What mathematical models can or cannot do in glioma description and understanding

Angélique Perrillat-Mercerot; Alain Miranville; Nicolas Bourmeyster; Carole Guillevin; Mathieu Naudin; Rémy Guillevin

doi:10.3934/dcdss.2020184

Article Contents

2020, Volume 13, Issue 8: 2165-2193. Doi: 10.3934/dcdss.2020184

This issue Previous Article Gevrey solutions of singularly perturbed differential equations, an extension to the non-autonomous case Next Article Paradoxes of vulnerability to predation in biological dynamics and mediate versus immediate causality

What mathematical models can or cannot do in glioma description and understanding

1.
Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7348 SP2MI, Equipe DACTIM-MIS, Boulevard Marie et Pierre Curie - Téléport 2, F-86962 Chasseneuil Futuroscope Cedex, France
2.
Université de Poitiers, Laboratoire Signalisation et Transports Ioniques Membranaires, ERL CNRS 7003, Equipe 4CS, CHU de Poitiers, 2 Rue de la Milétrie, F-86021 Poitiers, France
3.
Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7348 SP2MI, Equipe DACTIM-MIS, Boulevard Marie et Pierre Curie - Téléport 2, F-86962 Chasseneuil Futuroscope Cedex, France, CHU de Poitiers, 2 Rue de la Milétrie, F-86021 Poitiers, France

^* Corresponding author: Angélique Perrillat-Mercerot
^* Corresponding author: Angélique Perrillat-Mercerot

Received: February 2019

Early access: November 2019

Published: May 2020

The author would like to thank Dr Marie Flores and Jocelyne Attab for their help with the illustrations

Abstract / Introduction Full Text(HTML) Figure(4) / Table(1) Related Papers Cited by

Abstract

Our aim in this article is not to provide a review of the existing literature but to make a critical analysis on glioma behavior mathematical modeling. We present here mathematical modeling history, interests, modalities and limitations on the study of glioma growth. We also make a point on model consideration according to glioma comportments. We finally introduce medical imaging coupled with in silico models as the next step in glioma research. We do not claim completeness of the bibliography but we tried to cover a large amount of mathematical considerations for glioma behavior illustrated with selected representative papers.

Keywords:

Review,
model,
modeling,
tumor,
glioma.

Mathematics Subject Classification: 01-00, 34A30, 34A33, 35Q92, 92B05.

Citation:

Full Text(HTML)

Figure 1. Schematic representation of different scales involved in tumor dynamic. From the organ scale to the atomic scale, modifications lead to the emergence of a degenerated tissue. a) MRI reconstitution at the organ scale. b) anatomic MRI slice at the tissu scale c) Histology HES resolution x4 at the cellular scale d) MRS data at the molecular scale, the one on the top comes from the black voxel on image 1.b while the one on the bottom comes from the yellow voxel. They give an idea of the metabolic concentrations on the relative voxels.
All existing rights reserved to :
a, b, d : Carole Guillevin, Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7348, SP2MI, Equipe DACTIM-MIS, CHU de Poitiers, France.
c : Dr Marie Flores, Service de Pathologie, CHU de Poitiers, France.

Download: Full-size image PowerPoint slide

Figure 2. Schematic representation of different possible views of the tumor. In the discrete one (on the left), tumor cells are considered one by one while on the continuum representation (on the left) the tumor are considered on the whole. Medical imaging studies usually consider tumor as spheric dealing with mean tumor diameter (estimation on black).

Download: Full-size image PowerPoint slide

Figure 3. Example of overfitting problem studying the approach of a dataset with a parabolic shape including measure errors (figure a). Using linear regression with a polynomial of a lower degree implies high error on describing the known data (figure b). Using linear regression with a polynomial of higher degree reduces the error on describing the known data (figure c) and zoom out on figure d). However it may provide a regression with important variations and therefore cause problems to explain the phenomena or predict new data value

Download: Full-size image PowerPoint slide

Figure 4. Schematic representation of several behaviors of a brain tumor. Brain tumor can be seen as a multimodal object with different comportments needing different mathematical modeling. From top to bottom, from left to right : mutations and carcinogenesis (section 3.1), heterogeneity (section 3.3), energetic changeover (section 3.4), mechanical actions (section 3.6), neoangiogenesis (section 3.5), invasion (section 3.2) and therapy modeling (sections 3.8)

Download: Full-size image PowerPoint slide

Table 1. Glioma grades according to WHO classification.

