Nucleation and bubble growth are key steps in polymer foam processing. A model that combines heterogeneous nucleation with bubble growth dynamics and is used to estimate the effects of different parameters on bubble (or cell) size distributions in closed-cell foams is presented. The model is based on an expanding influence volume approach where the influence volumes are associated with bubbles such that the nucleation rate is negligible in the influence regions compared to the initial nucleation rate. The influence volumes expand as the bubbles grow. The equations governing the bubble growth are solved using Galerkin method. The initial conditions for bubble growth are obtained from the state of the pre-bubble nuclei (or clusters) at the upper bound of the critical cluster region predicted by the classical nucleation theory. Single bubble growth dynamics predicted by the Galerkin method compare well with the analytical solutions obtained in the initial growth period. The results indicate that the most sensitive parameters to final bubble size distribution are those associated with the nucleation process; growth dynamics can alter the distribution, but these effects are secondary.