A method framework of cruciate ligaments segmentation and reconstruction from MRI images

Abstract

Segmenting anterior and posterior cruciate ligaments (ACL/PCL) presents challenges in medical imaging due to diverse characteristics, including size, shape, and intensity. Our study uses superpixel-based spectral clustering for knee cruciate ligament segmentation in 2D DICOM slices, renowned for generating high-quality clusters. The proposed method addresses the challenges by (i) identifying the ligamentous region (ROI) through superpixel-based computation, (ii) extracting features (intensity-based, shape-based, geometric complexity, and Scale-Invariant Feature Transform) from the ROI, and (iii) segmenting knee ligament tissues using spectral clustering on the extracted features. Superpixel-based spectral clustering addresses the challenge of constructing a dense similarity matrix and significantly reduces the computational burden. Furthermore, 3D visualization of ligament structures is performed using the Visualization Toolkit (VTK). We evaluated our proposed approach on a dataset of knee MRI slices, assessing the results via the dice score, average surface distance (ASD), and root mean squared error (RMSE) metrics. Our method achieved an average dice score of 0.912 for ACL segmentation and 0.896 for PCL segmentation, outperforming other clustering methods. These scores showed an enhancement of 10.7% and 14.9% in segmentation accuracy for the ACL and PCL, respectively. Furthermore, reduced error margins were demonstrated with the mean ASD values of 1.60 and 1.78 and the mean RMSE values of 1.76 and 1.86 for ACL and PCL, respectively. These results show the effectiveness of the proposed method for cruciate ligament segmentation and its potential for increasing the segmentation accuracy and speed, offering significant advantages over manual segmentation by reducing time and expertise.

Keywords

ACL and PCL segmentation knee MRI superpixels spectral clustering feature extraction 3D reconstruction

1 Introduction

Cruciate ligaments are crucial anatomical structures as they, along with the collateral ligaments, provide stability to the knee joint.¹ With the articular surface, muscle function, and applied forces, these ligaments provide accurate arthrokinematics. In passive motion, the cruciate ligaments smoothly transition from rolling to sliding movements. During active motion, these ligaments prevent translations and reduce shear forces. They also control rotation in a flexed knee and, with the collateral ligaments, provide rotational stability for a knee in extension. Thus, cruciate ligaments are highly susceptible to injuries, especially among athletes.² Segmenting these ligaments in knee magnetic resonance (MR) images has become a key focus for diagnosing ligament injuries and planning surgical procedures. The existing literature extensively documented the manual segmentation of the cruciate ligaments, with the assistance of orthopedic experts, for clinical purposes.³ The process is very laborious and is associated with intra- and inter-observer variabilities, emphasizing the need for an automated segmentation method.⁴ Segmentation of cruciate ligament in knee MR imaging is challenging due to specific image characteristics. The signal densities of the ACL and PCL are often similar to the densities of other soft tissues, such as cartilage and meniscus, which causes over-segmentation and leakage of labels into adjacent regions.⁵ These issues must be resolved to enhance the effectiveness and reliability of cruciate ligament segmentation.

Magnetic resonance imaging (MRI) is the most commonly used technique in evaluating the knee joint, especially in the case of cruciate ligament ruptures.⁶ This imaging technique offers a detailed and non-invasive visualization of the entire length, morphology, and internal composition. T1-weighted sequences in knee MRI provide detailed images of anatomical features such as muscles, ligaments, and tendons, while T2-weighted sequences help identify damages, edema, and inflammation within the soft tissues and bones.⁷

Automatic segmentation can be done in two ways: supervised or unsupervised techniques. Supervised methods rely on massive datasets that are labeled correctly, which can be time-consuming due to manual annotation.⁸ However, unsupervised methods do not rely on training data and are thus able to handle various types of imaging sequences.⁹ Our study emphasizes the benefits of unsupervised clustering techniques, such as spectral clustering, as they provide a more efficient and faster solution while maintaining high accuracy. Spectral clustering aims to find optimal clustering solutions by utilizing graph Laplacian matrices to group the data into clusters. To relate these data points, eigenvalues and eigenvectors of these matrices are calculated and referred to as the ‘spectrum’ of the graph.¹⁰

The published literature on anterior and posterior cruciate ligament segmentation is relatively limited. Ho et al.¹¹ introduced a technique for ACL segmentation that uses morphological operations and the Chan–Vese active contours model,¹² while Lee et al.¹³ proposed an alternative technique using adaptive thresholding and graph cuts.¹⁴ Zarychta et al.¹⁵ presented the fuzzy C-means (FCM) approach for segmenting ACL and PCL. Liu et al.¹⁶ developed a fully automated system for detecting ACL tears. It uses CNNs to isolate the ACL in knee MRI scans, followed by another CNN for classifying injuries. Namiri et al.¹⁷ developed a deep learning pipeline for isolating the ACL ROI, detecting abnormalities, and assessing lesion severity with 3D and 2D CNNs. Flannery et al.¹⁸ designed an automated ACL segmentation model based on a 2D U-Net architecture. Additionally, Namiri et al.¹⁹ also introduced a 3D CNN model for detecting and classifying knee structures, including ACL lesions, menisci, bone marrow, and cartilage.

