Subject enrolment
Twenty subjects aged 75 ± 6 years (mean ± standard deviation, range 55–82 years, eight women) having asymptomatic plaques in the left carotid artery were retrospectively selected from the Rotterdam Study. The Rotterdam study is a population-based study of subjects aged ≥ 45 years investigating determinants of disease among the elderly [16]. The local Medical Ethics Committee approved the study and all participants provided informed consent. The study was performed in accordance with the declaration of Helsinki. All subjects underwent a two-dimensional ultrasound examination of the left carotid artery during which it was determined that WT was ≥ 2.5 mm at least at one location in the carotid artery [17].
MRI protocol
The carotid MRI examination, performed with a 1.5-T scanner (Signa Excite II; GE Healthcare, Milwaukee, WI, USA), included non-gated, time-averaged 3D phase-contrast imaging (field of view 180 × 180 mm2; 40 slices; spatial resolution 0.7 × 0.7 × 1 mm3, echo time 4.3 ms; repetition time 13 ms, velocity encoding 60 cm/s) and proton density-weighted echo planar imaging (field of view 130 × 70 mm2; 51 slices; spatial resolution 0.5 × 0.5 × 1.2 mm3; echo time 24.3 ms; repetition time 12,000 ms). The phase-contrast images were corrected for background phase offsets by subtraction of the velocity in static tissue (the sternocleidomastoid muscle).
MRI post-processing
The vessel lumen and wall were manually segmented in the proton density-weighted echo-planar images using ITK-snap (version 3.2, www.itksnap.org) [18]. The wall was converted to a mesh to enable 3D WT calculations. WT was quantified as the shortest distance between a mesh point and the outer wall (that included the adventitia). Rigid co-registration of the proton density-weighted echo-planar images and 3D flow images was performed in Elastix [19] to ensure that identical lumen segmentation was used for calculation of 3D WSS and 3D diameter.
WSS was quantified from the 3D flow images as previously described [8]. To derive the velocity gradient at the wall, smoothing splines were fitted through the rotated x- and y-velocity values using three equidistant points along the inward normal. The length of the inward normal used for WSS estimation was the radius of the vessel. All maps consisted of the common carotid artery (CCA), the bifurcation, the proximal ICA and the proximal external carotid artery.
To calculate the vessel diameter in 3D, the vascular modelling toolkit VMTK [20] was used to: 1) calculate the centreline of the lumen; 2) calculate the inward distance from the luminal surface to the centreline; and 3) multiply the distance to the centreline (i.e. the radius) by a factor of 2 to yield the diameter on each point of the vessel wall.
Cohort-averaged maps
Cohort-averaged maps for WSS, WT and diameter were created by: 1) creation of a shared geometry; 2) interpolation of the individual WSS, WT and diameter values to the shared geometry; and 3) averaging of WSS, WT and diameter on the shared geometry over the cohort [11].
Statistical analysis
Spearman’s rank correlation coefficient ρ was determined for the correlations between subject-specific WSS and WT, WT and diameter as well as WSS and diameter, using all pixels. A Fisher z-transformation was applied to normalize ρ, enabling a Student’s t-test to determine if the z-values averaged across subjects were significantly different from 0. p values lower than 0.05 were considered significant. Linear regression was performed as well and R2 reported. A linearity check was additionally performed.
For the assessment of ρ between the cohort-averaged maps using all pixels, the bootstrapping method was used [15]: the 20 individual WT, WSS and diameter maps were randomly sampled (with replacement) followed by 3D averaging. Such a cohort-averaged map thus contained the map of the same subject multiple times. Random sampling and averaging was repeated 1000 times (i.e. a bootstrapping size of 1000 was used). ρ was determined for each combination of randomly sampled and averaged WT and WSS maps (ρWSS − WT), for each combination of WT and diameter maps (ρWT − D) and for each combination of WSS and diameter maps (ρWSS − D), using all pixels. For these combinations, ρ was significant when the 95% confidence interval (CI) of the ρ values did not contain 0.
Linear regression for WT estimation based on individual maps
For each subject i, linear regression was performed for all pixels using Eq. 1:
$$ {WT}_i={\beta_0}_i+{\beta_1}_i\ast {WSS}_i+{\beta_2}_i\ast {D}_i+{\beta_3}_i\ast {WSS}_i\ast {D}_i+{\varepsilon}_i $$
(1)
where WTi is the 3D WT map, WSSi is the 3D WSS map and Di is the 3D diameter map. β0i is the intercept and β1i, β2i and β3i are the regression coefficients; εi is the residual.
The regression coefficients were used to compute the 3D WT map for each subject. The inclusion of β2i ∗ Di follows from the law of Laplace. The inclusion of β3i ∗ WSSi ∗ Di follows from the vessel calibre regulatory mechanisms of WSS [3, 4].
Linear regression for WT estimation based on cohort-averaged maps
Two linear regression analyses were performed for estimation of patient-specific 3D WT maps. First, linear regression was performed for all 1000 bootstrapped cohort-averaged WT, WSS and diameter maps using the equation for all pixels:
$$ {\overline{WT}}_b={\beta_0}_b+{\beta_1}_b\ast {\overline{WSS}}_b+{\beta_2}_b\ast {\overline{D}}_b+{\beta_3}_b\ast {\overline{WSS}}_b\ast {\overline{D}}_b+{\varepsilon}_b $$
(2)
where \( {\overline{WT}}_b \) is one bootstrap b of the cohort-averaged 3D WT map, \( {\overline{WSS}}_b \) is one bootstrap b of the cohort-averaged 3D WSS map and \( {\overline{D}}_b \) is one bootstrap b of the cohort-averaged 3D diameter map. β0b is the intercept and β1b, β2b and β3b are the regression coefficients per bootstrap. εb is the residual per bootstrap. Thus, 1000 β0, β1, β2 and β3 coefficients are obtained by solving this regression equation. The mean and 95% CI of the 1000 β0, β1, β2 and β3 are reported.
Second, the average of the 1000 βi (\( \overline{\beta_0} \), \( \overline{\beta_1} \) and \( \overline{\beta_2} \)) are used to predict the individual 3D WT maps from the individual 3D WSS and diameter maps for all pixels.
An overview of the workflow for predicting WT maps from cohort-averaged WSS and diameter maps is given in Fig. 1. To investigate the influence of WSS, diameter and the interaction factor WSS*D separately, the two-step linear regression process is repeated by including only WSS, diameter or the interaction term in Eqs. 1 to 4.
Comparison between estimated and measured WT maps
For the individual and cohort-averaged estimation results, Spearman’s ρ was determined using all pixels per subject to quantify the agreement between the estimated and the original WT map. Bland-Altman analysis was performed and the mean difference and limits of agreement averaged over the subjects reported. Furthermore, maximum WT, defined as the average of the top 5% of all values, was quantified for the original and predicted WT maps.