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Mice endplate segmentation from microCT data through graphbased trabecula recognition
EURASIP Journal on Image and Video Processingvolume 2019, Article number: 60 (2019)
Abstract
Though segmentation of spinal column from medical images have been intensively studied for decade, most of the works were concentrated on the segmentation of the vertebral body and arch, instead of the endplate. Recently, the increasing study on degeneration analysis of vertebra and intervertebral disc (IVD) make endplate segmentation as important as others. While the accurately segmentation of mice endplate from micro computed tomography (CT) images is challenging. The major difficulties include potential high system payload and poor runtime efficiency resulting from highresolution microCT data, highly complicated and variable shape of the vertebra tissues, and the ambiguous segmentation boundary due to the similarity of spongy structures inside both the endplate and its adjacent vertebral body. To solve the problems, the core idea of the proposed method is to identify trabeculae between the endplate and the body through a graphbased strategy. In addition, in order to reduce the data complexity, an endplatetargeted region of interest (ROI) extraction method is introduced according to the analysis of spatial relationship and variety of bone density of vertebra. Furthermore, shape priori of endplate in both twodimensional and threedimensional are extracted to assist in the segmentation. Finally, an iterative cutting procedure is implemented to produce the final result. Experiments were carried out which validate the performances of the proposed method in terms of effectiveness and accuracy.
Introduction
Demanded by imagebased spine assessment, biomechanical modeling and surgery simulation, segmentation of spinal column from medical images have been intensively studied for decades. In former researches, most of the works were concentrated on segmenting vertebral tissues such as the body and arch, while vertebral endplate segmentation were rarely mentioned. However, endplate is as important as the others. As a transitional zone between the intervertebral disc (IVD) and vertebra, it not only plays an important role in containing the adjacent disc, distributing applied loads evenly to the underlying vertebra, but also serves as a semipermeable interface that allows the transfer of water and solutes, preventing the loss of large proteoglycan molecules from the disc [1, 2]. Recently, it is also believed that there has a close relationship between the harm of endplate and osteoporosis [3]. Therefore, endplate segmentation could become an important prerequisite of vertebra/IVD degeneration analysis and many other applications, which is a major motivation of this work.
Up to now, endplate segmentation from computed tomography (CT) images accurately is still a challenging task in practice, which relies heavily on knowledge, experience, and manual works [4]. The main difficulties can be concluded as follows.

Potential high system payload and poor runtime efficiency. The average thickness of endplate is generally less than 80 μm; therefore, a highresolution imagery such as the microCT technique is required in order to achieve a highaccuracy segmentation. While, high resolution often leads to high system payload and poor runtime efficiency.

Highly complicated and variable shape of the vertebra tissues. The shape of vertebra including the endplate, body and arch are highly complicated and variable. For example, the mice vertebra mice we test have transverse processes relatively longer and of larger curvature comparing to the human’s. Therefore, mice vertebral body and arch are more likely to be interrupted with each other; it is more difficult to separate them properly. Besides, existing algorithms designed for human vertebrae may not be suitable for our case anymore.

Ambiguous segmentation boundary. Both the endplate and body are composed of spongy cancellous bone (see blue and green region marked in Fig. 1a), which makes the segmentation boundary ambiguous. Though there exists a gap area (labeled with a yellow dotted line in Fig. 1b) between them, however, the gap is considerably narrow and divided by trabeculae, which means it is difficult to find a proper continuous curve as a segmentation boundary.
As we mentioned above, though few works could be found on endplate segmentation, plenty of researches have been done to achieve detection and segmentation of vertebrae [5–9], spinal canals [10], and spinal cord [11].
For the purpose of separating the neighboring vertebrae roughly, Zhao et al. [12] proposed a neighboring point rating method based on constructing membership grade to form a virtual plane to cut adjacent processes. The method proposed by Kim et al. [9] searches a ray emitted from the stared point among the center axis of the spine and further construct a threedimensional (3D) surface by propagating the ray to detect the accurate gap area between two processes.
