  The nonrigid alignment between a pre-operative biomechanical model and an
intra-operative observation is a critical step to track the motion of a soft
organ in augmented surgery. While many elastic registration procedures
introduce artificial forces into the direct physical model to drive the
registration, we propose in this paper a method to reconstruct the surface
loading that actually generated the observed deformation. The registration
problem is formulated as an optimal control problem where the unknown is the
surface force distribution that applies on the organ and the resulting
deformation is computed using an hyperelastic model. Advantages of this
approach include a greater control over the set of admissible force
distributions, in particular the opportunity to choose where forces should
apply, thus promoting physically-consistent displacement fields. The
optimization problem is solved using a standard adjoint method. We present
registration results with experimental phantom data showing that our procedure
is competitive in terms of accuracy. In an example of application, we estimate
the forces applied by a surgery tool on the organ. Such an estimation is
relevant in the context of robotic surgery systems, where robotic arms usually
do not allow force measurements, and providing force feedback remains a
challenge.
