Elastography comprises a set of modalities that image the biomechanical properties of soft tissues for disease detection and diagnosis. Quasi-static ultrasound elastography, in particular, tracks sub-surface displacements resulting from an applied surface force. The local displacement information and measured surface loads may be used to compute a parametric summary of biomechanical properties; however, the inverse problem is under-determined, limiting most techniques to estimating a single linear-elastic parameter. We previously described a new method to develop mechanical models using a combination of computational mechanics and machine learning that circumvents the limitations associated with the inverse problem. The Autoprogressive method weaves together finite element analysis and artificial neural networks (ANNs) to develop empirical models of mechanical behavior using only measured force-displacement data. We are extending that work by incorporating spatial information with the material properties. Previously, the ANNs accepted only a strain vector input and computed the corresponding stress, meaning any spatial information was encoded in the finite element mesh. Now, using a pair of ANNs working in tandem with spatial coordinates included as part of the input, these new Cartesian ANNs are able to learn the spatially varying mechanical behavior of complex media. We show that a single Cartesian ANN is able to describe the same mechanical behavior of an object that previously required at least two ANNs. Furthermore, we show the new ANNs can learn complex material property distributions and reconstruct images of the Young's modulus distribution, not merely classify, filter, or otherwise process an existing image. For the first time, we present results using Cartesian neural networks within the Autoprogressive Method to form elastic modulus images.