3D anatomical variability assessment of the scoliotic spine using statistics on Lie groups

We present a method to analyse the variability of the spine shape using rigid transforms. The spine was expressed as a set of rigid transforms that superpose local coordinates systems of neighbouring vertebrae. Those transforms were computed from anatomical landmarks reconstructed in 3D using two radiographs. Since rigid transforms do not belong to a vector space, conventional mean and covariance could not be applied. The Frechet means and a generalized covariance computed in the exponential chart of the Frechet mean were used instead. Those statistics were computed for each intervertebral transforms of a group of 307 untreated scoliotic patients. The variability of inter-vertebral transforms was found to be inhomogeneous (lumbar vertebrae were more variable than the thoracic ones) and anisotropic (with maximal rotational variability in the coronal plane and maximal translational variability in the axial direction)