In vivo visualization of cardiovascular structures is
possible using medical images. However, one has to realize that the resulting 3D
geometries correspond to in vivo conditions. This entails an internal
stress state to be present in the in vivo measured geometry of
e.g. a blood vessel due to the presence of the blood pressure. In order
to correct for this in vivo stress, this paper presents an inverse method
to restore the original zero-pressure geometry of a structure, and to recover the
in vivo stress field of the final, loaded structure. The proposed
backward displacement method is able to solve the inverse problem iteratively using fixed
point iterations, but can be significantly accelerated by a quasi-Newton technique in
which a least-squares model is used to approximate the inverse of the Jacobian. The here
proposed backward displacement method allows for a straightforward implementation of the
algorithm in combination with existing structural solvers, even if the structural solver
is a black box, as only an update of the coordinates of the mesh needs to be
performed.