Numerical simulations in 3-dimensions of reaction–diffusion models for brain tumour growth

References

• F. Åström, Variational tensor-based models for image diffusion in non-linear domains, Ph.D. diss., Linköping University Electronic Press, 2015.
   Google Scholar
• D. Basanta, M. Simon, H. Hatzikirou, and A. Deutsch, Evolutionary game theory elucidates the role of glycolysis in glioma progression and invasion, Cell. Prolif. 41 (2008), pp. 980–987. doi: 10.1111/j.1365-2184.2008.00563.x
   PubMed Web of Science ®Google Scholar
• G. Bastay, V.A. Kozlov, and B.O. Turesson, Iterative methods for an inverse heat conduction problem, J. Inverse Ill-posed Probl. 9 (2001), pp. 375–388. doi: 10.1515/jiip.2001.9.4.375
   Google Scholar
• S. Benzekry, C. Lamont, A. Beheshti, A. Tracz, J.M. Ebos, L. Hlatky, and P. Hahnfeldt, Classical mathematical models for description and prediction of experimental tumor growth, PLoS. Comput. Biol. 10 (2014), pp. e1003800. doi: 10.1371/journal.pcbi.1003800
   PubMed Web of Science ®Google Scholar
• P.K. Burgess, P.M. Kulesa, J.D. Murray, and E.C. Alvord, The interaction of growth rates and diffusion coefficients in a three-dimensional mathematical model of gliomas, J. Neuropath. Exp. Neur. 56 (1997), pp. 704–713. doi: 10.1097/00005072-199706000-00008
   PubMed Web of Science ®Google Scholar
• P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North Holland, Amsterdam, 1978.
   Google Scholar
• O. Clatz, M. Sermesant, P.Y. Bondiau, H. Delingette, S.K. Warfield, G. Malandain, and N. Ayache, Realistic simulation of the 3-D growth of brain tumors in MR images coupling diffusion with biomechanical deformation, Med. Imag. IEEE T. 24 (2005), pp. 1334–1346. doi: 10.1109/TMI.2005.857217
   PubMed Web of Science ®Google Scholar
• C. Cocosco, V. Kollokian, R. Kwan, and A. Evans, Brainweb: Online interface to a 3d simulated brain database, Neuroimage 5 (1997), p. S425.
   Google Scholar
• D.L. Collins, A.P. Zijdenbos, V. Kollokian, J.G. Sled, N.J. Kabani, C.J. Holmes, and A.C. Evans, Design and construction of a realistic digital brain phantom, IEEE Trans. Med. Imag. 17 (1998), pp. 463–468. doi: 10.1109/42.712135
   PubMed Web of Science ®Google Scholar
• G.C. Cruywagen, D.E. Woodward, P. Tracqui, G.T. Bartoo, J. Murray, and E.C. Alvord, The modelling of diffusive tumours, J. Biol. Syst. 3 (1995), pp. 937–945. doi: 10.1142/S0218339095000836
   Google Scholar
• L.M. DeAngelis, Brain tumors, New Engl. J. Med. 344 (2001), pp. 114–123. doi: 10.1056/NEJM200101113440207
   PubMed Web of Science ®Google Scholar
• A. Drevelegas and N. Papanikolaou, Imaging Modalities in Brain Tumors, Springer, Berlin, Heidelberg, 2011.
