Quantitative analysis of spinal curvature in 3D: application to CT images of normal spine

This article has been downloaded from IOPscience. Please scroll down to see the full text article.

2008 Phys. Med. Biol. 53 1895

(http://iopscience.iop.org/0031-9155/53/7/006)

Download details:

IP Address: 147.26.11.80

The article was downloaded on 10/09/2013 at 08:55

Please note that terms and conditions apply.

View the table of contents for this issue, or go to the journal homepage for more

Home Search Collections Journals About Contact us My IOPscience

IOP PUBLISHING PHYSICS IN MEDICINE AND BIOLOGY

Phys. Med. Biol. 53 (2008) 1895–1908 doi:10.1088/0031-9155/53/7/006

Quantitative analysis of spinal curvature in 3D:application to CT images of normal spine

Tomaž Vrtovec, Boštjan Likar and Franjo Pernuš

University of Ljubljana, Faculty of Electrical Engineering, Tržaska 25, SI-1000 Ljubljana,Slovenia

E-mail: tomaz.vrtovec@fe.uni-lj.si, bostjan.likar@fe.uni-lj.si and franjo.pernus@fe.uni-lj.si

Received 4 October 2007, in final form 31 January 2008Published 10 March 2008Online at stacks.iop.org/PMB/53/1895

AbstractThe purpose of this study is to present a framework for quantitative analysisof spinal curvature in 3D. In order to study the properties of such complex3D structures, we propose two descriptors that capture the characteristics ofspinal curvature in 3D. The descriptors are the geometric curvature (GC) andcurvature angle (CA), which are independent of the orientation and size ofspine anatomy. We demonstrate the two descriptors that characterize the spinalcurvature in 3D on 30 computed tomography (CT) images of normal spineand on a scoliotic spine. The descriptors are determined from 3D vertebralbody lines, which are obtained by two different methods. The first method isbased on the least-squares technique that approximates the manually identifiedvertebra centroids, while the second method searches for vertebra centroids inan automated optimization scheme, based on computer-assisted image analysis.Polynomial functions of the fourth and fifth degree were used for the descriptionof normal and scoliotic spinal curvature in 3D, respectively. The mean distanceto vertebra centroids was 1.1 mm (±0.6 mm) for the first and 2.1 mm (±1.4 mm)for the second method. The distributions of GC and CA values wereobtained along the 30 images of normal spine at each vertebral level andshow that maximal thoracic kyphosis (TK), thoracolumbar junction (TJ) andmaximal lumbar lordosis (LL) on average occur at T3/T4, T12/L1 and L4/L5,respectively. The main advantage of GC and CA is that the measurements areindependent of the orientation and size of the spine, thus allowing objectiveintra- and inter-subject comparisons. The positions of maximal TK, TJ andmaximal LL can be easily identified by observing the GC and CA distributionsat different vertebral levels. The obtained courses of the GC and CA forthe scoliotic spine were compared to the distributions of GC and CA for thenormal spines. The significant difference in values indicates that the descriptorsof GC and CA may be used to detect and quantify scoliotic spinal curvatures.The proposed framework may therefore improve the understanding of spineanatomy and aid in the clinical quantitative evaluation of spinal deformities.

0031-9155/08/071895+14$30.00 © 2008 Institute of Physics and Engineering in Medicine Printed in the UK 1895

1896 T Vrtovec et al

1. Introduction

Quantitative analysis of spinal curvature is important for understanding the nature of normaland pathological spine anatomy, monitoring of the progression of spinal deformities, surgicalplanning and analysis of surgical results (Mac-Thiong et al 2000, Asazuma et al 2004, Dukeet al 2005). The Cobb technique (Cobb 1948) is the most established method for quantifyingspinal curvature in the coronal plane in the case of scoliotic deformities (Morrissy et al 1990,Shea et al 1998, Cheung et al 2002, Chockalingam et al 2002, Adam et al 2005, Stokes andAronsson 2006), as well as in the sagittal plane to measure lordosis and kyphosis (Bernhardtand Bridwell 1989, Korovessis et al 1998, Harrison et al 2000, Goh et al 2000, Pinel-Girouxet al 2006). Several limitations of this technique have been reported (Stokes et al 1987, Pollyet al 1996, Harrison et al 2000), and stimulated the development of new methods for measuringthe degree of spinal deformity (Harrison et al 2000, 2002, Goh et al 2000, Pinel-Girouxet al 2006). Most of the proposed methods proved to be too complex for routine use inclinical environment, and provided only two-dimensional (2D) geometric descriptors of spinaldeformity. It has already been emphasized that three-dimensional (3D) descriptors might yielda more complete assessment of 3D spinal curvatures (Stokes et al 1987, Villemure et al 2001,Duong et al 2006).

