The analysis of the local myocardium motion is of high interest for the diagnosis and monitoring of various cardiac diseases. Magnetic resonance imaging (MRI) provides non-invasive tools to investigate myocardial velocities during the cardiac cycle such as tissue phase mapping (TPM)[1]. TPM directly quantifies velocity resulting in time-resolved 3D vector fields as a 2D image series. Clinicians try to interpret the 3D velocity field over time to detect global or local motion abnormalities. During analysis observers distinguish between radial and longitudinal contraction and rotation motion, which can be independently impaired. Hence, the goal is to explicitly analyze the three velocities (radial, longitudinal, rotational), which can potentially serve as biomarkers.

Traditionally, visual analysis is performed by inspecting each slice or by using the AHA bullseye plot[2] for every timepoint. We propose a novel visual encoding of local myocardial velocity by mapping radial, longitudinal, and rotational velocities to glyph shapes within a barycentric 3D superquadric glyph space. Superquadrics were previously used to visualize myocardial strain in 2D[3] or depict myocardial diffusion[4], but without providing a 3D global perspective. Our approach supports such a global overview by displaying average local velocity according to the AHA model in a 3D layout. The superquadric glyphs provide a non-ambiguous representation of the velocities in the three motion directions.

TPM was acquired in three short-axis slices using 1.5T Siemens MR systems (Aera and Avanto). TPM consisted of a black-blood prepared cine phasecontrast sequence with three-directional velocity encoding of myocardial motion (venc=25cm/s, temp res=20.8ms, spatial res between 2x2x8mm). Spatio-temporal imaging acceleration (k-t GRAPPA) with net acceleration factor R_net=3.6 was employed which permitted data acquisition during breath-holding (breath-hold time=25 heartbeats/slice).

The myocardium was semi-automatically segmented. One contour was manually drawn for one time step which was propagated to the rest of the time series using the velocity field through Runge-Kutta integration. To distinguish endocardial and epicardial regions, the myocardium centerline was extracted using a skeletonization approach based on topology-preserving morphological thinning[5]. The 3D velocity vectors acquired in Euclidean coordinates $$$(v_x,v_y,v_z)$$$ were transformed to the left ventricle-centered cylindrical coordinate system $$$(v_r,v_φ,v_z)$$$ (Fig.1) [6,7]. The velocity components were normalized to [0,1] and projected on to the AHA bullseye plot.

In order to parameterize the contribution of $$$(v_r,v_φ,v_z)$$$ we used a continuous set of shapes (Fig.2) represented in the superquadric space by a pair of parameters $$$(\alpha,\beta)$$$ using the equation[8]:$$Glyph(θ,Φ)=\left(\begin{array}{c}\cos^{\alpha}(\theta)\sin^{\beta}(\phi)\\\sin^{\alpha}(\theta)\sin^{\beta}(\phi)\\\cos^{\beta}(\phi)\end{array}\right),\begin{array}{l}0\le\phi\le\pi,\\0\le\theta\le2\pi\end{array}$$While Kindlmann[8] considered a continuous interval of $$$(\alpha,\beta)$$$ to visualize tensors defined by two parameters, we design a custom shape space starting from three desired shapes to encode a triple of parameters.The vertices of the triangle defining the barycentric space correspond to a maximum of $$$v_r$$$,$$$v_\varphi$$$,or $$$v_z$$$. They are assigned the shapes of a cylinder, double pyramid, and cuboid, respectively.These custom shapes were chosen because of their distinctive characteristics that we associate with the intensity of one of the three parameters. Moreover, they have an increased ability to point 3D direction, as they avoid rotational symmetry along horizontal and through-plane axes. For a smooth, intuitive transition, we additionally specified the glyph shapes at the midpoints of the triangle edges, where two parameters have equally large values, while the third is close to 0 ($$$v_r=v_l\gg v_\varphi$$$). If all three parameters are equal, the natural representation is the sphere.

The rest of the shape space can be filled continuously using barycentric interpolation.For this, we use the squared normalized cylindrical velocity components $$$(v^2_r,v^2_\varphi,v^2_z)$$$ as barycentric coordinates in the shape triangle (Fig.2) to compute the location of the glyph in the $$$(\alpha,\beta)$$$ parameter space. As the shape lies in one of the six sub-triangles, to determine the exact values of $$$(\alpha,\beta)$$$ we simply perform barycentric interpolation in the respective sub-triangle.

We generate the glyphs according to the previous section and place them at the center of the AHA segments in a 3D layout, pointing in the direction of the velocity. A cone is used to additionally show the motion direction. The resulting glyphs are scaled by the magnitude of the local velocity vector and also color-coded using a blue-red color map to improve 3D perception. Fig.3 shows the 3D glyph configuration for key time points in the cardiac cycle for a healthy volunteer.

The velocity direction and magnitude are easily perceived using the cones. Additionally we provide the explicit encoding of radial, longitudinal, and rotational motion, which allows us to visually assess the contribution of each component. Moreover, by using super-quadric glyphs, the glyph's shape and can be non-ambiguously perceived independent of the viewing angle. In future, we will perform a user study to evaluate our method.