   Google Scholar
• L. Fass, Imaging and cancer: A review, Mol. Oncol. 2 (2008), pp. 115–152. doi: 10.1016/j.molonc.2008.04.001
   PubMed Web of Science ®Google Scholar
• J. Ferlay, I. Soerjomataram, R. Dikshit, S. Eser, C. Mathers, M. Rebelo, D.M. Parkin, D. Forman, and F. Bray, Cancer incidence and mortality worldwide: Sources, methods and major patterns in globocan 2012, Int. J. Cancer 136 (2015), pp. E359–E386. doi: 10.1002/ijc.29210
   PubMed Web of Science ®Google Scholar
• A. Gholami, A. Mang, and G. Biros, An inverse problem formulation for parameter estimation of a reaction–diffusion model of low grade gliomas, J. Math. Biol. 72 (2016), pp. 409–433. doi: 10.1007/s00285-015-0888-x
   PubMed Web of Science ®Google Scholar
• A. Giese, L. Kluwe, B. Laube, H. Meissner, M.E. Berens, and M. Westphal, Migration of human glioma cells on myelin, Neurosurgery 38 (1996), pp. 755–764. doi: 10.1227/00006123-199604000-00026
   PubMed Web of Science ®Google Scholar
• M.L. Goodenberger and R.B. Jenkins, Genetics of adult glioma, Cancer Genet. 205 (2012), pp. 613–621. doi: 10.1016/j.cancergen.2012.10.009
   PubMed Web of Science ®Google Scholar
• D. Hanahan and R.A. Weinberg, Hallmarks of cancer: The next generation, Cell 144 (2011), pp. 646–674. doi: 10.1016/j.cell.2011.02.013
   PubMed Web of Science ®Google Scholar
• C. Hogea, C. Davatzikos, and G. Biros, An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects, J. Math. Biol. 56 (2008), pp. 793–825. doi: 10.1007/s00285-007-0139-x
   PubMed Web of Science ®Google Scholar
• Ibsr, Internet Brain Segmentation Repository. Available at www.nitrc.org/projects/ibsr/ (accessed March 2016).
   Google Scholar
• R. Jaroudi, Inverse mathematical models for brain tumour growth, Licentiate thesis, Linköping University Electronic Press, 2017.
   Google Scholar
• R. Jaroudi, G. Baravdish, F. Åström, and B.T. Johansson, Source localization of reaction–diffusion models for brain tumors, German Conference on Pattern Recognition, Springer, 2016, pp. 414–425.
   Google Scholar
• R. Jaroudi, G. Baravdish, B.T. Johansson, and F. Åström, Numerical reconstruction of brain tumours, Inverse Probl. Sci. Eng. 27 (2019), pp. 278–298. doi: 10.1080/17415977.2018.1456537
   Web of Science ®Google Scholar
• S. Jbabdi, E. Mandonnet, H. Duffau, L. Capelle, K.R. Swanson, M. Pélégrini-Issac, R. Guillevin, and H. Benali, Simulation of anisotropic growth of low-grade gliomas using diffusion tensor imaging, Magn. Reson. Med. 54 (2005), pp. 616–624. doi: 10.1002/mrm.20625
   PubMed Web of Science ®Google Scholar
• J. Kačur, Method of Rothe in Evolution Equations, B. G. Teubner, Leipzig, 1985.
   Google Scholar
• A. Kansal, S. Torquato, G. Harsh, E. Chiocca, and T. Deisboeck, Simulated brain tumor growth dynamics using a three-dimensional cellular automaton, J. Theoret. Biol. 203 (2000), pp. 367–382. doi: 10.1006/jtbi.2000.2000
   PubMed Web of Science ®Google Scholar
• P. Kleihues, P.C. Burger, and B.W. Scheithauer, The new who classification of brain tumours, Brain Pathol. 3 (1993), pp. 255–268. doi: 10.1111/j.1750-3639.1993.tb00752.x
   PubMed Web of Science ®Google Scholar
• E. Konukoglu, O. Clatz, B.H. Menze, B. Stieltjes, M.A. Weber, E. Mandonnet, H. Delingette, and N. Ayache, Image guided personalization of reaction–diffusion type tumor growth models using modified anisotropic eikonal equations, IEEE. Trans. Med. Imaging 29 (2010), pp. 77–95. doi: 10.1109/TMI.2009.2026413
   PubMed Web of Science ®Google Scholar
• O.A. Ladyženskaja, V.A. Solonnikov, and N.N. Ural'ceva, Linear and quasilinear equations of parabolic type, Translated from the Russian by S. Smith, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968.
   Google Scholar
• J.L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York-Heidelberg, 1972, Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 182.