The literature highlights a significant concern regarding knee cruciate ligament injuries, affecting approximately 1 in 3000 individuals.²⁰ Despite extensive efforts to improve segmentation techniques, accurately segmenting cruciate ligaments remains a challenging and ongoing area of research.²¹

Our study presents a novel approach for segmenting cruciate ligaments in knee MRI images using superpixel-based spectral clustering. The method applies spectral grouping techniques on superpixels to accurately identify ligamentous tissues. Superpixels are created by calculating central tendency values within equally sized image blocks, incorporating localized spatial information to enhance noise resistance during clustering. Subsequently, relevant features are extracted from the identified ROI, and spectral clustering is applied to segment the knee ligament tissues based on these features. The segmentation results were used for 3D reconstruction, enhancing the visualization and evaluation of cruciate ligament structures, potentially improving clinical utility for surgical planning and diagnostics. The workflow is illustrated in Figure 1. The manuscript is structured as follows: Section 2 describes the methodology, Section 3 discusses experimental results and analysis, and Section 4 concludes the study.

Figure 1.

Flow diagram of the proposed methodology.

2 Proposed methodology

Our method focuses on precisely segmenting knee cruciate ligaments within the Region of Interest (ROI) in MRI data rather than processing the entire image. We generate superpixels using the Central Tendency Value (CTV) within uniformly sized image blocks to enhance computational efficiency and utilize localized image characteristics. These superpixels serve as nodes in the spectral clustering process, facilitating the delineation of the ROI, explicitly focusing on the knee cruciate ligaments. After classifying the superpixels as ligament or non-ligament, we identify the primary block of ligament-related superpixels as the “ligament block”. The Local Binary Pattern (LBP) feature extraction method identifies neighboring blocks that resemble the ligament block, thereby refining the ROI. Additionally, we extract relevant features from the identified ROI, including intensity-based characteristics, shape-based attributes, geometrical complexity properties, and Scale-Invariant Feature Transform (SIFT) descriptors. Finally, spectral clustering is applied to the ROI to segment the knee cruciate ligament structures. Detailed explanations are provided in the subsequent sections of this paper.

2.1 Preprocessing

The preprocessing phase is essential for preparing images by correcting distortions and enhancing critical features. Since we are using unsupervised segmentation, which lacks reference points or manual labeling, it introduces challenges such as noise and intensity irregularities that can affect clustering results. To address these issues, our method includes NLM filtering²² and N4ITK Bias field correction²³ techniques. After preprocessing, we identify the ROI, a crucial step detailed in the following section.

2.2 Identification of Region of Interest (ROI)

Image resizing is performed to reduce the ROI compared to the original image, optimizing it for precise ligament segmentation. Non-ligament areas are excluded to enhance the segmentation accuracy during ROI identification. The procedure involves analyzing sagittal MRI slices with a 256 × 256 pixels resolution. Initially, we reduced the area threefold, resulting in an average 2D ROI size of around 100 × 100 pixels for both the ACL and PCL. This resized ROI is then mapped onto all MRI slices, forming a comprehensive ROI that includes both cruciate ligaments, as shown in Figure 2. The proposed method focuses on the crucial ligament region, which is then processed further. Spectral clustering is applied to this refined ROI, optimizing the construction of a dense similarity matrix. This optimization enhances spectral clustering accuracy without compromising quality. The ROI identification process includes several steps: superpixel computation, superpixel segmentation, identification of the ligament block, and identification of neighboring blocks.

Figure 2.

Knee MRI Series showing the Sagittal Plane's cruciate ligament (ACL & PCL) regions.

2.2.1 Superpixel computation

Our method initiates by partitioning the MRI image into small, equally sized blocks. The partitioning strategy utilizes the CTV²⁴ concept, representing the data's tendency to cluster around a central point. We calculate the CTV in each block using statistical measures such as the mean and median, which serve as attributes for the superpixels within each partition. Specifically, the average pixel value within a block determines its mean value.

