2.1.

Overview

Our fiber analysis method consists of the following two schemes: (1) estimation of fiber orientation and (2) fiber tracking.

We did scheme (1) for each voxel in the input CT volume to estimate the fiber orientation around the voxels to quantitatively analyze the fiber orientation statistics.

We did scheme (2) on the entire CT volume to produce trajectories that follow the fibers. Scheme (1) must be performed during the tracking process. The results of scheme (2) are useful for qualitatively visualizing how fibers flow in the entire LV.

2.2.

Fiber Orientation Estimation

Structure tensor (ST) analysis is commonly used for estimating the cardiac fiber orientation in $\mu \mathrm{CT}$ volumes.^11,12 First, for each volume, we apply a Gaussian smoothing filter with standard deviation ${\sigma }_{\mathrm{P}}$ to smooth the intensity gradients and empirically set ${\sigma }_{\mathrm{P}}=20\mu \mathrm{m}$.

ST $\mathbf{T}(\mathbf{x})$ at voxel $\mathbf{x}$ is defined:

2.3.

Fiber Tracking

We randomly generated $N$ initial points in the mask of the LV region. From each initial point, fiber tracking was done by an iterative process. First (iteration $k=0$), we estimated fiber direction vector ${\mathbf{f}(\mathbf{x}}_{\mathbf{0}})$ at each initial point $x_{\mathbf{0}}\in {\mathbf{R}}^3$ using the ST analysis described in Sec. 2.2. Since the fibers are running in both directions, ${\mathbf{f}(\mathbf{x}}_0)$ and $-{\mathbf{f}(\mathbf{x}}_0)$, fiber tracking was also performed for both directions. We calculated the endpoint coordinates of the trajectories at the $k$’th iteration (iteration $k>0$):

3.1.

RCT and $\mu \mathrm{CT}$ Scanning

RCT and $\mu \mathrm{CT}$ volumes of a rabbit heart were obtained by the following sequence: (1) harvesting a heart, (2) ethanol fixation, (3) RCT scanning, (4) contrast enhancement, (5) rinse, and (6) $\mu \mathrm{CT}$ scanning. Fixation was performed once using ethanol. ${\mathrm{I}}_2\mathrm{KI}$ was used for the contrast enhancement for the $\mu \mathrm{CT}$ scanning. Ethanol was used again in preparation for the $\mu \mathrm{CT}$ scanning for rinsing excess ${\mathrm{I}}_2\mathrm{KI}$ to reduce the artifact.

The following are the specimen preparation and scanning procedures. We scanned one $\mu \mathrm{CT}$ and one RCT volume of a rabbit heart (Fig. 1) under the IRB approval of Nagoya University. We harvested the heart of a Japanese white rabbit (10-week-old male) just after euthanasia with a KCl injection into the aortic arch and obtained a heart specimen. The following is the RCT scanning procedure: (1) ethanol fixation: the heart was fixated with an 80% ethanol water solution since ethanol fixation effectively improves the tissue contrast better than formalin fixation for the other phase-contrast imagings of hearts.¹³ (2) RCT scanning: RCT scanning was performed using the synchrotron system developed by Ando et al.’s group [Fig. 2(a)] at the High Energy Accelerator Research Organization (KEK) (Japan).¹⁴ The synchrotron system used for RCT scanning cost about 177 million USD (1 USD = 110 JPY).¹⁵ The RCT scanning specifications are listed in Table 1. Axial and coronal slices of the RCT volume are shown in Fig. 3(a).

Fig. 1

Rabbit heart: longest axis is about 20 mm.

Fig. 2

Machines that scanned rabbit heart in Fig. 1: (a) RCT and (b) $\mu \mathrm{CT}$.

Table 1

RCT scanning specifications.

Fig. 3

Axial and coronal slices: (a) RCT and (b) $\mu \mathrm{CT}$ volumes. Fibers on RCT volume look clearer than those of $\mu \mathrm{CT}$ volume. Registration is required for comparison due to different heart positions.

After RCT scanning, we scanned the same heart specimen in the following manner. We introduced an additional staining process for $\mu \mathrm{CT}$ scanning. (1) Contrast enhancement: We stained the rabbit heart with a 7.5% ${\mathrm{I}}_2\mathrm{KI}$ solution for one day. (2) Rinse: the heart was briefly rinsed in an 80% ethanol solution. (3) $\mu \mathrm{CT}$ scanning. Table 2 shows the scanning specification. Our scanner’s field of view (FOV) was limited: $1024\times 1024\times 548\text{voxels}$ at $17\times 17\times 17\mu {\mathrm{m}}^3/\text{voxel}$ resolution. It has a stitch-scanning mode to cover larger FOVs. We used this feature to cover the entire heart (three consecutive scans), although not every volume was aligned well in the stitching mode. Furthermore, ring artifacts on the $\mu \mathrm{CT}$ volume were quite obvious. We used TomoPy¹⁶ to reduce the ring artifacts, which are commonly observed on $\mu \mathrm{CT}$ volumes. Examples of the axial and coronal slices of the $\mu \mathrm{CT}$ volume are shown in Fig. 3(b).

