Recovering vector displacement estimates in quasistatic elastography using sparse relaxation of the momentum equation.

We consider the problem of estimating the 2D vector displacement field in a heterogeneous elastic solid deforming under plane stress conditions. The problem is motivated by applications in quasistatic elastography. From precise and accurate measurements of one component of the 2D vector displacement field and very limited information of the second component, the method reconstructs the second component quite accurately. No a priori knowledge of the heterogeneous distribution of material properties is required. This method relies on using a special form of the momentum equations to filter ultrasound displacement measurements to produce more precise estimates. We verify the method with applications to simulated displacement data. We validate the method with applications to displacement data measured from a tissue mimicking phantom, and in-vivo data; significant improvements are noticed in the filtered displacements recovered from all the tests. In verification studies, error in lateral displacement estimates decreased from about 50% to about 2%, and strain error decreased from more than 250% to below 2%.

Direct Error in Constitutive Equation Formulation for Plane stress Inverse Elasticity Problem.

We present a new computational formulation for inverse problems in elasticity with full field data. The formulation is a variant of an error in the constitutive equation formulation, but allows direct solution for the modulus field and accommodates discontinuous strain fields. The development of the formulation is motivated by the relatively poor performance of current direct formulations, reported so far in literature, in dealing with discontinuities in the strain and material property distribution. The formulation relies on minimizing the error in the constitutive equation, and a momentum equation constraint. Numerical results on model problems show that the formulation is capable handling discontinuous, and noisy strain fields, and also converging with mesh refinement for continuous and discontinuous material property distributions. The application to reconstruct the elastic modulus distribution in solid breast tumors is shown.

Noninvasive In-Vivo Quantification of Mechanical Heterogeneity of Invasive Breast Carcinomas.

Heterogeneity is a hallmark of cancer whether one considers the genotype of cancerous cells, the composition of their microenvironment, the distribution of blood and lymphatic microvasculature, or the spatial distribution of the desmoplastic reaction. It is logical to expect that this heterogeneity in tumor microenvironment will lead to spatial heterogeneity in its mechanical properties. In this study we seek to quantify the mechanical heterogeneity within malignant and benign tumors using ultrasound based elasticity imaging. By creating in-vivo elastic modulus images for ten human subjects with breast tumors, we show that Young's modulus distribution in cancerous breast tumors is more heterogeneous when compared with tumors that are not malignant, and that this signature may be used to distinguish malignant breast tumors. Our results complement the view of cancer as a heterogeneous disease on multiple length scales by demonstrating that mechanical properties within cancerous tumors are also spatially heterogeneous.