Automatic segmentation of mandibular canal in cone beam CT images using conditional statistical shape model and fast marching

Abstract

Purpose

Accurate segmentation of the mandibular canal in cone beam CT data is a prerequisite for implant surgical planning. In this article, a new segmentation method based on the combination of anatomical and statistical information is presented to segment mandibular canal in CBCT scans.

Methods

Generally, embedding shape information in segmentation models is challenging. The proposed approach consists of three main steps as follows: At first, a method based on low-rank decomposition is proposed for preprocessing. Then, a conditional statistical shape model is trained, and mandibular bone is segmented with high accuracy. In the final stage, fast marching with a new speed function is utilized to find the optimal path between mandibular and mental foramen. Fast marching tries to find the darkest tunnel close to the initial segmentation of the canal, which was obtained with conditional SSM model. In this regard, localization of mandibular canal is performed more accurately.

Results

The method is applied to the identification of mandibular canal in 120 sets of CBCT images. Conditional statistical model is evaluated by calculating the compactness capacity, specificity and generalization ability measures. The capability of the proposed model is evaluated in the segmentation of mandibular bone and canal. The framework is effective in noisy scans and is able to detect canal in cases with mild bone resorption.

Conclusion

Quantitative analysis of the results shows that the method performed better than two other recent methods in the literature. Experimental results demonstrate that the proposed framework is effective and can be used in computer-guided dental implant surgery.

Appendix A: Performance evaluation metrics

To investigate the statistical behavior of conditional SSM, we utilize compactness, specificity and generalization measures. Compactness is an estimation of parameters required to generate a valid instance of the modeled object [37]. The compactness of the shape model is calculated as the cumulated variance for the first $i=1,\ldots ,M$ modes:

$$\begin{aligned} C(\tau )=\frac{\mathop {\sum }\nolimits _{i=1}^{\tau } {{\lambda }_i } }{\mathop {\sum }\nolimits _{i=1}^M {{\lambda } _i } }\times 100, \end{aligned}$$

(10)

where $C(\tau )$ and ${\lambda }_i$ are the compactness capacity and i-th largest eigenvalue, respectively. Generalization of a model measures the ability to represent unseen instances of the object class modeled [37], and it is defined as follows:

$$\begin{aligned} G(\tau )=\frac{1}{N_s }\sum _{k=1}^{N_s } {\left\| {r_k ({\tau })-t_k } \right\| } , \end{aligned}$$

(11)

where $N_s$ is the number of training data, $t_k$ is the training sample that is eliminated in leave-one-out procedure, and $r_k$ is the reconstructed shape using ${\tau }$ parameters. The specificity of a shape model is described as how much it can represent valid instances of the modeled class of object [37] and it is formulated as following:

$$\begin{aligned} S(\uptau )=\frac{1}{N_r }\sum _{k=1}^{N_r } {\mathop {\min }\limits _k } \left\| {s_k (\uptau )-{t}^{\prime }_k } \right\| , \end{aligned}$$

(12)

where $s_k (\uptau )$ is an arbitrary sample constructed by $\uptau$ parameters, $N_r$ is the number of data, and ${t}^{\prime }_k$ is the closest sample in training datasets to $s_k (\uptau )$. Leave-one-out crossvalidation [38] is employed to compare the segmented mandible bone with the gold standard. To create the gold standard dataset, each mandible and the mandibular canal was manually segmented by two radiologists in a slice-by-slice fashion. In order to compare the automatic segmentation results with the gold standard, two criteria are employed: (1) Dice’s coefficient [39] and (2) average symmetric surface distance [40]. Dice’s coefficient measures the overlap between the automatic segmentation result and reference manual annotations. This similarity measure is defined as following:

$$\begin{aligned} D(A,M)=\frac{2\left| {A\cap M} \right| }{\left| A \right| +\left| M \right| }, \end{aligned}$$

(13)

where A and M are segmentation results obtained by automatic segmentation and gold standard, respectively. This criterion is one of the most well-known methods in evaluating different segmentation methods.

Average symmetric surface distance (ASSD) [40] is defined as the space between two segmentations A and M in millimeters. If we assume that $S_\mathrm{A}$ and $S_{\mathrm{M}}$ are surface voxels of A and M, the Euclidean distance for each surface voxel of $S_{\mathrm{A}}$ to the closest surface voxel of $S_{\mathrm{M} }$ is calculated. To preserve symmetry, the same process is applied for the surface voxels of $S_{\mathrm{M}}$ to $S_{\mathrm{A}}$. Therefore, ASSD is expressed as the average of all stored distances as follows:

$$\begin{aligned} \mathrm{ASSD}(A,M)= & {} \frac{1}{\left| {S_A } \right| +\left| {S_M } \right| }\nonumber \\&\times \left( \sum _{s_A \in S_A } d(s_A ,S_M )+\sum _{s_M \in S_M} {d(s_M ,S_A )} \right) , \nonumber \\ \end{aligned}$$

(14)

$$\begin{aligned} d(v,S)= & {} \mathop {\min }\limits _{s\in S} \left\| {v-s} \right\| , \end{aligned}$$

(15)

where d is the shortest distance of voxel v to surface S, $\left\| . \right\|$ and $\left| . \right|$ represent vector norm and number of vertices, respectively. ASSD provides a volumetric-based evaluation criterion for the assessment of segmentation result.