Table 2

Specifications of μCT scanning.

This work used a desktop-type $\mu \mathrm{CT}$ scanner, inspeXio SMX-90CT Plus (Shimadzu, Japan) [Fig. 2(b)], which is a low-end, desktop type. Its catalog price is $\sim \mathrm{236,000}$ USD ($1\mathrm{USD}=110\mathrm{JPY}$). Ethanol fixation¹³ is also suitable for $\mu \mathrm{CT}$ scanning in combination with contrast enhancement. Other $\mu \mathrm{CT}$ cardiac imaging works^12,17 use high-end, much more expensive $\mu \mathrm{CT}$ scanners than ours. In those works, contrast enhancement continued for several days by staining the specimens in an iodine-potassium iodide (${\mathrm{I}}_2\mathrm{KI}$) solution. For instance, one trial by Stephenson et al.¹⁷ stained a rabbit heart in a 7.5% ${\mathrm{I}}_2\mathrm{KI}$ solution for 3 days with the Metris X-Tec custom 320-kV bay system with 155-kV tube voltage and $150\text{-}\mu \mathrm{A}$ tube current. However, directly using the same protocols as these Refs. 12 and 17 for our scanner caused artifacts since our scanner has lower x-ray energy.

RCT has superior soft tissue contrast to $\mu \mathrm{CT}$. This means that RCT can depict different soft tissues in different intensities although $\mu \mathrm{CT}$ depicts such soft tissues in the same intensities. Phase-contrast x-ray imaging including RCT has been developed for better soft tissue contrast.

3.2.

Registration of $\mu \mathrm{CT}$ and RCT Volumes

To compare the fiber analysis results, we registered the RCT volume as $\mu \mathrm{CT}$. The heart’s $\mu \mathrm{CT}$ and RCT volumes were cropped and rotated manually using the MITK Workbench 2016.11.¹⁸ The LV is entirely covered with a slight margin around it and roughly aligned between the two volumes whose size and resolution were adjusted into $900\times 980\times 1080$ and $18\times 18\times 18\mu \mathrm{m}/\text{voxels}$ respectively. The coordinate system of these volumes is shown in Fig. 4. Since the parts of the surrounding regions such as RV were also included, the processing target region was specified by masking. The mask of the LV region (LV mask) was segmented semiautomatically using the MITK Workbench 2016.11¹⁸ on the $\mu \mathrm{CT}$ volume. Then, we applied nonrigid registration to the RCT volume to align it with the $\mu \mathrm{CT}$ volume. We used deedsBCV, which is open-source software published by Heinrich et al.¹⁹

Fig. 4

Coordinate system and position of ventricles: Axial planes ($x-y$ plane) cut axis along base and apex into rounds. On axial planes, RV is shown on left of LV. Outside ratio is illustrated in magnified part. Outside ratio becomes 0% at endocardium side and 100% at epicardium side.

Figures 5(a) and 5(b) show the axial and coronal slices of the registration results. In Fig. 5(c), the axial slices of the two registered volumes are shown as one figure after being merged to resemble a checkerboard. Clearly, the RCT volume was successfully registered to the $\mu \mathrm{CT}$ volume. As shown in Fig. 5(c), the boundaries of the LV and the image patterns shown in both volumes were successfully aligned.

Fig. 5

Axial and coronal slices of registered volumes: Registration results of (a) RCT and (b) $\mu \mathrm{CT}$ with LV mask (red line); (c) checkerboard-like scheme visualization of these volumes.

4.1.

Overview

We evaluated how our fiber analysis method produced precise results from the $\mu \mathrm{CT}$ volume by comparing them with the RCT volume results. We performed fiber tracking for each registered volume to compare the tracking results obtained from the RCT and $\mu \mathrm{CT}$ volumes. We analyzed the fiber orientation statistics on multiple axial slices. Detailed analysis was conducted on one of those axial slices around the central part of the LV and focused on fiber orientations. The 3-D visualization of the fibers was performed by fiber tracking (Sec. 2.3).

4.2.

Fiber Orientation Statistics

4.2.2.