Let P₁, P₂, … , and P_n represent the image partitions, where ‘n’ represents the partitions count. For each partition P_j, we compute its CTV, denoted as $μ_{j}$ , using the mean pixel intensities. Let $I_{i j}$ represent the intensity of i^th pixel in partition P_j, where i ranges from 1 to $k_{j}$ , the number of pixels in partition P_j. The CTV, or the mean pixel intensity for partition P_j, is calculated as in equation (1)

μ_{j} = \frac{1}{k_{j}} \sum_{i = 1}^{k_{j}} I_{i j}

(1)

where

μ_{j}

provides a central intensity measure for partition P_j. This process is repeated for all ‘n’ partitions, and each partition's mean value is crucial for the subsequent superpixel representation.

The median-based CTV, denoted as Med_i, characterizes the central value within each partition P_i. This computation involves the following steps:

Sorting of Pixel Intensities: The pixel values within P_i, corresponding to the i^th block of the MRI image I, are sorted.

Median-based CTV Calculation: The CTV value, Med_i, is determined using the median, and the formula is given in Eq. (2) and Eq. (3).

\begin{aligned} M e d_{i} & = \frac{1}{2} (P_{i}^{(\frac{n}{2})} + P_{i}^{(\frac{n}{2} + 1)}) i f n i s e v e n \end{aligned}

(2)

\begin{aligned} M e d_{i} & = P_{i}^{(\frac{n + 1}{2})} i f n i s o d d \end{aligned}

(3)

where n represents the number of pixels in partition P_i, and the pixel values

P_{i}^{(\frac{n}{2})}, P_{i}^{(\frac{n}{2} + 1)}

and

P_{i}^{(\frac{n + 1}{2})}

, are used to compute used to compute the median-based CTV. After calculating the CTVs, the image is reconstructed using superpixels, which are then applied to spectral clustering for accurate segmentation.

2.2.2 Superpixel segmentation

Spectral clustering²⁵ is used to analyze the superpixel data derived from the image blocks and distinguish superpixels relevant to knee ligaments from those of other structures. Figure 3 visually represents the results of the superpixel segmentation, with regions shaded in black indicating the presence of knee ligament structures.

Figure 3.

Visualization of superpixel segmentation results achieved through spectral clustering analysis of image blocks.

2.2.3 Identification of the ligament block

We initiate the process by determining the image block containing a ligament superpixel. Given the possibility of ligaments extending across block boundaries, our methodology systematically identifies neighboring blocks that exhibit congruent features to the ligament-containing block. Furthermore, we used a feature extraction method based on a LBP to precisely identify neighboring blocks sharing analogous characteristics with ligament block L.

2.2.4 Identification of the neighboring blocks

Local Binary Pattern (LBP)²⁶ is a widely used method for feature extraction that holds significance in various applications, including image segmentation, image retrieval, texture classification, and phase recognition. This operator captures local spatial relationships and delineates nuances in gray tone contrast. The process involves partitioning the image into uniform cells containing eight neighboring pixels. The central pixel is assessed against its eight neighbors, resulting in an eight-bit binary value for each pixel. These values are then transformed into decimals and saved in the LBP mask. The transformation from binary to decimal is concisely expressed in Eq. (4).

L B P_{r a d i u s, s a m p l i n g p o int s} (x, y) = \sum_{s t e p = 0}^{sampling points - 1} f u n c (i n t e n s i t y (x, y) - i n t e n s i t y (x_{n e i g h b o r}, y_{n e i g h b o r}))^{2 \times s t e p}

(4)

The expression

LB P_{radius, sampling points} (x, y)

represents the calculation of local binary patterns, using sampling points neighbors arranged in a circular pattern with a radius around a pixel

(x, y)

intensity (x, y)

represents the grayscale level of the central pixel, while

intensity (x_{neighbor}, y_{neighbor})

corresponds to the grayscale level within the symmetric neighborhood.

The function $t (w)$ defines a thresholding operation, defined in Eq. (5), where $w$ represents the intensity contrast between the central pixel $intensity (x, y)$ and its neighboring pixel $intensity (x_{neighbor}, y_{neighbor})$ . The thresholding function $t (w)$ yields 1 for $w \geq 0$ and 0 for $w \leq 0$ .

t (w) = {\begin{array}{lc} 1 & i f w \geq 0 \\ 0 & i f w \leq 0 \end{array}

(5)

Then, we compute the histogram based on the decimal values of the blocks, followed by optimal normalization. The feature vector is constructed by concatenating the histograms of the blocks. Furthermore, the LBP histogram of a specific block is compared with the histograms of its eight neighboring blocks to define the ROI. The block identified earlier (C) serves as the central block, while its eight neighbors are labeled as K₁, K₂, … K₈ (Figure 4). The Bhattacharya similarity metric,²⁷ defined in Eq. (6), is used to evaluate the similarity between the histograms of the central block (C) and its neighboring blocks.

