Appearance 1: three nested layers of arterial sub-compartments from orthogonal tissue cuts

In an orthogonal cross-section, four arterial sub-compartments are visible: adventitia (excluded from this analysis because outside boundary cannot be separated from interstitial collagen histologically), media, intima, and lumen. Denoting the outer contours of media, intima, and lumen as $C_m$, $C_i$, and $C_l$, respectively, we first calculate the luminal center of mass ($\mathrm{CoM}$) by leveraging the spatial moments of $C_l$, as shown in Eq. (1)

As depicted in Fig. 1, for a given radial measurement at angle $\alpha$, we cast three rays $R_{\alpha -\beta }$, $R_{\alpha }$, and $R_{\alpha +\beta }$, from $\mathrm{CoM}$ at angle $\alpha -\beta$, $\alpha$, and $\alpha +\beta$, respectively, and calculate their intersections with $C_i$ as $P_{\alpha -\beta }^i$, $P_{\alpha }^i$, and $P_{\alpha +\beta }^i$. In cases of multiple intersections, we choose the farthest. From $P_{\alpha }^i$, the line perpendicular to the vector $(P_{\alpha -\beta }^i,P_{\alpha +\beta }^i)$ was identified as $L_{\alpha }$, and its intersections with $C_m$ and $C_l$ were calculated as the points $P_{\alpha }^m$ and $P_{\alpha }^l$, respectively. In cases of multiple intersections, we choose the closest for $C_m$ and $C_l$. Media and intima thickness at angle $\alpha$, denoted as $T_{\mathrm{media}}^{\alpha }$ and $T_{\mathrm{intima}}^{\alpha }$, respectively, were calculated as the Euclidean distance of $P_{\alpha }^m-P_{\alpha }^i$ and $P_{\alpha }^i-P_{\alpha }^l$, respectively. Figure 2(a) illustrates the radial measurements at various polar coordinates within the artery.

Fig. 1

Illustration for the radial measurement at angle $\alpha$. The media, intima, and lumen outer contours are depicted in red, green, and blue, respectively. Intersections between $C_i$ and rays $R_{\alpha -\beta }$, $R_{\alpha }$, and $R_{\alpha +\beta }$ are marked as $P_{\alpha -\beta }^i$, $P_{\alpha }^i$, and $P_{\alpha +\beta }^i$. $P_{\alpha }^m$ and $P_{\alpha }^l$ mark the intersections of the line perpendicular to the vector $(P_{\alpha -\beta }^i,P_{\alpha +\beta }^i)$ with $C_m$ and $C_l$. Media and intima thickness at angle $\alpha$, denoted as $T_{\mathrm{media}}^{\alpha }$ and $T_{\mathrm{intima}}^{\alpha }$, respectively, were calculated as the Euclidean distance of $P_{\alpha }^m-P_{\alpha }^i$ and $P_{\alpha }^i-P_{\alpha }^l$, respectively.

Fig. 2

Illustrating examples of radial measurements at various polar coordinates. The first column shows images of the arteries cropped from trichrome-stained WSI, sourced from the NEPTUNE digital pathology repository, overlaid with geometric annotations. Columns 2 to 6 present auxiliary lines illustrating thickness measurements at angles 0, 70, 140, 210, and 280 deg. (a) Appearance 1: three nested layers of arterial sub-compartments from orthogonal issue cuts. Cross-sectional view of a complete artery. Thickness is measured at all five specified angles. (b) Appearance 2: open lumen from edge tissue cuts. Incomplete artery at the edge of the biopsy core. Measurements at angles 210 and 280 deg are discarded as they fall within the range of open lumen. (c) Appearance 3: multiple areas of lumen/intima from tangential tissue cuts. Serpiginous artery cuts through two adjacent points, resulting in a dual lumina connected by a media layer. The radial measurement at angle 280 deg was excluded due to its intersection with other lumen and intima regions.