Angle difference of $\mu \mathrm{CT}$ from RCT

We define the angle difference of $\mu \mathrm{CT}$ from RCT ${\theta }_1$:

Eq. (3)

$${\theta }_1={\cos }^{-1}\{{\mathbf{f}}^{\mu }(\mathbf{x})·{\mathbf{f}}^{\mathrm{R}}(\mathbf{x})\}(0\le {\theta }_1\le \pi ),$$

where ${\mathbf{f}}^{\mu }(\mathbf{x})$ and ${\mathbf{f}}^{\mathrm{R}}(\mathbf{x})$ represent the unit vectors of the fiber orientations estimated from the $\mu \mathrm{CT}$ and RCT volumes [Fig. 6(a)], respectively. Assuming the orientation from RCT is the ground-truth, the angle difference of $\mu \mathrm{CT}$ from RCT represents the estimation error on $\mu \mathrm{CT}$.

Fig. 6

Definitions of angles: (a) angle difference of $\mu \mathrm{CT}$ from RCT ${\theta }_1$ and (b) inclination angle ${\theta }_2$.

To evaluate how the fiber orientations estimated from $\mu \mathrm{CT}$ volumes are different from those of the RCT, we computed the average and standard deviations of the angle difference of $\mu \mathrm{CT}$ from RCT at 100-slice intervals and plotted them on a graph. We also visualized the angle differences of $\mu \mathrm{CT}$ from RCT on sample points on an axial slice around the central area.

4.3.

3-D Visualization of Fibers

We performed 3-D visualization using open-source software ParaView 5.3.0²¹ for each registered volume to qualitatively compare the fiber trajectories from the RCT and $\mu \mathrm{CT}$ volumes in the entire LV. All the points of the trajectories were colored to show the inclination angle. We showed all the tracking results. We trimmed them and showed whether for the sagittal slices, the tracking was done properly in the entire LV. Since ParaView crashed when we directly opened the RCT or $\mu \mathrm{CT}$ volumes, we downsampled these volumes twice by cubic interpolation before opening them.

5.1.

Fiber Orientation Statistics

Figure 7 shows the mean and standard deviation of the angle differences of $\mu \mathrm{CT}$ from RCT on axial slices throughout the LV, most of which had mean angle differences of $\mu \mathrm{CT}$ from RCT around 20 deg. For instance, their mean and standard deviations were $21.8\pm 20.5\mathrm{deg}$ on an axial slice around the central part. Figure 8 shows the angle differences of $\mu \mathrm{CT}$ from RCT on a manually selected slice ($\text{depth}=8.85\text{\hspace{0.17em}mm}$. see Fig. 7). In Fig. 8, fiber orientations at a sample point are represented as two cylinders. A white cylinder shows the fiber orientation estimated from the RCT volume. The colored cylinder shows fiber orientation estimated from the $\mu \mathrm{CT}$ volume, colored based on their angle difference of $\mu \mathrm{CT}$ from RCT.

Fig. 7

Mean and standard deviation of angle differences of $\mu \mathrm{CT}$ from RCT at sample points on each axial slice: target axial slices were selected at 100-slice intervals along $Z$-axis (longest axis from apex to base) of RCT and $\mu \mathrm{CT}$ volumes. Each point on graph shows mean angle differences of $\mu \mathrm{CT}$ from RCT of a sample point on slice, and error bars represent standard deviation. Sample points on each slice were defined by a radial search scheme, explained in Sec. 4.2. Results do not greatly vary throughout the entire LV. Slice located at depth 8.85 mm is used for further evaluation in Figs. 8Fig. 9–10 and indicated by arrow.

Fig. 8

Angle differences of $\mu \mathrm{CT}$ from RCT on manually selected slice (depth = 8.85 mm, see Fig. 7): colored cylinders show fiber orientations estimated from $\mu \mathrm{CT}$ volume, colored based on angle difference of $\mu \mathrm{CT}$ from RCT. Fiber orientations estimated from RCT volume are also shown as white cylinders.

Figure 9 also shows the estimated fiber orientations. Cylinders show estimated fiber orientation, colored based on their inclination angles.

Fig. 9

Fiber orientations on manually selected slice (depth = 8.85 mm. see Fig. 7) with coloring based on inclination angle: (a) RCT and (b) $\mu \mathrm{CT}$. Cylinders show estimated fiber orientations; colors represent inclination angles.

The relationship of the inclination angles measured in the RCT and $\mu \mathrm{CT}$ volumes is shown in Fig. 10. Each circle in Fig. 10 is gray-scale coded based on the outside ratio. The inclination angles estimated from the RCT and $\mu \mathrm{CT}$ volumes had a correlation coefficient (CC) of 0.63. No significant difference was observed by Spearman’s significant test: $p<2.2\times {10}^{-16}$. This shows $\mu \mathrm{CT}$ produced fiber analysis results that resembled those of RCT. The inclination angles of RCT and the outside ratio also show a significant correlation: $p=2.4\times {10}^{-6}$ with a CC of $-0.48$. Those of the $\mu \mathrm{CT}$ and the outside ratio are $p=1.2\times {10}^{-7}$ and showed a correlation with a CC of $-0.53$.

