A Robust Segmentation Framework forSpine Trauma Diagnosis

Poay Hoon Lim1, Ulas Bagci2 and Li Bai1

1 School of Computer Science, University of Nottingham, UK2 Radiology and Imaging Sciences, National Institutes of Health, USA

Abstract. Accurate three-dimensional (3D) image segmentation tech-niques have become increasingly important for medical image analysisin general, and for spinal vertebrae image analysis in particular. Thecomplexity of vertebrae shapes, gaps in the cortical bone and internalboundaries pose significant challenge for image analysis. In this paper,we describe a level set image segmentation framework that integratesprior shape knowledge and local geometrical features to segment bothnormal and fractured spinal vertebrae. The prior shape knowledge iscomputed via kernel density estimation whereas the local geometricalfeatures is captured through an edge-mounted Willmore energy. Whilethe shape prior energy draws the level set function towards possible shapeboundaries, the Willmore energy helps to capture the detail shape andcurvature information of the vertebrae. Experiment on CT images ofnormal and fractured spinal vertebrae demonstrate promising results in3D segmentation.

1 Introduction

Accurate 3D spinal vertebrae image segmentation techniques are important toolsto assist the diagnosis and treatment of spinal disorders such as spine trauma[1, 2]. Severe spine injury can result in life threatening and chronological prob-lems unless treated promptly and properly. In any spinal injury, the possibility ofspinal fracture must be examined immediately. Image segmentation of spinal ver-tebrae in 3D allows detection, measurement, and monitoring of the fracture(s),and facilitates biomechanics analysis of the spinal column.

Despite an increasing interest in spinal vertebrae segmentation in recentyears, accurate 3D segmentation methods for diseased or fractured vertebrae arestill lacking. There are some existing works in the literature for 2D or 3D segmen-tations, however they often require user intervention or fall short in achievinghigh accuracy [3–10]. Segmentation of diseased or fractured vertebrae has beenattempted recently [11–13], however, these are either in 2D or focused only onvertebral body rather than the whole 3D spinal vertebrae.

Traumatic injury of the spine often correlates with morphometric features inimages. Segmentation of the whole vertebra in 3D would facilitate the detectionof fractured vertebra and the assessment of the severity of the fracture. Forexample, the highlighted volumetric region of vertebrae could assist physicians

2 P.H. Lim et al.

in performing visual inspection of vertebral fractures, determining its stabilityand measuring quantitatively the fractured vertebrae.

This work extends the spinal vertebrae segmentation method presented in[14] to segment fractured vertebrae. In this case, high variability of fracturedvertebral shape is largely captured by the embedded Willmore flow, while priorshape energy comes into action only when encountering inhomogeneous imageintensity distribution.

2 Segmentation Framework

It is well-known that level set methods have advantages such as flexibility in deal-ing with topological change, easy extension into higher dimensions, as well aseasy integration of prior knowledge and region statistic. The segmentation frame-work presented here has made use of these properties. The framework combinesthe kernel density estimation technique and Willmore flow to incorporate priorshape knowledge and local geometrical features from images into the level setmethod. Whilst the prior shape model provides much needed prior knowledgewhen information is missing from the image, the edge-mounted Willmore flowhelps to capture the local geometry and smoothes the evolving level set surface.

The level set method embeds an interface in a higher dimensional functionφ (the signed distance function) as a level set φ = 0 [15]. The evolution ofthe level set function φ(t) is governed by ∂φ

∂t + F |∇φ| = 0, where F is thespeed function. Based on the variational framework, an energy function E(φ) isdefined in relation to the the speed function. The minimization of such energygenerates the Euler-Lagrange equation, and the evolution of the equation isthrough calculus of variation:

∂φ

∂t= −∂E(φ)

∂φ.

In this work, the fusion of energies whereby a shape prior distribution estimatorEs and an edge-mounted Willmore energy Ew0

is employed:

E(φ) = λEs + Ew0 ,

where λ (0 < λ ≤ 1) is the weight parameter, which is tuned to suit the segmen-tation of normal and abnormal spinal vertebrae.

2.1 Computing Prior Shape Energy via Kernel Density Estimation

Kernel density estimation (KDE) is a nonparametric approach for estimatingthe probability density function of a random variable. Without assuming theprior shapes are Gaussian distributed, KDE presents advantage in estimatingthe shape distribution even with a small number of training set, in addition tomodeling shapes with high complexity and structure. In this study, we adoptedthe prior shape energy formulation discussed by Cremers et al. [16].

