The local material stiffness of tissues is a well-known indicator of pathology, with locally stiffer tissue related to the possible presence of an abnormal growth in otherwise compliant tissue. Elastography is a non-invasive technique for measuring displacement distributions in loaded tissues within a medical imaging context. From these measured displacement fields, estimated for local strain have been made using well-studied techniques, but the calculation of elastic modulus has been difficult. In this study we show a method for estimating local tissue elastic modulus that gives numerically stable and robust results in test cases, and that is numerically efficient. The method assumes the tissue is isotropic and it requires an independent estimate of tissue Poisson's ratio, but the method reaches a stable result when the estimated Poisson's ratio is in error, and the resulting estimates are not very sensitive to the assumed value.