   Google Scholar
• J.L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. II, Springer-Verlag, New York-Heidelberg, 1972, Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 182.
   Google Scholar
• D.N. Louis, A. Perry, G. Reifenberger, A. von Deimling, D. Figarella-Branger, W.K. Cavenee, H. Ohgaki, O.D. Wiestler, P. Kleihues, and D.W. Ellison, The 2016 world health organization classification of tumors of the central nervous system: A summary, Acta. Neuropathol. 131 (2016), pp. 803–820. doi: 10.1007/s00401-016-1545-1
   PubMed Web of Science ®Google Scholar
• M.C. Mabray, R.F. Barajas, and S. Cha, Modern brain tumor imaging, Brain Tumor Res. Treat. 3 (2015), pp. 8–23. doi: 10.14791/btrt.2015.3.1.8
   PubMedGoogle Scholar
• M. Marušić, Ž. Bajzer, J. Freyer, and S. Vuk-Pavlović, Analysis of growth of multicellular tumour spheroids by mathematical models, Cell. Prolif. 27 (1994), pp. 73–94. doi: 10.1111/j.1365-2184.1994.tb01407.x
   PubMed Web of Science ®Google Scholar
• J.D. Murray, Mathematical Biology. I An Introduction Interdisciplinary Applied Mathematics V. 17, Springer-Verlag, New York, 2001.
   Google Scholar
• J.D. Murray, Mathematical Biology. II Spatial Models and Biomedical Applications Interdisciplinary Applied Mathematics V. 18, Springer-Verlag, New York, 2002.
   Google Scholar
• J. Nocedal and S. Wright, Numerical Optimization, Springer Series in Operations Research and Financial Engineering, Springer New York, 2006.
   Google Scholar
• A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
   Google Scholar
• I. Rekik, S. Allassonnière, O. Clatz, E. Geremia, E. Stretton, H. Delingette, and N. Ayache, Tumor growth parameters estimation and source localization from a unique time point: Application to low-grade gliomas, Comput. Vis. Image Underst. 117 (2013), pp. 238–249. doi: 10.1016/j.cviu.2012.11.001
   Web of Science ®Google Scholar
• Release name, Male subject, t1-weighted brain scan: 788_6. Available at www.nitrc.org/frs/shownotes.php?release_id=2305 (accessed March 2016).
   Google Scholar
• T. Roubíček, Nonlinear Partial Differential Equations with Applications, International Series of Numerical Mathematics, Vol. 153, Birkhäuser Verlag, Basel, 2005.
   Google Scholar
• L.A. Shepp and B.F. Logan, The Fourier reconstruction of a head section, IEEE. Trans. Nucl. Sci. 21 (1974), pp. 21–43. doi: 10.1109/TNS.1974.6499235
   Web of Science ®Google Scholar
• K.R. Swanson, E. Alvord, and J. Murray, A quantitative model for differential motility of gliomas in grey and white matter, Cell. Prolif. 33 (2000), pp. 317–329. doi: 10.1046/j.1365-2184.2000.00177.x
   PubMed Web of Science ®Google Scholar
• K.R. Swanson, E. Alvord, and J. Murray, Virtual brain tumours (gliomas) enhance the reality of medical imaging and highlight inadequacies of current therapy, Br. J. Cancer 86 (2002), pp. 14–18. doi: 10.1038/sj.bjc.6600021
   PubMed Web of Science ®Google Scholar
• P. Tracqui, From passive diffusion to active cellular migration in mathematical models of tumour invasion, Acta. Biotheor. 43 (1995), pp. 443–464. doi: 10.1007/BF00713564
   PubMed Web of Science ®Google Scholar
• D. Woodward, J. Cook, P. Tracqui, G. Cruywagen, J. Murray, and E. Alvord, A mathematical model of glioma growth: The effect of extent of surgical resection, Cell Prolif. 29 (1996), pp. 269–288. doi: 10.1111/j.1365-2184.1996.tb01580.x
   PubMed Web of Science ®Google Scholar