Effective particle methods for Fisher-Kolmogorov equations: Theory and applications to brain tumor dynamics

Abstract Extended systems governed by partial differential equations can, under suitable conditions, be approximated by means of sets of ordinary differential equations for global quantities capturing the essential features of the systems dynamics. Here we obtain a small number of effective equations describing the dynamics of single-front and localized solutions of Fisher–Kolmogorov type equations. These solutions are parametrized by means of a minimal set of time-dependent quantities for which ordinary differential equations ruling their dynamics are found. A comparison of the finite dimensional equations and the dynamics of the full partial differential equation is made showing a very good quantitative agreement with the dynamics of the partial differential equation. We also discuss some implications of our findings for the understanding of the growth progression of certain types of primary brain tumors and discuss possible extensions of our results to related equations arising in different modeling scenarios.

[1]  K. Fulling,et al.  The value of radiation therapy in addition to surgery for astrocytomas of the adult cerebrum , 1985, Cancer.

[2]  Desmond Curran,et al.  Prognostic factors for survival in adult patients with cerebral low-grade glioma. , 2002, Journal of clinical oncology : official journal of the American Society of Clinical Oncology.

[3]  A. Karim,et al.  A randomized trial on dose-response in radiation therapy of low-grade cerebral glioma: European Organization for Research and Treatment of Cancer (EORTC) Study 22844. , 1996, International journal of radiation oncology, biology, physics.

[4]  Akira Yamaura,et al.  Treatment of low-grade oligodendroglial tumors without radiotherapy , 2004, Neurology.

[5]  K Hendrickson,et al.  Predicting the efficacy of radiotherapy in individual glioblastoma patients in vivo: a mathematical modeling approach , 2010, Physics in medicine and biology.

[6]  H Duffau,et al.  Improving the time‐machine: estimating date of birth of grade II gliomas , 2012, Cell proliferation.

[7]  B. Scheithauer,et al.  The 2007 WHO classification of tumours of the central nervous system , 2007, Acta Neuropathologica.

[8]  Jack F Douglas,et al.  Propagating waves of self-assembly in organosilane monolayers , 2007, Proceedings of the National Academy of Sciences.

[9]  Nikolaos K. Uzunoglu,et al.  Simulating chemotherapeutic schemes in the individualized treatment context: The paradigm of glioblastoma multiforme treated by temozolomide in vivo , 2006, Comput. Biol. Medicine.

[10]  James D. Murray.  Mathematical Biology: I. An Introduction , 2007 .

[11]  Angelo Vulpiani,et al.  Dynamical Systems Approach to Turbulence , 1998 .

[12]  Michael C. Joiner,et al.  Basic Clinical Radiobiology , 2009 .

[13]  Peter Grindrod,et al.  Patterns and Waves: The Theory and Applications of Reaction-Diffusion Equations , 1991 .

[14]  Luc Taillandier,et al.  Quantitative morphological magnetic resonance imaging follow-up of low-grade glioma: a plea for systematic measurement of growth rates. , 2012, Neurosurgery.

[15]  Jonathan A. Sherratt.  On the transition from initial data to travelling waves in the Fisher-KPP equation , 1998 .

[16]  D. Bortz,et al.  Propagation of fronts in the Fisher-Kolmogorov equation with spatially varying diffusion. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  Alexander R A Anderson,et al.  Quantifying the Role of Angiogenesis in Malignant Progression of Gliomas: in Silico Modeling Integrates Imaging and Histology Nih Public Access Author Manuscript Introduction , 2011 .

[18]  John J. Tyson,et al.  On Traveling Wave Solutions of Fisher's Equation in Two Spatial Dimensions , 1999, SIAM J. Appl. Math..

[19]  Sara Cuenda,et al.  Length scale competition in nonlinear Klein-Gordon models: a collective coordinate approach. , 2005, Chaos.

[20]  Tracy Batchelor,et al.  Low-grade gliomas in adults. , 2006, The oncologist.

[21]  M.W.Kalinowski.  Nonlinear Waves , 2016, 1611.10114.

[22]  Geirmund Unsgård,et al.  Comparison of a strategy favoring early surgical resection vs a strategy favoring watchful waiting in low-grade gliomas. , 2012, JAMA.

[23]  Víctor M. Pérez-García,et al.  The Method of Moments for Nonlinear Schrödinger Equations: Theory and Applications , 2007, SIAM J. Appl. Math..

[24]  Johan Pallud,et al.  Dynamic imaging response following radiation therapy predicts long-term outcomes for diffuse low-grade gliomas. , 2012, Neuro-oncology.

[25]  M. Williams,et al.  Basic clinical radiobiology , 1994, British Journal of Cancer.

[26]  N. Rashevsky,et al.  Mathematical biology , 1961, Connecticut medicine.

[27]  K. Swanson,et al.  A mathematical modelling tool for predicting survival of individual patients following resection of glioblastoma: a proof of principle , 2007, British Journal of Cancer.

[28]  V. Volpert,et al.  Reaction-diffusion waves in biology. , 2009, Physics of life reviews.

[29]  N. Shigesada,et al.  Biological Invasions: Theory and Practice , 1997 .

[30]  Alwyn C. Scott,et al.  Nonlinear Science: Emergence and Dynamics of Coherent Structures , 1999 .

