Mia Mojica^{1}, Mihaela Pop^{2}, Maxime Sermesant^{3}, and Mehran Ebrahimi^{1}

In this study we successfully created the first high-resolution cardiac Diffusion Tensor (DT) imaging-based fiber atlas from porcine hearts. Furthermore, we laid the foundation of a framework for building a statistical cardiac atlas by computing an average cardiac geometry from a small database (N=8) of explanted porcine hearts without the need for selecting landmarks, transforming the diffusion tensors of subjects, and obtaining the associated average DT field and fiber architecture. The fiber atlas can be used for personalized electro-mechanical simulations of cardiac function (in anisotropic models when DT data is not available), and aid in diagnosis or therapy guidance procedures.

A diagram of the workflow is presented in Figure 1. First, the
diffusion-weighted (DW) MRI studies were performed on a dedicated 1.5T GE Signa
Excite scanner using a small database of $$$N=8$$$ explanted healthy pig hearts, with
approval from our institute. In the current study, we used the following MR
parameters: TE=35 ms, TR=700 ms, ETL=2, b=0 for the unweighted MR images, and
b=500 s/mm^{2} when the seven diffusion gradients were applied,
respectively^{4}. Image resolution was ~0.5x0.5x1.6 mm.

The geometries of the 3D anatomical volumes $$$T_i ~(i = 1,\ldots,N)$$$ were first normalized to generate a mean cardiac
volume. This was done through an iterative groupwise registration scheme. At
every groupwise iteration, all the subjects were registered to the current reference
volume via multilevel nonparametric registration^{5}. The resulting
transformations from the pairwise registration step were used to update the reference
geometry $$$I_{\text{mean}}^{n}$$$.

The diffusion tensors associated with each subject were subsequently transformed using the transformations aligning the subjects to the average geometry; this was done to project the tensors onto a common reference frame, which would ultimately enable us to extract tensor statistics for the population. We used a Finite Strain reorientation method to transform the tensors. Given a deformation field $$$y$$$, the rotation component of the local affine transformation given by $$$A = I + \nabla y$$$ was used to reorient the tensor at every voxel. It was obtained by solving for $$$R$$$ in the polar decomposition of $$$A$$$: $$$A = RU$$$, while the reoriented tensor $$$A(D)$$$ was given by $$$A(D) = RDR^T$$$. Finally, we used the Log-Euclidean metric to calculate the average diffusion tensor field $$$\overline{D}_\text{log}(X)$$$: $$$\overline{D}_\text{log}(X) = \exp\left[\displaystyle\frac{1}{N}\sum_{i=1}^{N}\log\left(D_i\left(X\right)\right)\right]$$$, where $$$D_i(X)$$$ refers to the reoriented tensor of the $$$i^\text{th}$$$ subject.

The groupwise
registration algorithm took only 8 iterations until it converged to the stable
average geometry shown in Figure 2(a). The error evolution of the groupwise algorithm
(measured in terms of change in average intensity values between consecutive
reference geometries) is shown in Figure 2(b).
In Figure 3, we
present the average tensor field and the results of the Finite Strain
reorientation method. Note that the geometric features and the local
orientation of the diffusion tensor fields were preserved. This makes Finite
Strain an ideal method for inter-subject DT-MRI registration^{3}.

The average cardiac fiber architecture is shown in Figure 4 in three views: anterior (A), posterior (P), and lateral (L). Fiber tractography was performed on the average DT field using MedInria (http://med.inria.fr) to find the end-to-end pathway of the principal direction of diffusion given by the first eigenvector of the diffusion tensors.

1. Piuze, E., *et al*. Atlases of
cardiac fiber differential geometry. In *International
Conference on Functional Imaging and Modeling of the Heart*, pages 442–449.
Springer, 2013.

2. Lombaert H., *et al.* Human atlas of the cardiac fiber architecture: study on a healthy population.
*IEEE transactions on medical imaging*,
31(7):1436–1447, 2012.

3. Peyrat, J.M., *et al*. A computational framework for the statistical analysis of cardiac
diffusion tensors: application to a small database of canine hearts. *IEEE transactions on medical imaging*,
26(11):1500–1514, 2007.

4. Pop, M., *et al*. Quantification of fibrosis in infarcted swine
hearts by ex vivo late gadolinium-enhancement and diffusion-weighted MRI methods.
*Physics in medicine and biology*,
58(15):5009, 2013

5. Modersitzki, J. *FAIR: Flexible algorithms for image registration*, volume 6. SIAM, 2009