  Tomographic image reconstruction is well understood if the specimen being
studied is stationary during data acquisition. However, if this specimen
changes during the measuring process, standard reconstruction techniques can
lead to severe motion artefacts in the computed images. Solving a dynamic
reconstruction problem therefore requires to model and incorporate suitable
information on the dynamics in the reconstruction step to compensate for the
motion.
  Many dynamic processes can be described by partial differential equations
which thus could serve as additional information for the purpose of motion
compensation. In this article, we consider the Navier-Cauchy equation which
characterizes small elastic deformations and serves, for instance, as a model
for respiratory motion. Our goal is to provide a proof-of-concept that by
incorporating the deformation fields provided by this PDE, one can reduce the
respective motion artefacts in the reconstructed image. To this end, we solve
the Navier-Cauchy equation prior to the image reconstruction step using
suitable initial and boundary data. Then, the thus computed deformation fields
are incorporated into an analytic dynamic reconstruction method to compute an
image of the unknown interior structure. The feasibility is illustrated with
numerical examples from computerized tomography.
