Evaluation of the adjoint equation based algorithm for elasticity imaging

Abstract

Recently a new adjoint equation based iterative method was proposed for evaluating the spatial distribution of the elastic modulus of tissue based on the knowledge of its displacement field under a deformation. In this method the original problem was reformulated as a minimization problem, and a gradient-based optimization algorithm was used to solve it. Significant computational savings were realized by utilizing the solution of the adjoint elasticity equations in calculating the gradient. In this paper, we examine the performance of this method with regard to measures which we believe will impact its eventual clinical use. In particular, we evaluate its abilities to (1) resolve geometrically the complex regions of elevated stiffness; (2) to handle noise levels inherent in typical instrumentation; and (3) to generate three-dimensional elasticity images. For our tests we utilize both synthetic and experimental displacement data, and consider both qualitative and quantitative measures of performance. We conclude that the method is robust and accurate, and a good candidate for clinical application because of its computational speed and efficiency.

Elastic modulus imaging: some exact solutions of the compressible elastography inverse problem

“…Several researchers have modeled the inverse elasticity problem for soft tissues by choosing a constant Poisson's ratio, ν, close to 1/2 [5,6,7]. A value of ν = 1/2 corresponds to perfect incompressibility.…”

A generalized computationally efficient inverse characterization approach combining direct inversion solution initialization with gradient-based optimization

“…The adjoint method requires only two numerical solutions of BVPs to calculate the necessary gradient, the given BVP and a corresponding adjoint BVP, with both having the same approximate computational expense. Therefore, the adjoint method represents a substantial computational savings in comparison to alternate methods, such as finite difference methods, which require at least N + 1 BVP solutions, or direct differentiation of the BVP, which requires N BVP solutions, where N is the number of unknown parameters in the optimization problem [4]. Particularly for generalized (e.g., finite element-type) parameterizations of the unknown property with large numbers of parameters to be determined, the adjoint method, or something similar, is a necessity for practical applicability.…”

“…are becoming ever more popular in a variety of fields in science and engineering. In particular, applications in the characterization of material property distributions span interest areas from civil engineering (e.g., structural damage characterization [1,2]) to medicine (e.g., tissue characterization for disease diagnosis [3,4]), where quantitative estimation of a variety of material parameters can provide critical information relating to the state of the system. A common structure of quantitative inverse material characterization approaches is to couple a numerical representation of the system forward problem (e.g., a finite element representation of the system response given the material properties) with some type of optimization to estimate the material properties that lead to a "best match" between the response estimated by the forward numerical representation and the available experimentally measured response.…”

“…[8,9] and those that use gradient-based optimization (e.g., Newton's method, conjugate gradient, etc.) [4,10]. Non-gradient-based methods often have more significant global search capabilities in comparison to the gradient-based optimization approaches, which can become trapped in local minima (i.e., an inaccurate solution).…”

“…Unfortunately, as stated, gradient-based methods are local in nature, and unless the initial estimate of the unknown property field provided is relatively accurate, the final solution estimate is likely to be inaccurate. Furthermore, limitations in the amount of measurement data and/or high-dimensional parameterizations of the unknown field often leads to complicated (non-convex) error surfaces for the optimization, and while regularization approaches can somewhat relieve this challenge [14,4,15], the importance of the initial estimate accuracy is increased significantly.…”