Appearance 2: open lumen from edge tissue cuts

When the kidney biopsy needle penetrates the kidney parenchyma, it may cut through the arteries, appearing incomplete at the edge of the kidney biopsy core and resulting in “partial open lumens.” For arteries with open lumens, we first identify the radial samples falling within the range of the open lumen, which are samples with intima and media thickness values below a pre-defined threshold. We then exclude samples both within and adjacent to these open lumen areas. This is achieved by employing a moving window technique, whereby we discard all samples within a certain window if the count of open lumen samples exceeds a specific limit. Figure 2(b) illustrates the radial measurements for an artery with the open lumen at the edge of the tissue, and radial measurements at angles 210 and 280 deg were discarded as they fall within the range of open lumen.

Appearance 3: multiple areas of lumen/intima from tangential tissue cuts

Multiple areas of lumen or intima can arise due to various factors, such as tissue folding, wrinkling during the cutting or mounting process, and bifurcation points in the artery where a single vessel splits into multiple smaller vessels. In cases where multiple lumens or intima are present, the lumen–intima–media combination with the largest intima and lumen area is selected for thickness measurements. Samples, where $R_{\alpha }$ intersects with other lumens or intima within the artery, and adjacent samples within a pre-defined window are discarded. Figure 2(c) illustrates the radial measurements for an artery with multiple areas of lumen/intima. In this instance, only the lower intima–lumen region was utilized for analysis, and the radial measurement at angle $280\mathrm{deg}$ was discarded due to its intersection with other lumen and intima regions.

Filtering

To address the challenges posed by the distorted shapes and varying thicknesses in arterial measurements, we implemented a two-step filtering process. First, a moving median filter was applied to mitigate the impact of outliers, where each thickness measurement is replaced by the median value of its neighbors within a specific window size. Subsequently, a moving average filter was employed to further smooth the measurements and reduce noise, where each thickness measurement is replaced with the average value of its neighbors within the same window size.

Missing value imputation

During the radial measurement process, instances may arise where $L_{\alpha }$ does not intersect with $C_l$, leading to missing measurement values. Instead of removing these sample points, we applied a weighted imputation method to the radial measurements of media and intima thickness. Given the missing value $l_i$ at angle $i$ in the list of measurements $l$, we first identified its closest non-missing values in the left and right, denoted as $l_{\mathrm{left}}$ and $l_{\mathrm{right}}$, respectively. Then, we impute the missing value by taking a weighted average of $l_{\mathrm{left}}$ and $l_{\mathrm{right}}$, as shown in Eq. (2)

Normalization

The radial measurements of media and intima thickness are based on Euclidean distances among pixel coordinates, with values dependent on image resolution and artery size. To account for these factors, we normalize the media and intima thickness values by dividing them by the artery wall thickness median value.

Figure 3 provides an example of the radial measurements before and after steps of filtering, missing value imputation, and normalization for an artery with missing values at angle $\alpha \in [174\mathrm{deg},176\mathrm{deg}]\cup [180\mathrm{deg},181\mathrm{deg}]$.

Fig. 3

Illustration of the application of filtering, missing value imputation, and normalization. (a) Artery: an artery with overlaid sub-component annotations. (b) Annotations: annotations of the outer contours of the media, intima, and lumen in distinguishable colors of red, green, and blue, respectively. Measurements at angle $\alpha \in [174°,176°]\cup [180°,181°]$ are missing, marked by white radial lines. (c) Media thickness: radial measurements of media thickness, presented as a function of angle, before and after application of the processing steps. (d) Intima thickness: radial measurements of intima thickness, before and after the processing stages.

Global features

We calculate the global statistical features, including the average, median, and variance for each set of measurements.

Local features

We extracted local features by analyzing the peak properties within each series of measurements. Peaks are identified as local maxima satisfying the width greater than a pre-determined threshold ${\theta }_w$. The maximum height and prominence of the peak properties are utilized as local features for analysis. Figure 4 provides an example of the detected peaks at angles 43, 227, and 89 deg for intima thickness, media thickness, and the intima-to-media ratio, respectively.

Fig. 4

Illustration of local feature extraction. (a) Artery: an artery with overlaid sub-component annotations. (b) Annotations: annotations of the outer contours of the media, intima, and lumen in distinguishable colors of red, green, and blue, respectively. Radial lines highlight angles 43, 227, and 89 deg, corresponding to local peaks in measurements of intima thickness, media thickness, and intima–media ratio, respectively. (c) Thickness measurements: radial measurements for intima thickness, media thickness, and intima–media ratio, with the peaks denoted by $X$ markers at angles 36, 226, and 84 deg, respectively.