B C (H (C), H (K_{i})) = \sum [H (C) - H (K_{i})]

(6)

The coefficient $BC (H (C), H (K_{i}))$ , provides a measure of dissimilarity between the histogram of the identified block (C) and the histograms of its neighboring blocks $K_{i}$ , where i ranges from 1 to 8 (Figure 5). A higher BC coefficient indicates more significant dissimilarity and vice versa. The newly identified area is designated as an ROI when regions exhibit similarity. Identifying neighboring blocks is iterated until no resemblance is observed between the neighboring and the target block. Finally, the resulting region is marked as an ROI, as depicted in Figure 6. The algorithm for ROI identification is outlined below.

Figure 4.

Histograms of LBP values for the central and eight neighboring blocks.

Figure 5.

Bhattacharyya coefficients with neighboring blocks.

Figure 6.

Extraction of Region of Interest representing the cruciate ligaments (ACL & PCL) from MRI data.

2.3 Feature extraction

In our proposed method, we have used different image processing techniques to extract diverse features. Subsequent sections provide detailed descriptions of these extracted features.

2.3.1 Pixel intensity features

The proposed method used grayscale images and derived pixel intensity features²⁸ relevant to ligament segmentation. These features are computed within selected regions known as superpixels. Six statistical features, namely Angular Second Moment (ASM), Contrast, Homogeneity, Variance, Entropy, and Standard Deviation, are extracted to capture crucial information for accurate ligament segmentation.²⁹ ASM values (0.2 to 0.8) indicate texture uniformity, essential for distinguishing ligament regions from surrounding tissues. Contrast values (50 to 150) capture subtle intensity variations for precise boundary delineation. Homogeneity values (0.6 to 1) reflect spatial coherence, enhancing segmentation fidelity. Variance values (100 to 400) quantify intensity variations, aiding structural characterization. Entropy values (3 to 7) assess texture randomness, revealing fine structural details. Standard Deviation values (10 to 30) quantify intensity dispersion, refining ligament characterization. The statistical distributions of these features are shown in Figure 7

Figure 7.

Statistical distribution of intensity-based features across superpixel regions.

2.3.2 Shape-Based features

The shape-based features³⁰ involve the intricate analysis of curvature characteristics obtained from gradient computations in specific directions, denoted as $G_{x}$ and $G_{y}$ . Additionally, we calculate the image normal at each pixel $(x, y)$ using Eq. (7)

N_{(x, y)} = \frac{1}{\sqrt{G_{x}^{2} + G_{y}^{2}}} (G_{x} G_{y})

(7)

The curvature in 2D images reveals the normal divergence, computed as in Eq. (8).

Curvature = \frac{G_{x x} G_{y} + G_{y y} G_{x} - 2 G_{x x} G_{x} G_{y}}{{(G_{x}^{2} + G_{y}^{2})}^{3 / 2}}

(8)

where

G_{xx}

and

G_{yy}

stand as the secondary derivatives, with

intensity (x, y)

denoting the image intensity. The average curvature value across the pixels within a superpixel represents the curvature for each superpixel. In Figure 8(a), the boxplot illustrates the distribution of curvature values within superpixels, while in Figure 8(b), the histogram provides a visual representation of the frequency distribution of these values.

Figure 8.

Curvature distribution analysis in superpixel regions: (a) distribution of curvature values and (b) frequency distribution of curvature values.

2.3.3 Geometric complexity features

The segmentation-based Fractal Texture Analysis (SFTA)³¹ algorithm is used to extract geometric complexity characteristics. The initial step involves image segmentation through the Otsu algorithm, which partitions the images into binary patches using multilevel thresholding (n_t). Each binary channel undergoes edge detection,³² generating image boundaries utilized for fractal feature extraction. These attributes encapsulate area, intensity, and fractal dimension calculated from the binary edge channels. The area feature quantifies the count of edge pixels within the selected superpixel. Similarly, the intensity feature denotes the mean intensity of image pixels corresponding to the superpixel's edge pixels. The fractal dimension, reflecting image structural complexity, is calculated using Equation (9):

D_{f} = lim_{δ \to 0} \frac{\log N (δ)}{l o g \frac{1}{δ}}

(9)

Here, N(

δ

) measures the hypercube count with specific dimensions and length ε. Utilizing the box-counting algorithm³³ an approximation of the fractal distance is derived from the binary images.

