  Elasticity images map biomechanical properties of soft tissues to aid in the
detection and diagnosis of pathological states. In particular, quasi-static
ultrasonic (US) elastography techniques use force-displacement measurements
acquired during an US scan to parameterize the spatio-temporal stress-strain
behavior. Current methods use a model-based inverse approach to estimate the
parameters associated with a chosen constitutive model. However, model-based
methods rely on simplifying assumptions of tissue biomechanical properties,
often limiting elastography to imaging one or two linear-elastic parameters.
  We previously described a data-driven method for building neural network
constitutive models (NNCMs) that learn stress-strain relationships from
force-displacement data. Using measurements acquired on gelatin phantoms, we
demonstrated the ability of NNCMs to characterize linear-elastic mechanical
properties without an initial model assumption and thus circumvent the
mathematical constraints typically encountered in classic model-based
approaches to the inverse problem. While successful, we were required to use a
priori knowledge of the internal object shape to define the spatial
distribution of regions exhibiting different material properties.
  Here, we introduce Cartesian neural network constitutive models (CaNNCMs)
that are capable of using data to model both linear-elastic mechanical
properties and their distribution in space. We demonstrate the ability of
CaNNCMs to capture arbitrary material property distributions using
stress-strain data from simulated phantoms. Furthermore, we show that a trained
CaNNCM can be used to reconstruct a Young's modulus image. CaNNCMs are an
important step toward data-driven modeling and imaging the complex mechanical
properties of soft tissues.
