<p>The mathematical modelling of the reaction and diffusion mechanism of lipophilic toxic compounds in the mammalian cell is a challenging task because of its considerable complexity and variation in the architecture of the cell. The heterogeneity of the cell regarding the enzyme distribution participating in the bio-transformation, makes the modelling even more difficult. In order to reduce the complexity of the model, and to make it less computationally expensive and numerically treatable, Homogenization techniques have been used. The resulting complex system of Partial Differential Equations (PDEs), generated from the model in 2-dimensional axi-symmetric setting is implemented in Comsol Multiphysics. The numerical results obtained from the model show a nice agreement with the in vitro cell experimental results. The model can be extended to more complex reaction systems and also to 3-dimensional space. For the reduction of complexity and computational cost, we have implemented a model of mixed PDEs and Ordinary Differential Equations (ODEs). We call this model as Non-Standard Compartment Model. Then the model is further reduced to a system of ODEs only, which is a Standard Compartment Model. The numerical results of the PDE Model have been qualitatively verified by using the Compartment Modeling approach. The quantitative analysis of the results of the Compartment Model shows that it cannot fully capture the features of metabolic system considered in general. Hence we need a more sophisticated model using PDEs for our homogenized cell model.</p> / Computational Modelling of the Mammalian Cell and Membrane Protein Enzymology

The mathematical modelling of the reaction and diffusion mechanism of lipophilic toxic compounds in the mammalian cell is a challenging task because of its considerable complexity and variation in the architecture of the cell. The heterogeneity of the cell regarding the enzyme distribution participating in the bio-transformation, makes the modelling even more difficult. In order to reduce the complexity of the model, and to make it less computationally expensive and numerically treatable, Homogenization techniques have been used. The resulting complex system of Partial Differential Equations (PDEs), generated from the model in 2-dimensional axi-symmetric setting is implemented in Comsol Multiphysics. The numerical results obtained from the model show a nice agreement with the in vitro cell experimental results. The model can be extended to more complex reaction systems and also to 3-dimensional space. For the reduction of complexity and computational cost, we have implemented a model of mixed PDEs and Ordinary Differential Equations (ODEs). We call this model as Non-Standard Compartment Model. Then the model is further reduced to a system of ODEs only, which is a Standard Compartment Model. The numerical results of the PDE Model have been qualitatively verified by using the Compartment Modeling approach. The quantitative analysis of the results of the Compartment Model shows that it cannot fully capture the features of metabolic system considered in general. Hence we need a more sophisticated model using PDEs for our homogenized cell model. / Computational Modelling of the Mammalian Cell and Membrane Protein Enzymology

La Tomographie par Emission de Positons (TEP) au 2-[18]-Fluoro-2-désoxy-D-glucose (FDG) est une méthode d’imagerie fonctionnelle particulièrement utilisée en oncologie afin de quantifier le métabolisme glucidique des lésions tumorales. En routine clinique, l’analyse quantitative de ces images est réalisée à l’aide de la valeur de fixation normalisée, notée SUV. Des approches de quantification plus élaborées existent dans la littérature, mais elles requièrent généralement de multiples prélèvements sanguins et/ou l’acquisition d’un examen TEP d’au moins 50 minutes. De ce fait, elles sont difficilement applicables en routine clinique. Notre travail a porté sur le développement d’une approche non-invasive, nommée ParaPET, basée sur les travaux initiaux de [Barbolosi et al.2016], permettant la détermination d’une cartographie de biomarqueurs dynamiques et requérant une unique acquisition TEP de durée limitée. Notre approche intègre plusieurs améliorations, parmi lesquelles l’élaboration d’un nouveau modèle d’estimation de la concentration moyenne de FDG et de l’erreur de mesure associée, basé sur un protocole de reconstructions TEP multiples utilisant un rééchantillonnage temporel des données, la détermination de la concentration d’activité sanguine de FDG à l’aide des images TEP de l’aorte, et la caractérisation des lésions tumorales à l’échelle du voxel. Notre approche a été évaluée sur une base de données de 31 patients atteints d’un cancer bronchique non à petites cellules (CBNPC) que nous avons construite au préalable. Notre analyse a porté sur la détermination du biomarqueur Ki , représentant le débit net entrant de FDG. Nos résultats ne montrent pas de différence significative dans l’estimation de Ki entre notre approche et la méthode de référence, l’analyse graphique de Patlak [Patlak et al. 1983]. Nous avons également montré l’existence d’une forte corrélation (R2 ¸ 0,87) entre les images de Ki et de SUV. Cependant, ces images ne sont pas identiques, et peuvent apporter des informations supplémentaires, par exemple pour les régions nécrotiques. Enfin, nous avons étudié la variation relative de Ki (¢(K¤ i )) et de SUVmax (¢(SUVmax )) entre les examens pré- et per-thérapeutiques. Nous avons constaté une corrélation médiocre entre ¢(K¤ i ) et ¢(SUVmax ) (R2 = 0,60) sur l’ensemble de la plage de variation, mais une corrélation plus importante à partir des valeurs de ¢(SUVmax ) ¸ 40 %. Il conviendrait d’approfondir la signification et l’intérêt médical associé aux faibles variations de SUV et de Ki . Au final, l’approche ParaPET permet une détermination simplifiée des paramètres cinétiques de la fixation du FDG, qui enrichiront les caractéristiques tumorales pouvant présenter un intérêt pour la radiomique. / Positron Emission Tomography (PET) with 2-[18]-Fluoro-2-deoxy-D-glucose (FDG) is a functional imaging technique especially used in oncology to quantify glucose metabolism of tumour lesions. In clinical routine, quantitative analysis of these images is carried out using the standardized uptake value (SUV). More sophisticated quantification approaches have been proposed in the literature, but they requiremultiple blood samples and/or at least a 50 minutes PET acquisition. As a result, they are difficult to implement in clinical routine. Our work focused on the development of a non-invasive approach, named ParaPET, based on the initial work of [Barbolosi et al. 2016], allowing the determination of 3D maps of dynamic biomarkers and only requiring a PET scan of a limited duration. Our approach includes several improvements, including the development of a new model for the estimation of the FDG activity concentration and the associated measurement error, based on amultiple PET reconstruction protocol using temporal data resampling, the determination of the blood FDG activity concentration using PET aorta images, and the characterisation of tumour lesions at a voxel level. Our approach was evaluated on a database of 31 patients with non-small cell lung cancer (NSCLC) treated by chemo-radiation therapy, that we previously constructed. Our analysis focused on thedetermination of the biomarker Ki , the net influx of FDG in the lesion. Our results show no significant difference in the Ki estimate between our approach and the reference method, the Patlak graphical analysis [Patlak et al. 1983]. We also have shown the existence of a strong correlation between Ki and SUV images (R2 ¸ 0,87).However, these images are not identical, and may provide additional information, for example for necrotic regions. Finally, we studied the relative variation of Ki (¢(K¤ i )) and SUVmax (¢(SUVmax )) between pre- et per-therapeutic PET scans. We have found a poor correlation between ¢(K¤i ) et ¢(SUVmax ) (R2 = 0,60) over the entire range of variation, but a higher correlation from ¢(SUVmax ) values ¸ 40 %. The meaning and the medical interest associated with small variations of SUV and Ki should be further investigated. To conclude, our ParaPET approach allows a simplified determination of kinetic parameters of FDG uptake, which will enhance the tumour characteristics that may be of interest for radiomics.