In this paper, we develop a method to predict the energy deposition in tumours undergoing ultrasound-hyperthermia treatments. Energy depositions are calculated using an exact, analytic solution to the problem of ultrasound scattering from a spherical tumour. The biological tissues are modelled as solid, lossy elastic media so that (i) transverse-wave modes, in addition to longitudinal-wave modes, are considered, and (ii) mode coupling is fully accounted for during the scattering. The model tumour is of arbitrary size and no restrictions are placed on its material parameters relative to the surrounding tissue. Simpler analytic results are given for tumours that differ in densities and rigidities only slightly from the surrounding tissue. We briefly discuss how the above analysis could be extended to more complex systems, i.e., irregularly shaped tumours.