Analysis of a finite volume method for a cross-diffusion model in population dynamics

The main goal of this paper is to propose a convergent finite volume method for a reactionâeuro"diffusion system with cross-diffusion. First, we sketch an existence proof for a class of cross-diffusion systems. Then the standard two-point finite volume fluxes are used in combination with a nonlinear positivity-preserving approximation of the cross-diffusion coefficients. Existence and uniqueness of the approximate solution are addressed, and it is also shown that the scheme converges to the corresponding weak solution for the studied model. Furthermore, we provide a stability analysis to study pattern-formation phenomena, and we perform two-dimensional numerical examples which exhibit formation of nonuniform spatial patterns. From the simulations it is also found that experimental rates of convergence are slightly below second order. The convergence proof uses two ingredients of interest for various applications, namely the discrete Sobolev embedding inequalities with general boundary conditions and a space-time $L^1$ compactness argument that mimics the compactness lemma due to Kruzhkov. The proofs of these results are given in the Appendix.

Intravoxel incoherent motion diffusion-weighted imaging in the liver: comparison of mono-, bi- and tri-exponential modelling at 3.0-T.

PURPOSE: To determine whether a mono-, bi- or tri-exponential model best fits the intravoxel incoherent motion (IVIM) diffusion-weighted imaging (DWI) signal of normal livers. MATERIALS AND METHODS: The pilot and validation studies were conducted in 38 and 36 patients with normal livers, respectively. The DWI sequence was performed using single-shot echoplanar imaging with 11 (pilot study) and 16 (validation study) b values. In each study, data from all patients were used to model the IVIM signal of normal liver. Diffusion coefficients (Di ± standard deviations) and their fractions (fi ± standard deviations) were determined from each model. The models were compared using the extra sum-of-squares test and information criteria. RESULTS: The tri-exponential model provided a better fit than both the bi- and mono-exponential models. The tri-exponential IVIM model determined three diffusion compartments: a slow (D1 = 1.35 ± 0.03 × 10(-3) mm(2)/s; f1 = 72.7 ± 0.9 %), a fast (D2 = 26.50 ± 2.49 × 10(-3) mm(2)/s; f2 = 13.7 ± 0.6 %) and a very fast (D3 = 404.00 ± 43.7 × 10(-3) mm(2)/s; f3 = 13.5 ± 0.8 %) diffusion compartment [results from the validation study]. The very fast compartment contributed to the IVIM signal only for b values ≤15 s/mm(2) CONCLUSION: The tri-exponential model provided the best fit for IVIM signal decay in the liver over the 0-800 s/mm(2) range. In IVIM analysis of normal liver, a third very fast (pseudo)diffusion component might be relevant. KEY POINTS: ? For normal liver, tri-exponential IVIM model might be superior to bi-exponential ? A very fast compartment (D = 404.00 ± 43.7 × 10 (-3)  mm (2) /s; f = 13.5 ± 0.8 %) is determined from the tri-exponential model ? The compartment contributes to the IVIM signal only for b ≤ 15 s/mm (2.)

Diffusion-weighted MR imaging in musculoskeletal diseases: Current concepts.

MR imaging is currently regarded as a pivotal technique for the assessment of a variety of musculoskeletal conditions. Diffusion-weighted MR imaging (DWI) is a relatively recent sequence that provides information on the degree of cellularity of lesions. Apparent diffusion coefficient (ADC) value provides information on the movement of water molecules outside the cells. The literature contains many studies that have evaluated the role of DWI in musculoskeletal diseases. However, to date they yielded conflicting results on the use and the diagnostic capabilities of DWI in the area of musculoskeletal diseases. However, many of them have showed that DWI is a useful technique for the evaluation of the extent of the disease in a subset of musculoskeletal cancers. In terms of tissue characterization, DWI may be an adjunct to the more conventional MR imaging techniques but should be interpreted along with the signal of the lesion as observed on conventional sequences, especially in musculoskeletal cancers. Regarding the monitoring of response to therapy in cancer or inflammatory disease, the use of ADC value may represent a more reliable additional tool but must be compared to the initial ADC value of the lesions along with the knowledge of the actual therapy.

Structured sparsity for spatially coherent fibre orientation estimation in diffusion MRI.

