A pseudo-dynamic sub-optimal filter for elastography under static loading and measurements

We propose a pseudo-dynamic form of a sub-optimal Kalman filter for elastography of plane-strain models of soft tissues under strictly static deformations and partial measurements. Since the tissue material is nearly incompressible and is thus prone to volumetric locking via standard displacement-based finite element formulations, we use a Cosserat point approach for deriving the static equilibrium equations. A pseudo-dynamical form of the equilibrium equations, with added noise and appropriate augmentation by the discretized shear modulus as additional states, is then adopted as the process equation such that its steady-state solution approaches the static response of the plane-strain model. A fictitious noise of small intensity is also added to the measurement equation and, following linearization of the process equation, a Kalman filter is applied to reconstruct the shear modulus profile. We present several numerical experiments, some of which also bring forth the relative advantages of the proposed approach over a deterministic reconstruction based on a quasi-Newton search.