Mathematical and numerical modeling of elastic waves propagation in the heart

The objective of this thesis is to develop a rigorous mathematical and numerical background for the extension and dissemination of imaging modalities by ultrasound, i.e. elastic waves, applied to the cardiac settings. The problems treated will concerned the topics of mathematical modeling, numerical analysis and scientific computing. More precisely we plan to define a linearized model for the propagation of elastic waves in the heart, study approximations of these models and define adapted numerical methods for the discretization of the resulting partial differential equations. Our end-objective is to be able to do high performance computing of the underlying multi-scale and multi-physics phenomena. Context and positioning of the thesis Cardiovascular diseases are the leading worldwide cause of death and the second in France. One of the precursor symptoms of these diseases is the local modification of the mechanical properties of the heart. Therefore, patient-specific monitoring of these mechanical properties has a strong potential to help medical doctors in the construction of a reliable diagnosis. Among many existing medical imaging modalities we want to develop a mathematical and numerical background for a new transient ultrasound elastography modality: the 3D shear-wave imaging with 3D ultrafast ultrasound (as experimented in [2] on living tissue). The principle of this modality is to use emitted and back-scattered ultra-sound (pressure waves) at a very high frequency to image the propagation of a generated shear waves in the tissue. The propagation speed of these shear wave can then be correlated with the mechanical properties of the probed medium at a (really) fine scale. These properties can be recovered in a second step using for instance data assimilation based strategies, or more specifically observer-based strategies. Experimental preliminary studies presented in [1] show the potential of the extension of transient elastography to cardiac imaging. However there does not exist in the applied mathematic literature studies of this modality for the application in a cardiac setting. It can be explained since realistic mechanical models for the heart contraction are relatively new (see [3]) and are of fundamental importance to define a relevant modeling background for the elastic wave propagation. The objective of this proposal is to develop a rigorous mathematical and numerical background for the extension of shear-wave imaging with ultrafast ultrasound techniques to the cardiac settings. More precisely, in this proposal oriented towards applied mathematics, we propose to study the elastic waves propagation (pressure and shear waves propagation) in the beating heart and wish to offer some well-defined mathematical modeling and numerical strategy as well as scientific computations of the involved phenomena. This will hopefully improve the dissemination of the transient ultrasound elastography technique, and this project will present a major step forward in the modeling and simulation of a multi-scale (pressure and shear waves) and multi-physics phenomenon (beating heart and wave propagation) for life sciences. Scientific and technical description We aim at developing a rigorous mathematical and numerical background for the extension and improvement of a transient ultrasound elastography modality in the context of non-invasive cardiac imaging. Some intrinsic difficulties related to our application have to be addressed. More precisely: • The heart contraction, in its fastest phase, has a velocity of the same order of magnitude than the speed of the shear wave propagation. Therefore the transient elastographic measurements (directly related to the propagating shear wave) are coupled with the heart mechanics.