Statistical Descriptions of Scaphoid and Lunate Bone Shapes

ARTICLE IN PRESS

Journal of Biomechanics 43 (2010) 1463–1469

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Statistical descriptions of scaphoid and lunate bone shapes

Martijn van de Giessen a,b,n, Mahyar Foumani c, Geert J. Streekstra b, Simon D. Strackee c, Mario Maas d, Lucas J. van Vliet a, Kees A. Grimbergen b, Frans M. Vos a,d a Delft University of Technology, Quantitative Imaging, Lorentzweg 1, 2628 CJ Delft, The Netherlands b Academic Medical Center, Department of Biomedical Engineering and Physics, The Netherlands c Academic Medical Center, Department of Plastic and Reconstructive Surgery, The Netherlands d Academic Medical Center, Department of Radiology, The Netherlands article info abstract

Article history: Diagnosing of injuries of the wrist bones is problematic due to a highly complex and variable geometry. Accepted 3 February 2010 knowledge of variations of healthy bone shapes is essential to detect wrist pathologies, developing prosthetics and investigating biomechanical properties of the wrist joint. Keywords: In previous literature various methods have been proposed to classify different scaphoid and lunate Wrist types. These classifications were mainly qualitative or were based on a limited number of manually Scaphoid determined surface points. The purposes of this study are to develop a quantitative, standardized Lunate description of the variations in the scaphoid and lunate and to investigate whether it is feasible to Active shape model divide carpal bones in isolated shape categories based on statistical grounds. Anatomy The shape variations of the scaphoid and lunate were described by constructing a statistical shape Classification model (SSM) of healthy bones. SSM shape parameters were determined that describe the deviation of each shape from the mean shape. The first five modes of variation in the SSMs describe 60% of the total variance of the scaphoid and 57% of the lunate. Higher modes describe less than 5% of the variance per mode. The distributions of the parameters that characterize the bone shape variations along the modes do not significantly differ from a normal distribution. The SSM provides a description of possible shape variations and the distribution of scaphoid and lunate shapes in our population at an accuracy of approximately the voxel size (0.3 Â 0.3 Â 0.3 mm3). The developed statistical shape model represents the previously qualitatively described variations of scaphoid and lunate. However, strict classifications based on shape differences are not feasible on statistical grounds. & 2010 Elsevier Ltd. All rights reserved.

1. Introduction recover functional wrist mobility depends on the estimated non- pathological bone shapes (Crisco et al., 2005; Craigen and Stanley, The scaphoid and lunate bones (Fig. 1) play important roles in 1995). Similarly, studying bone shape variations is needed for the the mobility of the wrist (Craigen and Stanley, 1995; Garcia-Elias design of prostheses (Gordon et al., 2002; Gupta and Al-Moosawi, et al., 1995). Knowledge of variations of the healthy bone shapes 2002). is essential for diagnosing wrist pathologies, e.g. arthritis Studies of the anatomical variations currently focus on (Moritomo et al., 2000) or fractures (Compson et al., 1994). This qualitative descriptions, such as visual classifications in different is important in order to distinguish the pathology from a bone shapes (Moritomo et al., 2000; Antunã-Zapico, 1966; fysiological shape variation. Moreover, relations have been Watson et al., 1996; Viegas et al., 1990) or quantitative measures found between the bone shapes and their kinematics that describe global shape characteristics, such as distances (Nakamura et al., 2000). Therefore, assessing if variations in between anatomical landmarks (Compson et al., 1994; Ceri motion patterns correspond to naturally occurring bone shape et al., 2004; Patterson et al., 1995) or volume differences (Crisco variations or to ligament injury requires knowledge of the shape et al., 2005; Patterson et al., 1995). Previous studies describe of the bones. In the area of reconstructive surgery, the ability to shape variations based on radiographs, e.g. Kauer (1980), Watson et al. (1996). Gupta and Al-Moosawi (2002) deemed bone type classifications based on radiographs unreliable after studying CT

n scans of 50 volunteers. However, they did not look at the three- Corresponding author at: Delft University of Technology, Quantitative Imaging, dimensional bone shape but only studied individual slices and Lorentzweg 1, 2628 CJ Delft, The Netherlands. Tel.: +31 15 278 2031; fax: +31 15 278 6740. therefore their measurements depend on the orientation of the E-mail address: [email protected] (M. van de Giessen). bone in the scan and the selection of the slice.

