Histotripsy is a form of therapeutic ultrasound that liquefies tissue mechanically via acoustic cavitation. The medium elasticity dictates the degree of bubble expansion, a key factor in efficacy of histotripsy therapy. An analytic model to predict histotripsy-induced bubble expansion based on the medium surface tension, viscosity, and inertia was extended to include the influence of medium elasticity. Good agreement was observed between the predictions of the analytic model and those computed via numerical integration of the Yang/Church model for shock-scattering-induced cavitation activity (shock scattering histotripsy), and cavitation activity nucleated from purely tensile pulses (microtripsy). The influence of medium elasticity on bubble expansion was dependent on the insonation type. The maximum bubble size steadily decreased with increasing medium elasticity when the medium elastic modulus was greater than 425 kPa for microtripsy pulses. For a shock scattering histotripsy, the maximum bubble size was independent of medium elasticity when the elastic modulus less than 2.95 MPa. Bubble growth was completely suppressed for both insonation types when the medium elastic modulus was greater than 20 MPa. Negligible changes in the predicted maximum bubble size were observed with finite-strain elastic models (linear, Kelvin-Voight, neo-Hookean), but were reduced by more than 60% with strain-hardening elastic models (Fung and Gent). Changes in cell viability adjacent to the bubble wall were evident, but limited to a distance of 44 mm. Furthermore, increases in the medium elasticity decreased the distance from the bubble wall over which cell viability was influenced. These results highlight the importance of the type of histotripsy insonation scheme based on the tissue elasticity, as well as provide an upper medium elasticity over which histotripsy can be utilized effectively for tissue liquefaction.