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Modeling the behavior of biological tissues and organs often necessitates theknowledge of their shape in the absence of external loads. However, when theirgeometry is acquired in-vivo through imaging techniques, bodies are typicallysubject to mechanical deformation due to the presence of external forces, andthe load-free configuration needs to be reconstructed. This paper addressesthis crucial and frequently overlooked topic, known as the inverse elasticityproblem (IEP), by delving into both theoretical and numerical aspects, with aparticular focus on cardiac mechanics. In this work, we extend Shield's seminalwork to determine the structure of the IEP with arbitrary materialinhomogeneities and in the presence of both body and active forces. Theseaspects are fundamental in computational cardiology, and we show that they maybreak the variational structure of the inverse problem. In addition, we showthat the inverse problem might be ill-posed, even in the presence of constantNeumann boundary conditions and a polyconvex strain energy functional. We thenpresent the results of extensive numerical tests to validate our theoreticalframework, and to characterize the computational challenges associated with adirect numerical approximation of the IEP. Specifically, we show that thisframework outperforms existing approaches both in terms of robustness andoptimality, such as Sellier's iterative procedure, even when the latter isimproved with acceleration techniques. A notable discovery is that multigridpreconditioners are, in contrast to standard elasticity, not efficient, anddomain decomposition methods provide a much more reliable alternative. Finally,we successfully address the IEP for a full-heart geometry, demonstrating thatthe IEP formulation can compute the stress-free configuration in real-lifescenarios where Sellier's algorithm proves inadequate.

Reconstructing relaxed configurations in elastic bodies: Mathematical formulation and numerical methods for cardiac modeling