Fig. 10

Relationship of inclination angles measured in RCT and $\mu \mathrm{CT}$ volumes. We manually selected slice (depth = 8.85 mm, see Fig. 7) and plot inclination angles measured on selected slice in this figure. Each circle is gray-scale coded based on outside ratio. Positive correlation is clearly observed between inclination angles estimated from RCT and $\mu \mathrm{CT}$ volumes. We can also observe positive inclination angles in outside area (epicardium area).

5.2.

3-D Visualization of Fibers

Figure 11 shows the fiber trajectories cropped along the coronal plane and a sagittal slice of the $\mu \mathrm{CT}$ or RCT volumes. Colors showing the inclination angles are red inside the LV and green outside it. These color tendencies visually confirm the correspondence of the outside ratio and the inclination angles. However, from the $\mu \mathrm{CT}$ results [Fig. 11(b)], some fiber tracking results were flat and densely gathered. This tendency was not observed in the RCT results [Fig. 11(a)]. These incorrect tracking results from the $\mu \mathrm{CT}$ volume were caused by the joints produced by the scanning procedure, as explained in Sec. 3.1.

Fig. 11

Fiber tracking results with sagittal slice. Colors represent inclination angles. Two viewpoints were defined: one for observing endocardium and another for epicardium. (a) RCT: tracking was performed properly in entire LV. (b) $\mu \mathrm{CT}$: although closely resembling RCT results in (a), flat tracking results, densely gathered in joints, were produced due to scanning procedure, explained in Sec. 3.1.

6.1.

Fiber Orientation Statistics

The $\mu \mathrm{CT}$ visually had lower contrast for the heart shown in Figs. 3 and 5. The fiber orientation estimations were not very similar to those of the RCT volume, which had an average error around 20 deg (Fig. 7). The average error values were increased by outliers, like the red bars in Fig. 8.

In the part magnified in Fig. 8, many outliers are observed. These errors were caused by an iodine solution artifact (having a higher absorption of x-ray) used for contrast enhancement of $\mu \mathrm{CT}$ imaging (Fig. 8). This iodine solution created a strong artifact in a slice plane. A tracking algorithm traced it and produced in-plane (flat) tracking.

The colors of the points in Fig. 9 suggest that the inclination angles computed from both the RCT and $\mu \mathrm{CT}$ volumes were positive inside and negative outside the LV. This tendency was already proved through anatomical studies,⁵ and the results of both the $\mu \mathrm{CT}$ and RCT volumes followed it.

We used nonrigid registration to compensate for the deformation of the specimen at the RCT and $\mu \mathrm{CT}$ scanning times. Our scanning procedures were performed in the following order: RCT scanning, iodine staining, and $\mu \mathrm{CT}$ scanning, as explained in Sec. 3.1. Iodine staining caused a slight contraction of the heart. There were some changes in the specimen sizes and small structures between the two CT volumes.

6.2.

3-D Visualization of Fibers

Fiber tracking allows intuitive understanding of fiber running orientations in 3-D space. The tendency of inclination angles, correlated to the outside ratio, was also visually observed in the fiber tracking results from both the RCT and $\mu \mathrm{CT}$ volumes (Fig. 11). Most of the fiber tracking results inside the LV were red, and most of those outside were green. The trajectories were visually smooth from both volumes.

Figure 11 shows the fiber tracking results from the base to the apex. One large difference between the RCT and $\mu \mathrm{CT}$ volumes is apparent. On the results from the $\mu \mathrm{CT}$ volume [Fig. 11(b)], flat tracking results are densely gathered. Since our $\mu \mathrm{CT}$ scanner had a limited FOV, the rabbit heart was scanned by dividing it into three parts (Sec. 3.1). Since the images of the three scanning results were not precisely aligned, their joints were followed by tracking. Correction processes for such mistracking are required in the future.

We found that it is possible to estimate fiber orientation well on $\mu \mathrm{CT}$ volumes. Our fiber orientation estimation procedures were useful for fiber tracking in the entire LV, although the results must be scrutinized for errors between two scans. $\mu \mathrm{CT}$, which is a promising modality for cardiac imaging and useful for observing cardiac fibers, is commonly used by many companies and institutes for industrial purposes. Our work shows an application for cardiac imaging, which presents imaging protocols and their usefulness for observing cardiac fibers.