A common strategy to reduce the data complexity is by limiting the algorithm applied to a relatively small region of interest (ROI) [13]. In order to identify the ROI, Athertya et al. [5] extract the Harris corners [14] among the possible vertebra region to detect the vertebra range. However, the detection needs several interactions and training in advance, meanwhile the accuracy of detection depends on the selection of multistage seed points. Cheng et al. [6] detect the vertebra region by a probabilistic map computed from a voxelwise classifier and use mean shift algorithm to estimate the ROI after annotations.
In our work, we also need to isolate individual vertebra from a given vertebral column, to extract the ROI for data complexity reduction, while more importantly, to segment the endplate, and we made it in a endplatetargeted way. Specifically, taking microCT images as input, we aim to develop an intelligent framework to accurately segment the endplate from the others. And the core idea is to recognize the trabeculae (e.g., marked by red circles in Fig. 1b) within the narrow gap formed by the endplate and its adjacent vertebral body by a graphbased algorithm. The proposed framework can be separated into four parts, which are (1) preprocessing, (2) priori shape extraction, (3) gap trabecula detection, and (4) trabecula cutting and refining. Particularly, we firstly isolate each vertebra and identify the ROI for endplate in a topdown basis (Section 2.1). Secondly, shape priori in both 2D and 3D are introduced to offer constraints for later graph cuts based trabeculae recognition (Section 2.2). After that, a graph will be constructed based on mask skeletonization before the graph cuts based trabeculae recognition (Section 2.3). Finally, we find an optimal cutting strategy to remove the redundant connections properly (Section 2.4). An accurate result can be achieved after these processes, which is ready for subsequent operations such as the 3D reconstruction.
Material and methods
The experimental data we used were collected from the animal laboratory of spinal surgery of the Xiangya Hospital. MicroCT technique is adopted due to the highquality imagery requirement. To capture the data, mice of different ages were fixed in a slot scanned by a Bruker SkyScan1172 microCT scanner. The inplane resolution of the image is 7.27 μm × 7.27 μm, and the slice spacing is 7.27 μm. Figure 2 demonstrates a typical data of the inputs, which contains 4 lumbar vertebrae.
Preprocessing
The aim of the preprocessing is to get proper ROI for the endplates. In this work, a topdown strategy which consists of three steps as shown in Fig. 3 is proposed to achieve the goal from coarse to fine.
The first step is to isolate the vertebra from each others. Other than finding separation lines or surfaces such as the method proposed by Kim et al. [9], thanks to the high resolution of microCT image and unconnected spatial relationship among different vertebrae, this step can easily be done by 3D region growing on the mask resulting from a Otsu threshold.
The second step finds out ROI for endplate within each axial slices. Due to the shape complexity mentioned above, it is normally impossible to completely cut off the vertebral arch by planes as proposed in existed methods [12, 15]. To remove useless regions as many as possible, an optimal cylinder will be computed automatically in our method to meet the quasicylindrical shape of the vertebral body as shown in the right of Fig. 4. The cylinder can be obtained by topology analysis of the vertebra beginning from both the top and bottom part to the center part in axial planes slice by slice as indicated by blue arrows in Fig. 4. Because in this way, we could find two regions indicated by R_{1} and R_{2} in Fig. 4 which cover the entire endplate. Then, based on the projection of bone regions within R_{1} and R_{2}, we can easily find a circle (e.g., C_{1} in Fig. 4) which can be used to define the target cylinder.
The last step produces the finest ROI by further confirming the proper range of the axial slices. It travels each slice in the way as described in the second step, while other than analyzing the topology, bone density ρ(I_{i}) of each slice i will be calculated according to Eq. 1, and then the variation of bone density will be used to find the rightful indexes.
where A_{F}(I_{i}) is the area (measured by number of pixels) of the entire vertebral region within slice I_{i}, while N_{F}(I_{i}) denotes the region where identified as the bone tissue within A_{F}(I_{i}) (e.g., the colored region of slice in Fig. 5a).
