Computational time

The convex max-flow algorithm was implemented using parallel computing architec- ture (CUDA, NVIDIA Corp., Santa Clara, CA). The user interface for initialization, preprocessing and cost calculation were performed using Matlab (Mathworks Inc., Natick, MA). The experiments were conducted on a Quad core Windows workstation with 3.0 GHz, 32GB RAM and a GPU of NVDIA GTX670. Initialization for our algorithm required a mean time of 21.0_±2.7 s, including the time required to run the

(a) (b)

(c) (d)

Figure 4.6: Example segmentations of a T1-weighted 3T image: (a) Initial surface for LIB obtained using region growing, (b) algorithm generated AB and LIB sur- faces. Example segmentations of a T1-weighted 1.5T image: (c) Initial surface for LIB obtaining using region growing, (d) algorithm generated AB and LIB segmented surfaces.

2D max-flow algorithm, which took 6.4 s (2 s for the max-flow computation and 4.4 s for the cost computation using a nonoptimized Matlab code). The 2D algorithm required on average 4 iterations to obtain the final result. The average computational time for one iteration of the max-flow solver was 4 s. For a 3T image, the time for convergence of the 3D algorithm was_≈26 s (6 s for the max-flow computation and 20 s for the cost computation using a nonoptimized Matlab code), which was achieved within 5-12 iterations for a single 3D MR image. For a 1.5T image, the convergence time of the 3D algorithm was _≈5 s (1.4 s for the max-flow computation and 3.6 s for the cost computation using a nonoptimized Matlab code), which was achieved within 10 iterations.

(a) (b) (c) (d)

Figure 4.7: Visual surface comparisons of some example algorithm-generated AB and LIB surfaces to manually generated surfaces. The manually segmented surface is shown in blue color whereas the algorithm generated surface is shown in purple color. (a) AB surface segmented from a T1-weighted 3T MR image, (b) LIB surface segmented from the same T1-weighted 3T MR image, (c) AB surface segmented from a T1-weighted 1.5T MR image, and (d) LIB surface segmented from the same T1- weighted 1.5T MR image.

4.4.2

Accuracy

Segmentation results of two example 3D MR images are shown in Fig. 4.6. The carotid AB and LIB surfaces generated using the proposed algorithm for 3T and 1.5T MR images are shown in Fig. 4.6(b) and (d), whereas the initial surfaces gener- ated from region growing method are shown in Fig. 4.6(a) and (c). The comparison of the algorithm-generated AB and LIB surfaces to the manual segmentations are shown in Fig. 4.7. The manually segmented surface is rendered in blue, whereas the algorithm surface is rendered in purple. Greater disagreement between the algorithm and manual delineations is present at the bifurcation of the carotid AB surface by visual comparison. Figure4.8 shows the slice-by-slice comparisons of the algorithm to the manual segmentations for the carotid AB and LIB of the CCA, ICA, and ECA. Algorithm segmentations are shown in green continuous contours, whereas manual segmentations corresponds to the red dashed lines. As observed in Fig. 4.8, the algorithm segmentations are in good agreement with the manual segmentations.

The performance results of the algorithm for 12 3T MR images are shown in Table 4.5. The algorithm yielded more than 90% DSCs for the AB and LIB of the CCA and ICA. The DSC of _≈93% yielded for the CCA AB and LIB is the highest.

Table 4.4: Confidence intervals (CI) and Pearson correlation coefficients for comput- ing VWV.

Artery Pearson r p-value 95% CI (mm3) p-value CCA 0.82 0.001 -80.5–67.3 0.998 3T ICA 0.76 0.006 -46.8–26.7 0.634 ECA 0.74 0.004 -13.2–20.7 0.560 CCA 0.94 <0.001 -16.5–54.8 0.279 1.5T ICA 0.87 <0.001 15.9–54.7 0.020 ECA 0.73 <0.001 4.7–39.5 0.015

In addition, the lowest and highest standard deviations were reported for the CCA and for the ECA, respectively. The absolute volume errors follows a similar trend, where the lowest volume errors were obtained for the CCA, even though the actual difference in mm3 is higher for the CCA, due to its larger size than the ICA and ECA. The volume errors (δV_E) obtained for the algorithm were small and negative (_≤ 35 mm3).

The RMSEs for the segmentation and their standard deviation are sub-millimeter for both the AB and LIB of the CCA, ICA, and ECA. For example, RMSEs for the AB and LIB of the CCA were 0.5 mm and 0.3 mm, respectively, which is equivalent to 2.5 and 1.5 times the width of a voxel (0.2 mm). The MAXD errors reported for the LIB were lower than for the AB for all three sections of the CA.

