26, 215–243 (2015)

    Article  Google Scholar 

  18. Frigeri, S., Rocca, E., Sprekels, J.: Optimal distributed control of a nonlocal Cahn–Hilliard/Navier–Stokes system in two dimensions. SIAM J. Control Optim. 54(1), 221–250 (2016)

    MathSciNet  Article  Google Scholar 

  19. Garcke, H., Lam, K.F.: Global weak solutions and asymptotic limits of a Cahn–Hilliard–Darcy system modelling tumour growth. AIMS Math. 1(3), 318–360 (2016)

    Article  Google Scholar 

  20. Garcke, H., Lam, K.F.: Analysis of a Cahn–Hilliard system with non zero Dirichlet conditions modelling tumour growth with chemotaxis. Discret. Contin. Dyn. Syst. (2017). arXiv:1604.00287

  21. Garcke, H., Lam, K.F.: Well-posedness of a Cahn–Hilliard system modelling tumour growth with chemotaxis and active transport. Eur. J. Appl. Math. 28(2), 284–316 (2017)

    MathSciNet  Article  Google Scholar 

  22. Garcke, H., Lam, K.F., 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)

    MathSciNet  Article  Google Scholar 

  23. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin (2001)

    MATH  Google Scholar 

  24. Grisvard, P.: Elliptic Problems on Nonsmooth Domains. Monographs and Studies in Mathematics, vol. 24. Pitman, Boston (1985)

  25. Hartl, R.F., Sethi, S.P.: A note on the free terminal time transversality condition. Z. Oper. Res. 27, 203–208 (1983)

    MathSciNet  MATH  Google Scholar 

  26. Hawkins-Daarud, A., Prudhomme, S., van der Zee, K.G., Oden, J.T.: Bayesian calibration, validation, and uncertainty quantification of diffuse interface models of tumor growth. J. Math. Biol. 67, 1457–1485 (2013)

    MathSciNet  Article  Google Scholar 

  27. Hawkins-Daarud, A., van der Zee, K.G., Oden, J.T.: Numerical simulation of a thermodynamically consistent four-species tumor growth model. Int. J. Numer. Methods Biomed. Eng. 28, 3–24 (2012)

    MathSciNet  Article  Google Scholar 

  28. Hintermüller, M., Keil, T., Wegner, D.: Optimal control of a semidiscrete Cahn–Hilliard–Navier–Stokes system with non-matched fluid densities. Preprint (2015). arXiv:1506.03591

  29. Hintermüller, M., Wegner, D.: Distributed optimal control of the Cahn–Hilliard system including the case of a double-obstacle homogeneous free energy density. SIAM J. Control Optim. 50(1), 388–418 (2012)

    MathSciNet  Article  Google Scholar 

  30. Hintermüller, M., Wegner, D.: Distributed and Boundary Control Problems for the Semidiscrete Cahn–Hilliard/Navier–Stokes System with Nonsmooth Ginzburg–Landau Energies. Isaac Newton Institute Preprint Series No. NI14042-FRB (2014)

  31. Hintermüller, M., Wegner, D.: Optimal control of a semidiscrete Cahn–Hilliard–Navier–Stokes system. SIAM J. Control Optim. 52(1), 747–772 (2014)

    MathSciNet  Article  Google Scholar 

  32. Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Mathematical Modelling: Theory and Applications, vol. 23. Springer, Dordrecht (2009)

  33. Jang, T., Kwon, H.D., Lee, J.: Free terminal time optimal control problem of an HIV model based on a conjugate gradient method. Bull. Math. Biol. 73, 2408–2429 (2011)

    MathSciNet  Article  Google Scholar 

  34. Jiang, J., Wu, H., Zheng, S.: Well-posedness and long-time behavior of a non-autonomous Cahn–Hilliard–Darcy system with mass source modeling tumor growth. J. Differ. Equ. 259(7), 3032–3077 (2015)

    MathSciNet  Article  Google Scholar 

  35. Lenhart, S., Workman, J.T.: Optimal Control Applied to Biological Models. Mathematical and Computational Biology. Chapman and Hall/CRC, London (2007)

