Direct EIT Jacobian calculations for conductivity change and electrode movement

Electrical impedance tomography (EIT) is very sensitive to deformations of the medium boundary shape. For lung imaging, breathing and changes in posture move the electrodes and change the chest shape, resulting in image artefacts. Several approaches have been proposed to improve the reconstructed images; most methods reconstruct both the boundary deformation and conductivity change from the measured data. These techniques require the calculation of the 'movement Jacobian', reflecting measurement changes due to the boundary deformation. Previous papers have calculated this Jacobian using perturbation techniques, which are slow (requiring multiple solutions of the forward problem) and become inaccurate with increasing finite element model size. This effect has limited reconstruction algorithms for deformable media to mostly 2D. To address this problem, we propose a direct method to calculate the Jacobian, based on a formulation of the derivatives of the finite element system matrix with respect to geometry changes. An illustrative example of these calculations is given, as well as a comparison between the proposed method and a perturbation method. Results show this method is approximately 300 times faster; and for larger model sizes, the perturbation method begins to diverge from those from the direct method proposed.