Carlos E. Rivas

Quantitative ultrasonic elastography for gel dosimetry.

Advanced radiation techniques such as intensity-modulated radiotherapy (IMRT) for complex geometries in which targets are close to organs at risk have been introduced in radiation therapy, creating a need for procedures that allow easy three-dimensional (3-D) measurement of dose for verification purposes. Polymer gels that change their material properties when irradiated have been suggested for such use. For example, the change in their magnetic properties has been thoroughly investigated with magnetic resonance imaging (MRI). Also, we have previously shown that the mechanical stiffness, i.e., Youngs modulus, of these gels changes with dose. This finding prompted us to assess whether we can image a radiation-induced stiffness distribution with quantitative ultrasound elastography and whether the stiffness distribution is correlated with the dose distribution. A methacrylic-acid-based gel was loaded with scatterers to create an ultrasound echoic signal. It was irradiated to create a rod-like region of increased stiffness with a 10 x 10 mm(2) cross-section. The gel block was compressed in a frame that restricted the movement of the gel to planes orthogonal to the long axis of the irradiated region and ultrasonic echo data were acquired in the central plane during compression. This simplified irradiation pattern and experimental set-up were designed to approximate plane-strain conditions and was chosen for proof of concept. The movement of the gel was tracked from ultrasound images of a different compressional state using cross-correlation, enabling a displacement map to be created. The shear modulus was reconstructed using an inverse algorithm. The role of the magnitude of the regularization parameter in the inverse problem and the boundary conditions in influencing the spatial distribution of stiffness and, thus, final dose contrast was investigated through parametric studies. These parameters were adjusted using prior knowledge about the stiffness in parts of the material, e.g., the background was not irradiated and therefore its stiffness was homogeneous. It was observed that a suitable choice for these reconstruction parameters was essential for a quantitative application of stiffness measurement such as dosimetry. The dose contrast and distribution found with the optimal parameters were close to those obtained with MRI. Initial results reported in this article are encouraging and indicate that with ongoing refinement of ultrasound elastography techniques and accompanying inverse algorithms, this approach could play an important role in gel dosimetry.

Adjoint-weighted variational formulation for the direct solution of plane stress inverse elasticity problems

We describe a novel variational formulation of the inverse elasticity problem given interior data. The strong form of this problem is governed by equations of pure advective transport. To address this problem, we generalize the adjoint-weighted variational equation (AWE) formulation, originally developed for flow of a passive scalar. Here, we shall study the properties of the AWE formulation in the context of inverse plane stress elasticity imaging. We show that the solution of the AWE formulation is equivalent to that of the strong form when both are well posed. We prove that the Galerkin discretization of the AWE formulation leads to a stable, convergent numerical method, and prove optimal rates of convergence.

A stabilized forward elasticity finite element formulation yields a stable and convergent inverse elasticity solution.

“Elastography” refers to the procedure of using ultrasound to image tissue deformation. Images showing tissue deformation are sensitive to distributions of tissue mechanical properties. The mechanical properties themselves may be quantified by solving an inverse elasticity problem. An iterative approach to solving the inverse problem can be formulated by repeated solutions of the forward problem. That is, the shear modulus distribution sought is that which predicts a displacement field most consistent with the measured displacement field. Here we show that given plane displacement measurements for plane stress elasticity, the continuous elastic equilibrium equation uniquely determines the modulus distribution. On the other hand, we shall also demonstrate that the discrete elasticity equations from standard FEM discretization does not, even at the limit of infinite mesh refinement. We diagnose the problem as an underenforcement of the elasticity equations. With this knowledge, we have been able to design n...

Imaging of dose distributions using polymer gels based on radiation induced changes in stiffness

Previously, dose determination based on radiation induced stiffness difference measurements has received no or very little attention. Here, a preliminary evaluation of a combined system for dosimetry based on radiation sensitive gels, ultrasonic elastography and a plane strain inverse algorithm is presented. A block of gel was irradiated along one of its axes producing stiff rod-like regions. The dose distribution found with quantitative ultrasound elastography was compared with a reference dose distribution measured with magnetic resonance imaging. In these early results, the high dose areas were clearly detected, while noise in the ultrasound measurement and strong regularisation in the inverse computing introduced shape distortions, noise in the dose estimates and problems estimating the correct dose contrast. Improvements in the experimental setup and inverse computing are possible, for example by acquisition of transversal ultrasound data, which could essentially reduce the noise and restrict direct influence of the experimental boundary condition on the dose estimation by providing additional information for inverse computing. Based on the preliminary results and the potential for improvement it is concluded that further investigations should follow to establish the potential of the rapidly developing field of elastography for measuring radiation dose based on radiation induced changes in stiffness.

A Stabilized B-Splines FEM Formulation for the Solution of an Inverse Elasticity Problem Arising in Medical Imaging

Soft tissue pathologies are often associated with changes in mechanical properties. For example, breast and other tumors usually present as stiff lumps. Imaging the spatial distribution of the mechanical properties of tissues thus reveals information of diagnostic value. Doing so, however, typically requires the solution of an inverse elasticity problem. In this work we consider the inverse elasticity problem for an incompressible material in plane stress, formulated and solved as a constrained optimization problem. We formulate this inverse problem enforcing high order continuity for our variables. Driven by the requirements for the strong and weak solutions to this problem, we assume that our data field (i.e. the measured displacement) is in H2 and our parameter distribution (i.e. the sought shear modulus distribution) is in H1 . This high order regularity requirement for the data is incompatible with standard FEM. We solve this problem using a FEM formulation that is novel in two respects. First, we employ quadratic b-splines that enforce C1 continuity in our displacement field, consistent with the variational requirements of the continuous problem. Second, we include Galerkin-least-squares (GLS) stabilization in the iterative optimization formulation. GLS adds consistent stability to the discrete formulation that otherwise violates an ellipticity condition that is satisfied by the continuous problem. Computational examples validate this formulation and demonstrate numerical convergence with mesh refinement.Copyright