  36. Lowengrub, J.S., Titi, E., Zhao, K.: Analysis of a mixture model of tumor growth. Eur. J. Appl. Math. 24, 691–734 (2013)

    MathSciNet  Article  Google Scholar 

  37. Oden, J.T., Hawkins, A., Prudhomme, S.: General diffuse-interface theories and an approach to predictive tumor growth modeling. Math. Models Methods Appl. Sci. 58, 723–763 (2010)

    MathSciNet  MATH  Google Scholar 

  38. Palanki, S., Kravaris, C., Wang, H.Y.: Optimal feedback control of batch reactors with a state inequality constraint and free terminal time. Chem. Eng. Sci. 49(1), 85–97 (1994)

    Article  Google Scholar 

  39. Raymond, J.P., Zidani, H.: Pontryagin’s principle for time-optimal problems. J. Optim. Theory Appl. 101(2), 375–402 (1999)

    MathSciNet  Article  Google Scholar 

  40. Raymond, J.P., Zidani, H.: Time optimal problems with boundary controls. Differ. Integral Equ. 13(7–9), 1039–1072 (2000)

    MathSciNet  MATH  Google Scholar 

  41. Rocca, E., Sprekels, J.: Optimal distributed control of a nonlocal convective Cahn–Hilliard equation by the velocity in three dimensions. SIAM J. Control Optim. 53(3), 1654–1680 (2015)

    MathSciNet  Article  Google Scholar 

  42. Roubíček, T.: Nonlinear Partial Differential Equations with Applications. International Series of Numerical Mathematics, vol. 153. Birkhäuser Verlag, Basel (2005)

  43. Simon, J.: Compact sets in space \(L^{p}(0, T;B)\). Ann. Mat. Pura Appl. 146(1), 65–96 (1986)

    Article  Google Scholar 

  44. Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods, and Applications. Graduate Studies in Mathematics, vol. 112. AMS, Providence (2010)

  45. Wise, S.M., Lowengrub, J.S., Frieboes, H.B., Cristini, V.: Three-dimensional multispecies nonlinear tumor growth—I: model and numerical method. J. Theor. Biol. 253(3), 524–543 (2008)

    MathSciNet  Article  Google Scholar 

  46. Zhao, X., Duan, N.: Optimal control of the sixth-order convective Cahn–Hilliard equation. Bound. Value Probl. 2014, 206–222 (2014)

    MathSciNet  Article  Google Scholar 

  47. Zhao, X., Liu, C.: Optimal control problem for viscous Cahn–Hilliard equation. Nonlinear Anal. 74, 6348–6357 (2011)

    MathSciNet  Article  Google Scholar 

  48. Zhao, X., Liu, C.: Optimal control of the convective Cahn–Hilliard equation. Appl. Anal. 92(5), 1028–1045 (2013)

    MathSciNet  Article  Google Scholar 

  49. Zhao, X., Liu, C.: Optimal control for the convective Cahn–Hilliard equation in 2D case. Appl. Math. Optim. 70, 61–82 (2014)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The financial support of the FP7-IDEAS-ERC-StG #256872 (EntroPhase) and of the Project Fondazione Cariplo-Regione Lombardia MEGAsTAR “Matematica d’Eccellenza in biologia ed ingegneria come accelleratore di una nuona strateGia per l’ATtRattività dell’ateneo pavese” is gratefully acknowledged.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Kei Fong Lam.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Garcke, H., Lam, K.F. & Rocca, E. Optimal Control of Treatment Time in a Diffuse Interface Model of Tumor Growth. Appl Math Optim 78, 495–544 (2018). https://doi.org/10.1007/s00245-017-9414-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00245-017-9414-4

Keywords

  • Tumor growth
  • Cancer treatment
  • Free terminal time
  • Distributed optimal control
  • Cahn–Hilliard equation
  • Well-posedness

Mathematics Subject Classification

  • 35K61
  • 49J20
  • 49K20
  • 92C50
  • 97M60