The effect of this method lies in the fact that, as shown in Fig. 5a, the top layer of endplate is much denser than the other parts. For example, ρ(I_{a})>ρ(I_{b})>ρ(I_{c})>ρ(I_{d}). With the computed data, a chart, as illustrated in Fig. 5b, can be used to find the desirable range. The horizontal axis of the chart is the slice index, the vertical axis is the bone density accordingly. For the case shown in Fig. 5b, P1 and P2 will be chosen as the bounds of the range. Because, along with the direction of the horizontal axis, ρ(I_{P1}) reaches the first peak and the variation of bone density around I_{P2} become very small.
Shape priori extraction
It is well known that shape priori could be very useful because the constraints they represent could be important guidance for the segmentation. As for endplate segmentation, there are two kinds of shape priori according to our observation. The first one is that there appears a concave gap between the endplate and the adjacent body, which can be observed from its 3D geometry. We use the yellow dotted line in Fig. 6a to show this 3D shape priori. The second one is the shape similarity between the internal and external edge of endplate, which can be approximated by the green and red dotted line in Fig. 6b respectively.
As we mentioned, the core idea of our method is to find and cut the trabeculae within the narrow gap within 2D image as shown in Fig. 6b; therefore, it is better for us to have some priori knowledge of the 2D gap, which described by a gap line L (i.e., the green dotted line shown in Fig. 6b) in our method. To identify L, two terminal points (i.e., T_{1} and T_{2}) and the curve between them are used.
Fortunately, with the first shape priori, an optimal 3D loop can be found by a harmonic field based method introduced in Section 2.2.1, which is useful for locating T_{1} and T_{2}. While, with the second shape priori, a fitting method is presented in Section 2.2.2 to approximate the shape of L.
Terminal point identification
In our method, we solve the 2D terminal point identification problem from a 3D point of view, namely, a harmonic field based method as presented in our previous works [16, 17] is adopted to find an optimal 3D loop laying on the vertebra mesh surface reconstructed from ROI introduced above. Figure 7a demonstrates such a reconstructed mesh model.
Generally speaking, a harmonic field Ø is a scalar field attached to each mesh vertexes which satisfies △Ø=0, where △ is the Laplacian operator. Basic steps of the harmonic field based method include (1) designating a proper weighting scheme, (2) calculating Ø by a least square sense, and (3) choosing a desirable line from the isolines uniformly sampled from Ø. We adopt most part of the method from our former works, from which details such as the methodology and parameter setting can be found. Major differences between this method and the previous one lays in the following two aspects.
Firstly, when adopting least square method to solve the Poisson equation △Ø=0, we use boundary constraints as illustrated in Fig. 7b, where the blue and red spheres indicate mesh vertexes of Dirichlet condition [18] equal to 1 and 0 respectively. Figure 7c shows all the isoline candidates.
Secondly, we proposed a ranking method to select the optimal isoline from the candidates through a score function Score(I_{i}) of the ith isoline I_{i} defined by Eq. 2, which takes two factors including the field gradient magnitude G(I_{i}) and the shape variance among the local region V(I_{i}) into consideration.
where G(I_{i}) and V(I_{i}) meets Eqs. 3 and 4 respectively. These two factors reflect both the variance along the individual isoline and among the local isoline region; therefore, they can offer more accurate variance for evaluating the final score.
Let f_{j} be the jth triangle where I_{i} path through, then l_{j} and g(f_{j}) denote the length and corresponding gradient of I_{i} within f_{j} respectively. For the shape variance factor V(I_{i}), we take the isoline I_{i} as the center, and select the previous and next k isolines as the local region, I_{i+k} is the previous kth isoline, I_{i−k} is the next kth isoline. l_{max} is the maximum length of the candidate isolines. The Gaussian convolution is used to design the different weight for neighboring isolines with different distances from the center isoline, and it makes the measure insensitive to the choice of k. In our experiment, we set k to 6 and σ to 2.
