A fully automated human knee 3D MRI bone segmentation using the ray casting technique

Abstract

This study aimed at developing a fully automated bone segmentation method for the human knee (femur and tibia) from magnetic resonance (MR) images. MR imaging was acquired on a whole body 1.5T scanner with a gradient echo fat suppressed sequence using an extremity coil. The method was based on the Ray Casting technique which relies on the decomposition of the MR images into multiple surface layers to localize the boundaries of the bones and several partial segmentation objects being automatically merged to obtain the final complete segmentation of the bones. Validation analyses were performed on 161 MR images from knee osteoarthritis patients, comparing the developed fully automated to a validated semi-automated segmentation method, using the average surface distance (ASD), volume correlation coefficient, and Dice similarity coefficient (DSC). For both femur and tibia, respectively, data showed excellent bone surface ASD (0.50 ± 0.12 mm; 0.37 ± 0.09 mm), average oriented distance between bone surfaces within the cartilage domain (0.02 ± 0.07 mm; −0.05 ± 0.10 mm), and bone volume DSC (0.94 ± 0.05; 0.92 ± 0.07). This newly developed fully automated bone segmentation method will enable large scale studies to be conducted within shorter time durations, as well as increase stability in the reading of pathological bone.

Appendix: Bone Localization Procedure

The Ray Casting technique requires some prior knowledge including an approximation of the bone location to set the observers in each image. This appendix describes the autonomous procedure.

The analysis of the intensity histogram H shows a mixture of two probability densities, one centered in the low values, and the other in the high values. Because of its stability and robustness, the Otsu’s algorithm [25] was chosen to analyze the histogram, giving a decomposition of the histogram as a sum of two Gaussian distributions N (μ _d , σ _d ) and $N(\mu_{{_{b} }} ,\sigma_{{_{b} }} )$. The histogram’s dark tissue peak is μ _d and bright tissue peak is $\mu_{{_{b} }}$. Values inferior to the dark peak are called very dark and values superior to the bright peak, very bright. Otsu’s algorithm also provides a decision threshold s of separation between dark and bright, and mixture parameters α _d and α _b , allowing the histogram to be written:

$$ H(I) \cong \alpha_{d} N(\mu_{d} ,\sigma_{d} ) + \alpha_{b} N(\mu_{{_{b} }} ,\sigma_{{_{b} }} ){\kern 1pt} \;with\;{\kern 1pt} \alpha_{d} + \alpha_{b} = 1 $$

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An algorithm was designed to capture IFemur and ITibia on the MR image where the bones, femur and tibia (dark), are surrounded by cartilage and muscle (very bright), further surrounded by fat (dark), then skin (bright), and finally the background (very dark).

One has to start from Ω and evaluate the following set in order of decreasing size, where Leg is the leg, and MCart is the cartilage and muscle set:

$$ \Upomega \supset Leg \supset MCart \supset \left\{ {IFemur \cup ITibia} \right\} $$

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Let us define H _convex (X) the convex hull of a finite set X. Considering the decision threshold s, the anatomical convexity of the leg in each axial slice, and the brightness of the tissues, each axial subset of the leg can be approximated by

$$ Leg_{x = h} \equiv H_{convex} \{ (y,z)\quad I_{x = h} (y,z) \ge s\} $$

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In order to have an appropriate cartilage and muscle set, surrounding femur and tibia, the selection relies on bright intensity and on convexity properties. Thus, MCart can be written as:

$$ MCart_{x = h} \equiv H_{convex} \{ (y,z) \in Leg_{x = h}^{{}} ,\quad I_{x = h} (y,z) \ge \mu_{{_{b} }} \} $$

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Figure 7a, b shows two sagittal slices of the intermediate results, the Leg _z=t set (Fig. 7a) and the MCart _z=t set (Fig. 7b).

Fig. 7

a, b A sagittal slice of the image restricted to the Leg _z=t set (a) and restricted to the smaller MCart _z=t set (b). The black parts of the slices denote the outside of each set while parts with image content denote the inside of each set. c A representative sagittal slice and d the sets Leg _z=t , MCart _z=t , IFemurandITibia, respectively, in dark, medium, light and bright gray

Finally, the bone part inside this set can be identified by the very dark tissue inside MCart such that:

$$ IFemurTibia_{x = h} \equiv ((y,z) \in MCart_{x = h} ,\quad I_{x = h} (y,z) \le \mu_{d} ) $$

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Femur and tibia sets are easily defined by separating the two largest components of FemurTibia as shown in Fig. 7c and d, where the largest is the set Femur and the second largest is the set Tibia. To facilitate the decomposition, an opening morphology operator ∘ is used [12] with structuring disk D of small diameter, e.g. five pixels. The set FemurTibia admits the following decomposition:

$$ IFemurTibia \circ D = IFemur \cup ITibia \cup \cdots Other\,small\,objects $$

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