INTRODUCTION Les mathématiques formelles, c’est-à-dire des démonstrations rédigées dans un langage tel qu’un ordinateur soit capable de les vérifier, sont un champ d’études depuis plus de quarante ans mais n’ont commencé à attirer une communauté importante d’utilisateurs que très récemment. Les assistants de preuve, qui sont des logiciels qui aident à écrire ce genre de démonstrations, intéressent à la fois la communauté de la certification (logicielle, matérielle) et la communauté mathématique. Cette dernière a, dans un passé récent, produit des démonstrations d’une taille telle qu’elle commence à craindre que la relecture par les pairs ne soit plus suffisante pour garantir la validité d’une démonstration . Même s’ils ont atteint un certain niveau de maturité, les assistants de preuve sont encore difficiles à utiliser et les rendre plus accessibles, pour permettre qu’une communauté bien plus importante les utilise, est une préoccupation de plus en plus grande parmi les chercheurs du domaine. Cette thèse a pour objectif de faire quelques pas dans cette direction dans le cadre de Coq , un assistant de preuve dont la communauté d'utilisateurs et l'équipe de développement sont extrêmement actifs et qui est parmi les plus utilisés au monde.

LITERATURE LIST Acquisition Devices - - - - - - - - - - - - - Basics - - - - - - - First-Order Analysis - Resolution , - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Aberrations - - Calibration - - - - - - Reviews - - - Microscopy - - - - - - - - - - - Camera arrays - - - - - Rendering - - - - - Various applications -

This paper proposes a framework to simulate patient specific structural Magnetic Resonance Images (MRIs) from the available time-points of Alzheimer’s Disease(AD) subjects. We use a biophysical model of brain deformation due to atrophy that can generate biologically plausible deformation for any given desired volume changes at the voxel level of the brain MRI. Large number of brain regions are segmented in 45 AD patients and the atrophy rates per year are estimated in these regions from two extremal available scans. Assuming linear progression of atrophy, the volume changes in scans closest to the middle time-point images from the baseline scans are computed. These atrophy maps are prescribed to the baseline images to simulate the middle time-point images by using the biophysical model of brain deformation. The volume changes from the baseline image to the real middle time-point are compared to the volume changes in the simulated middle time-point images. This present framework also allows to introduce desired atrophy patterns at different time-points to simulate non-linear progression of atrophy. This opens a way to use a biophysical model of brain deformation to evaluate methods that study the temporal progression and spatial relationships of atrophy evolution in AD. KEYWORDS: Alzheimer’s disease, biophysical modeling, biomechanical simulation THIS IS A PRE-PRINT OF THE FOLLOWING PUBLISHED ARTICLE: Bishesh Khanal, Marco Lorenzi, Nicholas Ayache, Xavier Pennec. Simulating Patient Specific Multiple Time-point MRIs From a Biophysical Model of Brain Deformation in Alzheimer’s Disease. Workshop on Computational Biomechanics for Medicine - X, Oct 2015, Munich, France. 2015.

THIS IS A PRE-PRINT (A WORKING PAPER) OF THE FOLLOWING PUBLISHED PAPER: Bishesh Khanal, Marco Lorenzi, Nicholas Ayache, Xavier Pennec. A biophysical model of brain deformation to simulate and analyze longitudinal MRIs of patients with Alzheimer’s disease. NeuroImage, Elsevier, 2016, 134, pp.35-52. We propose a framework for developing a comprehensive biophysical model that could predict and simulate realistic longitudinal MRIs of patients with Alzheimer’s Disease (AD). The framework includes three major building blocks: i) Atrophy generation ii) Brain deformation iii) Realistic MRI generation. Within this framework, this paper focuses on a detailed implementation of the brain deformation block with a carefully designed biomechanics-based tissue loss model. For a given baseline brain MRI, the model yields a deformation field imposing the desired atrophy at each voxel of the brain parenchyma while allowing the CSF to expand as required to globally compensate for the locally prescribed volume loss. Our approach is inspired by biomechanical principles and involves a system of equations similar to Stokes equations in fluid mechanics but with the presence of a non-zero mass source term. We use this model to simulate longitudinal MRIs by prescribing complex patterns of atrophy. We explore the influence of different spatial distributions of atrophy on the image appearance . The proposed framework could help understand the implications of different model assumptions, regularization choices and spatial priors for the detection and measurement of brain atrophy from longitudinal brain MRIs. Keywords: biophysical model, Alzheimer’s disease, simulation of atrophy, longitudinal MRIs simulation, longitudinal modeling