A detailed numerical treatment of the boundary conditions imposed by the skull on a diffusion-reaction model of glioma tumor growth. Clinical validation aspects

Applied Mathematics and Computation

Volumn 218, Issue 17, 2012, Pages 8779-8799

  Giatili, Stavroula G ^a   Stamatakos, Georgios S ^a  

^a INSTITUTE OF COMMUNICATION AND COMPUTER SYSTEMS   (Greece)

Author keywords

Boundary conditions; Diffusion; Glioblastoma multiforme; Glioma; In silico oncology; Multiscale cancer modeling

Indexed keywords

ACTIVE FIELD; BIOLOGICAL PHENOMENA; BIOLOGICAL PROBLEMS; BIOMEDICAL RESEARCH; BOUNDARY GEOMETRY; BRAIN-IMAGING DATA; COMPUTATIONAL PROBLEM; CONJUGATE GRADIENT; CRANK-NICOLSON; DIFFUSION-REACTION; DIFFUSION-REACTION MODEL; DIFFUSIVE BEHAVIOR; DIFFUSIVE GROWTH; GLIOBLASTOMA MULTIFORME; GLIOMA; MACROSCOPIC BEHAVIORS; MATHEMATICAL TREATMENTS; MULTISCALES; NEUMANN BOUNDARY CONDITION; NUMERICAL EXPERIMENTATIONS; NUMERICAL FORMULATION; NUMERICAL SOLUTION; NUMERICAL TREATMENTS; TUMOR GROWTH;

BOUNDARY CONDITIONS; CONJUGATE GRADIENT METHOD; DIFFUSION;

TUMORS;

EID: 84859430088     PISSN: 00963003     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.amc.2012.02.036     Document Type: Article

References (42)