According to the value of the obtained sores, we rank the candidate isolines and select the one with the highest value as the final 3D gap line. Figure 7d shows the result (i.e., the blue line) of the proposed ranking method, which is a closed and smooth curve locating among the desirable concave 3D gap region. With this loop, it is easily for us to find the corresponding pixels within the 2D image which can be used as the terminal points of the 2D gap line.
2D gap line identification
After the identification of terminal points T_{1} and T_{2} (e.g., the red points shown in Fig. 8), the next step is to determine the shape of the gap line L with a proposed fitting method demonstrated in Fig. 8, which is based on the similarity of shape between the internal and external edge of an endplate. Specifically, let \(\vec {z}\) denotes the direction of Z axis, we first locate T1′ and T2′ which are offset points from T_{1} and T_{2} along with \(\vec {z}\) by distance D, where D is the average distance travel from T_{1} and T_{2} to the top contour of endplate along with \(\vec {z}\). After that, we can find more endplate top contour points (e.g., the yellow points in Fig. 8) which have the same space in Y axis between T1′ and T2′. Then, taking the yellow points as controllers, a spline L^{′} (e.g., the yellow dotted line in Fig. 8) could be generated. Finally, we can get the L by pushing back distance D along with the opposite direction of Z axis.
Trabecula detection
Introduced by Boykov et al. [19, 20], graph cuts theory has become a powerful tool for medical image segmentation [21–23]. There are two major steps for graph cutbased segmentation, which are graph construction and cost function designation. Traditionally, all pixels and their neighborhood relationship in the image will be used as vertexes and edges to construct a graph, and pixel intensities are used to determine the weight. One problem with this method is that when dealing with highresolution images, the graph scale would be explosive.
The graph cuts framework is also adopted in our method, however, aiming for trabeculae detection instead of image segmentation, the proposed method is different from the traditional one in the following two aspects. (1) Only key pixels are treated as the vertexes for graph construction, which can be extracted by a skeletonization of the mask image within the coronal or sagittal view of the ROI (Section 2.3.1). (2) The shape prior introduced in Section 2.2 are used to design the cost function (Section 2.3.2).
Graph construction
To address the graph sale explosion problem when constructing the graph, Linguraru et al. [21] generate a regular sampling of the organ’s surface before segmentation. A semisupervised learning method is proposed by Mahapatra et al. [22] to predict annotations combing with the global features and local image consistency for graph cut optimization. Pauchard et al. [23] introduced a multilevel banded graph cuts for fast segmentation. In our case, a skeletonizationbased method is proposed as shown in Fig. 9.
Specifically, with the mask depicted in Fig. 9b, a skeletonisation method proposed by Cardenes [24] is firstly employed to thin the foreground bone tissues into 1pixel width skeletons as shown in 9c. Then the skeletons will be refined by keeping the largest connected component as shown in Fig. 9d. Finally, a graph G=(V,E) will be constructed according to the skeleton S, where V and E are the vertex set and edge set respectively.
According to the graph cuts theory, V is consisted of three kinds of nodes, i.e., the source node S, sink node T, and the rests denoted by N. In our method, N is formed by vertexes selected from S by checking every skeleton point p according to its 8neighborhood connectivity N_{d8}(p), namely, p belongs to N if and only if N_{d8}(p)>2 or N_{d8}(p)=1, which means p is a branch or terminal point of S. In additional, if there exists a edge E_{vu}∈E between vertex u and v (u,v∈N), then within S we can always find a path from u to v without passing any other vertex w∈N.