Grade Tumor bounds Growth Malignancy Necrosis

Ⅰ Precise Really slow No No

Ⅱ Imprecise Slow Suspect No

Ⅲ Blurry Fast Yes Possible

Ⅳ Metastasis Really fast Yes Yes

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	A. Agosti, P. F. Antonietti, P. Ciarletta, M. Grasselli and M. Verani, A Cahn-Hilliard-type equation with application to tumor growth dynamics, Math. Methods Appl. Sci., 40 (2017), 7598-7626. doi: 10.1002/mma.4548.
[2]	Z. Agur, L. Arakelyan, P. Daugulis and Y. Ginosar, Hopf point analysis for angiogenesis models, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 29-38. doi: 10.3934/dcdsb.2004.4.29.
[3]	M. Agus, D. Boges, N. Gagnon, P. J. Magistretti, M. Hadwiger and C. Cali, GLAM: Glycogen-derived Lactate Absorption Map for visual analysis of dense and sparse surface reconstructions of rodent brain structures on desktop systems and virtual environments, Computers & Graphics, 74 (2018), 85-98. doi: 10.1016/j.cag.2018.04.007.
[4]	P. M. Altrock, L. L. Liu and F. Michor, The mathematics of cancer: Integrating quantitative models, Nature Reviews Cancer, 15 (2015), 730. doi: 10.1038/nrc4029.
[5]	D. Ambrosi and F. Mollica, The role of stress in the growth of a multicell spheroid, J. Math. Biol., 48 (2004), 477-499. doi: 10.1007/s00285-003-0238-2.
[6]	L. K. Andersen and M. C. Mackey, Resonance in periodic chemotherapy: A case study of acute myelogenous leukemia, J. Theoretical Biol., 209 (2001), 113-130. doi: 10.1006/jtbi.2000.2255.
[7]	A. R. Anderson, M. Hassanein, K. M. Branch, J. Lu, N. A. Lobdell, J. Maier, D. Basanta, B. Weidow, A. Narasanna and C. L. Arteaga et al., Microenvironmental independence associated with tumor progression, Cancer Research, 69 (2009), 8797-8806. doi: 10.1158/0008-5472.CAN-09-0437.
[8]	A. R. Anderson and V. Quaranta, Integrative mathematical oncology, Nature Reviews Cancer, 8 (2008), 227-234. doi: 10.1038/nrc2329.
[9]	A. R. Anderson, A. M. Weaver, P. T. Cummings and V. Quaranta, Tumor morphology and phenotypic evolution driven by selective pressure from the microenvironment, Cell, 127 (2006), 905-915. doi: 10.1016/j.cell.2006.09.042.
[10]	L. Arakelyan, Y. Merbl and Z. Agur, Vessel maturation effects on tumour growth: Validation of a computer model in implanted human ovarian carcinoma spheroids, European J. Cancer, 41 (2005), 159-167. doi: 10.1016/j.ejca.2004.09.012.
[11]	R. P. Araujo and D. S. McElwain, A history of the study of solid tumour growth: The contribution of mathematical modelling, Bull. Math. Biol., 66 (2004), 1039-1091. doi: 10.1016/j.bulm.2003.11.002.
[12]	O. Arino, A. Bertuzzi, A. Gandolfi, E. Sánchez and C. Sinisgalli, A model with `growth retardation' for the kinetic heterogeneity of tumour cell populations, Math. Biosci., 206 (2007), 185-199. doi: 10.1016/j.mbs.2005.04.008.
[13]	A. Aubert and R. Costalat, Interaction between astrocytes and neurons studied using a mathematical model of compartmentalized energy metabolism, J. Cerebral Blood Flow & Metabolism, 25 (2005), 1476-1490. doi: 10.1038/sj.jcbfm.9600144.
[14]	D. Balding and D. McElwain, A mathematical model of tumour-induced capillary growth, J. Theoret. Biol., 114 (1985), 53-73. doi: 10.1016/S0022-5193(85)80255-1.
[15]	M. Baron, L. Bauchet, V. Bernier, L. Capelle, D. Fontaine, P. Gatignol, J. Guyotat, M. Leroy, E. Mandonnet and J. Pallud et al., Gliomes de grade Ⅱ, EMC-Neurol., 5 (2008), 1-17. doi: 10.1016/S0246-0378(08)46100-6.
[16]	B. Basse and P. Ubezio, A generalised age-and phase-structured model of human tumour cell populations both unperturbed and exposed to a range of cancer therapies, Bull. Math. Biol., 69 (2007), 1673-1690. doi: 10.1007/s11538-006-9185-6.
[17]	L. Bauchet, Epidemiology of diffuse low grade gliomas, in Diffuse Low-Grade Gliomas in Adults, Springer, 2017, 13–53. doi: 10.1007/978-3-319-55466-2_2.