2.3.4 SIFT features

The SIFT³⁴ method involves two core phases: keypoint detection and descriptor calculation. Key points are identified using the Degree of Gradient (DoG) method. The computation of $D (i, j, σ)$ requires the difference between smoothed images $L (i, j, k, σ)$ and $L (i, j, σ)$ . This is achieved by convolving Gaussian filters of different scales with the input image $I (i, j)$ , as detailed in Eq. (10) and Eq. (11).

\begin{aligned} D (i, j, σ) & = L (i, j, k, σ) - L (i, j, σ) \end{aligned}

(10)

\begin{aligned} L (i, i, σ) & = \frac{1}{2 π σ^{2}} e x p (- \frac{i^{2} + j^{2}}{2 σ^{2}}) * I (i, j) \end{aligned}

(11)

Multiple DoG images are generated across varying scales to capture image features at different resolutions. An essential step involves detecting local extrema within these DoG images, achieved by analyzing 26-pixel neighborhoods across three DoG images. A pixel qualifies as a key point if it stands as an extremum.³⁵

Furthermore, an essential aspect of SIFT features involves the key point descriptor, which relies on the gradient magnitude, denoted as $M (i, j)$ , and the orientation, denoted as $θ (i, j)$ , at each pixel within the smoothed image. Equations (12, 13, and 14) illustrate the calculations of key points.

\begin{aligned} M (i, j) & = \sqrt{f_{x} {(i, j)}^{2} + f_{y} {(i, j)}^{2}} \end{aligned}

(12)

\begin{aligned} θ (i, j) & = \tan^{- 1} (\frac{f_{y} (i, j)}{f_{x} (i, j)}) \end{aligned}

(13)

\begin{aligned} f_{x} (i, j) & = {\begin{cases} L (i + 1, j) - L (i - 1, j) \\ L (i, j + 1) - L (i, j - 1) \end{cases} \end{aligned}

(14)

Following this, a weighted histogram is developed using 24 directional bins derived from the gradient magnitude surrounding the key point's orientation within its immediate area. This histogram (Figure 9) typically demonstrates a prominent peak, often reaching around 85% or potentially exceeding this value, indicating the key point's dominant orientation. Subsequently, an adjacent region aligned with the key point's orientation undergoes precise rotation, resulting in descriptor formation.

Figure 9.

Histogram of gradient orientations surrounding SIFT key points, aiding in feature descriptor construction.

2.4 Spectral clustering

Spectral clustering, a widely used unsupervised clustering method based on graph theory,³⁶ organizes data by analyzing eigenvectors and eigenvalues of the Laplacian matrix.³⁷ This technique extracts significant information from the interconnections within the data. In our study, the image is represented as I₁, I₂, … , I_n, where I₁, I₂, … , I_n, denote individual pixels. This image is transformed into a weighted graph, denoted as G (N, R), where N represents the set of nodes and R indicates their relationships. The graph, depicted as a similarity matrix $S_{(u, v)}$ , calculates similarities between the u^th and v^th rows, according to Eq. (15).

S_{(u, v)} = {\begin{array}{ll} e x p^{- \frac{d {(I_{u} - I_{v})}^{2}}{2 r^{2}}}; & for u \neq v \\ 0; & for u \neq v \end{array}

(15)

where

d (I_{u} - I_{v})

represents the Euclidean distance measuring the dissimilarity between nodes

I_{u}

and

I_{v}

. The parameter

r

denotes the intensity of the neighborhood, often set to 1. The similarity matrix is converted to a normalized Laplacian matrix for eigenvector decomposition. Subsequently, applying the K-means algorithm on the top k eigenvectors facilitates the extraction of clusters representing original information.

Spectral clustering directly applied to the identified Cruciate Ligament (CL) ROI substantially enhances processing speed and segmentation accuracy. This method constructs a similarity matrix by leveraging all pixel values within the ROI. A smaller ROI minimizes the formation of dense similarity matrices, eliminating the need for any approximations during matrix construction. Our method upholds the inherent accuracy of spectral clustering in ligament segmentation.

3 Experimental results & discussion

3.1 Dataset

The proposed methodology was validated using a dataset comprising 50 healthy knee joint sample cases. This dataset consisted of 30 male and 20 female instances, focusing on both the cruciate ligaments (ACL & PCL). The MRI data were gathered from the 2nd Affiliated Hospital of Dalian Medical University, each scan containing 75 to 100 MR image slices. The images featured a resolution of 512 × 512 pixels per slice, with each pixel measuring 0.31 mm × 0.31 mm and a slice thickness of 1 mm.

3.2 Experiments

The proposed methodology was evaluated through visual analysis and accuracy measurements. To verify the precision of our segmentation technique, we applied three key metrics: the Dice similarity coefficient, average surface distance (ASD), and root mean square error. These metrics were validated against ground truth labels manually segmented by an expert orthopedist. The DSC evaluates the overlap between the segmented area (A) and the ground truth (B), as defined in Eq (16):

D S C = \frac{2 \times | A \cap B |}{| A | + | B |}

(16)

where

A

represents the segmented label,

B

denotes the ground truth label, and∣∣signifies the area of the label. The resultant dice values lie between 0 and 1, where higher values indicate enhanced segmentation accuracy.