Architecture design

We employed a nine-module convolutional neural network based on the common U-Net architecture with a dilated bottleneck, which has been proved effective in the segmentation of histologic structures on kidney biopsies.^26,34 The network’s input accepts $256\times 256\times 3$ red, green, blue (RGB) images, outputting a $256\times 256\times 4$ segmentation map for the background, media, intima, and lumen.

Model training

We used 648 artery images, derived from 20 WSIs from NEPTUNE and four WSIs from U MICHIGAN to train the DL model. To boost model generalization, we applied a data augmentation procedure (e.g., horizontal vertical flip and crop resize). In addition, we incorporated transfer learning by initializing the first four fully convolutional blocks of the network with pre-trained weights from the publicly available VGG16 model, originally trained on the ImageNet dataset. During the training process, a cross-entropy loss function was minimized based on Adam optimization to learn optimal model parameters. The training was run for 50 epochs with a batch size of 4 and an initial learning rate of 0.001.

Post-processing

After acquiring the DL segmentation output and applying an argmax operation, we obtain three distinct Boolean masks representing the segmented artery sub-compartments: media, intima, and lumen. To enhance the quality of these segmentation results, we implement a noise reduction step on each mask by removing small holes and objects with areas below 5% of the total positive pixels. From the cleaned segmentation masks, we identify the largest contour in each mask as the input for our proposed thickness measurement technique.

Individual feature correlation analysis

We first measure the association between each extracted feature and arteriosclerosis severities by calculating Kendall’s tau-b correlation coefficient, $\tau$. $\tau$ is a statistic used to measure the ordinal association between two measured quantities. Despite our features being continuous, they can be conceptualized as ordinal for this analysis due to their ranking nature. Note that $\tau$ takes values between $-1$ and $+1$. A value of $\pm 1$ indicates a perfect association between the feature and arteriosclerosis severities, whereas a coefficient closer to 0 suggests a weaker relationship.

Multi-variate feature evaluation

To evaluate the ability to quantify arteriosclerosis of our thickness-based features in comparison with the baseline area-based features, we employed the ridge regression estimator. We compute ridge regression estimators using three distinct feature sets: only area-based features, only thickness-based features, and a combination of both. The performance and contribution of these features were then assessed using the $R^2$ score of the corresponding estimator. $R^2$ score ranges between 0 and 1, denoting the effectiveness of the chosen features for quantifying arteriosclerosis.

Manual segmentation

We evaluated the feature values derived from thickness measurements using manual segmentation (Fig. 7). For the area-based features, the intima area feature exhibits a clear separability with $\tau =0.44$. For the proposed thickness-based features, the global features (average and median) derived from all three measurements present a comparative separability compared with the intima area feature, with $\left|\mathrm{\tau }\right|\ge 0.4$. Among the local features, the peak height from media and ratio measurements show a moderate association, with $\left|\mathrm{\tau }\right|\ge 0.3$.

Fig. 7

Violin plots representing the distribution of feature values for each arteriosclerosis severity score with associated Kendall’s coefficients, where features are derived from thickness measurement based on manual segmentation and normalized using min–max scaling. The top, middle, and bottom rows represent the features extracted from media thickness, intima thickness, and intima–media thickness ratio, respectively. The intima area feature exhibits a clear separability with $\tau =0.44$. Global thickness-based features (average and median derived from all three measurements) present a comparative separability ($\left|\mathrm{\tau }\right|\ge 0.4$). Local features (peak height from media and ratio measurements) present a moderate association, with $\left|\mathrm{\tau }\right|\ge 0.30$. (a) Media area ($\tau =-0.21$, $p=0.0106$). (b) Media average ($\tau =-0.44$, $p<0.0001$). (c) Media median ($\tau =-0.43$, $p<0.0001$). (d) Media variance ($\tau =-0.13$, $p=0.115$). (e) Media peak height ($\tau =-0.30$, $p=0.0002$). (f) Media peak prominence ($\tau =-0.22$, $p=0.0066$). (g) Intima area ($\tau =0.44$, $p<0.0001$). (h) Intima average ($\tau =0.42$, $p<0.0001$). (i) Intima median ($\tau =0.41$, $p<0.0001$). (j) Intima variance ($\tau =0.26$, $p=0.0012$). (k) Intima peak height ($\tau =0.21$, $p=0.0106$). (l) Intima peak prominence ($\tau =0.04$, $p=0.5799$). (m) Ratio intima/media area ($\tau =0.39$, $p<0.0001$). (n) Ratio average ($\tau =0.45$, $p<0.0001$). (o) Ratio median ($\tau =0.42$, $p<0.0001$). (p) Ratio variance ($\tau =0.17$, $p=0.0372$). (q) Ratio peak height ($\tau =0.32$, $p<0.0001$). (r) Ratio peak prominence ($\tau =0.19$, $p=0.0176$).

