The 4D trajectories may be built by composing motion vectors between pairs of surfaces found from consecutive time frames. Flow vectors running through the sample points of the surfaces over time are connected to form trajectories that track each individual point's motion over the entire cardiac cycle.
Then, motion and function indices for LV regional function analysis can be derived from these trajectories, including endocardial-epicardial surface motion measures, LV thickening measures, and LV cross-wall strain measures, etc. Each of these measures are recorded for each point trajectory, indexed by , on each heart image data set, indexed by . A subscript will also be used to differentiate algorithm-derived measures() from gold-standard-derived measures(). The gold standard measurements for validation purpose will be derived from the trajectories of the implanted markers described in the acute animal model. Thus, correlations will be observed between the shape-tracked trajectories and the marker-derived trajectories.
Some of the motion and function measures we have proposed on either endocardial or epicardial surface(single surface measures) include: i). the sum of the magnitude(length) of the flow vectors(path length), ; ii). the shape of the path(compare to a linear trajectory), ; and iii). the peak velocity within the path, . Here, is the displacement vector, is the curvature of the space curve representing the trajectory, of point of heart image data set at time .
The LV thickening measures, the cross-wall measures indexed by the subscript , is computed using measures of cross-wall vectors which relate points on the endocardial surface to corresponding points on the corresponding epicardial surface. The magnitude of each of these connecting vectors at frame , , is then taken as a measure of thickness, i.e . Since the trajectories of the points on endocardial and epicardial surfaces will have been found by either the shape-based motion tracking algorithm() or the implanted markers(), the connecting vectors can be tracked for every time frame in the cardiac cycle. By observing these vector length changes, thus the LV thickness changes, over all the frames, the following spatially localized endocardial-epicardial 4D thickening measures can be computed for each connected set of paired points. Existence of a state of strain is implicit in non-rigid body motion. As the body deforms from one configuration to another, the matter in the neighborhood of any given point is rotated, translated, and deformed, and hence strained. The different state of strain of myocardium is a significant clinical indicator of heart health. To compute the strain measure from the LV motion trajectories, we construct a certain number of cross--wall cube--like six--faced polyhedra on different parts of the LV. Each of these cubes consists of four points on the endocardium and another four on the epicardium. After the development of a locally defined coordinate system to eliminate any rotational or translational rigid motion components of the cube on a global scale, and by following the deformation of the cube from the trajectories of the eight vertices, normal and shear strain components can be found, which in turn indicates the compressive and torsional components of the cube's non-rigid motion, respectively.