[18]	R. E. Bellman, Mathematical Methods in Medicine, Series in Modern Applied Mathematics, 1, World Scientific Publishing Co., Singapore, 1983. doi: 10.1142/0028.
[19]	S. Benzekry, C. Lamont, D. Barbolosi, L. Hlatky and P. Hahnfeldt, Mathematical modeling of tumor-tumor distant interactions supports a systemic control of tumor growth, Cancer Research, 77 (2017), 5183-5193. doi: 10.1158/0008-5472.CAN-17-0564.
[20]	S. Benzekry, A. Tracz, M. Mastri, R. Corbelli, D. Barbolosi and J. M. Ebos, Modeling spontaneous metastasis following surgery: An in vivo-in silico approach, Cancer Research, 76 (2016), 535-547. doi: 10.1158/0008-5472.CAN-15-1389.
[21]	F. Billy, B. Ribba, O. Saut, H. Morre-Trouilhet, T. Colin, D. Bresch, J.-P. Boissel, E. Grenier and J.-P. Flandrois, A pharmacologically based multiscale mathematical model of angiogenesis and its use in investigating the efficacy of a new cancer treatment strategy, J. Theoret. Biol., 260 (2009), 545-562. doi: 10.1016/j.jtbi.2009.06.026.
[22]	A. Bratus, I. Yegorov and D. Yurchenko, Dynamic mathematical models of therapy processes against glioma and leukemia under stochastic uncertainties, Meccanica dei Materiali e delle Strutture, 6 (2016), 131-138.
[23]	H. Byrne and L. Preziosi, Modelling solid tumour growth using the theory of mixtures, Math. Medicine Biol.: J. IMA, 20 (2003), 341-366. doi: 10.1093/imammb/20.4.341.
[24]	H. M. Byrne, Dissecting cancer through mathematics: From the cell to the animal model, Nature Reviews Cancer, 10 (2010), 221-230. doi: 10.1038/nrc2808.
[25]	H. Byrne, T. Alarcon, M. Owen, S. Webb and P. Maini, Modelling aspects of cancer dynamics: A review, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 364 (2006), 1563-1578. doi: 10.1098/rsta.2006.1786.
[26]	H. Byrne and M. A. J. Chaplain, Growth of nonnecrotic tumors in the presence and absence of inhibitors, Math. Biosciences, 130 (1995), 151-181. doi: 10.1016/0025-5564(94)00117-3.
[27]	M. Chaplain, Avascular growth, angiogenesis and vascular growth in solid tumours: The mathematical modelling of the stages of tumour development, Math. Comput. Modell., 23 (1996), 47-87. doi: 10.1016/0895-7177(96)00019-2.
[28]	G. Cheng, J. Tse, R. K. Jain and L. L. Munn, Micro-environmental mechanical stress controls tumor spheroid size and morphology by suppressing proliferation and inducing apoptosis in cancer cells, PLoS one, 4 (2009). doi: 10.1371/journal.pone.0004632.
[29]	E. B. Claus and P. M. Black, Survival rates and patterns of care for patients diagnosed with supratentorial low-grade gliomas, Cancer: Interdisciplinary Internat. J. Amer. Cancer Soc., 106 (2006), 1358-1363. doi: 10.1002/cncr.21733.
[30]	C. A. Cocosco, V. Kollokian, R. K.-S. Kwan, G. B. Pike and A. C. Evans, Brainweb: Online interface to a 3D MRI simulated brain database, in NeuroImage, Citeseer, 1997.
[31]	E. A. Codling, M. J. Plank and S. Benhamou, Random walk models in biology, J. Royal Soc. Interface, 5 (2008), 813-834. doi: 10.1098/rsif.2008.0014.
[32]	V. Cristini, X. Li, J. S. Lowengrub and S. M. Wise, Nonlinear simulations of solid tumor growth using a mixture model: Invasion and branching, J. Math. Biol., 58 (2009), 723-763. doi: 10.1007/s00285-008-0215-x.
[33]	E. De Angelis and L. Preziosi, Advection-diffusion models for solid tumour evolution in vivo and related free boundary problem, Math. Models Methods Appl. Sci., 10 (2000), 379-407. doi: 10.1142/S0218202500000239.
[34]	L. G. de Pillis, W. Gu and A. E. Radunskaya, Mixed immunotherapy and chemotherapy of tumors: Modeling, applications and biological interpretations, J. Theoret. Biol., 238 (2006), 841-862. doi: 10.1016/j.jtbi.2005.06.037.
[35]	L. Demetrius and J. Tuszynski, Quantum metabolism explains the allometric scaling of metabolic rates, J. Royal Soc. Interface, 7 (2010), 507-514. doi: 10.1098/rsif.2009.0310.