DL segmentation

Our DL segmentation model effectively delineates the intra-arterial compartments, achieving Dice scores of 0.97 for the background, 0.84 for the media, 0.78 for the intima, and 0.86 for the lumen. Subsequently, we evaluated the feature values derived from thickness measurements based on the segmentation model (Fig. 8). For baseline area-based features, the intima area exhibits a clear separability with $\tau =0.43$. For the proposed thickness-based features, six global features (average and median from all three measurements) and one local feature (ratio peak height) present a more robust association compared with the intima area feature. It is notable that the ratio average displayed a strong separability, with $\tau =0.52$.

Fig. 8

Violin plots representing the distribution of feature values for each arteriosclerosis severity score with associated Kendall’s coefficients, where features are derived from thickness measurement based on deep learning segmentation and normalized using min–max scaling. The top, middle, and bottom rows represent the features extracted from media thickness, intima thickness, and intima–media thickness ratio, respectively. Six global features (average and median from all three measurements) and one local feature (ratio peak height) present a more robust association compared with the intima area feature. (a) Media area ($\tau =-0.27$, $p=0.0009$). (b) Media average ($\tau =-0.50$, $p<0.0001$). (c) Media median ($\tau =-0.50$, $p<0.0001$). (d) Media variance ($\tau =-0.06$, $p=0.4829$). (e) Media peak height ($\tau =-0.24$, $p=0.0023$). (f) Media peak prominence ($\tau =-0.09$, $p=0.2489$). (g) Intima area ($\tau =0.43$, $p<0.0001$). (h) Intima average ($\tau =0.45$, $p<0.0001$). (i) Intima median ($\tau =0.47$, $p<0.0001$). (j) Intima variance ($\tau =0.22$, $p=0.0052$). (k) Intima peak height ($\tau =0.27$, $p=0.0009$). (l) Intima peak prominence ($\tau =0.21$, $p=0.0096$). (m) Ratio intima/media area ($\tau =0.41$, $p<0.0001$). (n) Ratio average ($\tau =0.52$, $p<0.0001$). (o) Ratio median ($\tau =0.48$, $p<0.0001$). ($p$) Ratio variance ($\tau =0.21$, $p=0.0084$). (q) Ratio peak height ($\tau =0.45$, $p<0.0001$). (r) Ratio peak prominence ($\tau =0.23$, $p=0.0044$).

These results suggest that the proposed thickness-based features especially global features can individually effectively contribute to the characterization of arteriosclerosis severity. When the score is 0, the distributions of intima and media thickness are well-separated, with the media exhibiting considerably larger values, indicating a normal artery. As the score increases, there is a noticeable shift in the intima thickness distribution toward the right, signaling the progression of arteriosclerosis severities. Examples of the intima–media thickness distributions for arteries with arteriosclerosis scores ranging from 0 to 3 are illustrated in Fig. 9.

Fig. 9

Distribution of intima–media thickness measurements across arteriosclerosis severity scores. (a) Artery example with score 0: intima and media thickness distributions are distinct, with media values substantially larger. (b) Artery example with score 1: intima and media thickness distributions overlap, resulting in similar thickness values for both. (c) Artery example with score 2: intima thickness values marginally surpass those of the media. (d) Artery example with score 3: intima and media thickness distributions separated again, but with intima values significantly exceeding media values.