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August  2020, 13(8): 2165-2193. doi: 10.3934/dcdss.2020184

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

Received  February 2019 Published  August 2020 Early access  November 2019

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

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.

Citation: Angélique Perrillat-Mercerot, Alain Miranville, Nicolas Bourmeyster, Carole Guillevin, Mathieu Naudin, Rémy Guillevin. What mathematical models can or cannot do in glioma description and understanding. Discrete and Continuous Dynamical Systems - S, 2020, 13 (8) : 2165-2193. doi: 10.3934/dcdss.2020184
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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.
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).
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
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)
Table 1.  Glioma grades according to WHO classification.
Grade Tumor bounds Growth Malignancy Necrosis
Precise Really slow No No
Imprecise Slow Suspect No
Blurry Fast Yes Possible
Metastasis Really fast Yes Yes
Grade Tumor bounds Growth Malignancy Necrosis
Precise Really slow No No
Imprecise Slow Suspect No
Blurry Fast Yes Possible
Metastasis Really fast Yes Yes
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