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model has been extensively used to describe statistical shape variations in many other applications, e.g. hearts (Frangi et al., 2002) and brain structures (Shen et al., 2001). 15 000 corresponding points on all registered bone surfaces were automatically established by subsampling the segmented bone surfaces (Van de Giessen et al., 2009a). This number of points gives a dense sampling of the bone surface, since it is in the order of the number of voxels lying on the bone surfaces. Principal component analysis (PCA) (Webb, 2002) is employed to study the multidimensional variations of the corresponding coordinate points across the 50 bones. For each shape a data vector is constructed, containing the ordered three-dimensional coordinates of its surface points; which amounts to 15 000 Â 3=45 000 elements per data vector in this application. PCA then consists of computing the covariance matrix of these data vectors and determining its eigenvalues and eigenvectors. The eigenvectors describe the directions of variation, called ‘modes of variation’ in an SSM. The associated eigenvalues denote the variance in the corresponding direction. This means that modes of variation with a high variance describe a larger part of the total shape variation between the Fig. 1. Wrist with bone names. The rectangles denote the transverse, sagittal and bones than modes of variation with a small variance. The number of elements in coronal planes. each data vector is larger than the number of shapes and therefore the covariance matrix has as many non-zero eigenvalues as the number of shapes minus one, in this case 49. Not all of these 49 modes of variation describe meaningful shape In this paper we study the shape variations of the lunate and variations. Typically, only the ﬁrst modes with large eigenvalues (i.e. large scaphoid using three-dimensional statistical shape models (SSM) variance) are of interest. The modes with smaller eigenvalues mainly describe (Cootes et al., 1995; Brett and Taylor, 2000; Vos et al., 2004; noise that emanates from scanning (CT) and random point sampling. How many Van de Giessen et al., 2009a). These models describe the most modes to choose depends on the application at hand (Section 2.4.1). Note that the modes of variation are orthogonal and thus statistically important morphological variations and their statistical distribution. independent. Each mode of variation is a dimension of the multi-dimensional The variations as found by the SSM will be related to previous distribution that describes similar shapes to the 50 example bones. New bone reports. The feasibility of classifying the lunate and scaphoid in shapes can be generated with a weighted summation of the eigenvectors of the different types will be investigated by looking at the statistical covariance matrix. The (positive or negative) weights describe the location of the distribution of bone shapes in a population of 50 healthy wrists. shape in the multi-dimensional distribution. As long as the weights are chosen within three times the standard deviation (SD), i.e. three times the square root of the eigenvalue of a mode, the shapes are usually assumed to be valid. The boundary of three times the SD is derived from the normal distribution in which 2. Materials and methods 99.7% of the samples in the distribution lie between À3 SD and +3 SD.

