QUANTITATIVE THREE DIMENSIONAL ELASTICITY IMAGING

Neoplastic tissue is typically highly vascularized, contains abnormal concentrations of extracellular proteins (e.g. collagen, proteoglycans) and has a high interstitial fluid pres- sure compared to most normal tissues. These changes result in an overall stiffening typical of most solid tumors. Elasticity Imaging (EI) is a technique which uses imaging systems to measure relative tissue deformation and thus noninvasively infer its mechanical stiffness. Stiffness is recovered from measured deformation by using an appropriate mathematical model and solving an inverse problem. The integration of EI with existing imaging modal- ities can improve their diagnostic and research capabilities. The aim of this work is to develop and evaluate techniques to image and quantify the mechanical properties of soft tissues in three dimensions (3D). To that end, this thesis presents and validates a method by which three dimensional ultrasound images can be used to image and quantify the shear modulus distribution of tissue mimicking phantoms. This work is presented to motivate and justify the use of this elasticity imaging technique in a clinical breast cancer screening study. The imaging methodologies discussed are intended to improve the specificity of mammography practices in general. During the development of these techniques, several issues concerning the accuracy and uniqueness of the result were elucidated. Two new algorithms for 3D EI are designed and characterized in this thesis. The first provides three dimensional motion estimates from ultrasound images of the deforming ma- terial. The novel features include finite element interpolation of the displacement field, inclusion of prior information and the ability to enforce physical constraints. The roles of regularization, mesh resolution and an incompressibility constraint on the accuracy of the measured deformation is quantified. The estimated signal to noise ratio of the measured displacement fields are approximately 1800, 21 and 41 for the axial, lateral and eleva- tional components, respectively. The second algorithm recovers the shear elastic modulus distribution of the deforming material by efficiently solving the three dimensional inverse problem as an optimization problem. This method utilizes finite element interpolations, the adjoint method to evaluate the gradient and a quasi-Newton BFGS method for optimiza- tion. Its novel features include the use of the adjoint method and TVD regularization with piece-wise constant interpolation. A source of non-uniqueness in this inverse problem is identified theoretically, demonstrated computationally, explained physically and overcome practically. Both algorithms were test on ultrasound data of independently characterized tissue mimicking phantoms. The recovered elastic modulus was in all cases within 35% of the reference elastic contrast. Finally, the preliminary application of these techniques to tomosynthesis images showed the feasiblity of imaging an elastic inclusion.

Verification with Natural Contexts: Soundness of Safe Compositional Network Sketches

In research areas involving mathematical rigor, there are numerous benefits to adopting a formal representation of models and arguments: reusability, automatic evaluation of examples, and verification of consistency and correctness. However, accessibility has not been a priority in the design of formal verification tools that can provide these benefits. In earlier work [30] we attempt to address this broad problem by proposing several specific design criteria organized around the notion of a natural context: the sphere of awareness a working human user maintains of the relevant constructs, arguments, experiences, and background materials necessary to accomplish the task at hand. In this report we evaluate our proposed design criteria by utilizing within the context of novel research a formal reasoning system that is designed according to these criteria. In particular, we consider how the design and capabilities of the formal reasoning system that we employ influence, aid, or hinder our ability to accomplish a formal reasoning task – the assembly of a machine-verifiable proof pertaining to the NetSketch formalism. NetSketch is a tool for the specification of constrained-flow applications and the certification of desirable safety properties imposed thereon. NetSketch is conceived to assist system integrators in two types of activities: modeling and design. It provides capabilities for compositional analysis based on a strongly-typed domain-specific language (DSL) for describing and reasoning about constrained-flow networks and invariants that need to be enforced thereupon. In a companion paper [13] we overview NetSketch, highlight its salient features, and illustrate how it could be used in actual applications. In this paper, we define using a machine-readable syntax major parts of the formal system underlying the operation of NetSketch, along with its semantics and a corresponding notion of validity. We then provide a proof of soundness for the formalism that can be partially verified using a lightweight formal reasoning system that simulates natural contexts. A traditional presentation of these definitions and arguments can be found in the full report on the NetSketch formalism [12].