Automatic segmentation of left ventricle cavity from short-axis cardiac magnetic resonance images

Abstract

In this paper, a computational framework is proposed to perform a fully automatic segmentation of the left ventricle (LV) cavity from short-axis cardiac magnetic resonance (CMR) images. In the initial phase, the region of interest (ROI) is automatically identified on the first image frame of the CMR slices. This is done by partitioning the image into different regions using a standard fuzzy c-means (FCM) clustering algorithm where the LV region is identified according to its intensity, size and circularity in the image. Next, LV segmentation is performed within the identified ROI by using a novel clustering method that utilizes an objective functional with a dissimilarity measure that incorporates a circular shape function. This circular shape-constrained FCM algorithm is able to differentiate pixels with similar intensity but are located in different regions (e.g. LV cavity and non-LV cavity), thus improving the accuracy of the segmentation even in the presence of papillary muscles. In the final step, the segmented LV cavity is propagated to the adjacent image frame to act as the ROI. The segmentation and ROI propagation are then iteratively executed until the segmentation has been performed for the whole cardiac sequence. Experiment results using the LV Segmentation Challenge validation datasets show that our proposed framework can achieve an average perpendicular distance (APD) shift of 2.23 ± 0.50 mm and the Dice metric (DM) index of 0.89 ± 0.03, which is comparable to the existing cutting edge methods. The added advantage over state of the art is that our approach is fully automatic, does not need manual initialization and does not require a prior trained model.

Appendix: derivation of the partial derivatives of J cs-fcm with respect to i c and j c

The partial derivative of J _cs-fcm (refer to Eq. (6)) with respect to i _c is given by

$$\begin{aligned} &\frac{{\partial J_{cs - fcm} }}{{\partial i_{c} }} = 0 \hfill \\ &\Rightarrow \frac{\partial }{{\partial i_{c} }}\left(\sum\limits_{{x_{ij} \in I}} {\sum\limits_{k = 1}^{K} {(u_{k|ij} )^{m} \cdot \hat{d}_{kij} } } \right) = 0 \hfill \\ &\Rightarrow \frac{\partial }{{\partial i_{c} }}\left(\sum\limits_{{x_{ij} \in I}} {\sum\limits_{k = 1}^{K} {(u_{k|ij} )^{m} \alpha f(i,j,s)} } \right) = 0 \hfill \\ &\Rightarrow \frac{\partial }{{\partial i_{c} }}\left(\sum\limits_{{x_{ij} \in I}} {\sum\limits_{k = 1}^{K} {(u_{k|ij} )^{m} \alpha \left[ {\frac{{\sqrt {(i - i_{c} )^{2} + (j - j_{c} )^{2} } }}{r}} \right]^{{\beta_{k} }} } } \right) = 0 \hfill \\ \end{aligned}$$

For LV region, if the values of β = 2, then we have

$$\begin{aligned} &\frac{\partial }{{\partial i_{c} }}\left( {\sum\limits_{{x_{ij} \in I}} {(u_{k|ij} )^{m} \alpha \left[ {\frac{{(i - i_{c} )^{2} + (j - j_{c} )^{2} }}{{r^{2} }}} \right]} } \right) = 0 \hfill \\ &\quad \Rightarrow \frac{\alpha }{{r^{2} }}\left( {\sum\limits_{{x_{ij} \in I}} {(u_{k|ij} )^{m} [ - 2(i - i_{c} )]} } \right) = 0 \hfill \\ & \quad \Rightarrow \sum\limits_{{x_{ij} \in I}} {(u_{k|ij} )^{m} i} = \sum\limits_{{x_{ij} \in I}} {(u_{k|ij} )^{m} i_{c} } \hfill \\ & \quad \Rightarrow i_{c} = \frac{{\sum\nolimits_{{x_{ij} \in I}} {i \cdot u_{k|ij}^{m} } }}{{\sum\nolimits_{{x_{ij} \in I}} {u_{k|ij}^{m} } }} \hfill \\ \end{aligned}$$

Similarly, the partial derivative of J _cs-fcm (refer to Eq. (6)) with respect to j _c is given by

$$\frac{{\partial J_{\text{cs} - \text{fcm}} }}{{\partial j_{c} }} \Rightarrow j_{c} = \frac{{\sum\nolimits_{{x_{ij} \in I}} {j \cdot u_{k|ij}^{m} } }}{{\sum\nolimits_{{x_{ij} \in I}} {u_{k|ij}^{m} } }}$$

This derives Eq. (8) for the updating of i _c and j _c .