2.1. Image acquisition 2.4.1. Number of modes: shape variation vs. noise To test how many modes describe shape variations in a single bone (e.g. the For building the statistical model, 50 CT scans of wrists were selected. Over a scaphoid) and how many describe noise, an SSM was constructed based on period of two years we consecutively included all wrists scanned in our hospital 49 bones, leaving a randomly selected test bone out. Subsequently, the model was that were deemed healthy by an experienced musculoskeletal radiologist. Among fit to the test bone using a selection of the n modes with the highest variance. As the exclusion criteria were: fractures, arthritis, presence of bone fragments, and error measure, the mean distance between the surface points predicted by the bone fusions. The images were acquired on an Mx8000 Quad CT scanner (Philips model and the surface points on the test bone was computed. The experiment was Medical Systems; Best; The Netherlands). The acquisition parameters were: carried out for n=5,10, y, 45 modes of variation and repeated 10 times with collimation 2 Â 0.5 mm, tube voltage 120 kV, effective dose mAs 75, rotation time different test shape selections. 0.75 s per 3601, pitch 0.875, slice thickness 0.6 mm; the scans were made in ‘ultra high resolution’ mode (i.e. small focal spot size). Tomographic reconstructions 2.4.2. Number of training shapes were made with convolution kernel E, a field of view of 150 mm, a slice increment To assess how many shapes are needed for an accurate description of all of 0.3 mm and a matrix of 512 Â 512 pixels. The voxel (three dimensional pixel) possible shape variations, a similar experiment as in the previous section was sizes differed between scans, but all deviated less than 10% from a size of performed. An SSM was constructed, based on a selection of N bones of the same 0.3 Â 0.3 Â 0.3 mm3. All scans were resampled to isotropic voxels of 0.3 mm using type, i.e. ‘training shapes’. Then a test bone was selected from the bones not used linear interpolation. for the SSM and the SSM was then fit to the surface of the test bone using all, i.e. the number of selected bones minus one, modes of variation. As error measure, the 2.2. Segmentation mean distance between the surface points predicted by the model and the surface points on the test bone was computed. This experiment was carried out for The carpal bones were segmented using a level set segmentation (Sethian, N=5,10, y, 45 shapes and repeated 10 times with different shape selections. 1999; Caselles et al., 1997). Using marching cubes (Lorensen and Cline, 1987), the triangulated bone surfaces were extracted from the segmentation result, with 2.5. Distribution of shapes approximately 20 000 surface points per bone. All segmented bone surfaces were labeled semi-automatically. This method was found to be reproducible with a precision in the order of a tenth of the voxel size for different CT scans of the same The 50 bones in the SSM are characterized by their shape parameters, i.e. carpal bones, while the accuracy was visually judged to be in the order of the voxel weights for the modes of variations. The distributions of these parameters give size (Van de Giessen et al., 2009b). information about the prevalence of different shape variants, e.g. a distribution with two peaks might signify two separate types of bone shapes and a skewed distribution shows that one bone shape is found more often than the mean bone 2.3. Registration shape. These distributions are studied by investigating the histograms of these All bones of the same type were aligned and scaled using an unbiased iterative shape parameters and by testing if these distributions are normally distributed. closest point registration, similar to the method described by Chen and Medioni Using a Kolmogorov–Smirnoff test, the probabilities (P-values) are obtained that (1992). During this registration, position, orientation and size differences between the the shape parameters come from a normal distribution with unit variance. 50 bones were minimized such that the only remaining differences between the bones can be attributed to variations in shape. A subselection of 1000 points, randomly selected from the 20 000 surface points and uniformly distributed over the 3. Results bone surface was used to represent each bone during the registration process. In similar data, this registration was found to give registration errors in the order of 3.1. Shape model evaluation 0.05 mm, far below the voxel size (0.3 mm isotropic) (VandeGiessenetal.,2009b).

2.4. Statistical shape model (SSM) For both the scaphoid and the lunate, the mean distance between the segmented surfaces and the surfaces predicted by the The SSMs of bones presented in this paper are point distribution models SSM, i.e. ‘the error’, decreased as the number of modes increased constructed in a similar way as an active shape model (Cootes et al., 1995). Such a (Fig. 2). This is to be expected, as more degrees of freedom, ARTICLE IN PRESS

M. van de Giessen et al. / Journal of Biomechanics 43 (2010) 1463–1469 1465 i.e. modes, allow for a better fit. For the scaphoid, the mean error is The significance of modes is conventionally measured by the smaller than the voxel size of the original CT scans (approximately variance represented by each of the modes, normalized by the total 0.3 Â 0.3 Â 0.3 mm3) for 15 modes or more and for the lunate for variance of the model (percentages in Tables 1 and 2). Modes six and 5 modes and more. This implies that modes higher than these higher described less than 5% of the variance per mode for both the numbers mainly describe noise. For both bones the errors converge scaphoid and lunate. Based on this measure, we chose to describe to approximately 0.25 mm, which is just below the voxel size. only the first five modes of variation of the bones in Section 3.2.

0.5 0.5

0.4 0.4

0.3 0.3

0.2 0.2 Mean error (mm) Mean error (mm) 0.1 0.1

0 0 0 10 20 30 40 50 0 10 20 30 40 50 Number of modes Number of modes

Fig. 2. Mean distance between real surfaces and predicted surfaces by the SSM as a function of the number of modes of variation (Zero modes corresponds to the mean shape). The error bars denote the standard deviation (10 experiments). The dotted line indicates the voxel size. (a) scaphoid and (b) lunate.

Table 1 Five most important modes of variation of the scaphoid.