ASD is calculated as the average Euclidean distance between two areas: the segmented area $A$ and the ground truth area $B$ . It is expressed as in Eq (17):

A S D = \frac{1}{| P (A) | + | P (B) |} \sum_{P_{A} \in P (A)} δ (p_{A}, P (B)) + \sum_{p_{B} \in P (B)} δ (p_{B}, P (A))

(17)

Where

P (A)

and

P (B)

represent the sets of surface points

p_{A}

and

p_{B\;}

in areas

A

and

B,

respectively. The term

δ (u, T)

represents the Euclidean distance between point

u

and surface

T

calculated as

δ (u, T) = min_{q \in Q (T)} | | u - q | |^{2}

RMSE evaluates the root mean square distance between two areas: the segmented area $A$ and the ground truth area $B$ . Mathematically, it's represented as in Eq (18):

RMSE = \frac{1}{| P (A) | + | P (B) |} \sqrt{(\sum_{p_{A} \in P (A)} δ {(p, P (B))}^{2} + \sum_{p_{B} \in P (B)} δ {(p_{B}, P (A))}^{2})}

(18)

Our research focuses on precisely segmenting cruciate ligaments within the 2D DICOM slice series. The process begins by preprocessing these slices via a 3D Slicer,³⁸ rectifying biases, and standardizing the images into a uniform 256 × 256 grayscale format as our baseline for segmentation.

Using spectral clustering, each 2D slice is segmented with optimized parameters, utilizing 3 clusters to capture potential variations in ligament structure. Affinity matrix thresholds of approximately 0.6 ensure moderate similarity, allowing distinct yet connected segments. Correspondingly, clustering thresholds around 0.4 ensure a balanced partitioning. The superpixel-based method involves fine-tuning the superpixel size to around 40, generating moderate-sized superpixels for a balanced granularity in segmentation. Emphasizing spatial coherence with a regularization factor of around 0.6 optimizes delineation among the surrounding tissue complexities.

Furthermore, our method includes partitioning each image into various block sizes to compute CTV, including mean and median, to generate a set of superpixels essential for precise ligament delineation. The assessment of optimal block size selection is crucial. Our experiments indicate that a block size of 8 achieves the most accurate delineation of cruciate ligaments within the ROI. Larger block sizes, such as 16 and 32, introduce excessive pixels, compromising precision in the segmentation process. Figure 10(a) highlights dice score performance comparison between mean and median CTV across block sizes. Additionally, mean and median computations exhibit slightly quicker processing times, as shown in Figure 10(b).

Figure 10.

Evaluation of the proposed method with different block sizes and central tendency values (a) Dice score results for ROI segmentation. (b) ROI segmentation processing time per image.

Moreover, LBP feature extraction refines ROI precision by identifying neighboring blocks resembling crucial ligamentous superpixel structures. Accurate ROI segmentation using spectral clustering incorporates finely tuned parameters: setting k-nearest neighbors to 10 for constructing the similarity matrix and calibrating sigma to 0.4 for the Gaussian similarity distance metric. In MRI image diagnostics with Non-Local Means (NLM) filtering, experimental parameters include a search ratio of 7, a similarity window ratio of 2, and a sigma of 4 to optimize results.

Our method incorporates the normalized Laplacian matrix and clusters its eigenvectors using the K means ++ algorithm. Initially, cluster centers are randomly selected, followed by iterative refinement based on the proximity probability of the closest points about the squared distance from existing clusters. Given the inherent non-deterministic nature of the K means ++ algorithm, we iteratively execute the process ten times. Subsequently, among the ten generated outcomes, we choose the clustering configuration that optimally aligns with our specified objective function.

We compare our proposed method with the KASP,³⁹ Nystrom,⁴⁰ SCAR,⁴¹ and PRcut⁴² methods. All five approaches, including ours, focus on reducing the computational complexity of constructing dense similarity matrices for spectral clustering.

