A Flux-Limited Model for Glioma Patterning with Hypoxia-Induced Angiogenesis

We propose a model for glioma patterns in a microlocal tumor environment under the influence of acidity, angiogenesis, and tissue anisotropy. The bottom-up model deduction eventually leads to a system of reaction-diffusion-taxis equations for glioma and endothelial cell population densities, of which the former infers flux limitation both in the self-diffusion and taxis terms. The model extends a recently introduced [34] description of glioma pseudopalisade formation, with the aim of studying the effect of hypoxia-induced tumor vascularization on the establishment and maintenance of these histological patterns which are typical for high grade brain cancer. Numerical simulations of the population level dynamics are performed to investigate several model scenarios containing this and further effects.

[1]  D. Burini,et al.  A multiscale view of nonlinear diffusion in biology: From cells to tissues , 2019, Mathematical Models and Methods in Applied Sciences.

[2]  Avner Friedman,et al.  A mathematical model for pattern formation of glioma cells outside the tumor spheroid core. , 2009, Journal of theoretical biology.

[3]  E. Kostelich,et al.  Virtual glioblastoma: growth, migration and treatment in a three‐dimensional mathematical model , 2009, Cell proliferation.

[4]  K. Swanson,et al.  The biology and mathematical modelling of glioma invasion: a review , 2017, Journal of The Royal Society Interface.

[5]  A. Klar,et al.  Modeling glioma invasion with anisotropy- and hypoxia-triggered motility enhancement: from subcellular dynamics to macroscopic PDEs with multiple taxis , 2020, Mathematical Models and Methods in Applied Sciences.

[6]  Jian Wang,et al.  Chemoresistance caused by the microenvironment of glioblastoma and the corresponding solutions. , 2019, Biomedicine & pharmacotherapy = Biomedecine & pharmacotherapie.

[7]  Salvatore Torquato,et al.  Modeling the effects of vasculature evolution on early brain tumor growth. , 2006, Journal of theoretical biology.

[8]  Pawan Kumar,et al.  Multiscale modeling of glioma pseudopalisades: contributions from the tumor microenvironment , 2020, Journal of Mathematical Biology.

[9]  Pieter Wesseling,et al.  Histologic classification of gliomas. , 2016, Handbook of clinical neurology.

[10]  R. Hartmann-Petersen,et al.  NCAM regulates cell motility. , 2002, Journal of cell science.

[12]  Z. Ou-Yang,et al.  Behaviors of Glioblastoma Cells in in Vitro Microenvironments , 2019, Scientific Reports.

[13]  Luigi Preziosi,et al.  Modeling cell movement in anisotropic and heterogeneous network tissues , 2007, Networks Heterog. Media.

[14]  B. Scheithauer,et al.  Histopathology, classification, and grading of gliomas , 1995, Glia.

[15]  Alfonso Caiazzo,et al.  Multiscale modelling of palisade formation in gliobastoma multiforme. , 2014, Journal of theoretical biology.

[16]  T. Hillen.  M5 mesoscopic and macroscopic models for mesenchymal motion , 2006, Journal of mathematical biology.

[17]  C. Engwer,et al.  A multiscale model for glioma spread including cell-tissue interactions and proliferation. , 2015, Mathematical biosciences and engineering : MBE.

[18]  C. Schaller,et al.  MATHEMATICAL MODELLING OF GLIOBLASTOMA TUMOUR DEVELOPMENT: A REVIEW , 2005 .

[19]  Christina Surulescu,et al.  Mathematical modeling of glioma invasion: acid- and vasculature mediated go-or-grow dichotomy and the influence of tissue anisotropy , 2020, Appl. Math. Comput..

[20]  Michael Berens,et al.  A mathematical model of glioblastoma tumor spheroid invasion in a three-dimensional in vitro experiment. , 2007, Biophysical journal.

[21]  Christina Surulescu,et al.  A Multiscale Modeling Approach to Glioma Invasion with Therapy , 2017 .

[22]  Q. Long,et al.  Mathematical Modelling of a Brain Tumour Initiation and Early Development: A Coupled Model of Glioblastoma Growth, Pre-Existing Vessel Co-Option, Angiogenesis and Blood Perfusion , 2016, PloS one.

[23]  N. Bellomo,et al.  On the Interaction Between Soft and Hard Sciences: the Role of Mathematical Sciences , 2020 .

[24]  M. Tate,et al.  Biology of angiogenesis and invasion in glioma , 2009, Neurotherapeutics.

[25]  Nicola Bellomo,et al.  A Quest Towards a Mathematical Theory of Living Systems , 2017 .

[26]  N. Bellomo,et al.  A degenerate chemotaxis system with flux limitation: Maximally extended solutions and absence of gradient blow-up , 2016, 1605.01924.

[27]  M. Winkler.  Singular structure formation in a degenerate haptotaxis model involving myopic diffusion , 2017, 1706.05211.

[28]  K. Painter,et al.  Anisotropic diffusion in oriented environments can lead to singularity formation , 2012, European Journal of Applied Mathematics.