A Robust Segmentation Framework for Spine Trauma Diagnosis 3

The density estimation is formulated as a sum of Gaussian of shape dissimi-larity measures d2(φ, φi), i = 1, 2, . . . , N :

P (φ) ∝ 1

N

N∑i=1

e−d2(φ,φi)

2σ2 ,

where the shape dissimilarity measure d2(φ, φi) is defined as

d2(φ, φi) =

∫Ω

1

2(H(φ)−H(φm))

2dx,

σ2 =1

N

N∑i=1

minj 6=i

d2(φi, φj).

and H(φ) is the Heaviside function. By maximizing the conditional probability

P (φ|I) =P (I|φ)P (φ)

P (I).

and considering the shape energy as

Es(φ) = − logP (φ|I),

the variational with respect to φ becomes

∂Es∂φ

=

∑Ni=1 αi

∂∂φd

2(φ, φi)

2σ2∑Ni=1 αi

=

N∑i=1

e−d2(φ,φi)

2σ2

2σ2∑Ni=1 αi

(2δ(φ)

[H(φ)−H(φi(x− µφ))

]

+

∫ [H(φ(ξ)−H(φi(ξ − µφ))

]δφ(ξ)

(x− µφ)T∇φ(ξ)∫Hφdx

dξ

),

where µφ is the centroid of φ and αi = exp(− 1

2σ2 d2(φ, φi)

)is the weight factor

for i = 1, 2, . . . , N .

2.2 Computing Local Geometry Energy via Willmore Flow

Willmore energy is a function of mean curvature, which is a quantitative measureof how much a given surface deviates from a sphere. It is formulated as

Ew =1

2

∫M

h2dA,

where M is a d-dimensional surface embedded in Rd+1 and h the mean curvatureon M [17]. For image segmentation, the Willmore energy provides an internal

4 P.H. Lim et al.

energy that gives a useful description of a region, where the effect of edge indi-cator is not significant. In these regions, smoothness of the shape of the curveshould be maintained and extended, which can be regarded as a weak form ofinpainting [18].

As a geometric functional, the Willmore energy is defined on the geometricrepresentation of a collection of level sets. Its gradient flow can be well repre-sented by defining a suitable metric, the Frobenius norm, on the space of thelevel sets. Frobenius norm is a convenient choice as it is equivalent to the l2-normof a matrix and more importantly it is computationally attainable. As Frobe-nius norm is an inner-product norm, the optimization in the variational methodcomes naturally.

Based on the formulation by Droske and Rumpf [18], the Willmore flow orthe variational form for the Willmore energy with respect to φ is

∂Ew∂φ

= −‖∇φ‖(∆Mh+ h(t)

(‖S(t)‖22 −

1

2h(t)2

)),

where ∆Mh = ∆h − h ∂h∂n −∂2h∂n2 is the Laplacian Beltrami operator on h with

n = ∇φ‖∇φ‖ , S = (I − n⊗ n)(∇×∇)φ is the shape operator on φ and ‖S‖2 is the

Frobenius norm of S.In order to ensure the smoothing effect work successfully around the con-

structed surface and not affecting the desired edge of vertebrae, the Willmoreflow is coupled with the edge indicator function g(I) = 1

1+|∇Gσ∗I|2 , where Gσ is

the Gaussian filter with standard deviation σ:

∂Ew0

∂φ= g(I)

∂Ew∂φ

.

3 Experiments and Results

Experiments have been conducted on CT images of spinal vertebrae for 2Dand 3D segmentation. The dataset consists of 20 CT images of normal and4 CT images of fractured spinal vertebrae images of patients aged 18 to 66years. The images were acquired from various CT scanners such as a 32-detectorrow Siemens definition, 64-detector row Philips Brilliance and 320-detector rowToshiba Aquilion. The in plane resolutions for these sagittal images range from0.88 mm to 1.14 mm, with consistent slice thickness of 2 mm. Original imagesfor these images have fixed sizes of 512x512, with number of slices varying from45 to 98. For 3D segmentation, a torus is set manually surrounding the spinalcanal as the initial contour. The level set method is then implemented using anarrow band scheme [19] with a re-initialization algorithm [20].

