Quantitative elasticity imaging seeks to retrieve spatial maps of elastic moduli of tissue. Unlike strain, which is commonly imaged in compression elastography, elastic moduli are intrinsic properties of tissue and so this approach reconstructs images that are largely operator and system independent, enabling objective, longitudinal and multi-site diagnoses. Recently, novel quantitative elasticity imaging approaches to compression elastography have been developed. These methods use a calibration layer with known mechanical properties to sense the stress at the tissue surface, which combined with strain, is used to estimate the tissue's elastic moduli by assuming homogeneity in the stress field. However, this assumption is violated in mechanically heterogeneous samples. We present a more general approach to quantitative elasticity imaging that overcomes this limitation through an efficient iterative solution of the inverse elasticity problem using adjoint elasticity equations. We present solutions for linear elastic, isotropic and incompressible solids; however, this method can be employed for more complex mechanical models. We retrieve the spatial

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