This project concerns the computational solution of inverse problems formulated as partial differential equation (PDE)-constrained optimization problems with interior data. The areas addressed are twofold. First, we present a novel software architecture designed to solve inverse problems constrained by an elliptic system of PDEs. These generally require the solution of forward and adjoint problems, evaluation of the objective function, and computation of its gradient, all of which are approximated numerically using finite elements. The creation of specialized "layered"' elements to perform these tasks leads to a modular software structure that improves code maintainability and promotes functional interoperability between different software components. Second, we address issues related to forward model definition in the presence of boundary condition (BC) uncertainty. We propose two variational formulations to accommodate that uncertainty: (a) a Bayesian formulation that assumes Gaussian measurement noise and a minimum strain energy prior, and (b) a Lagrangian formulation that is completely free of displacement and traction BCs. This work is motivated by applications in the field of biomechanical imaging, where the mechanical properties within soft tissues are inferred from observations of tissue motion. In this context, the constraint PDE is well accepted, but considerable uncertainty exists in the BCs. The approaches developed here are demonstrated on a variety of applications, including simulated and experimental data. We present modulus reconstructions of individual cells, tissue-mimicking phantoms, and breast tumors.

Cell modulus and prestress are important determinants of cell behavior. This study creates new software tools to compute the modulus and prestress distribution within a living cell. As input, we have a sequence of images of a cell plated on a substrate with fluorescently labeled fibronectin dots. The cell generates focal adhesions with the dots and thus deforms the substrate. A sequence of images of the cell and the fibronectin dots shows their deformation. We tested three different ways to track the movement of the fluorescent fibronectin dots. We demonstrated the accuracy and the adaptability of each method on a sequence of test images with a rigid movement. We found the best method for dot tracking is a combination of successive dot identification and digital image correlation. The dot deformation provides a measure of traction forces acting on the cell. From traction forces thus inferred, we use FEM to compute the stress distribution within a cell. We consider two approaches. The first is based on the assumption that the cell has homogeneous elastic properties. This is straightforward and requires only the cell being meshed and the linear elasticity problem solved on that mesh. Second, we relaxed the homogeneity assumption. We used previously published correlations between prestress and modulus to iteratively update the modulus and prestress distributions within the cell. A novel feature of this work is the implicit reconstruction of the modulus distribution without a measured displacement field, and the reconstruction of the prestress distribution accounting for intracellular inhomogeneity.

We consider the design of finite element methods for inverse problems with full-field data governed by elliptic forward operators. Such problems arise in applications in inverse heat conduction, in mechanical property characterization, and in medical imaging. For this class of problems, novel finite element methods have been proposed (Barbone et al., 2010) that give good performance, provided the solutions are in the H^1(Ω) function space. The material property distributions being estimated can be discontinuous, however, and therefore it is desirable to have formulations that can accommodate discontinuities in both data and solution. Toward this end, we present a mixed variational formulation for this class of problems that handles discontinuities well. We motivate the mixed formulation by examining the possibility of discretizing using a discontinuous discretization in an irreducible finite element method, and discuss the limitations of that approach. We then derive a new mixed formulation based on a least-square error in the constitutive equation. We prove that the continuous variational formulations are well-posed for applications in both inverse heat conduction and plane stress elasticity. We derive a priori error bounds for discretization error, valid in the limit of mesh refinement. We demonstrate convergence of the method with mesh refinement in cases with both continuous and discontinuous solutions. Finally we apply the formulation to measured data to estimate the elastic shear modulus distributions in both tissue mimicking phantoms and in breast masses from data collected in vivo.