Source Localization of Reaction-Diffusion Models for Brain Tumors

  • Rym JaroudiEmail author
  • George Baravdish
  • Freddie Åström
  • B. Tomas Johansson
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9796)


We propose a mathematically well-founded approach for locating the source (initial state) of density functions evolved within a nonlinear reaction-diffusion model. The reconstruction of the initial source is an ill-posed inverse problem since the solution is highly unstable with respect to measurement noise. To address this instability problem, we introduce a regularization procedure based on the nonlinear Landweber method for the stable determination of the source location. This amounts to solving a sequence of well-posed forward reaction-diffusion problems. The developed framework is general, and as a special instance we consider the problem of source localization of brain tumors. We show numerically that the source of the initial densities of tumor cells are reconstructed well on both imaging data consisting of simple and complex geometric structures.


Source Term Forward Problem Gray Matter Region Ordinary Differential Equation Model Tumor Cell Density 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    IBSR, Internet Brain Segmentation Repository. Accessed Mar 2016
  2. 2.
    Release Name: Male Subject, T1-Weighted Brain Scan: 788_6. Accessed Mar 2016
  3. 3.
    Bellomo, N., de Angelis, E.: Selected Topics in Cancer Modeling Genesis Evolution Immune Competition and Therapy. Springer Science & Business Media, Berlin (2008)zbMATHGoogle Scholar
  4. 4.
    Cannon, J.R.: Determination of an unknown heat source from overspecified boundary data. SIAM J. Numer. Anal. 5(2), 275–286 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chaplain, M.: Avascular growth, angiogenesis and vascular growth in solid tumours: the mathematical modelling of the stages of tumour development. Math. Comput. Model. 23(6), 47–87 (1996)CrossRefzbMATHGoogle Scholar
  6. 6.
    Clatz, O., Sermesant, M., Bondiau, P.Y., Delingette, H., Warfield, S.K., Malandain, G., Ayache, N.: Realistic simulation of the 3-D growth of brain tumors in MR images coupling diffusion with biomechanical deformation. IEEE Trans. Med. Imaging 24(10), 1334–1346 (2005)CrossRefGoogle Scholar
  7. 7.
    D’haeyer, S., Johansson, B.T., Slodička, M.: Reconstruction of a spacewise-dependent heat source in a time-dependent heat diffusion process. IMA J. Appl. Math. 79(1), 33–53 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Giese, A., Kluwe, L., Laube, B., Meissner, H., Berens, M.E., Westphal, M.: Migration of human glioma cells on myelin. Neurosurgery 38(4), 755–764 (1996)CrossRefGoogle Scholar
  9. 9.
    Hanke, M., Neubauer, A., Scherzer, O.: A convergence analysis of the Landweber iteration for nonlinear ill-posed problems. Numer. Math. 72(1), 21–37 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Isakov, V.: Inverse Problems for Partial Differential Equations. Springer, New York (1998)CrossRefzbMATHGoogle Scholar
  11. 11.
    Jbabdi, S., Mandonnet, E., Duffau, H., Capelle, L., Swanson, K.R., Pélégrini-Issac, M., Guillevin, R., Benali, H.: Simulation of anisotropic growth of low-grade gliomas using diffusion tensor imaging. Magn. Reson. Med. 54(3), 616–624 (2005)CrossRefGoogle Scholar
  12. 12.
    Johansson, B.T., Lesnic, D.: A procedure for determining a spacewise dependent heat source and the initial temperature. Appl. Anal. 87(3), 265–276 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Konukoğlu, E., Clatz, O., Bondiau, P.-Y., Delingette, H., Ayache, N.: Extrapolating tumor invasion margins for physiologically determined radiotherapy regions. In: Larsen, R., Nielsen, M., Sporring, J. (eds.) MICCAI 2006. LNCS, vol. 4190, pp. 338–346. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  14. 14.
