Elastostatics of a spherical inclusion in homogeneous biological media

A three-dimensional spherical inclusion model that approximates a lesion bonded to a tissue matrix is proposed for biomedical elastography. Analytical formulae describing spatial strain and stress distributions generated in infinite media by uniform loading are given under a linear, homogeneous, isotropic elasticity assumption. Strain and stress distributions are also calculated using finite-element analysis (FEA) for a variety of cases to determine the effects of shear modulus distribution, external loading conditions (uniform stress versus uniform displacement), compressor size and matrix dimensions on the elastostatics of the tissue. Analytical strain and stress predictions are shown to agree with the FEA results to within 10% accuracy provided that the matrix dimensions are at least ten times that of the inclusion. Also for these cases, uniform-stress boundary conditions can be equivalently represented by uniform displacement of the boundary. Spherical inclusions exhibit a lower efficiency for transferring elastic shear modulus contrast into strain contrast than cylindrical or planar inclusions. Additional compression will increase the strain contrast. However, large compressions also lead to increases in ultrasonic signal decorrelation and strain and stress concentrations in the homogeneous matrix around the inclusion. Although strain concentrations may help describe the boundaries of the inclusion more clearly, they also increase the risk of damaging the tissue. Understanding the strain and stress distributions in a biological tissue containing a lesion is necessary for optimizing the experimental configurations and consequently improving the diagnostic values of elasticity imaging.