The role of microstructure in elasticity imaging

Abstract

Modeling the microstructure explicitly while solving a macroscopic inverse problem is computationally not feasible. Therefore we utilize a simple homogenization strategy. We begin from the response of a single collage fiber, and calculate the strain energy stored in it. Then we use this strain energy stored in a single fiber and an assumed distribution of fiber along all orientations in order to determine the total strain energy density of the tissue. The end result is a mechanical model has two parameters of interest: concentration of fibers and their tortuosity.

We have applied this approach to ten patients with five benign and five malignant breast tumors. We noted that malignant tumors appear heterogeneous in Young's modulus images. By creating high resolution modulus images, we have quantified this heterogeneity through a parameter and demonstrated that this heterogeneity is consistent with the current understand of biomechanics of cancer, and can be potentially used to diagnose breast cancer. We also have proved that some of other microstructural markers obtained have the potential to be used to diagnose malignancy.

We have developed and implemented this approach in NLACE: Nonlinear Adjoint Coefficient Estimator. We have verified different aspects of the algorithm by comparisons with analytical solutions and noisy synthetic data. We have validated the mathematical model with data from tissue-mimicking gelatin-agar co-gels and proved that this approach is able to accurately predicted the spatial distribution of tissue microstructure from tissue macroscopic response.

Thereafter we use the microstructure-based constitutive model in an inverse problem to determine homogenized microscopic parameters. We pose the inverse problem as a minimization problem where we seek a distribution of material parameters that minimizes the deference between the measured and predicted displacement fields. We solve the minimization problem using a gradient-based algorithm. We evaluate the gradient vector efficiently by using the adjoint equations and a continuation strategy in the material parameters. At each iteration of the minimization algorithm we are required to solve forward nonlinear elastic problems.

It has been long known that disease alters tissue microstructure, and in turn affects the macroscopic mechanical response of tissue. By developing and utilizing microstructure-based constitutive models in inverse problems that are driven by macroscopic measurements, we aim to infer these changes in microstructure and therefore infer the health of tissue.

As an example, cancerous tumors display a microstructure that it altered in several ways. First, desmoplasia leads to the formation of dense, collagen-rich stroma around the tumor. Second, as the tumor becomes more malignant, the collagen fibers develop from being tortuous to straight, rod-like. Finally, they grow in a very heterogeneous manner, and display a "stellate" or "spiculated" tumor margin. In this research we develop methods to capture these microstructural changes in-vivo by using elasticity imaging. We accomplish this by first developing a microstructure-based constitutivemodel and then using it in an inverse problem to determine homogenized microscopic parameters.