[36]	L. A. Demetrius, J. F. Coy and J. A. Tuszynski, Cancer proliferation and therapy: The Warburg effect and quantum metabolism, Theoret. Biol. Medical Modell., 7 (2010). doi: 10.1186/1742-4682-7-2.
[37]	L. A. Demetrius and D. K. Simon, An inverse-Warburg effect and the origin of Alzheimer's disease, Biogerontology, 13 (2012), 583-594. doi: 10.1007/s10522-012-9403-6.
[38]	D. Dingli, M. D. Cascino, K. Josić, S. J. Russell and Ž. Bajzer, Mathematical modeling of cancer radiovirotherapy, Math. Biosci., 199 (2006), 55-78. doi: 10.1016/j.mbs.2005.11.001.
[39]	H. Duffau and L. Taillandier, New individualized and dynamic therapeutic strategies in DLGG, in Diffuse Low-Grade Gliomas in Adults, Springer, Cham, 2017,609–624. doi: 10.1007/978-3-319-55466-2_28.
[40]	A. d'Onofrio and A. Gandolfi, Tumour eradication by antiangiogenic therapy: Analysis and extensions of the model by Hahnfeldt et al.(1999), Math. Biosci., 191 (2004), 159-184. doi: 10.1016/j.mbs.2004.06.003.
[41]	A. Ergun, K. Camphausen and L. M. Wein, Optimal scheduling of radiotherapy and angiogenic inhibitors, Bull. Math. Biol., 65 (2003), 407-424. doi: 10.1016/S0092-8240(03)00006-5.
[42]	M. E. Fernández-Sánchez, S. Barbier, J. Whitehead, G. Béalle, A. Michel, H. Latorre-Ossa, C. Rey, L. Fouassier, A. Claperon and L. Brullé et al., Mechanical induction of the tumorigenic $\beta$–catenin pathway by tumour growth pressure, Nature, 523 (2015), 92-95. doi: 10.1038/nature14329.
[43]	U. Forys, Y. Kheifetz and Y. Kogan, Critical-point analysis for three-variable cancer angiogenesis models, Math. Biosci. Eng., 2 (2005), 511-525. doi: 10.3934/mbe.2005.2.511.
[44]	H. B. Frieboes, X. Zheng, C.-H. Sun, B. Tromberg, R. Gatenby and V. Cristini, An integrated computational/experimental model of tumor invasion, Cancer Research, 66 (2006), 1597-1604. doi: 10.1158/0008-5472.CAN-05-3166.
[45]	M. Gałach, Dynamics of the tumor-immune system competition-the effect of time delay, Int. J. Appl. Math. Comput. Sci., 13 (2003), 395-406.
[46]	H. Garcke, K. F. Lam, E. Sitka and V. Styles, A Cahn–Hilliard–Darcy model for tumour growth with chemotaxis and active transport, Math. Models Methods Appl. Sci., 26 (2016), 1095-1148. doi: 10.1142/S0218202516500263.
[47]	R. Gatenby, K. Smallbone, P. Maini, F. Rose, J. Averill, R. Nagle, L. Worrall and R. Gillies, Cellular adaptations to hypoxia and acidosis during somatic evolution of breast cancer, British J. Cancer, 97 (2007), 646-653. doi: 10.1038/sj.bjc.6603922.
[48]	R. A. Gatenby and T. L. Vincent, An evolutionary model of carcinogenesis, Cancer Research, 63 (2003), 6212-6220.
[49]	R. A. Gatenby, T. L. Vincent and R. J. Gillies, Evolutionary dynamics in carcinogenesis, Math. Models Methods Appl. Sci., 15 (2005), 1619-1638. doi: 10.1142/S0218202505000911.
[50]	L. Gay, A.-M. Baker and T. A. Graham, Tumour cell heterogeneity, F1000 Res., 5 (2016). doi: 10.12688/f1000research.7210.1.
[51]	C. Gerin, Modelisation et etude histologique de gliomes diffus de bas grade, Ph.D thesis, Université Paris-Diderot in Paris, 2012.
[52]	H. Gomez, Quantitative analysis of the proliferative-to-invasive transition of hypoxic glioma cells, Integrative Biol., 9 (2017), 257-262. doi: 10.1039/C6IB00208K.
[53]	L. Graziano and L. Preziosi, Mechanics in tumor growth, in Modeling of Biological Materials, Model. Simul. Sci. Eng. Technol., Birkhäuser Boston, Boston, MA, 2007,263–321. doi: 10.1007/978-0-8176-4411-6_7.
[54]	H. Greenspan, Models for the growth of a solid tumor by diffusion, Studies in Appl. Math., 51 (1972), 317-340. doi: 10.1002/sapm1972514317.
[55]	H. Greenspan, On the growth and stability of cell cultures and solid tumors, J. Theoret. Biol., 56 (1976), 229-242. doi: 10.1016/S0022-5193(76)80054-9.