Cost function design
The graph cuts is driven by cost functions as the weight of the graph. Derived from the basic cost function templet [25], the graph cuts is defined as an energy minimization problem. For a vertex set N and a label set L, the goal is to find a mapping f:N→L such that the boundary of an object can be detected by minimizing energy function E(f) defined in Eq. 5.
where E_{d} is a data term defined as a cluster likelihood of nodes for the object, while the region cost is the sum of a data penalty term E_{d}. On the other hand, E_{s} is a boundary term that denotes a shape boundary penalty term of the two adjacent vertexes labeled by different labels. λ controls the balance between region and shape boundary constraints. In order to better incorporate the endplate shape priori into segmentation, we redesigned both E_{d} and E_{s} as follows.
In our data term, the source node S represents endplate regions, and the sink node T represents the body region. Take the gap line introduced above as a initial cut curve C_{cut}, the new data term meets Eq. 6.
where d(v,C_{cut}) is the distance from vertex v to C_{cut}, which could be positive or negative depending on which side v lie on. d_{max}=max(d(v,C_{cut})),(v∈N), and d_{min}=min(d(v,C_{cut})),(v∈N). In this way, the data term measures the confidence degree belonging to labels with signed distance of nodes related to the initial cut curve, namely, the larger d(v,C_{cut}), the higher confidence degree v has, and vice versa.
Though both vertebral body and endplate exist in trabeculae, their shape are different. Specifically, trabeculae inside the body seem longer and their growth directions are inconsistent, while trabeculae among endplate seem much shorter and their growth direction are similar to the principal direction. Therefore, these shape characteristic can help us to tell the two kinds of trabeculae, and the corresponding boundary term can be defined in Eq. 7.
where \(\overrightarrow n_{1}(v,u)\) represents the vector starting from node v to node u, \(\overrightarrow n\) is a normalized principle vector of the vertebra, length(v,u) denotes the length of the skeleton path between v and u, \({\text {arccos}}(\frac {\overrightarrow n_{1}(v,u) \times \overrightarrow n}{\overrightarrow n_{1}(v,u) \cdot \overrightarrow n})\) is the intersection angle between vector \(\overrightarrow n_{1}\) and vector \(\overrightarrow n\), which is among the range \([0, \frac {\pi }{2}]\), β is a weighting factor for balancing the intersection angle and the length of edge. The design of the boundary term assign a higher weight to edge with shorter length and with direction similar to principle direction, which makes the minimal cut passes the edge standing for trabecula with such characteristic. With the help of the proposed graph cuts method, all the trabeculae between endplate and vertebra body can be found successfully.
Trabecula cutting
As a result, the graph cuts proposed above labeled the vertexes into two sets, and the edges between the two sets stand for the trabeculae between the endplate and vertebral body in our method; therefore, in order to obtain the final segmentation result, we have to further cut them. Our basic idea is to generate a proper straight line to achieve the purpose, while the main difficulty is the position and angle of the straight line. To solve the problem, an iterative searching strategy is proposed.
Firstly, we initialize a cut line described as a purple line in Fig. 10a which passes through the trabecula represented by the white edge on the graph. The cut line locates at the middle of the white edge and forms two points p_{1} and p_{2} when extending to the boundary of bone tissue (e.g., the white points in Fig. 10). The distance d_{1} between two ends is p_{1}−p_{2}.
Next, we move the cut line towards the endplate at uniform velocity. As the line moves, each move generates a updated twoends distance d_{i} (i=2,3,⋯⋯). When the cut line touches the endplate, the △d_{i}=d_{i}−d_{i−1} will be increased dramatically. So we stop the movement when △d_{i} meet the Eq. 8.
where we set λ to 20 empirically.
Finally, the cut line will be rotated clockwise with a constant angle θ iteratively to get the proper angle. During each iteration, the distance d_{i}(j∗θ),(j=1,2,⋯⋯,2Π/θ) at the jth rotation will be compared with d_{i−1}, namely, d_{i}(j∗θ)−d_{i−1}. The angle with minimal variation is the optimal direction, which is the most parallel to the boundary of endplate, and the interface trabecula can be cut completely according to such a line as described in Fig. 10c.