We propose a novel formulation to solve the problem of intra-voxel reconstruction of the fibre orientation distribution function (FOD) in each voxel of the white matter of the brain from diffusion MRI data. The majority of the state-of-the-art methods in the field perform the reconstruction on a voxel-by-voxel level, promoting sparsity of the orientation distribution. Recent methods have proposed a global denoising of the diffusion data using spatial information prior to reconstruction, while others promote spatial regularisation through an additional empirical prior on the diffusion image at each q-space point. Our approach reconciles voxelwise sparsity and spatial regularisation and defines a spatially structured FOD sparsity prior, where the structure originates from the spatial coherence of the fibre orientation between neighbour voxels. The method is shown, through both simulated and real data, to enable accurate FOD reconstruction from a much lower number of q-space samples than the state of the art, typically 15 samples, even for quite adverse noise conditions.

Quantitative Analysis of Myelin and Axonal Remodeling in the Uninjured Motor Network After Stroke.

Contralesional brain connectivity plasticity was previously reported after stroke. This study aims at disentangling the biological mechanisms underlying connectivity plasticity in the uninjured motor network after an ischemic lesion. In particular, we measured generalized fractional anisotropy (GFA) and magnetization transfer ratio (MTR) to assess whether poststroke connectivity remodeling depends on axonal and/or myelin changes. Diffusion-spectrum imaging and magnetization transfer MRI at 3T were performed in 10 patients in acute phase, at 1 and 6 months after stroke, which was affecting motor cortical and/or subcortical areas. Ten age- and gender-matched healthy volunteers were scanned 1 month apart for longitudinal comparison. Clinical assessment was also performed in patients prior to magnetic resonance imaging (MRI). In the contralesional hemisphere, average measures and tract-based quantitative analysis of GFA and MTR were performed to assess axonal integrity and myelination along motor connections as well as their variations in time. Mean and tract-based measures of MTR and GFA showed significant changes in a number of contralesional motor connections, confirming both axonal and myelin plasticity in our cohort of patients. Moreover, density-derived features (peak height, standard deviation, and skewness) of GFA and MTR along the tracts showed additional correlation with clinical scores than mean values. These findings reveal the interplay between contralateral myelin and axonal remodeling after stroke.

Cognitive anatomy of the temporal lobe: Effect of personality in population with mild cognitive impairment & Functional specialization for memory systems in healthy individuals.