[56]	C. Guillevin, R. Guillevin, A. Miranville and A. Perrillat-Mercerot, Analysis of a mathematical model for brain lactate kinetics, Math. Biosci. Eng., 15 (2018), 1225-1242. doi: 10.3934/mbe.2018056.
[57]	C. Guiot, N. Pugno and P. P. Delsanto, Elastomechanical model of tumor invasion, Appl. Phys. Lett., 89 (2006). doi: 10.1063/1.2398910.
[58]	P. Hahnfeldt, D. Panigrahy, J. Folkman and L. Hlatky, Tumor development under angiogenic signaling: A dynamical theory of tumor growth, treatment response, and postvascular dormancy, Cancer Research, 59 (1999), 4770-4775.
[59]	H. L. Harpold, E. C. Alvord Jr. and K. R. Swanson, The evolution of mathematical modeling of glioma proliferation and invasion, J. Neuropathology & Experimental Neurology, 66 (2007), 1-9. doi: 10.1097/nen.0b013e31802d9000.
[60]	T. E. Harris, The Theory of Branching Processes, Dover Publications, Inc., Mineola, NY, 2002.
[61]	H. Hatzikirou, D. Basanta, M. Simon, K. Schaller and A. Deutsch, `Go or grow': The key to the emergence of invasion in tumour progression?, Math. Med. Biol., 29 (2012), 49-65. doi: 10.1093/imammb/dqq011.
[62]	P. Herzmark, K. Campbell, F. Wang, K. Wong, H. El-Samad, A. Groisman and H. R. Bourne, Bound attractant at the leading vs. the trailing edge determines chemotactic prowess, Proceedings of the National Academy of Sciences, 104 (2007), 13349-13354. doi: 10.1073/pnas.0705889104.
[63]	E. Höring, P. N. Harter, J. Seznec, J. Schittenhelm, H.-J. Bühring, S. Bhattacharyya, E. von Hattingen, C. Zachskorn, M. Mittelbronn and U. Naumann, The "go or grow'' potential of gliomas is linked to the neuropeptide processing enzyme carboxypeptidase E and mediated by metabolic stress, Acta Neuropathologica, 124 (2012), 83-97. doi: 10.1007/s00401-011-0940-x.
[64]	P. Huneman and S. Dutreuil, Considérations épistémologiques sur la modélisation mathématique en biologie, 2013.
[65]	Y. Iwasa, M. A. Nowak and F. Michor, Evolution of resistance during clonal expansion, Genetics, 172 (2006), 2557-2566. doi: 10.1534/genetics.105.049791.
[66]	K. Jaeckle, P. Decker, K. Ballman, P. Flynn, C. Giannini, B. Scheithauer, R. Jenkins and J. Buckner, Transformation of low grade glioma and correlation with outcome: An NCCTG database analysis, J. Neuro-Oncology, 104 (2011), 253-259. doi: 10.1007/s11060-010-0476-2.
[67]	Y. Jiang, J. Pjesivac-Grbovic, C. Cantrell and J. P. Freyer, A multiscale model for avascular tumor growth, Biophysical Journal, 89 (2005), 3884-3894. doi: 10.1529/biophysj.105.060640.
[68]	R. Jolivet, J. S. Coggan, I. Allaman and P. J. Magistretti, Multi-timescale modeling of activity-dependent metabolic coupling in the neuron-glia-vasculature ensemble, PLoS Comput Biol, 11 (2015). doi: 10.1371/journal.pcbi.1004036.
[69]	A. Jones, H. Byrne, J. Gibson and J. Dold, A mathematical model of the stress induced during avascular tumour growth, J. Math. Biol., 40 (2000), 473-499. doi: 10.1007/s002850000033.
[70]	J. Kassis, D. A. Lauffenburger, T. Turner and A. Wells, Tumor invasion as dysregulated cell motility, Seminars in Cancer Biology, 11 (2001), 105-117. doi: 10.1006/scbi.2000.0362.
[71]	A. Kaznatcheev, J. G. Scott and D. Basanta, Edge effects in game-theoretic dynamics of spatially structured tumours, J. Royal Society Interface, 12 (2015). doi: 10.1098/rsif.2015.0154.
[72]	E. Khain and L. M. Sander, Generalized Cahn-Hilliard equation for biological applications, Phys. Rev. E, 77 (2008). doi: 10.1103/PhysRevE.77.051129.
[73]	S. Khajanchi, Modeling the dynamics of glioma-immune surveillance, Chaos Solitons Fractals, 114 (2018), 108-118. doi: 10.1016/j.chaos.2018.06.028.
[74]	Y. Kim, M. A. Stolarska and H. G. Othmer, A hybrid model for tumor spheroid growth in vitro. Ⅰ: Theoretical development and early results, Math. Models Methods Appl. Sci., 17 (2007), 1773-1798. doi: 10.1142/S0218202507002479.