Results and discussion
In this section, experiments designed to evaluate the performance of the proposed method will be presented. All experiments were carried out on a common personal computer with a Intel Core i3 processor (3.5 GHz) and 8 GB memory.
Experiment for shape prior extraction
In Section 2.2, we proposed two kinds of shape prior, which are the 3D gap line on the mesh surface and the 2D gap line within CT slices. Since the 3D gap line serves as the foundation of the 2D gap line detection, its extraction performance is firstly assessed.
Shape prior extraction results
We test effect of the proposed shape prior extraction method by employing it for all vertebra in our data set. The (1) generated harmonic field, (2) isoline candidates and (3) resulting 3D gap line of a randomly selected vertebra are shown in the first row of Fig. 11 from left to right respectively. Since the 3D gap line will be further transformed into two terminal points of the 2D gap line, Fig. 11 also shows the corresponding terminal points (i.e., the red dots) within some typical slices, whose index are labeled on the topright of the slice image. From the figures, we can see that both the 3D gap line and terminal points are correctly detected.
Parameter evaluation of 3D gap extraction
During the selection of the best isoline, there is a parameter k in Eq. 4 standing for the range of considered local region. As we mentioned, the extraction result is insensitive to the value of k, and we set it to 6 empirically. In order to prove that, another experiment is carried out, where we vary the value of k for the optimal 2D gap line selection and check out for the differences.
Specifically, let G_{a} be the selected gap line when k equals to a, then we can have 7 gap lines under the conditions that k varies from 3 to 9 (i.e., G_{3},G_{4},⋯⋯,G_{9}). In order to see the differences between them, we calculate the \(DCD(G_{a}, G_{6}), (a\in [3, 9] \bigwedge a \neq 6)\) which is the directional cut discrepancy (DCD) metric as employed in our previous work [16] to act as the mean errors comparing G_{a} with G_{6} in millimeters. We randomly selected eight vertebra from the data set and extract eight gap lines for each of them. We calculate the DCD data for these eight cases, and record them in Table 1.
From the table, we can see that on the one hand, most of the DCD data are zero, which means we can have the same gap line with k varies slightly for most of the cases. On the other hand, all nonzero data are very small, which means even if the selected gap line is different from the desired one under some circumstances, the mean errors between them is within a reasonable tolerance. In summary, the gap line extraction method is effective and efficient, the result is tolerable to the choice of k.
Segmentation results
The next experiment is to test the effect of the proposed method by segmenting every vertebra in our data set. The experimental results show that most of the endplates (93% exactly) can be successfully segmented, the reason for a few failed cases is mainly due to the great damage of the target vertebra.
Among the successful ones, segmentation results of a randomly selected vertebra are shown in both 2D (Fig. 12) and 3D (Fig. 13). Figure 12 shows the segmentation results in terms of 2D slices. The first and second doublerow slices belong to the two endplates of selected vertebra respectively. The slices were uniformly selected from different sagittal views. Figure 13 shows the segmentation results in terms of reconstructed 3D meshes, namely, the first and second row depict two endplates of the target vertebra in different perspective views, while the third row shows the relationship between the endplates and the rest parts of the vertebra. From the figures, we can conclude that the endplates were successfully segmented using the proposed method.
Quantitative results
There are many approaches proposed for human’s vertebra segmentation; however, there is no study on mice endplate segmentation from microCT data except our study; therefore, no other methods can be found for comparison. Instead, we compared our results with the ground truth which are manually segmented by experienced experts with an image editing software slicebyslice. In order to carry out the comparisons, four metrics are used to quantitatively evaluate the segmentation accuracy, which include volume difference (VD, μm^{3}), dice similarity coefficient (DC, %), and two surface distance metrics, i.e., average symmetric surface distance (ASSD, μm), maximum symmetric surface distance (MSSD, μm) [26, 27].
