Fig. Patient-specific aorta case. On the most left, the half-transparent white aorta represents the fixed image; while the blue represents the moving image. Three salient deformed regions appear in this case: the compression on descending aorta (Region A), the expansion on ascending aorta (Region B), and the longitudinal shrink on distal descending aorta (Region C). The directional components are linearly interpolated to the vertex points of the 3D aortic surface for display. Expansion of aorta is visualized in red, compression blue, and no general deformation in green. Compared to Original Det, the other three types of Directional Jaocobian can well preserve the deformation in a specific direction while removing that in other directions

Traditional cross-sectional imaging surveillance techniques are focused on the measurement of maximal aortic diameter, which could severely suffer from human error and unsatisfactory aorta position. Only recently, the Jacobian-based methods are proposed for aortic 3D deformation measurement. Although Jacobian analysis can well capture the 3-dimensional changes in the aortic wall geometry, there are existing evidence suggests that the deformation of the aorta in specific directions is more important for clinicians. However, current techniques do not allow measuring the directional component of deformation. 

Here we propose a method to decompose the Jacobian into directional components, which can provide radiologists with finer-grained and unambiguous deformation measurements in any specified directions, e.g., cross-sectional(2D). In other words, this technique allows isolation and removal of any directional change, e.g., longitudinal stretch or compression. 

We rigorously prove the theory of decomposition of Jacobian into directional components. Also, the experiment on cylinders and real aorta shows clear discrepancies between the original Jacobian and its directional components, with the directional component being able to remove the longitudinal change while maintaining the radial/cross-sectional deformation. This technique can significantly benefit the imaging surveillance and clinical decision-making.