Identification of Some Unknown Parameters in an Aggressive–Invasive Cancer Model Using Adjoint Approach

  • M. GarshasbiEmail author
  • M. Abdolmanafi


This study deals with the identification of unknown parameters in an important class of continuous models of aggressive–invasive cancers. This model is obtained as a system of non-linear reaction–diffusion equations from the evolution of cancer cells based on population dynamics. The proposed identification problem is formulated as an optimal control problem with PDEs’ constraint in which an objective functional is defined with respect to the patient and experimental data and an adjoint problem is derived to develop an iterative procedure. A gradient-based iteration method is established to solve this optimal control problem. In each iteration, two systems of nonlinear initial and boundary-value problems (namely, direct and adjoint problems) should be solved. To this end, a nonstandard finite-difference approach is proposed. The robustness of the numerical approach is examined by a test problem without and with noisy data. The numerical results are in good agreement with the results in the literature.


Cancer tumor identification problem optimal control iterative method adjoint problem 

Mathematics Subject Classification

65M06 92B05 35K20 



We would like to express our great appreciation to the respected reviewers for their critical and useful comments that improved this paper substantially.


  1. 1.
    Casciari, J.J., Sotirchos, S.V., Sutherland, R.M.: Variations in tumor cell growth rates and metabolism with oxygen concentration, glucose concentration, and extracellular pH. J. Cell. Physiol. 15, 386–394 (1992)CrossRefGoogle Scholar
  2. 2.
    Pettet, G., Please, C.P., Tindall, M.J., McElwain, D.: The migration of cells in multicell tumor spheroids. Bull. Math. Biol. 63, 231–257 (2001)CrossRefGoogle Scholar
  3. 3.
    Roose, T., Chapman, S.J., Maini, P.K.: Mathematical models of avascular tumor growth. SIAM Rev. 49, 179–208 (2007)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Mahmood, M.S., Mahmood, S., Dobrota, D.: Formulation and numerical simulations of a continuum model of avascular tumor growth. Math. Biosci. 231, 159–171 (2011)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Mahmood, M.S., Mahmood, S., Dobrota, D.: A numerical algorithm for avascular tumor growth model. Math. Comput. Simul. 80, 1269–1277 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    De Pillis, L.G., Rudonskaya, R.: The dynamics of an optimally controlled tumor model: a case study. Math. Comput. Model. 37(11), 1221–1244 (2003)CrossRefGoogle Scholar
  7. 7.
    Solis, F.J., Delgadillo, S.E.: Evolution of a mathematical model of an aggressive-invasive cancer under chemotherapy. Comput. Math. Appl. 69, 545–558 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Tosin, A.: Initial/boundary-value problems of tumor growth within a host tissue. J. Math. Biol. 66, 163–202 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Knopoff, D.A., Fernandez, D.R., Torres, G.A., Turner, C.V.: Adjoint method for a tumor growth PDE-constrained optimization problem. Comput. Math. Appl. 66, 1104–1119 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Gatenby, R.A., Gawlinski, E.T.: A reaction–diffusion model of cancer invasion. Cancer Res. 56, 5745–5753 (1996)Google Scholar
  11. 11.
    Gatenby, R.A., Gawlinski, E.T., Gmitro, A.F., Kaylor, B., Gillies, R.J.: Acid-mediated tumor invasion: a multidisciplinary study. Cancer Res. 66, 5216–5223 (2006)CrossRefGoogle Scholar
  12. 12.
    Martin, N.K., Gaffney, E.A., Gatenby, R.A., Maini, P.K.: Tumourstromal interactions in acidmediated invasion: a mathematical model. J. Theor. Biol. 267, 461470 (2010)Google Scholar
  13. 13.
    McGillen, J.B., Gaffney, E.A., Martin, N.K., Maini, P.K.: A general reactiondiffusion model of acidity in cancer invasion. J. Math. Biol. 68, 1199–224 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Conde, I.R., Chaplain, M.A.J., Anderson, A.R.A.: Mathematical modelling of cancer cell invasion of tissue. Math. Comput. Model. 47, 533–545 (2008)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kirsch, A.: An Introduction to the Mathematical Theory of Inverse Problems. Springer Science, Berlin (2011)CrossRefGoogle Scholar
  16. 16.
    Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)CrossRefGoogle Scholar
  17. 17.
    Borggaard, J., Burns, J.: A PDE sensitivity equation method for optimal aerodynamic design. J. Comput. Phys. 136, 366–389 (1997)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Schattler, H., Ledzewicz, U.: Optimal Control for Mathematical Models of Cancer Therapies. Interdisciplinary Applied Mathematics. Springer Science, Berlin (2015)CrossRefGoogle Scholar
  19. 19.
    Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Springer Science, Berlin (2009)zbMATHGoogle Scholar
  20. 20.
    Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer Academic Pub, Norwell (1996)CrossRefGoogle Scholar
  21. 21.
    Mickens, R.E.: Nonstandard Finite Difference Models of Differential Equations. World Scientific, Singapore (1994)zbMATHGoogle Scholar
  22. 22.
    Mickens, R.E.: Advances in the Applications of Nonstandard Finite Diffference Schemes. World Scientific, Singapore (2005)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of MathematicsIran University of Science and TechnologyTehranIran

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