    Ladyženskaja, O.A., Solonnikov, V.A., Uralceva, N.N.: Linear and Quasilinear Equations of Parabolic Type, vol. 23. American Mathematical Society, Providence (1968). Translated from the Russian by S. Smith. Translations of Mathematical MonographsGoogle Scholar
  15. 15.
    Lions, J.L., Magenes, E.: Non-homogeneous Boundary Value Problems and Applications, vol. II. Springer, New York, Heidelberg (1972). Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 182CrossRefzbMATHGoogle Scholar
  16. 16.
    Marušić, M., Bajzer, Ž., Freyer, J., Vuk-Pavlović, S.: Analysis of growth of multicellular tumour spheroids by mathematical models. Cell Prolif. 27(2), 73–94 (1994)CrossRefzbMATHGoogle Scholar
  17. 17.
    Mueller, W., Hartmann, C., Hoffmann, A., Lanksch, W., Kiwit, J., Tonn, J., Veelken, J., Schramm, J., Weller, M., Wiestler, O.D., et al.: Genetic signature of oligoastrocytomas correlates with tumor location and denotes distinct molecular subsets. Am. J. Pathol. 161(1), 313–319 (2002)CrossRefGoogle Scholar
  18. 18.
    Murray, J.D.: Mathematical Biology II Spatial Models and Biomedical Applications Interdisciplinary Applied Mathematics, vol. 18. Springer, New York (2001)Google Scholar
  19. 19.
    Nocedal, J., Wright, S.: Numerical Optimization. Springer Series in Operations Research and Financial Engineering. Springer, New York (2006)zbMATHGoogle Scholar
  20. 20.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44. Springer, New York (1983)CrossRefzbMATHGoogle Scholar
  21. 21.
    Rekik, I., Allassonnière, S., Clatz, O., Geremia, E., Stretton, E., Delingette, H., Ayache, N.: Tumor growth parameters estimation and source localization from a unique time point: application to low-grade gliomas. Comput. Vis. Image Underst. 117(3), 238–249 (2013)CrossRefGoogle Scholar
  22. 22.
    Roubíček, T.: Nonlinear Partial Differential Equations with Applications. International Series of Numerical Mathematics, vol. 153. Birkhäuser Verlag, Basel (2005)zbMATHGoogle Scholar
  23. 23.
    Schuster, T., Kaltenbacher, B., Hofmann, B., Kazimierski, K.S.: Regularization Methods in Banach Spaces. Radon Series on Computational and Applied Mathematics, vol. 10. Walter de Gruyter GmbH & Co. KG, Berlin (2012)CrossRefzbMATHGoogle Scholar
  24. 24.
    Shepp, L.A., Logan, B.F.: The fourier reconstruction of a head section. IEEE Trans. Nucl. Sci. 21(3), 21–43 (1974)CrossRefGoogle Scholar
  25. 25.
    Swanson, K.R., Alvord, E., Murray, J.: A quantitative model for differential motility of gliomas in grey and white matter. Cell Prolif. 33(5), 317–329 (2000)CrossRefGoogle Scholar
  26. 26.
    Swanson, K.R., Alvord, E., Murray, J.: Virtual brain tumours (gliomas) enhance the reality of medical imaging and highlight inadequacies of current therapy. Br. J. Cancer 86(1), 14–18 (2002)CrossRefGoogle Scholar
  27. 27.
    Tracqui, P.: From passive diffusion to active cellular migration in mathematical models of tumour invasion. Acta Biotheor. 43(4), 443–464 (1995)CrossRefGoogle Scholar
  28. 28.
    Tuan, N.H., Trong, D.D.: On a backward parabolic problem with local Lipschitz source. J. Math. Anal. Appl. 414(2), 678–692 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Zeidler, E.: Nonlinear Functional Analysis and Its Applications. II/B: Nonlinear Monotone Operators. Springer, New York (1990). Translated from the German by the author and Leo F. BoronCrossRefzbMATHGoogle Scholar
  30. 30.
    Zlatescu, M.C., TehraniYazdi, A., Sasaki, H., Megyesi, J.F., Betensky, R.A., Louis, D.N., Cairncross, J.G.: Tumor location and growth pattern correlate with genetic signature in oligodendroglial neoplasms. Cancer Res. 61(18), 6713–6715 (2001)Google Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Rym Jaroudi
    • 1
    Email author
  • George Baravdish
    • 1
  • Freddie Åström
    • 2
  • B. Tomas Johansson
    • 1
    • 3
  1. 1.Linköping UniversityLinköpingSweden
  2. 2.Heidelberg Collaboratory for Image ProcessingHeidelberg UniversityHeidelbergGermany
  3. 3.Aston UniversityBirminghamUK

Personalised recommendations