Automated liver segmentation from a postmortem CT scan based on a statistical shape model

Abstract

Purpose

Automated liver segmentation from a postmortem computed tomography (PMCT) volume is a challenging problem owing to the large deformation and intensity changes caused by severe pathology and/or postmortem changes. This paper addresses this problem by a novel segmentation algorithm using a statistical shape model (SSM) for a postmortem liver.

Methods

The location and shape parameters of a liver are directly estimated from a given volume by the proposed SSM-guided expectation–maximization (EM) algorithm without any spatial standardization that might fail owing to the large deformation and intensity changes. The estimated location and shape parameters are then used as a constraint of the subsequent fine segmentation process based on graph cuts. Algorithms with eight different SSMs were trained using 144 in vivo and 32 postmortem livers, and the segmentation algorithm was tested on 32 postmortem livers in a twofold cross validation manner. The segmentation performance is measured by the Jaccard index (JI) between the segmentation result and the true liver label.

Results

The average JI of the segmentation result with the best SSM was 0.8501, which was better compared with the results obtained using conventional SSMs and the results of the previous postmortem liver segmentation with statistically significant difference.

Conclusions

We proposed an algorithm for automated liver segmentation from a PMCT volume, in which an SSM-guided EM algorithm estimated the location and shape parameters of a liver in a given volume accurately. We demonstrated the effectiveness of the proposed algorithm using actual postmortem CT volumes.

Appendix: Calculation of the gradient \(\nabla f({\varvec{t}}')\) and \(\nabla g({\varvec{\alpha }}')\)

Because of the difficulties faced in the analytical computation of the derivative of $\phi$ with respect to ${\varvec{t}}$, the gradient $\nabla f({\varvec{t}}')$ is obtained by difference approximation.

$$\begin{aligned} \nabla f({\varvec{t}}') = \frac{1}{\delta } \left[ \begin{array}{c} f({\varvec{t}}' + \varDelta _x)-f({\varvec{t}}') \\ f({\varvec{t}}' + \varDelta _y)-f({\varvec{t}}') \\ f({\varvec{t}}' + \varDelta _z)-f({\varvec{t}}') \end{array} \right] \end{aligned}$$

(33)

where $\varDelta _x = [\delta ,0,0]^\top$, $\varDelta _y = [0,\delta ,0]^\top$, and $\varDelta _z = [0,0,\delta ]^\top$ with $\delta = 1$ voxel. The partial derivative of $g({\varvec{\alpha }}) = Q\left( {\varTheta ',{\varvec{\alpha }},{\varvec{t}}'}\big |{\varTheta ',{\varvec{\alpha }}',{\varvec{t}}'}\right)$ in terms of $\alpha _j$ used in Eq. (25) is derived as follows:

(34)

$$\begin{aligned}= & {} \sum _{i=1}^{N}{ \sum _{k=1}^{K}{ \left\{ z^i_k\frac{\partial }{\partial \alpha _j} \ln {p_i(x_i= k| {\varvec{\alpha }}, {\varvec{t}}')} \right\} } } \end{aligned}$$

(35)

$$\begin{aligned}= & {} \sum _{i=1}^{N} \left\{ z^i_1 \frac{\partial }{\partial \alpha _j} \ln {p_i(x_i= 1 | {\varvec{\alpha }}, {\varvec{t}}')} + \sum _{k=2}^{K}{ z^i_k\frac{\partial }{\partial \alpha _j} \ln {p_i(x_i= k| {\varvec{\alpha }}, {\varvec{t}}')} } \right\} \nonumber \\\end{aligned}$$

(36)

$$\begin{aligned}= & {} \sum _{i=1}^{N} \left\{ z^i_1 \frac{\partial }{\partial \alpha _j} \ln {A_i({\varvec{\alpha }},{\varvec{t}})} + \sum _{k=2}^{K}{ z^i_k\frac{\partial }{\partial \alpha _j} \ln {\frac{1 - A_i({\varvec{\alpha }},{\varvec{t}})}{K- 1}} } \right\} \end{aligned}$$

(37)

(38)

(39)

$$\begin{aligned}= & {} a \cdot \sum _{i=1}^{N} \left\{ \sum _{k=1}^{K} z^i_kA_i({\varvec{\alpha }},{\varvec{t}}) - z^i_1 \right\} u_j(\varvec{r}^i-{\varvec{t}}). \end{aligned}$$

(40)

In Eq. (39), we used the fact that, for an arbitrary function h(v), the following equation holds for the derivative of the logarithm of the sigmoid function $\ln {\varsigma _a(h(v))}$:

$$\begin{aligned} \frac{\partial }{\partial v} \ln {\varsigma _a(h(v))} = \{\varsigma _a(h(v)) - 1\} \cdot a \cdot h'(v), \end{aligned}$$

(41)

and for the derivative of $\ln \{1-\varsigma _a(h(v))\}$:

$$\begin{aligned} \frac{\partial }{\partial v} \ln {\{1-\varsigma _a(h(v))\}} = \varsigma _a(h(v)) \cdot a \cdot h'(v). \end{aligned}$$

(42)