We develop a modified solid-on-solid model for studying heteroepitaxial growth of Ge on a 2 x 1 reconstructed Si(001) surface. The model is implemented using lattice-based kinetic Monte Carlo simulations using an atomistic model of elasticity. Our simulations show that the films are grown via Stranski-Krastanov mode with about 2 film layers wetting the substrate below three-dimensional islands. The island shapes evolve from rectangular-based at low coverage(thin films) to square-based at higher coverage. Their size distribution changes from bimodal to unimodal at higher coverage, and subsequently the surface is covered by uniform-sized islands. For simulations performed at lower deposition flux, the shape transition is observed for lower coverage. Our results are consistent with experiments and emphasize the role of the 2 x 1 reconstruction of the underlying Si(001) surface on the observed behavior. This is the only atomistic model that has been shown to reproduce features of both submonolayer growth and the Stranski-Krastanov growth transition at higher coverage. (C) 2016 Elsevier B.V. All rights reserved.