The KASP method utilizes cluster centers derived from K-means clustering to approximate the similarity matrix. However, in the Nystrom method, spectral clustering doesn't directly generate the original similarity matrix; instead, it randomly selects pixel points from the provided matrix. The SCAR method accelerates spectral clustering by robustifying the similarity matrix construction, improving resilience against noise, and enhancing computational efficiency. On the other hand, the PRcut method leverages power iteration to approximate eigenvalues and eigenvectors, optimizing the clustering process by reducing the number of required computations while maintaining high segmentation accuracy. These techniques offer an approximation of the original similarity matrix, enhancing segmentation accuracy while reducing computational load. To conduct this comparison, we use available codes for KASP,³⁹ Nystrom,⁴⁰ SCAR,⁴¹ and PRcut⁴² methods, evaluating their performance alongside our proposed method using our dataset. Table 1 illustrates the dice scores obtained by our method compared to other methods. Our proposed method consistently demonstrates superior results, leveraging full consideration of pixels within the ROI during similarity matrix construction with a block size of 8. Furthermore, Figure 11 and Figure 12 provide a comparative analysis of ASD and RMSE values among our proposed method, KASP, Nystrom, SCAR, and PRcut methods. Additionally, Figure 13 and Figure 14 visually illustrate the accurate segmentation of cruciate ligament structures based on our proposed method.

Figure 11.

Comparative analysis for ACL and PCL in terms of ASD between proposed and other methods based on spectral clustering.

Figure 12.

Comparative analysis for ACL and PCL in terms of RMSE between proposed and other methods based on spectral clustering.

Figure 13.

Stages of the anterior cruciate ligament segmentation process (a) Original images with initialized seed points (b) Generated mask delineating potential ligament regions using spectral clustering (c) superimposition of the mask on the original image, showing identified regions (d) Final segmentation result, illustrating precise ACL delineation.

Figure 14.

Stages of the posterior cruciate ligament segmentation process (a) Original images with initialized seed points (b) Generated mask delineating potential ligament regions using spectral clustering (c) Superimposition of the mask on the original image, showing identified regions (d) Final segmentation result, illustrating precise PCL delineation.

Table 1.

Dice score comparison between proposed and other methods based on spectral clustering.

Method	Patient 1		Patient 2		Patient 3		Patient 4		Patient 5		Mean
Method	ACL	PCL	ACL	PCL	ACL	PCL	ACL	PCL	ACL	PCL	ACL	PCL
KASP ³⁹	0.85	0.81	0.82	0.78	0.87	0.83	0.83	0.79	0.84	0.80	0.844	0.802
Nystrom ⁴⁰	0.86	0.82	0.81	0.75	0.83	0.80	0.80	0.76	0.81	0.77	0.824	0.780
SCAR ⁴¹	0.88	0.84	0.84	0.77	0.81	0.81	0.77	0.78	0.83	0.79	0.840	0.798
PRcut ⁴²	0.91	0.87	0.88	0.83	0.90	0.87	0.87	0.84	0.88	0.85	0.888	0.852
Proposed	0.92	0.88	0.89	0.84	0.94	0.90	0.90	0.87	0.91	0.89	0.912	0.896

Table 2 presents the comparison of the dice score between our proposed method and conventional clustering methods, such as FCM,⁴³ K-means,⁴⁴ K-medoids,⁴⁵ and DBSCAN.⁴⁶ Our method demonstrates significantly higher dice scores, indicating its superior performance. This advantage derives from our spectral clustering algorithm, which constructs a similarity matrix based on precise distance metrics. This approach efficiently captures intricate data patterns and complex cluster configurations, distinguishing it from the non-deterministic nature of the K-means algorithm and FCM, resulting in comparatively lower dice scores. The K-medoids algorithm is more resilient to noise and outliers than K-means. However, it still struggles to capture complex cluster structures because it relies on medoid selection, resulting in lower performance metrics. DBSCAN excels in identifying clusters of arbitrary shapes and handling noise, yet it struggles with datasets exhibiting varying densities, leading to lower Dice scores than our method. Our proposed method consistently outperforms these traditional clustering techniques by maintaining high accuracy through the precise construction of the similarity matrix, which comprehensively considers the pixel relationships within the ROI.

Table 2.

Dice score comparison between proposed and other clustering methods.

Method	Patient 1		Patient 2		Patient 3		Patient 4		Patient 5		Mean
Method	ACL	PCL	ACL	PCL	ACL	PCL	ACL	PCL	ACL	PCL	ACL	PCL
FCM ⁴³	0.75	0.71	0.77	0.73	0.76	0.72	0.78	0.74	0.74	0.70	0.760	0.720
K-means ⁴⁴	0.78	0.73	0.81	0.76	0.79	0.74	0.82	0.77	0.80	0.75	0.800	0.750
K-medoids ⁴⁵	0.76	0.72	0.79	0.75	0.77	0.73	0.80	0.76	0.78	0.74	0.780	0.740
DBSCAN ⁴⁶	0.77	0.74	0.75	0.72	0.79	0.76	0.76	0.73	0.78	0.75	0.770	0.740
Proposed	0.92	0.88	0.89	0.84	0.94	0.90	0.90	0.87	0.91	0.89	0.912	0.896

Moreover, Figure 15 and Figure 16 depict the comparison of ASD and RMSE scores between our method and other clustering methods, further highlighting the efficacy of our approach.