[75]	N. L. Komarova, Spatial stochastic models for cancer initiation and progression, Bull. Math. Biol., 68 (2006), 1573-1599. doi: 10.1007/s11538-005-9046-8.
[76]	N. L. Komarova, Stochastic modeling of loss-and gain-of-function mutations in cancer, Math. Models Methods Appl. Sci., 17 (2007), 1647-1673. doi: 10.1142/S021820250700242X.
[77]	N. L. Komarova, A. Sengupta and M. A. Nowak, Mutation–selection networks of cancer initiation: Tumor suppressor genes and chromosomal instability, J. Theoret. Biol., 223 (2003), 433-450. doi: 10.1016/S0022-5193(03)00120-6.
[78]	N. Kronik, Y. Kogan, V. Vainstein and Z. Agur, Improving alloreactive CTL immunotherapy for malignant gliomas using a simulation model of their interactive dynamics, Cancer Immunology, Immunotherapy, 57 (2008), 425-439. doi: 10.1007/s00262-007-0387-z.
[79]	V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis, Bull. Math. Biol., 56 (1994), 295-321. doi: 10.1016/S0092-8240(05)80260-5.
[80]	S. K. Kyriacou, C. Davatzikos, S. J. Zinreich and R. N. Bryan, Nonlinear elastic registration of brain images with tumor pathology using a biomechanical model [MRI], IEEE Transactions on Medical Imaging, 18 (1999), 580-592. doi: 10.1109/42.790458.
[81]	C. A. La Porta and S. Zapperi, The Physics of Cancer, Cambridge University Press, 2017. doi: 10.1017/9781316271759.
[82]	C. A. La Porta, S. Zapperi and J. P. Sethna, Senescent cells in growing tumors: Population dynamics and cancer stem cells, PLoS Comput. Biol., 8 (2012), 13pp. doi: 10.1371/journal.pcbi.1002316.
[83]	J.-B. Lagaert, Modélisation de la croissance tumorale: estimation de paramètres d'un modèle de croissance et introduction d'un modèle spécifique aux gliomes de tout grade, Ph.D thesis, Université Sciences et Technologies-Bordeaux I, 2011.
[84]	L. A. Liotta, J. Kleinerman and G. M. Saidel, Quantitative relationships of intravascular tumor cells, tumor vessels, and pulmonary metastases following tumor implantation, Cancer Research, 34 (1974), 997-1004.
[85]	J. S. Lowengrub, H. B. Frieboes, F. Jin, Y.-L. Chuang, X. Li, P. Macklin, S. M. Wise and V. Cristini, Nonlinear modelling of cancer: Bridging the gap between cells and tumours, Nonlinearity, 23 (2010), R1–R91. doi: 10.1088/0951-7715/23/1/R01.
[86]	P. Macklin, When seeing isn't believing: How math can guide our interpretation of measurements and experiments, Cell Syst., 5 (2017), 92-94. doi: 10.1016/j.cels.2017.08.005.
[87]	P. Macklin, S. McDougall, A. R. Anderson, M. A. Chaplain, V. Cristini and J. Lowengrub, Multiscale modelling and nonlinear simulation of vascular tumour growth, J. Math. Biol., 58 (2009), 765-798. doi: 10.1007/s00285-008-0216-9.
[88]	E. Mandonnet, Biomathematical modeling of DLGG, in Diffuse Low-Grade Gliomas in Adults, Springer, 2017,651–664.
[89]	E. Mandonnet, Dynamics of DLGG and clinical implications, in Diffuse Low-Grade Gliomas in Adults, Springer, 2017,287–306. doi: 10.1007/978-3-319-55466-2_16.
[90]	E. Mandonnet, J.-Y. Delattre, M.-L. Tanguy, K. R. Swanson, A. F. Carpentier, H. Duffau, P. Cornu, R. Van Effenterre, E. C. Alvord Jr. and L. Capelle, Continuous growth of mean tumor diameter in a subset of grade Ⅱ gliomas, Annals of Neurology, 53 (2003), 524-528. doi: 10.1002/ana.10528.
[91]	I. Manini, F. Caponnetto, A. Bartolini, T. Ius, L. Mariuzzi, C. Di Loreto, A. P. Beltrami and D. Cesselli, Role of microenvironment in glioma invasion: What we learned from in vitro models, Internat. J. Molecular Sciences, 19 (2018). doi: 10.3390/ijms19010147.
[92]	P. Mazzocco, C. Barthélémy, G. Kaloshi, M. Lavielle, D. Ricard, A. Idbaih, D. Psimaras, M.-A. Renard, A. Alentorn and J. Honnorat et al., Prediction of response to temozolomide in low-grade glioma patients based on tumor size dynamics and genetic characteristics, CPT: Pharmacometrics Syst. Pharmacology, 4 (2015), 728-737. doi: 10.1002/psp4.54.