Mode À3 SD Mean Shape +3 SD Var. Description

1 29% Height of the waist

2 12% Length of the tubercle

3 8% Volume ratio between proximal and distal poles (arrow denotes the largest part)

4 6% Orientation and length of the distal ridge

5 5% Anteroposterior intrascaphoid angle (denoted by black lines)

The orientation of the scaphoid is different for each mode to show the variations. ARTICLE IN PRESS

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Table 2 Five most important modes of variation of the lunate.

Mode À3 SD Mean Shape +3 SD Var. Description

1 25% Ratio between width and height, viewed in the sagittal plane

2 11% Angle between sides adjacent to scaphoid and triquetrum and height of the lunate along the long axis of the capitate

3 9% Skewness in the coronal plane

4 7% Higher volar or dorsal bone end (arrow denotes higher side)

5 5% Extra facet (denoted by arrow) adjacent to hamate

The orientation of the lunate is different for each mode to show the variations.

0.5 0.5

0.4 0.4

0.3 0.3

0.2 0.2 Mean error (mm) Mean error (mm) 0.1 0.1

0 0 0 10 20 30 40 50 0 10 20 30 40 50 Number of training shapes Number of training shapes

Fig. 3. Mean distance between real surfaces and predicted surfaces by the SSM as a function of the number of shapes in the model. The error bars denote the standard deviation (10 experiments). The dotted line indicates the voxel size. (a) scaphoid and (b) lunate.

For an increasing number of training shapes, the SSM was 3.2. Bone shape variations better able to approximate shapes that were not in the model (Fig. 3). 25 shapes were sufficient to describe the scaphoid with a The five modes of variation that describe the largest amount of mean error below the voxel size. For the lunate, the mean error variation of the scaphoid and the lunate are described in Tables 1 dropped below the voxel size for only 10 shapes or more. From and 2, respectively. These five modes describe 60% of the variance this, we concluded that 50 shapes are adequate to construct of the scaphoid and 57% of the variance of the lunate. Modes six statistical shape models of the bones with a similar accuracy as and higher describe more subtle variations, distributed over the the voxel size in the CT scans. complete bones surfaces. ARTICLE IN PRESS

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30 30 30 30 30

20 20 20 20 20 Count Count Count Count Count 10 10 10 10 10

0 0 0 0 0 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 Standard deviation (σ) Standard deviation (σ) Standard deviation (σ) Standard deviation (σ) Standard deviation (σ)

Fig. 4. Distribution of shape parameters of the ﬁrst ﬁve modes of the scaphoid. (a) Mode 1, (b) Mode 2, (c) Mode 3, (d) Mode 4 and (e) Mode 5.

30 30 30 30 30

20 20 20 20 20 Count Count Count Count Count 10 10 10 10 10

Fig. 5. Distribution of shape parameters of the ﬁrst ﬁve modes of the lunate. (a) Mode 1, (b) Mode 2, (c) Mode 3, (d) Mode 4 and (e) Mode 5.

Table 3 -3 SD Mean shape +3 SD P-values resulting from a Kolmogorov–Smirnoff test on the similarity between shape parameter distributions and a normal distribution with the same standard deviation.

