This thesis presents a new continuum model for predicting results from atom probe tomography (APT) experiments. Unlike previous atomic-scale approaches, this continuum approach makes realistic specimen simulation computationally tractable, enabling direct quantitative comparison with experiment. Here, specimen evaporation resembles a moving boundary problem, where the electrostatic field at the surface and in the surrounding vacuum must be solved throughout time. Tracking the surface implicitly as a contour in a scalar field ensures numerical stability of surface faceting and evaporation of voids (cavities). The electric field is efficiently solved by a boundary integral equation. Proof-of-concept submodels for laser-assisted evaporation, crystallographic faceting, and material dielectricity are also considered. Predictions are shown to agree well with experiment. The application of the model to directly performing data reconstruction is also considered. A protocol is proposed where the specimen model for an APT analysis is calibrated against experimental data by solving a generalised image registration problem. Here, model parameters are constrained by maximising the similarity between the model and experimental data, measured using mutual information (MI). Through this proposed method, large-scale trajectory distortions are shown to be corrected for two experimental semiconductor devices: a multilayer structure and a fin field-effect transistor (FinFET) structure.