[93]	L. M. Merlo, J. W. Pepper, B. J. Reid and C. C. Maley, Cancer as an evolutionary and ecological process, Nature Reviews Cancer, 6 (2006), 924-935. doi: 10.1038/nrc2013.
[94]	C. Metzner, C. Mark, J. Steinwachs, L. Lautscham, F. Stadler and B. Fabry, Superstatistical analysis and modelling of heterogeneous random walks, Nature Commun., 6 (2015). doi: 10.1038/ncomms8516.
[95]	F. Michor, M. A. Nowak and Y. Iwasa, Evolution of resistance to cancer therapy, Current Pharmaceutical Design, 12 (2006), 261-271. doi: 10.2174/138161206775201956.
[96]	T. Mikkelsen, S. A. Enam and R. M. L., Invasion in malignant glioma, Youman's Neurological Surgery, 687–713.
[97]	E. Moeendarbary, L. Valon, M. Fritzsche, A. R. Harris, D. A. Moulding, A. J. Thrasher, E. Stride, L. Mahadevan and G. T. Charras, The cytoplasm of living cells behaves as a poroelastic material, Nature Materials, 12 (2013), 253-261. doi: 10.1038/nmat3517.
[98]	J. Monod, On chance and necessity, in Studies in the Philosophy of Biology, Springer, 1974,357–375. doi: 10.1007/978-1-349-01892-5_20.
[99]	P. A. P. Moran, Random processes in genetics, Proc. Cambridge Philos. Soc., 54 (1958), 60-71. doi: 10.1017/S0305004100033193.
[100]	M. Naudin, B. Tremblais, C. Guillevin, R. Guillevin and C. Fernandez-Maloigne, Diffuse low grade glioma NMR assessment for better intra-operative targeting using fuzzy logic, in International Conference on Advanced Concepts for Intelligent Vision Systems, Springer, 2018,200–210. doi: 10.1007/978-3-030-01449-0_17.
[101]	C. Nieder, A. Grosu and M. Molls, A comparison of treatment results for recurrent malignant gliomas, Cancer Treatment Reviews, 26 (2000), 397-409. doi: 10.1053/ctrv.2000.0191.
[102]	B. Novák and J. J. Tyson, A model for restriction point control of the mammalian cell cycle, J. Theoret. Biol., 230 (2004), 563-579. doi: 10.1016/j.jtbi.2004.04.039.
[103]	J. Pallud, E. Mandonnet, H. Duffau, M. Kujas, R. Guillevin, D. Galanaud, L. Taillandier and L. Capelle, Prognostic value of initial magnetic resonance imaging growth rates for World Health Organization grade Ⅱ gliomas, Annals of Neurology, 60 (2006), 380-383. doi: 10.1002/ana.20946.
[104]	A. Perrillat-Mercerot, N. Bourmeyster, C. Guillevin, R. Guillevin and A. Miranville, Mathematical modeling of substrates fluxes and tumor growth in the brain, Acta Biotheoretica, 67 (2019), 149-175. doi: 10.1007/s10441-019-09343-1.
[105]	K. A. Rejniak and A. R. Anderson, Hybrid models of tumor growth, Wiley Interdisciplinary Reviews: Systems Biology and Medicine, 3 (2011), 115-125. doi: 10.1002/wsbm.102.
[106]	V. Rigau, Towards an intermediate grade in glioma classification, in Diffuse Low-Grade Gliomas in Adults, Springer-Verlag, London, 2017,101–108. doi: 10.1007/978-3-319-55466-2_5.
[107]	R. Rockne, J. Rockhill, M. Mrugala, A. Spence, I. Kalet, K. Hendrickson, A. Lai, T. Cloughesy, E. Alvord Jr. and K. Swanson, Predicting the efficacy of radiotherapy in individual glioblastoma patients in vivo: A mathematical modeling approach, Physics in Medicine & Biology, 55 (2010), 3271-3285. doi: 10.1088/0031-9155/55/12/001.
[108]	E. T. Roussos, J. S. Condeelis and A. Patsialou, Chemotaxis in cancer, Nature Reviews Cancer, 11 (2011), 573-587. doi: 10.1038/nrc3078.
[109]	R. G. Sargent, Verification and validation of simulation models, Proceedings of the 2009 Winter Simulation Conference (WSC), 2009,162–176. doi: 10.1109/WSC.2011.6147750.
[110]	E. Sherer, R. Hannemann, A. Rundell and D. Ramkrishna, Analysis of resonance chemotherapy in leukemia treatment via multi-staged population balance models, J. Theoret. Biol., 240 (2006), 648-661. doi: 10.1016/j.jtbi.2005.11.017.
[111]	K. Smallbone, R. A. Gatenby, R. J. Gillies, P. K. Maini and D. J. Gavaghan, Metabolic changes during carcinogenesis: Potential impact on invasiveness, J. Theoret. Biol., 244 (2007), 703-713. doi: 10.1016/j.jtbi.2006.09.010.