Résumé: L'impact de la maladie d'Alzheimer (MA) est dévastateur pour la vie quotidienne de la personne affectée, avec perte progressive de la mémoire et d'autres facultés cognitives jusqu'à la démence. Il n'existe toujours pas de traitement contre cette maladie et il y a aussi une grande incertitude sur le diagnostic des premiers stades de la MA. La signature anatomique de la MA, en particulier l'atrophie du lobe temporal moyen (LTM) mesurée avec la neuroimagerie, peut être utilisée comme un biomarqueur précoce, in vivo, des premiers stades de la MA. Toutefois, malgré le rôle évident du LMT dans les processus de la mémoire, nous savons que les modèles anatomiques prédictifs de la MA basés seulement sur des mesures d'atrophie du LTM n'expliquent pas tous les cas cliniques. Au cours de ma thèse, j'ai conduit trois projets pour comprendre l'anatomie et le fonctionnement du LMT dans (1) les processus de la maladie et dans (2) les processus de mémoire ainsi que (3) ceux de l'apprentissage. Je me suis intéressée à une population avec déficit cognitif léger (« Mild Cognitive Impairment », MCI), à risque pour la MA. Le but du premier projet était de tester l'hypothèse que des facteurs, autres que ceux cognitifs, tels que les traits de personnalité peuvent expliquer les différences interindividuelles dans le LTM. De plus, la diversité phénotypique des manifestations précliniques de la MA provient aussi d'une connaissance limitée des processus de mémoire et d'apprentissage dans le cerveau sain. L'objectif du deuxième projet porte sur l'investigation des sous-régions du LTM, et plus particulièrement de leur contribution dans différentes composantes de la mémoire de reconnaissance chez le sujet sain. Pour étudier cela, j'ai utilisé une nouvelle méthode multivariée ainsi que l'IRM à haute résolution pour tester la contribution de ces sous-régions dans les processus de familiarité (« ou Know ») et de remémoration (ou « Recollection »). Finalement, l'objectif du troisième projet était de tester la contribution du LTM en tant que système de mémoire dans l'apprentissage et l'interaction dynamique entre différents systèmes de mémoire durant l'apprentissage. Les résultats du premier projet montrent que, en plus du déficit cognitif observé dans une population avec MCI, les traits de personnalité peuvent expliquer les différences interindividuelles du LTM ; notamment avec une plus grande contribution du neuroticisme liée à une vulnérabilité au stress et à la dépression. Mon étude a permis d'identifier un pattern d'anormalité anatomique dans le LTM associé à la personnalité avec des mesures de volume et de diffusion moyenne du tissu. Ce pattern est caractérisé par une asymétrie droite-gauche du LTM et un gradient antéro-postérieur dans le LTM. J'ai interprété ce résultat par des propriétés tissulaires et neurochimiques différemment sensibles au stress. Les résultats de mon deuxième projet ont contribué au débat actuel sur la contribution des sous-régions du LTM dans les processus de familiarité et de remémoration. Utilisant une nouvelle méthode multivariée, les résultats supportent premièrement une dissociation des sous-régions associées aux différentes composantes de la mémoire. L'hippocampe est le plus associé à la mémoire de type remémoration et le cortex parahippocampique, à la mémoire de type familiarité. Deuxièmement, l'activation correspondant à la trace mnésique pour chaque type de mémoire est caractérisée par une distribution spatiale distincte. La représentation neuronale spécifique, « sparse-distributed», associée à la mémoire de remémoration dans l'hippocampe serait la meilleure manière d'encoder rapidement des souvenirs détaillés sans interférer les souvenirs précédemment stockés. Dans mon troisième projet, j'ai mis en place une tâche d'apprentissage en IRM fonctionnelle pour étudier les processus d'apprentissage d'associations probabilistes basé sur le feedback/récompense. Cette étude m'a permis de mettre en évidence le rôle du LTM dans l'apprentissage et l'interaction entre différents systèmes de mémoire comme la mémoire procédurale, perceptuelle ou d'amorçage et la mémoire de travail. Nous avons trouvé des activations dans le LTM correspondant à un processus de mémoire épisodique; les ganglions de la base (GB), à la mémoire procédurale et la récompense; le cortex occipito-temporal (OT), à la mémoire de représentation perceptive ou l'amorçage et le cortex préfrontal, à la mémoire de travail. Nous avons également observé que ces régions peuvent interagir; le type de relation entre le LTM et les GB a été interprété comme une compétition, ce qui a déjà été reporté dans des études récentes. De plus, avec un modèle dynamique causal, j'ai démontré l'existence d'une connectivité effective entre des régions. Elle se caractérise par une influence causale de type « top-down » venant de régions corticales associées avec des processus de plus haut niveau venant du cortex préfrontal sur des régions corticales plus primaires comme le OT cortex. Cette influence diminue au cours du de l'apprentissage; cela pourrait correspondre à un mécanisme de diminution de l'erreur de prédiction. Mon interprétation est que cela est à l'origine de la connaissance sémantique. J'ai également montré que les choix du sujet et l'activation cérébrale associée sont influencés par les traits de personnalité et des états affectifs négatifs. Les résultats de cette thèse m'ont amenée à proposer (1) un modèle expliquant les mécanismes possibles liés à l'influence de la personnalité sur le LTM dans une population avec MCI, (2) une dissociation des sous-régions du LTM dans différents types de mémoire et une représentation neuronale spécifique à ces régions. Cela pourrait être une piste pour résoudre les débats actuels sur la mémoire de reconnaissance. Finalement, (3) le LTM est aussi un système de mémoire impliqué dans l'apprentissage