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While ultrasound- and magnetic resonance-based elastography techniques have proved to be powerful biomedical imaging tools, most approaches assume isotropic material properties. In this paper, a general framework is developed for tomography-based anisotropic elastography. An anatomically well- motivated piece-wise homogeneous model is proposed to represent a class of biological objects consisting of different regions. With established tomography modality, static displacements are measured on the entire external and internal boundaries, and the force distribution is recorded on part of the external surface. A principle is proposed to identify the anisotropic elastic moduli of the constituent regions with the obtained boundary measurements. The reconstruction procedure is optimization-based with minimizing an objective function that measures the difference between the predicted and observed displacements. Analytic gradients of the objective function with respect to the elastic moduli are calculated using an adjoint method, and are utilized to significantly improve the numerical efficiency. Simulations are performed to identify the elastic moduli in a breast phantom consisting of soft tissue and a hard tumor. For isotropic phantom, one set of the boundary measurements enables unique reconstruction results for the tissue and tumor. For anisotropic phantom, however, multiple sets of the measurements corresponding to different deformation modes become necessary.