Table 2 shows the segmentation accuracy of the proposed method in terms of the metrics described above for the segmentation results mentioned in Section 3.2. For better analysis, we classified the statistical results of endplates according to their anatomical positions, namely, the L4U, L4B, L5U, and L5B stand for the upside/downside endplate of the fourth/fourth lumbar vertebra respectively. As shown in Table 2, four types of endplate’s Dice coefficient are all more than 90%, the average DC is 91.33±2.8%. The average ASSD of the endplates is 3.02±0.04μm, which is smaller than 4 μm. L5U has the largest MSSD, which is less than 8.5 μm.
As the VD data shows, the concave angle of upside endplate is typically less than that of downside endplate within the same vertebra. For example, L4U with VD less than that of L4D. It can be interpreted by the fact that the surface of upside endplate is usually more flat than that of the downside one. In additional, endplate of L5 has higher Dice coefficient than the that of L4, which is mainly because, comparing with L4, L5 is closer to the ischium, and the concave angle of L5 is typically smaller in general. According to the results illustrated above, we can draw the conclusion that our method is accurate and complies with the actual anatomic shape feature of mice endplate.
Computational efficiency
As another important issue, the efficiency of the proposed method can be assessed by system time consumptions. Therefore, time consumption for each process of the proposed method including the preprocess, skeletonization, shape priori and graph cuts, are recorded. Figure 14 shows 8 cases of our records. For each case, the input data contains 6 endplates represented by 610 slices in total.
According to Fig. 14, the average time consumption for the preprocess, skeletonization, shape priori and graph cuts are 4.13±0.83 s, 24.63±4.27 s, 32.25±3.88 s, and 62.88±4.42 s respectively. And it takes around 20 s to segment one endplate using the proposed method, which is approved by the doctors.
Conclusions and future work
In this paper, an intelligent framework is proposed to segment each endplate from microCT data captured from the vertebral column of mice. The innovations of the proposed method can be summarized as the following three aspects. Firstly, this work is one of few researches which focus on endplate segmentation. Secondly, we solve the endplate segmentation as a recognition problem of trabeculae within 2D gap area formed by target endplate and its adjacent vertebral body by a graphbased strategy. Thirdly, both 3D and 2D shape priori of the vertebra are used to guide for the segmentation, which are extracted by a harmonic fieldbased ranking method and a splinefitting method respectively.
To assess the proposed method, experiments for shape prior extraction, accuracy and efficiency evaluation, and demonstrations of the segmentation results are presented and discussed in details, which proved the effective and efficiency of this work. However, there still have some works that could be done in the future including improvement of the accuracy and efficiency by incorporating more reliable shape priori and optimizing the graph cutrelated procedures, since it consumes half of the total time as shown in the experiment.
Abbreviations
 3D:

Threedimensional
 ASSD:

Average symmetric surface distance
 CT:

Computed tomography
 DC:

Dice similarity coefficient
 DCD:

Directional cut discrepancy
 IVD:

Intervertebral disc
 MSSD:

Maximum symmetric surface distance
 ROI:

Region of interest
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Acknowledgements
The authors gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation.
Funding
This work is supported by the National Natural Science Foundation of China (61872085, 61772556), Scientific Research Project of Fujian University of Technology (GYZ160130, GYZ160138, GYZ160066), the Natural Science Foundation of Fujian Province (2017J05098, 2017H0003, 2018Y3001), Project of Fujian Education Department Funds (JK2017029, JZ160461), Project of Fujian Provincial Education Bureau (JAT160328, JA15325) and Project of Science and Technology Development Center, Ministry of Education (2017A13025).
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SJL and ZZ contributed equally to the design and plan of the algorithm. ZZ collects all the materials and related literatures. SJL offers feasible experimental scheme. JSP and SHL offer the microCT data and guide the research. All authors read and approved the final manuscript.
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Correspondence to JengShyang Pan.
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Keywords
 Vertebral endplate
 Trabecular bone
 Segmentation
 Harmonic field
 Graph cuts
 Microcomputed tomography