et qui peut interagir avec les GB par une compétition. Nous avons aussi mis en évidence une interaction dynamique de type « top -down » et « bottom-up » entre le cortex préfrontal et le cortex OT. En conclusion, les résultats peuvent donner des indices afin de mieux comprendre certains dysfonctionnements de la mémoire liés à l'âge et la maladie d'Alzheimer ainsi qu'à améliorer le développement de traitement. Abstract: The impact of Alzheimer's disease is devastating for the daily life of the affected patients, with progressive loss of memory and other cognitive skills until dementia. We still lack disease modifying treatment and there is also a great amount of uncertainty regarding the accuracy of diagnostic classification in the early stages of AD. The anatomical signature of AD, in particular the medial temporal lobe (MTL) atrophy measured with neuroimaging, can be used as an early in vivo biomarker in early stages of AD. However, despite the evident role of MTL in memory, we know that the derived predictive anatomical model based only on measures of brain atrophy in MTL does not explain all clinical cases. Throughout my thesis, I have conducted three projects to understand the anatomy and the functioning of MTL on (1) disease's progression, (2) memory process and (3) learning process. I was interested in a population with mild cognitive impairment (MCI), at risk for AD. The objective of the first project was to test the hypothesis that factors, other than the cognitive ones, such as the personality traits, can explain inter-individual differences in the MTL. Moreover, the phenotypic diversity in the manifestations of preclinical AD arises also from the limited knowledge of memory and learning processes in healthy brain. The objective of the second project concerns the investigation of sub-regions of the MTL, and more particularly their contributions in the different components of recognition memory in healthy subjects. To study that, I have used a new multivariate method as well as MRI at high resolution to test the contribution of those sub-regions in the processes of familiarity and recollection. Finally, the objective of the third project was to test the contribution of the MTL as a memory system in learning and the dynamic interaction between memory systems during learning. The results of the first project show that, beyond cognitive state of impairment observed in the population with MCI, the personality traits can explain the inter-individual differences in the MTL; notably with a higher contribution of neuroticism linked to proneness to stress and depression. My study has allowed identifying a pattern of anatomical abnormality in the MTL related to personality with measures of volume and mean diffusion of the tissue. That pattern is characterized by right-left asymmetry in MTL and an anterior to posterior gradient within MTL. I have interpreted that result by tissue and neurochemical properties differently sensitive to stress. Results of my second project have contributed to the actual debate on the contribution of MTL sub-regions in the processes of familiarity and recollection. Using a new multivariate method, the results support firstly a dissociation of the subregions associated with different memory components. The hippocampus was mostly associated with recollection and the surrounding parahippocampal cortex, with familiarity type of memory. Secondly, the activation corresponding to the mensic trace for each type of memory is characterized by a distinct spatial distribution. The specific neuronal representation, "sparse-distributed", associated with recollection in the hippocampus would be the best way to rapidly encode detailed memories without overwriting previously stored memories. In the third project, I have created a learning task with functional MRI to sudy the processes of learning of probabilistic associations based on feedback/reward. That study allowed me to highlight the role of the MTL in learning and the interaction between different memory systems such as the procedural memory, the perceptual memory or priming and the working memory. We have found activations in the MTL corresponding to a process of episodic memory; the basal ganglia (BG), to a procedural memory and reward; the occipito-temporal (OT) cortex, to a perceptive memory or priming and the prefrontal cortex, to working memory. We have also observed that those regions can interact; the relation type between the MTL and the BG has been interpreted as a competition. In addition, with a dynamic causal model, I have demonstrated a "top-down" influence from cortical regions associated with high level cortical area such as the prefrontal cortex on lower level cortical regions such as the OT cortex. That influence decreases during learning; that could correspond to a mechanism linked to a diminution of prediction error. My interpretation is that this is at the origin of the semantic knowledge. I have also shown that the subject's choice and the associated brain activation are influenced by personality traits and negative affects. Overall results of this thesis have brought me to propose (1) a model explaining the possible mechanism linked to the influence of personality on the MTL in a population with MCI, (2) a dissociation of MTL sub-regions in different memory types and a neuronal representation specific to each region. This could be a cue to resolve the actual debates on recognition memory. Finally, (3) the MTL is also a system involved in learning and that can interact with the BG by a competition. We have also shown a dynamic interaction of « top -down » and « bottom-up » types between the pre-frontal cortex and the OT cortex. In conclusion, the results could give cues to better understand some memory dysfunctions in aging and Alzheimer's disease and to improve development of treatment.