Nachiket H Gokhale

Solution of inverse problems in elasticity imaging using the adjoint method

We consider the problem of determining the shear modulus of a linear-elastic, incompressible medium given boundary data and one component of the displacement field in the entire domain. The problem is derived from applications in quantitative elasticity imaging. We pose the problem as one of minimizing a functional and consider the use of gradient-based algorithms to solve it. In order to calculate the gradient efficiently we develop an algorithm based on the adjoint elasticity operator. The main cost associated with this algorithm is equivalent to solving two forward problems, independent of the number of optimization variables. We present numerical examples that demonstrate the effectiveness of the proposed approach.

Evaluation of the adjoint equation based algorithm for elasticity imaging.

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.

Linear and nonlinear elasticity imaging of soft tissue in vivo: demonstration of feasibility.

We establish the feasibility of imaging the linear and nonlinear elastic properties of soft tissue using ultrasound. We report results for breast tissue where it is conjectured that these properties may be used to discern malignant tumors from benign tumors. We consider and compare three different quantities that describe nonlinear behavior, including the variation of strain distribution with overall strain, the variation of the secant modulus with overall applied strain and finally the distribution of the nonlinear parameter in a fully nonlinear hyperelastic model of the breast tissue.

Elastic modulus imaging: on the uniqueness and nonuniqueness of the elastography inverse problem in two dimensions

We examine the uniqueness of an N-field generalization of a 2D inverse problem associated with elastic modulus imaging: given?N?linearly independent displacement fields in an incompressible elastic material, determine the shear modulus. We show that for the standard case, N=1, the general solution contains two arbitrary functions which must be prescribed to make the solution unique. In practice, the data required to evaluate the necessary functions are impossible to obtain. For N=2, on the other hand, the general solution contains at most four arbitrary constants, and so very few data are required to find the unique solution. For N=4, the general solution contains only one arbitrary constant. Our results apply to both quasistatic and dynamic deformations.

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

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.

Elastic nonlinearity imaging

Previous work has demonstrated improved diagnostic performance of highly trained breast radiologists when provided with B-mode plus elastography images over B-mode images alone. In those studies we have observed that elasticity imaging can be difficult to perform if there is substantial motion of tissue out of the image plane. So we are extending our methods to 3D/4D elasticity imaging with 2D arrays. Further, we have also documented the fact that some breast tumors change contrast with increasing deformation and those observations are consistent with in vitro tissue measurements. Hence, we are investigating imaging tissue stress-strain nonlinearity. These studies will require relatively large tissue deformations (e.g., > 20%) which will induce out of plane motion further justifying 3D/4D motion tracking. To further enhance our efforts, we have begun testing the ability to perform modulus reconstructions (absolute elastic parameter) imaging of in vivo breast tissues. The reconstructions are based on high quality 2D displacement estimates from strain imaging. Piecewise linear (secant) modulus reconstructions demonstrate the changes in elasticity image contrast seen in strain images but, unlike the strain images, the contrast in the modulus images approximates the absolute modulus contrast. Nonlinear reconstructions assume a reasonable approximation to the underlying constitutive relations for the tissue and provide images of the (near) zero-strain shear modulus and a nonlinearity parameter that describes the rate of tissue stiffening with increased deformation. Limited data from clinical trials are consistent with in vitro measurements of elastic properties of tissue samples and suggest that the nonlinearity of invasive ductal carcinoma exceeds that of fibroadenoma and might be useful for improving diagnostic specificity. This work is being extended to 3D.

Simultaneous elastic image registration and elastic modulus reconstruction

Ultrasound elastography, the imaging of soft tissues based on shear elastic modulus, is a growing imaging method. The technique relies on being able to image soft tissue while it is being deformed by a set of externally applied forces. Typically, block matching methods are used to obtain a dense estimate of the point-to-point displacement field in the field of view. This displacement field can then be used as input to an inverse problem to reconstruct the elastic modulus distribution. We describe several advantages of combining these steps into one, and show a practical methods to do so. In essence, we show how to regularize a nonrigid elastic registration problem via an unknown elastic modulus distribution.

Design of pentamode acoustic materials using homogenization and the adjoint approach.

Acoustic metamaterials need to be realized using subwavelength microstructures. By tailoring the microstructure of the underlying unit cell, different effective properties at the macroscopic scale may be achieved. These macroscopic properties can be related to the microstructure using homogenization theory. The problem of designing a pentamode acoustic metamaterial is formulated as the minimization of an objective functional representing the difference between the homogenized properties and the target pentamode acoustic metamaterial properties. A quasi‐Newton optimization approach is used to solve the optimization problem. Such an approach requires the gradient of the objective function with respect to the microstructural properties at every iteration, and hence the cost of gradient computation must be low. The gradient is calculated efficiently using the adjoint approach which requires the solution of only two (primal and dual) finite element problems, per homogenization deformation field, independent of...

Three‐dimensional ultrasound image registration and shear elastic modulus reconstruction

It is widely recognized that tissue pathologies often change biomechanical properties. For instance, neoplastic tissue is typically highly vascularized, contains abnormal concentrations of extracellular proteins (i.e., collagen, proteoglycans) and has a high interstitial fluid pressure compared to most normal tissues. The aim of this work is to develop and evaluate an ultrasound technique to quantitatively measure and image the mechanical properties of soft tissues in three dimensions. The intended application of our work is in the detection and diagnosis of breast cancer and other soft tissue pathologies. The specific goal of this project is the accurate measurement of the elastic shear modulus distribution of a tissue‐mimicking ultrasound phantom, using a 3‐D ultrasound imaging system. This requires the design and characterization of algorithms to provide three‐dimensional motion estimates from ultrasound images and to solve the three‐dimensional inverse problem to recover shear elastic modulus. Modulus...

Transformation acoustics: Theory, ray‐tracing, and finite element simulations.

Norris’ theory of transformation acoustics enables the realization of pentamode acoustic materials having anisotropic density and finite mass. Two additional extensions of the theory are considered, with a view toward demonstrating feasibility for practical applications. First, specializations of the theory developed by Norris [“Acoustic cloaking theory,” Proc. R. Soc. London, Ser. A 464, 2411–2434 (2008)] are considered in order to develop transformations corresponding to pentamode acoustic metamaterials with specified functional forms (constant or powerlaw) for density and stiffness. Ray‐tracing and finite element simulations of wave propagation through such pentamode acoustic media are presented. Next, the design of acoustic materials for objects composed of simple geometric shapes (e.g., a cylinder with spherical endcaps) is considered. It is shown that certain classes of transformations are well‐suited for the design of such shapes and validate the concept with finite element simulations.