1. J.D. Murray Mathematical Biology II: Spatial Models and Biomedical Applications third ed. 2003 Springer Verlag Inc. New York pp. 536-613
2. K.R. Swanson Quantifying glioma cell growth and invasion in vitro Math. Comput. Model. 47 2008 638 648
3. O. Clatz, P.Y. Bondiau, H. Delingette, M. Sermesant, S.K. Warfield, G. Malandain, and N. Ayache Brain tumor growth simulation. Institute National de Recherché en Informatique et en Automatique Rapp. Rech. 5187 2004
4. 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 IEEE Trans. Medical Imag. 24 10 2000 1334 1346
5. P. Macklin, and J.S. Lowengrub A new ghost cell/level set method for moving boundary problems: application to tumor growth J. Sci. Comput. 35 2008 266 299
6. O. Clatz, P. Bondiau, H. Delingette, G. Malandain, M. Sermesant, S. Warfield, and N. Ayache In silico tumor growth: application to glioblastomas MICCAI 3217 2004 337 345
7. S. Becker, J.O. Jungmann, A. Mang, and T.M. Buzug An Adaptive Landmark Scheme for Modeling Brain Deformation in Diffusion-Based Tumor Growth IFMBE Proc. 25 4 2010 41 44
8. P. McCorquodale, P. Collela, and H. Johansen A cartesian grid embedded boundary method for the heat equation in irregular domains J. Comput. Phys. 173 2001 620 635
9. A.R. Kansal, S. Torquato, G.R. Harsh, E.A. Chiocca, and T.S. Deisboeck Simulated brain tumor growth dynamics using a three-dimensional cellular automaton J. Theor. Biol. 203 2000 367 382
10. S.M. Wise, J.S. Lowengrub, H.B. Frieboes, and V. Cristini Three-dimensional multispecies nonlinear tumor growth - I Model and numerical method J. Theor. Biol. 253 2008 524 543
11. E. Konukoglou, O. Clatz, H. Delingette, and N. Ayache Personalization to Brain Gliomas Characterization and Radiotherapy Planning T. Deisboeck, G. Stamatakos, Multiscale Cancer Modelling 2011 Chapman & Hall/ CRC Press Boca Raton, FL 385 406
12. H.B. Frieboes, J.S. Lowengrub, S. Wise, X. Zheng, P. Macklin, E.L. Bearer, and V. Cristini Computer simulation of glioma growth and morphology Neuroimage 2007
13. A. Roniotis, K. Marias, V. Sakkalis, G. Stamatakos, M. Zervakis, Comparing finite elements and finite differences for developing diffusive models of glioma growth, in: Conf. Proc. IEEE Eng. Med. Biol. Soc., vol. 1, 2010, pp. 6797-6800.
14. S.G. Giatili, N.K. Uzunoglu, G.S. Stamatakos, An explicit boundary condition treatment of a diffusion-based glioblastoma tumor growth model, in: G. Stamatakos, D. Dionysiou (Eds.), in: Proc. 4th Int. Adv. Res. Workshop on In Silico Oncology and Cancer Investigation, Athens, Greece, 2010, pp. 86-88.
15. E. Milotti, and R. Chignola Emergent properties of tumor microenvironment in a real-life model of multicell tumor spheroids PLoS One 5 11 2010 e13942
16. G. Cruywagen, D. Woodward, P. Tracqui, G. Bartoo, J. Murray, and E. Alvord The modeling of diffusive tumours J. Biol. Syst. 3 1995 937 945
17. D.E. Woodward, J. Cook, P. Tracqui, G.C. Cruywagen, J.D. Murray, and E.C. Alvord A mathematical model of glioma growth: the effect of extent of surgical resection Cell. Prolif. 29 1996 269 288
18. 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. Neuropathol. Exp. Neurol. 56 1997 704
19. S. Ji, D.W. Roberts, A. Hartov, and K.D. Paulsen Brain-skull contact boundary conditions in an inverse computational deformation model Med. Image Anal. 13 2009 659 672
20. G.S. Stamatakos, D.D. Dionysiou, E.I. Zacharaki, N.A. Mouravliansky, K.S. Nikita, and N.K. Uzunoglu In silico radiation oncology: combining novel simulation algorithms with current visualization techniques Proc. IEEE, Special Issue on Bioinformatics: Advances and Challenges 90 11 2002 1764 1777
21. D.D. Dionysiou, G.S. Stamatakos, N.K. Uzunoglu, K.S. Nikita, and A. Marioli A four dimensional in vivo model of tumor response to radiotherapy: parametric validation considering radiosensitivity, genetic profile and fractionation J. Theor. Biol. 230 2004 1 20
22. G.S. Stamatakos, E.A. Kolokotroni, D.D. Dionysiou, E.Ch. Georgiadi, and C. Desmedt An advanced discrete state-discrete event multiscale simulation model of the response of a solid tumor to chemotherapy: mimicking a clinical study J. Theor. Biol. 266 2010 124 139
23. G. Stamatakos In Silico Oncology: PART I Clinically oriented cancer multilevel modeling based on discrete event simulation T.S. Deisboeck, G.S. Stamatakos, Multiscale Cancer Modeling 2011 Chapman & Hall, CRC Boca Raton, Florida, USA 9781439814406
24. G.S. Stamatakos, E.Ch. Georgiadi, N. Graf, E.A. Kolokotroni, and D.D. Dionysiou Exploiting clinical trial data drastically narrows the window of possible solutions to the problem of clinical adaptation of a multiscale cancer model PLoS One 6 3 2011 e17594
25. Incopera, Dewitt, Bergman, Lavine, Fundamentals of Heat and Mass Transfer, sixth ed., John Wiley & Sons, Unites States of America, 2007.
26. K.R. Swanson, C. Bridge, J.D. Murray, and E.C. Alvord Jr. Virtual and real brain tumors: using mathematical modeling to quantify glioma growth and invasion J. Neurol. Sci. 216 2003 1 10
27. K.R. Swanson, E.C. Alvord Jr., and J.D. Murray Dynamics of a model for brain tumors reveals a small window for therapeutic intervention Discrete Contin. Dynam. Syst. - Ser. B 4 1 2004 289 295
28. M. Dehgan Numerical solution of the three-dimensional advection-diffusion equation Appl. Math. Comput. 150 2004 5 19
29. D. Britz, O. Osterby, and J. Strutwolf Damping of Crank-Nicolson error oscillations Comput. Biol. Chem. 27 2003 253 263
30. B. Bradie, A friendly Introduction to Numerical Analysis, Pearson International ed., Unites States of America, 2006.
31. A. Jeffrey Applied Partial Differential Equations - An Introduction 2003 Academic Press. An imprint of Elsevier Science Unites States of America
32. L.A. Khan, and P.L.-F. Liu An operator-splitting algorithm for the three -dimensional diffusion equation Numer. Methods Partial Differ. Equat. 11 1995 617 624
33. H.S. Carslaw, and J.C. Jaeger Conduction of Heat in Solids second ed. 1959 Oxford University Press London
34. G. Steel, Basic Clinical Radiobiology, third ed., Arnold, London, 2002, pp. 7-22.
35. Image Processing and Analysis in Java, 2011. (accessed 20.11.11).
36. Corel, 2011. . (accessed 20.11.11).
37. Andras Csillag, Anatomy of the Living Human - Atlas of Medical Imaging, Koenemann, Koeln, 1999.
38. Multi-image Analysis GUI, 2011. (accessed 20.11.11).
39. K.R. Swanson, E.C. Alvord, and J.D. Murray Virtual brain tumors (gliomas) enhance the reality of medical imaging and highlight inadequacies of current therapy British J. Cancer 86 2002 14 18
40. A.D. Waldman, A. Jackson, S.J. Price, C.A. Clark, T.C. Booth, D.P. Auer, P.S. Tofts, D.J. Collins, M.O. Leach, and J.H. Rees Quantitative imaging biomarkers in neuro-oncology Nat. Rev. Clin. Oncol. 6 2009 445 454
41. Glioblastoma multiforme, 2011. http://en.wikipedia.org/wiki/Glioblastoma- multiforme (accessed 26.11.11).
42. N.G. Burnet, R. Jena, S.J. Jefferies, S.P. Stenning, and N.F. Kirkby Mathematical modelling of survival of glioblastoma patients suggests a role for radiotherapy dose escalation and predicts poorer outcome after delay to start treatment Clin. Oncol. 19 2006 93 103