The continuum equations presented here model the growth of epitaxial films in terms of a local edge density similar to \delh\ and surface concentration (number density) of adatoms, This model is more amenable to computations than existing models that feature discrete edges and solve continuum equations on each terrace; yet it offers a more detailed picture than continuum models that treat the surface height as the only dependent variable. This latter feature is especially important if one wishes to account for several species which may react on the surface of the film or at step edges to build complicated unit cells. The model is motivated by and compared with numerical solutions of rate equations which are derived from kinetic Monte-Carlo simulations. After introducing the model in a I + I dimensional setting, we extend it to a 2+1 dimensional setting assuming spatial derivatives become surface gradients. We also discuss extension for the case with multiple species.