Figure 15.

Comparative analysis for ACL and PCL regarding ASD between proposed and other clustering methods.

Figure 16.

Comparative analysis for ACL and PCL regarding RMSE between proposed and other clustering methods.

In addition to accurate segmentation, our method enhances the visualization of segmented cruciate ligament structures through 3D reconstruction techniques. By leveraging the Visualization Toolkit (VTK)⁴⁷ library, we reconstructed the segmented ACL and PCL into 3D models using the Marching Cubes algorithm.⁴⁸ This algorithm transforms segmented 2D slices into 3D surface representations, facilitating accurate visualization of the ligament contours. The resultant 3D reconstruction provides invaluable perspectives on spatial relationships and structural integrity of the cruciate ligaments, offering diverse insights for thorough clinical assessment. Figure 17 illustrates the 3D reconstruction of the ACL and PCL.

Figure 17.

3D reconstruction of the ACL and PCL using the Marching Cubes algorithm with the VTK library.

4 Conclusion

This research introduces a novel technique for automatically segmenting cruciate ligaments from 2D DICOM slices. Our method identifies the ROI through superpixel-based spectral clustering. These superpixels are generated by evaluating the CTV within image blocks. Optimal block size selection is crucial, with experiments indicating that a block size of 8 provides the most accurate delineation. The proposed method achieved an average Dice score of 0.912 for ACL and 0.896 for PCL, outperforming KASP, Nystrom, SCAR, and PRcut methods by 8.1%, 10.7%, 8.6%, and 2.7% for ACL and by 11.7%, 14.9%, 12.3%, and 5.2% for PCL, respectively. Additionally, we compared our method with other clustering techniques such as FCM, K-means, K-medoids, and DBSCAN. Our method improved segmentation accuracy by 20.0%, 14.0%, 17.0%, and 18.4% for ACL and 24.4%, 19.5%, 21.1%, and 21.1% for PCL, respectively. Moreover, our method demonstrated reduced ASD and RMSE values, with mean ASD values of 1.6 for ACL and 1.78 for PCL and mean RMSE values of 1.76 for ACL and 1.86 for PCL. The mean ASD and RMSE values for ACL and PCL segmentation were consistently reduced by up to 40% and 37%, respectively.

In conclusion, the proposed method is a viable and effective ACL and PCL segmentation technique, offering faster and more accurate results than manual methods. This is particularly beneficial for clinical trials and facilitates direct evaluations significantly since it can be scaled without requiring much training. Thus, the proposed method addresses important limitations of the manual segmentation process, which is time-consuming and requires a specialized workforce to advance orthopedic research and applications. However, it has some drawbacks and limitations while still being able to offer a competitive segmentation performance. A fundamental limitation is its dependence on image preprocessing techniques, which may introduce noise or artifacts, potentially affecting segmentation accuracy. Further investigation is needed to evaluate its performance in cases involving pathological conditions or anatomical variations. While our study focuses on cruciate ligament segmentation, future research could extend to the segmentation and geometrical modeling of other ligament structures, such as the patellar ligament. Furthermore, the clinical utility of our segmentation framework should be validated through rigorous clinical trials. These trials would evaluate the feasibility of integrating our method into routine clinical practice and its impact on surgical planning. We aim to enhance our segmentation method's robustness and clinical applicability by addressing these limitations.

Footnotes

Acknowledgements

This study was supported by the National Natural Science Foundation of China (Nos. 62372079, 61972440 and 61572101), the Fundamental Research Funds for the Central Universities of China (Nos. DUT22YG104 and DUT24YG129), the National Natural Science Foundation of Liaoning Province of China (Nos. 2022-YGJC-47 and 2022-YGJC-73), the Scientific Research Project of Educational Department of Liaoning Province of China (No. LZ2020031) and the Key Research and Development Projects of Liaoning Province of China (No. 2021JH2/10300025).

Ethical considerations

The study was conducted in accordance with the Declaration of Helsinki (as revised in 2013) and informed consent was taken from all the participants.

Authors contributions

The authors confirm contribution to the paper as follows: study conception and design: Ahsan Humayun and Bin Liu; data collection: Ahsan Humayun; analysis and interpretation of results: Ahsan Humayun, Mustafain Rehman, Zhipeng Zou and Luning Xu; draft manuscript preparation: Ahsan Humayun. All authors reviewed the results and approved the final version of the manuscript.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Data availability

The authors of this article can provide all the data supporting the reported results (MRI dicom series).

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