Table 4.6 shows performance results of the algorithm for 26 1.5T T1-weighted MR images. The algorithm yielded more than 90% DSCs for the AB and LIB of the carotid CCA and ICA. The DSC of 91%_±1.6% and 92%_±2.1% were the highest DSC for the CCA AB and LIB. The lowest DSCs were reported for the ECA AB and LIB, which were_≈87% and_≈86%. Similar to the previous results with 3T MR images, the algorithm underestimated the volumes relative to the manual segmentations except for the ECA AB.

The RMSEs were sub-millimeter for the 1.5T MR images. For example, RMSEs for the CCA AB and LIB were 0.6_±0.1 mm and 0.5_±0.2 mm, respectively. The algorithm also yielded small MAXD errors (0.9 to 2.1 mm). For example for the CCA, MAXD errors were 1.3_±0.4 mm and 1.3_±0.8 mm for the AB and LIB, respectively.

(a) Slice-by-slice comparisons for a T1w 3T image

(b) Slice-by-slice comparisons for a T1w 1.5T image

Figure 4.8: 2D slice-by-slice comparisons of algorithm-generated surface to manual segmentations for two example images. The algorithm generated surface is sliced on the same planes as the 2D manual segmentations. Algorithm segmentations are shown as a green continuous line whereas the manual segmentations are shown in red dashed lines.

(a) CCA (b) CCA

(c) ICA (d) ICA

(e) ECA (f) ECA

Figure 4.9: Bland-Altman plots [49] of the vessel wall volume (VWV) measurements. Confidence interval (CI) of mean of differences between the VWVs are also indicated by the green dotted line. 1st Column: Graph for 12 3T images, 2nd Column:

Table 4.5: Accuracy results for 12 T1-weighted 3T MR images using the proposed algorithm. DSC RMSE MAXD _|δV_{E| |}δV_P| δV_E δV_P (%) (mm) (mm) (mm3) (%) (mm3) (%) CCA AB 93.0_±1.9 0.5_±0.7 1.1_±0.9 83.0_±93.9 5.8_±5.4 ₋34.3_±122.6 ₋1.7_±8.0 LIB 93.3_±1.4 0.3_±0.2 0.8_±0.5 50.7_±33.8 7.4_±5.7 ₋27.7_±55.8 ₋4.0_±10.1 ICA AB 90.1_±3.3 0.5_±0.5 1.6_±1.4 61.0_±46.2 11.9_±7.4 ₋34.6_±69.8 ₋4.9_±14.5 LIB 90.4±3.2 0.2±0.1 0.6±0.2 25.1±17.4 14.5±6.0 −24.5±18.2 −14.1±6.7 ECA AB 88.4_±4.6 0.7_±0.7 2.0_±2.1 23.8_±24.2 10.1_±9.7 ₋13.0_±32.0 ₋4.2_±13.9 LIB 84.1_±6.0 0.2_±0.1 0.6_±0.1 17.2_±11.0 23.6_±10.9 ₋16.8_±11.8 ₋23.1_±12.2

Table 4.6: Accuracy results for 26 T1-weighted 1.5T MR images using the proposed algorithm. DSC RMSE MAXD _|δV_{E| |}δV_P| δV_E δV_P (%) (mm) (mm) (mm3) (%) (mm3) (%) CCA AB 91.3_±1.6 0.6_±0.1 1.3_±0.4 111.0_±88.0 6.7_±5.3 ₋16.0_±142.3 ₋1.1_±8.7 LIB 92.4_±2.1 0.5_±0.2 1.3_±0.8 91.6_±82.4 12.8_±11.2 ₋35.1_±119.3 ₋3.9_±17.1 ICA AB 91.1_±2.1 0.6_±0.4 2.1_±1.4 56.7_±40.5 10.0_±8.5 ₋7.9_±70.2 ₋0.1_±13.3 LIB 90.0_±4.4 0.3_±0.1 1.1_±0.6 48.0_±48.0 16.0_±12.4 ₋45.3_±50.6 ₋15.4_±13.8 ECA AB 87.4_±4.2 0.6_±0.5 2.0_±1.6 38.2_±24.6 13.9_±10.6 1.5_±46.1 3.5_±17.4 LIB 85.7_±6.6 0.3_±0.2 0.9_±0.5 28.4_±22.9 25.3_±14.6 ₋20.6_±30.4 ₋20.5_±25.6