It has been reported that the 3D segmentations of spinal vertebrae clearlyoutperform the other methods such as region growing, graph cut, the classicallevel set methods such as Chan-Vese and Caselles models as well as their combi-nations with shape priors [14]. While some methods can perform relatively wellin the 2D segmentation of spinal vertebrae, the majority of them fail badly when

A Robust Segmentation Framework for Spine Trauma Diagnosis 5

Fig. 1. An example of patient CT image with fractures on lumbar vertebrae L2 andL3.

extended to the 3D segmentation of an individual vertebra due to the highly com-plex shape and connected structure as well as the nonuniform image intensitydistribution in the posterior column of vertebra. Table 1 summarized 3D segmen-tation results on normal lumbar vertebrae using various approach, evaluated withground truth. It is worth noted the effectiveness of our segmentation framework,with an overall accuracy of 89.32±1.70% and 14.03±1.40mm based on Dice sim-ilarity coefficient and Hausdorff distance respectively, whilst the inter- and intra-observer variation agreements were 92.11±1.97%, 94.94±1.69%, 3.32±0.46mmand 3.80 ± 0.56mm. Segmentation results depend highly on available dataset.The intra- and inter-observer variation estimations were performed to verify thedifficulty of manual delineation in 3D using our dataset. We have shown that ourresults present no significant statistical differences (p > 0.05) when compared

6 P.H. Lim et al.

Fig. 2. Different views of 3D segmentation for fractured lumbar vertebrae L2 and L3as well as their projections in 2D (red) compared with manual segmentation (yellow)on axial, sagittal and coronal views.

with these observer estimations. The robustness of the proposed segmentationframework is demonstrated on the CT image of a patient with fractures on lum-bar vertebrae L2 and L3 as seen in Fig. 1. As shown in Fig. 2, the segmentationframework manages to capture the 3D shape of fractured vertebrae L2 and L3,despite the inhomogeneity, noise and missing edges appeared on these fracturedvertebra images. It enables individual segmentation of vertebrae without leakinginto the nearby connected vertebra. Segmentation results on fractured vertebraewere evaluated via visual inspection by radiologist.

4 Discussion and Conclusion

An accurate level set segmentation framework for segmenting spinal vertebraein 2D and 3D is presented in this study. The robustness of the framework isdemonstrated on CT images of fractured vertebrae. The framework combinesthe kernel density estimation technique and Willmore flow to incorporate priorshape knowledge and local geometrical features from images into the level setmethod. It is worth noted that fusion of these energies effectively translate theprior shape knowledge and local geometrical feature of spinal vertebrae into thelevel set segmentation framework. The Willmore flow driven level set segmen-tation demonstrates better regularization than the widely used mean curvature

A Robust Segmentation Framework for Spine Trauma Diagnosis 7

Table 1. Average DSC (%) and HD (mm) with standard deviation for segmentations ofnormal lumbar vertebrae (L1 to L5) using Chan-Vese (CV), Chan-Vese with prior shape(CV+S), Caselles (Ca), Caselles with prior shape (Ca+S), edge-mounted Willmore(W0), edge-mounted Willmore with prior shape (W0+S) energies, region growing (RG)and graph cut (GC) approach.

Method DSC(%) HD(mm)

CV 37.68±7.07 26.68±2.18CVS 45.09±7.54 25.31±2.38Ca 55.75±8.14 22.22±1.57CaS 71.12±2.72 18.39±1.15W 75.82±2.81 19.21±1.51WS 89.32±1.70 14.30±1.40RG 42.30±11.43 25.20±2.30GC 13.23±11.11 63.22±14.82

flow in level set segmentation. Unlike minimizing surface area by mean curva-ture flow in regularization, Willmore flow minimizes the bending energy whenperforming surface smoothing, which is more suitable for object with complexshape and structure. The segmentation algorithm was performed directly in a3D volumetric manner instead of sequentially to the slices of a 3D image. Thisallows the volumetric tissue connectivity be taken into consideration and hence,enables more meaningful representation of 3D anatomical shape and structure.Moreover, it forms a continuous, smooth 3D surface and without the post pro-cessing redundancy posed by the slice by slice segmentation approach. Moresamples of fractured vertebrae are needed to perform further evaluation on thesegmentation framework. Future work will integrate the algorithm into a patho-logical vertebrae characterization framework to yield an efficient computer aideddiagnosis platform for quantitative analysis of spinal vertebrae fracture and re-lated problems.

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