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In this article, we present a framework to estimate cardiac function parameters like local myocardium contractility using clinical MRI, a heart model and data assimilation. First, we build a generic anatomical model of the ventricles including muscle fibre orientations and anatomical subdivi- sions. Then, this model is deformed to fit a segmented MRI, using an affine registration method and a local deformable biomechanical model. An electromechanical model of the heart can be simulated on this mesh. Data assimilation makes it possible to estimate local contractility from given displacements. Presented results on simulated data and adjustment to clinical data are very promising. Current work on model calibration and estimation of patient parameters open up possibilities for clinical application of this framework. Resume. Dans cet article, nous presentons un cadre de travail pour estimer certains parametres de la fonction cardiaque (comme la contractilite locale) en utilisant l'IRM, un modele du myocarde et l'assimilation de donnees. Tout d'abord, nous detaillons la construction d'un modele generique du my- ocarde incluant l'orientation des fibres musculaires et les differents segments anatomiques. Ce modele peutdeforme pour s'ajuster aux images cliniques. Un modeleecanique du myocarde est ensuite presentee t simule. Enfin, l'assimilation de donnees permet d'estimer la contractilitel ocalea partir de donnees telles que les deplacements. Les resultats sur l'ajustement du modelea geometrie du patient et l'assimilation sur des donnees simulees sont tres encourageants. Les travaux en cours sur la calibration du modele et l'estimation de parametres du patient ouvrent de nouvelles possibilites pour l'application de ce cadre de travail dans un environnement clinique.

Les progrès réalisés en mathématiques appliquées permettent aujourd’hui d’envisager la simulation sur ordinateur de certains compartiments du système cardiovasculaire. Nous proposons de faire un point sur quelques modèles, en nous focalisant sur la simulation de l’écoulement du sang dans des artères déformables et sur la simulation de la contraction du myocarde sous l’effet de la propagation d’un signal électrique. Nous tentons également de présenter des applications possibles de ce type de travaux., In this article, we aim at giving a non-technical overview of some mathematical models currently used in the numerical simulation of the cardiovascular system. A hierarchy of models for blood flows in large arteries is briefly described as well as an electromechanical model for the heart. We discuss some possible applications of the numerical simulations of such models, for example the optimization of prostheses. We also address the issue of the data assimilation for the calibration of the models.

We propose a muscle chemo-mechanical model by which myosin heads - that can chemically bind to actin, thus creating so-called cross-bridges producing contraction forces in sarcomeres at the subcellular level - are considered as special chemical entities having internal mechanical variables pertaining to the actual geometric configuration. This provides a thermodynamical basis for modeling the complex interplay of chemical and mechanical phenomena at the sarcomere level. The resulting model is in the form of stochastic equations governing the dynamics of these microscopic mechanical variables in a Langevin framework. Equivalently, Fokker-Planck equations can be derived to describe the evolution of the associated probability densities. Under certain assumptions the corresponding moment equations can be closed, thus directly providing access to macroscopic quantities that can be incorporated in the overall constitutive equations of the muscle tissue. The underlying thermodynamical framework also enables the derivation of compatible numerical schemes, in particular in terms of energy balances. These modeling and discretization ingredients can be integrated in a global model of the cardiac system, to represent physiological and pathological phenomena in various medical applications.