Engineering simulations using boundary-value partial differential equations often implicitly assume that the uncertainty in the location of the boundary has a negligible impact on the output of the simulation. In this work, we develop a novel method for describing the geometric uncertainty in image-derived models and use a naive method for subsequently quantifying a simulation's sensitivity to that uncertainty. A Gaussian random field is constructed to represent the space of possible geometries, based on image-derived quantities such as pixel size, which can then be used to probe the simulation's output space. The algorithm is demonstrated with examples from biomechanics where patient-specific geometries are often segmented from low-resolution, three-dimensional images. These examples show the method's wide applicability with examples using linear elasticity and fluid dynamics. We show that important biomechanical outputs of these example simulations, namely maximum principal stress and wall shear stress, can be highly sensitive to realistic uncertainties in geometry.