[29]  Erwin G. Van Meir,et al.  Genetic and Biologic Progression in Astrocytomas and Their Relation to Angiogenic Dysregulation , 2002, Advances in anatomic pathology.

[30]  Daniel J Brat,et al.  'Pseudopalisading' Necrosis in Glioblastoma: A Familiar Morphologic Feature That Links Vascular Pathology, Hypoxia, and Angiogenesis , 2006, Journal of neuropathology and experimental neurology.

[31]  C. Surulescu,et al.  Global weak solutions to a strongly degenerate haptotaxis model , 2016, 1603.04233.

[32]  C. Engwer,et al.  Effective equations for anisotropic glioma spread with proliferation: a multiscale approach and comparisons with previous settings. , 2016, Mathematical medicine and biology : a journal of the IMA.

[33]  B. Perthame,et al.  The flux limited Keller–Segel system; properties and derivation from kinetic equations , 2018, Revista Matemática Iberoamericana.

[34]  Matthew P. Jacobson,et al.  Dysregulated pH: a perfect storm for cancer progression , 2011, Nature Reviews Cancer.

[35]  J. Lennerz,et al.  Neuropathology for the neuroradiologist: palisades and pseudopalisades. , 2006, AJNR. American journal of neuroradiology.

[36]  A. Czirók.  Endothelial cell motility, coordination and pattern formation during vasculogenesis , 2013, Wiley interdisciplinary reviews. Systems biology and medicine.

[37]  C. Haudenschild,et al.  Endothelial cell motility , 1984 .

[38]  Daniel J Brat,et al.  Pseudopalisades in Glioblastoma Are Hypoxic, Express Extracellular Matrix Proteases, and Are Formed by an Actively Migrating Cell Population , 2004, Cancer Research.

[39]  Milad Shamsi,et al.  Mathematical Modeling of the Function of Warburg Effect in Tumor Microenvironment , 2018, Scientific Reports.

[40]  R K Jain,et al.  Noninvasive measurement of interstitial pH profiles in normal and neoplastic tissue using fluorescence ratio imaging microscopy. , 1994, Cancer research.

[41]  J. D. de Groot,et al.  Antiangiogenic Therapy for Glioblastoma: Current Status and Future Prospects , 2014, Clinical Cancer Research.

[42]  Yang Kuang,et al.  Mathematically modeling the biological properties of gliomas: A review. , 2015, Mathematical biosciences and engineering : MBE.

[43]  D. Soll,et al.  Cofilin determines the migration behavior and turning frequency of metastatic cancer cells , 2007, The Journal of cell biology.

[44]  I. Date,et al.  Angiogenesis and invasion in glioma , 2011, Brain Tumor Pathology.

[45]  F. Feuerhake,et al.  Why one-size-fits-all vaso-modulatory interventions fail to control glioma invasion: in silico insights , 2016, Scientific Reports.

[46]  Juan Soler,et al.  MULTISCALE BIOLOGICAL TISSUE MODELS AND FLUX-LIMITED CHEMOTAXIS FOR MULTICELLULAR GROWING SYSTEMS , 2010 .

[48]  K. Painter,et al.  Transport and anisotropic diffusion models for movement in oriented habitats , 2013 .

[50]  Axel Klar,et al.  Higher-order models for glioma invasion: From a two-scale description to effective equations for mass density and momentum , 2018, Mathematical Models and Methods in Applied Sciences.

[51]  Víctor M. Pérez-García,et al.  Hypoxic Cell Waves Around Necrotic Cores in Glioblastoma: A Biomathematical Model and Its Therapeutic Implications , 2012, Bulletin of Mathematical Biology.

[52]  Joachim Weickert,et al.  Anisotropic diffusion in image processing , 1996 .

[53]  C. Surulescu,et al.  Modeling multiple taxis: Tumor invasion with phenotypic heterogeneity, haptotaxis, and unilateral interspecies repellence , 2020, Discrete & Continuous Dynamical Systems - B.

[54]  K. Painter,et al.  Mathematical modelling of glioma growth: the use of Diffusion Tensor Imaging (DTI) data to predict the anisotropic pathways of cancer invasion. , 2013, Journal of theoretical biology.

[55]  T. Hillen,et al.  Glioma follow white matter tracts: a multiscale DTI-based model , 2015, Journal of mathematical biology.

[56]  Leonardo F. Jurado,et al.  Histopathology , 2019, Fungal Infections of the Central Nervous System.

[57]  P. Cochat,et al.  Et al , 2008, Archives de pediatrie : organe officiel de la Societe francaise de pediatrie.

[58]  Daniel J Brat,et al.  Vaso-occlusive and prothrombotic mechanisms associated with tumor hypoxia, necrosis, and accelerated growth in glioblastoma , 2004, Laboratory Investigation.

[59]  N. Bellomo,et al.  Finite-time blow-up in a degenerate chemotaxis system with flux limitation , 2017 .

[60]  M. Plank,et al.  A mathematical model of tumour angiogenesis, regulated by vascular endothelial growth factor and the angiopoietins. , 2004, Journal of theoretical biology.