Solution of the nonlinear elasticity imaging inverse problem: the compressible case

Abstract

We have recently developed and tested an efficient algorithm for solving the nonlinear inverse elasticity problem for a compressible hyperelastic material. The data for this problem are the quasi-static deformation fields within the solid measured at two distinct overall strain levels. The main ingredients of our algorithm are a gradient based quasi-Newton minimization strategy, the use of adjoint equations and a novel strategy for continuation in the material parameters. In this paper we present several extensions to this algorithm. First, we extend it to incompressible media thereby extending its applicability to tissues which are nearly incompressible under slow deformation. We achieve this by solving the forward problem using a residual-based, stabilized, mixed finite element formulation which circumvents the Ladyzenskaya-Babuska-Brezzi condition. Second, we demonstrate how the recovery of the spatial distribution of the nonlinear parameter can be improved either by preconditioning the system of equations for the material parameters, or by splitting the problem into two distinct steps. Finally, we present a new strain energy density function with an exponential stress-strain behavior that yields a deviatoric stress tensor, thereby simplifying the interpretation of pressure when compared with other exponential functions. We test the overall approach by solving for the spatial distribution of material parameters from noisy, synthetic deformation fields.

Anisotropic behaviour of human gallbladder walls

“…Young's modulus (Guo et al 2010) or a general linear elasticity tensor (Raghupathy and Barocas 2010), can be determined directly from the force equilibrium equation iteratively (Moulton et al 1995;Govindjee and Mihalic 1998;Kauer et al 2002;Seshaiyer and Humphrey 2003;Bosisio et al 2007;Lei and Szeri 2007;Samani and Plewes 2007;Gokhale et al 2008;Karimi et al 2008;Li et al 2009;Balocco et al 2010;Kroon 2010). …”

An inverse finite element method for determining the anisotropic properties of the cornea

“…The gradient was computed using the adjoint method developed by Gokhale et al (2008) for inverse elasticity imaging (see also Oberai et al (2003)). The mathematical development of the adjoint method is reviewed briefly in the "Appendix".…”

“…The remaining term D κ L evaluated to, Gokhale et al (2008) showed that Eq. (20) is equivalent to the gradient vector of the objective functional.…”

Isotropic incompressible hyperelastic models for modelling the mechanical behaviour of biological tissues: a review

“…It could be used, for example, in surgery planning [74], for simulation of surgical procedures for training and learning purposes [10,74], to study virtual reality in clinical systems [107], in tissue engineering [3,29,60], and for finite element modelling of different organs [11,36,51]. Determining the elastic properties of soft biological materials under finite deformation is of great interest in elasticity imaging, where these material properties can be used to distinguish between malignant tissue and benign tissue and even for cancer classification [35,80,85].…”