[31]  Victor M. Perez-Garcia.  Self-similar solutions and collective coordinate methods for Nonlinear Schrodinger Equations , 2004 .

[32]  Roger Temam,et al.  The connection between infinite dimensional and finite dimensional dynamical systems : proceedings of the AMS-IMS-SIAM joint summer research conference held July 19-25, 1987, with support from the National Science Foundation and the Air Force Office of Scientific Research , 1989 .

[33]  Kristin R. Swanson,et al.  Dynamics of a model for brain tumors reveals a small window for therapeutic intervention , 2003 .

[34]  Juan Belmonte-Beitia,et al.  Bright solitary waves in malignant gliomas. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[35]  Nader Pouratian,et al.  Management of Low-Grade Glioma , 2010, Current neurology and neuroscience reports.

[36]  Hervé Delingette,et al.  Extrapolating glioma invasion margin in brain magnetic resonance images: Suggesting new irradiation margins , 2010, Medical Image Anal..

[37]  Boris A. Malomed,et al.  Chapter 2 - Variational methods in nonlinear fiber optics and related fields , 2002 .

[38]  Olivier Clatz,et al.  Biocomputing: numerical simulation of glioblastoma growth using diffusion tensor imaging , 2008, Physics in medicine and biology.

[39]  Niurka R. Quintero,et al.  Lagrangian formalism in perturbed nonlinear Klein-Gordon equations , 2004 .

[40]  L. Deangelis,et al.  Long-term outcome of low-grade oligodendroglioma and mixed glioma , 2000, Neurology.

[41]  P. K. Maini,et al.  Traveling Wave Phenomena in Some Degenerate Reaction-Diffusion Equations , 1995 .

[42]  Michalis Zervakis,et al.  Diffusive modelling of glioma evolution: a review , 2010 .

[43]  Jimmy Ruiz,et al.  Low-Grade Gliomas , 2009, Current treatment options in oncology.

[44]  I. Epstein,et al.  An Introduction to Nonlinear Chemical Dynamics , 1998 .

[45]  P K Maini,et al.  Modelling biological invasions: Individual to population scales at interfaces. , 2013, Journal of theoretical biology.

[46]  Shirley Dex,et al.  JR 旅客販売総合システム（マルス）における運用及び管理について , 1991 .

[47]  Mark J. Ablowitz,et al.  Explicit solutions of Fisher's equation for a special wave speed , 1979 .

[48]  Karima R. Khusnutdinova,et al.  Nonlinear waves in integrable and nonintegrable systems (Mathematical Modeling and Computation 16) By Jianke Yang , 2015 .

[49]  Jack Xin,et al.  Front Propagation in Heterogeneous Media , 2000, SIAM Rev..

[50]  Michel Peyrard,et al.  Physics of Solitons , 2006 .

[51]  Susan M. Chang,et al.  Role of extent of resection in the long-term outcome of low-grade hemispheric gliomas. , 2008, Journal of clinical oncology : official journal of the American Society of Clinical Oncology.

[52]  P. G. Kevrekidis,et al.  Nonlinear waves in Bose–Einstein condensates: physical relevance and mathematical techniques , 2008, 0805.0761.

[53]  Romain Joly,et al.  A STRIKING CORRESPONDENCE BETWEEN THE DYNAMICS GENERATED BY THE VECTOR FIELDS AND BY THE SCALAR PARABOLIC EQUATIONS , 2010, 1005.1727.

[54]  Yuri S. Kivshar,et al.  Dynamics of Solitons in Nearly Integrable Systems , 1989 .

[55]  P. Drazin,et al.  Solitons: An Introduction , 1989 .

[56]  Laurent Capelle,et al.  Continuous growth of mean tumor diameter in a subset of grade II gliomas , 2003, Annals of neurology.

[57]  A. N. Carvalho,et al.  Reduction of Infinite Dimensional Systems to Finite Dimensions: Compact Convergence Approach , 2013, SIAM J. Math. Anal..

[58]  M. Piérart,et al.  Long-term efficacy of early versus delayed radiotherapy for low-grade astrocytoma and oligodendroglioma in adults: the EORTC 22845 randomised trial , 2005, The Lancet.

[59]  Alicia Martínez-González,et al.  Delay effects in the response of low-grade gliomas to radiotherapy: a mathematical model and its therapeutical implications. , 2014, Mathematical medicine and biology : a journal of the IMA.

[60]  R. Guillevin,et al.  Simulation of anisotropic growth of low‐grade gliomas using diffusion tensor imaging , 2005, Magnetic resonance in medicine.

[61]  A R Bishop,et al.  Generalized traveling-wave method, variational approach, and modified conserved quantities for the perturbed nonlinear Schrödinger equation. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[62]  Norman F. Kirkby,et al.  A mathematical model of brain tumour response to radiotherapy and chemotherapy considering radiobiological aspects. , 2010, Journal of theoretical biology.

[63]  Albert Lai,et al.  Prognostic significance of growth kinetics in newly diagnosed glioblastomas revealed by combining serial imaging with a novel biomathematical model. , 2009, Cancer research.

[64]  G Powathil,et al.  Mathematical modeling of brain tumors: effects of radiotherapy and chemotherapy , 2007, Physics in medicine and biology.