2012 Proceedings - International Tissue Elasticity Conference

063 THE ADJOINT WEIGHTED EQUATION (AWE) METHOD FOR THE SOLUTION OF INVERSE
PROBLEMS OF INCOMPRESSIBLE ISOTROPIC ELASTICITY.
Uri Albocher 1 , Isaac Harari 1 , Paul E. Barbone 2 , Assad A. Oberai 3 .
1 Tel Aviv University, Tel Aviv, ISRAEL; 2 Boston University, Boston, MA, USA; 3 Rensselaer
Polytechnic Institute, Troy, NY, USA.
Background: In quantitative elastography, inverse problems of elasticity are solved for material property
distributions (e.g., shear modulus). Typically, the problems are solved using iterative optimization
approaches where the difference between a measured displacement field and a predicted displacement
field (calculated from the momentum equation) is minimized. These methods are considered robust and
handle partial displacement measurements well, but are computationally expensive. In cases where full
field measurements of all displacement components are available, direct (non–iterative) approaches can
be considered. Here, the measured displacement data are inserted into the momentum equation, and the
equation is solved for the sought material properties directly. This procedure does not involve any
iterations, and the solution is obtained from a single solving of the equation, providing a significant step
towards real time imaging.
Aims: In anticipation of improved methods to measure all displacement vector components, our purpose is
to create an efficient and robust tool for directly solving the inverse problem of two– and three–dimensional
incompressible isotropic linear elasticity.
Methods: We analyze the uniqueness of the inverse problem of three–dimensional incompressible
isotropic elasticity. We find that when two independent displacement fields are available the shear
modulus distribution can be completely determined if it is known at five distinct points. To solve the
inverse problem directly, we use the adjoint weighted equation (AWE) method, a novel formulation for
direct (non–iterative) solutions of inverse problems [1]. We use several simple model problems with
simulated data to test the ability of the AWE method to reconstruct the shear modulus distribution.
These include inclusion problems where a circular inclusion is embedded in a homogeneous background.
We also investigate the performance of the method in the presence of noise and consider options to
improve the reconstruction.
Results: We successfully solve the inverse problem of incompressible isotropic elasticity using the AWE
formulation and reconstruct the shear modulus. The reconstructions are most accurate when the
distribution of the shear modulus in the inclusion is continuous. When the shear modulus is
discontinuous some accuracy is lost, but the overall shape of the inclusion and the
inclusion/background interface are captured well. We also find that at low levels of noise in the
displacements, the AWE can still provide fair results. At higher levels of noise, closer to those obtained
using ultrasound equipment, we find that smoothing the displacement data and adding regularization to
the AWE can significantly improve the results.
Conclusions: The AWE method, as a direct solution approach, can provide fast reconstruction of material
properties compared to conventional optimization approaches. We find that with some adaptations, the
method can also handle noise making it a potential tool for real time imaging. The availability of this and
related direct inversion approaches motivates the development of techniques to measure all components
of the displacement field.
Acknowledgements: This research was supported by Grant No. 2004400 from the United States–Israel Binational
Science Foundation (BSF), Jerusalem, Israel.
References:
[1] P. E. Barbone, C. E. Rivas, I. Harari, U. Albocher, A. A. Oberai, Y. Zhang: Adjoint–Weighted Variational
Formulation for the Direct Solution of Inverse Problems of General Linear Elasticity with Full Interior Data. Int.
J. Numer. Meth. Eng., 9, pp. 1–23, 2009.
(a) (b)
Figure 1: Shear modulus reconstruction using the AWE
method of a circular inclusion inside a
homogeneous background. Solution in the entire
domain (a) and along the diagonal (b).
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