[112]	J. S. Smith and R. B. Jenkins, Genetic alterations in adult diffuse glioma: Occurrence, significance, and prognostic implications, Front Biosci., 5 (2000), 213-231. doi: 10.2741/smith.
[113]	N. R. Smoll, O. P. Gautschi, B. Schatlo, K. Schaller and D. C. Weber, Relative survival of patients with supratentorial low-grade gliomas, Neuro-Oncology, 14 (2012), 1062-1069. doi: 10.1093/neuonc/nos144.
[114]	S. L. Spencer, M. J. Berryman, J. A. Garcia and D. Abbott, An ordinary differential equation model for the multistep transformation to cancer, J. Theoret. Biol., 231 (2004), 515-524. doi: 10.1016/j.jtbi.2004.07.006.
[115]	K. R. Swanson, C. Bridge, J. Murray and E. C. Alvord Jr., Virtual and real brain tumors: Using mathematical modeling to quantify glioma growth and invasion, J. Neurological Sci., 216 (2003), 1-10. doi: 10.1016/j.jns.2003.06.001.
[116]	A. Swierniak, M. Kimmel and J. Smieja, Mathematical modeling as a tool for planning anticancer therapy, European J. Pharmacology, 625 (2009), 108-121. doi: 10.1016/j.ejphar.2009.08.041.
[117]	P. Swietach, R. D. Vaughan-Jones, A. L. Harris and A. Hulikova, The chemistry, physiology and pathology of pH in cancer, Phil. Trans. R. Soc. B, 369 (2014). doi: 10.1098/rstb.2013.0099.
[118]	Y. Tao and M. Wang, Global solution for a chemotactic-haptotactic model of cancer invasion, Nonlinearity, 21 (2008), 2221-2238. doi: 10.1088/0951-7715/21/10/002.
[119]	C. Tomasetti and D. Levy, Role of symmetric and asymmetric division of stem cells in developing drug resistance, Proceedings of the National Academy of Sciences, 107, 2010, 16766–16771. doi: 10.1073/pnas.1007726107.
[120]	P. Tracqui, Biophysical models of tumour growth, Reports on Progress in Physics, 72 (2009). doi: 10.1088/0034-4885/72/5/056701.
[121]	P. Tracqui, From passive diffusion to active cellular migration in mathematical models of tumour invasion, Acta Biotheoretica, 43 (1995), 443-464. doi: 10.1007/BF00713564.
[122]	P. Tracqui and M. Mendjeli, Modelling three-dimensional growth of brain tumours from time series of scans, Math. Models Methods Appl. Sci., 9 (1999), 581-598. doi: 10.1142/S0218202599000300.
[123]	A. Vazquez and Z. N. Oltvai, Molecular crowding defines a common origin for the warburg effect in proliferating cells and the lactate threshold in muscle physiology, PloS one, 6 (2011). doi: 10.1371/journal.pone.0019538.
[124]	G. Vilanova, I. Colominas and H. Gomez, A mathematical model of tumour angiogenesis: Growth, regression and regrowth, J. Royal Society Interface, 14 (2017). doi: 10.1098/rsif.2016.0918.
[125]	M. Villasana and A. Radunskaya, A delay differential equation model for tumor growth, J. Math. Biol., 47 (2003), 270-294. doi: 10.1007/s00285-003-0211-0.
[126]	R. Wasserman, R. Acharya, C. Sibata and K. Shin, A patient-specific in vivo tumor model, Math. Biosci., 136 (1996), 111-140. doi: 10.1016/0025-5564(96)00045-4.
[127]	S. M. Wise, J. S. Lowengrub, H. B. Frieboes and V. Cristini, Three-dimensional multispecies nonlinear tumor growth–I: Model and numerical method, J. Theoret. Biol., 253 (2008), 524-543. doi: 10.1016/j.jtbi.2008.03.027.
[128]	D. Wodarz and N. L. Komarova, Dynamics of Cancer: Mathematical Foundations of Oncology, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2014. doi: 10.1142/8973.
[129]	D. E. Woodward, J. Cook, P. Tracqui, G. Cruywagen, J. Murray and E. Alvord Jr., A mathematical model of glioma growth: The effect of extent of surgical resection, Cell Proliferation, 29 (1996), 269-288. doi: 10.1111/j.1365-2184.1996.tb01580.x.
[130]	P.-H. Wu, A. Giri, S. X. Sun and D. Wirtz, Three-dimensional cell migration does not follow a random walk, Proceedings of the National Academy of Sciences, 11, 2014, 3949–3954. doi: 10.1073/pnas.1318967111.