Bone Mode 1 Mode 2 Mode 3 Mode 4 Mode 5

Scaphoid 0.29 0.68 0.90 0.67 0.43 Lunate 0.45 0.94 0.31 0.96 0.37 Fig. 6. View of mode 5 of the scaphoid in a different orientation than in Table 1. The line denotes the location of the distal ridge. The curve shows the widening of the scaphoid at the side of the STT joint. 3.3. Distribution of shapes described variations in the height (Mode 1) of the waist of the The shapes used for the construction of the SSM are each scaphoid and also noticed that the tubercle varied in length described by a set of shape parameters, i.e. the parameters that (Mode 2). Ceri et al. (2004) noted that the proximal and distal pole describe how large the difference between a given shape and the of the scaphoid mostly have a similar volume, but that the mean shape is in the direction of each mode of variation. For the proximal and the distal pole were less developed in 19% and 23% five modes of the scaphoid and lunate the distributions of these of their study population, respectively. This observation corre- shape parameters, normalized to their standard deviations, are sponds to a mixture of modes 2 and 3 (Table 1). Variations in the shown in Figs. 4 and 5. path of the distal ridge as found by Compson et al. (1994) are The resulting P-values are listed in Table 3. All these found in mode 4. We described mode 5 to be a change in distributions do not significantly (P-values45%) differ from a angulation between the proximal and distal poles as previously Gaussian distribution. described by Amadio et al. (1989). While Amadio et al. (1989) reported a mean angle of 401 with a standard deviation of 41 for healthy wrists, we found a mean angle of 461 in our population 4. Discussion with a standard deviation of 8.31. When looking at the articulating surface with trapezium and trapezoid (Fig. 6), however, the shape The goals of this study were to describe the variations of the of the scaphotrapezotrapezoidal (STT) joint surface of the scaphoid and the lunate using a statistical shape model and to scaphoid changes in width as described by Moritomo et al. investigate whether distinct bone shapes can be distinguished (2000). Their proposed classification in three types is more from a statistical point of view. Most of the observed variations subjective than the angle measured by Amadio et al. (1989) that within the first five modes of the SSMs confirm previous findings describes the same mode of variation. both for the lunate and the scaphoid bone shapes. The possibility of classifying the scaphoid and lunate in different types was investigated by exploring the distribution of each shape variation 4.2. Shape variations of the lunate in an SSM (Figs. 4 and 5). Since the histograms of all modes of variations do not show multimodal distributions and all inter- Despite the large body of literature on variations of the lunate mediate shape are present, no evidence for distinct shape classes shape, we did not find previous research that describes the ratio is found. Therefore, strict shape classifications are arbitrary. between width and height in the sagittal plane (Mode 1) and the skewness in the coronal plane (Mode 3). However, variations in 4.1. Shape variations of the scaphoid the shape of the surface adjacent to the radius as reported in Antunã-Zapico (1966) and Taleisnik (1985) were also observed Based on manual measurements between landmarks on dried in this study as the second mode of variation. Types I (+3 S.D), II cadaver scaphoids, Ceri et al. (2004) and Compson et al. (1994) (Mean shape) and III (-3 S.D.) were all observed in his mode ARTICLE IN PRESS

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(Table 2), although type III is less pronounced than in Antunã- The statistical shape model that we described is crucial to Zapico’s description. The higher dorsal and volar bone sides as relate shape differences of wrist bones to specific motion patterns. found in mode 4 have been previously reported by Kauer (1980) In future research the values of the shape parameters will be (higher dorsal side) and by Watson et al. (1996) and were correlated with motion data, e.g. rotation and translation para- classified as D- and V-types, for higher dorsal and volar sides and meters. The SSMs will also be used to develop biomechanical a type N with approximately equal sides. Another classification, models of the wrist joint using varying bone geometries. They can proposed by Viegas et al. (1990) and Burgess (1990), based also be applied for the development of prosthetics such as in the on the absence (Type I) or presence (Type II) of a facet that previous study by Modgil et al. (2002). articulates with the hamate was confirmed as the fifth mode of variation in our SSM. The latter variation in the lunate is of importance since significant cartilage erosion at the proximal pole Conflict of interest statement of the hamate is evident at dissection in more than half of the type II lunates, but is not seen in type I lunates (Viegas et al., 1993; None of the authors has a financial interest or business Sagerman et al., 1995; Nakamura et al., 2001). Nakamura et al. relationship posing a conflict of interest concerning the research (2000) demonstrated that the kinematics of type lunates are described in this paper. different from those of type II lunates during radial-ulnar deviations. Most shape variations that were found in earlier studies are Acknowledgements also present in our statistical model. However, the distributions of shape parameters do not differ significantly from a normal The authors wish to thank for the support of the Technology distribution. This implies that the bone shapes come from a Foundation STW, The Netherlands under Grant 6563 and of single, but multi-dimensional, continuous distribution. Using the Philips Medical Systems BV, Best, The Netherlands. presented SSM it is possible to express the difference between each scaphoid or lunate and their mean bone shapes along a continuous scale using the shape parameters. We found evidence References that previously described shape classifications are all identifiable as extrema within the measured distributions. They are, however, Amadio, P.C., Berquist, T.H., Smith, D.K., Ilstrup, D.M., Cooney, W.P., Linscheid, R.L., not separable as distinct types, since all intermediate shapes are 1989. Scaphoid malunion. J. Hand Surg. Am. 14 (4), 679–687. also present. Antunã Zapico, J.M., 1966. Malacia del semilunar. Ph.D. Thesis, Universidad Valladolid. Previously described shape variations of scaphoid and lunate Brett, A.D., Taylor, C.J., 2000. 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