In this paper, we present a framework for quantitative analysis of spinal curvature in 3D.The observed 3D descriptors of spinal curvature are the geometric curvature (GC) and curvatureangle (CA), both of which are independent of the orientation of spine anatomy. Moreover,CA is also independent of the size of spine anatomy. The evaluation of the descriptors isstraightforward for a given 3D vertebral body line in a parametric form, which can be easilyobtained by identifying vertebra centroids. We also present an approach to the determinationof a 3D vertebral body line that is based on automated image analysis techniques and canreplace the identification of vertebra centroids, when applied to computed tomography (CT)images of the spine. We show results of applying the proposed framework to images of normalspine. Knowledge of the normal spine may improve the understanding of spinal deformities(Bernhardt and Bridwell 1989, Korovessis et al 1998, Harrison et al 2002, Berthonnaud et al2005, Roussouly et al 2005, Kouwenhoven et al 2006), while computer-assisted analysis ofspine images may allow a faster, more objective and more reliable evaluation of spine anatomy.The comparison of the results for the normal spines and the results obtained on a scolioticspine image indicates that the proposed framework may also be used to detect and quantifyspinal deformities. The terms in this paper follow the convention on 3D terminology of spinaldeformity, proposed by the Scoliosis Research Society (Stokes 1994).

2. Materials and methods

2.1. Images

CT spine images (Tomoscan AVE and MX 8000, Philips Medical Systems, The Netherlands)of 30 adult patients (gender and exact age was not considered relevant) that were treatedfor reasons other than spinal pathology were used in this study. The image voxel size was0.6 × 0.6 × 4 mm3. In addition to the images of normal spine, a CT image of a scoliotic spinewas used (voxel size 0.6 × 0.6 × 1 mm3). Reference anatomical landmarks, required for theproposed evaluation of the spinal curvature, were identified by an expert, well experiencedin the manipulation and shape analysis of 3D spine images. By using dedicated computersoftware that allowed simultaneous navigation through the axial, sagittal and coronal image

Quantitative analysis of spinal curvature in 3D 1897

(a) (b)

Figure 1. (a) Dedicated software for identification of vertebra centroids. (b) Geometricrepresentation of geometric curvature (GC) and curvature angle (CA).

planes, reference vertebra centroids from T1 to L5 were identified in 3D on each image andtheir spatial coordinates were recorded (figure 1(a)).

2.2. 3D vertebral body line

The 3D vertebral body line is a continuous curve that passes through vertebra centroids. Forthe given vertebra centroids, the 3D vertebral body line can be determined by fitting a curveto the spatial coordinates of vertebra centroids using the least-squares method (Bjorck1996). In order to ensure the continuity and preserve the generality of the description, werepresent the 3D vertebral body line with a parametric polynomial function in 3D. A curve C(figure 1(b)) that represents the 3D vertebral body line,

C(p) = [x(p), y(p), z(p)

], (1)

is parameterized by the arbitrary position p on the spine and defined as a polynomial functionof degree K:

C(p) =[

K∑k=0

ax,kpk,

K∑k=0

ay,kpk,

K∑k=0

az,kpk

], (2)

where ax,k, ay,k and az,k, k = 0, 1, . . . , K , are the coefficients of the polynomials x(p), y(p)

and z(p) that represent the sagittal, coronal and axial components of the 3D vertebral bodyline, respectively. The least-squares coefficients of the parametric 3D vertebral body line areobtained by minimizing the sum of square distances between the 3D vertebral body line andvertebra centroids. The optimal values are computed by relatively simple matrix operations,which can be performed by dedicated computer software such as Matlab (Mathworks Inc.,USA). We will refer to this method as the least-squares fitting (LSF) method.

Alternative approaches to the determination of the 3D vertebral body line can be usedwhen vertebra centroids are not available. In this study, we present an automated method for

1898 T Vrtovec et al

analysing CT spine images that is derived from the method used for curved planar reformation(CPR) of whole-length spine images (Vrtovec et al 2005). The method exploits the anatomicalproperty of the spine that vertebral bodies represent the largest bony regions in a spine imageand that regions with bone structures can be coarsely determined by intensity thresholding ofthe CT image. If the distance between an arbitrary point in the image and the nearest boneedge is positive when the point is located inside, and negative when the point is located outsidea bony region, then the distance is expected to be the largest at vertebra centroids. The 3Dvertebral line is initialized as a parametric polynomial function of the first degree in 3D, i.e.as a straight line, where vertebra centroids at T1 and L5 can serve as initialization points. Thepolynomial degree is gradually increased in an optimization scheme that searches for thosepolynomial coefficients that maximize the sum of distances between points located on the 3Dvertebral body line and the edges of the regions with bone structures. For more details on themethod, which was developed as a computer program in C++ language (C++ Builder, BorlandInc., USA), the reader is referred to Vrtovec et al (2005). We will refer to this method as theedge distance optimization (EDO) method.

Irrespective of the applied method for 3D vertebral body line extraction, the 3D vertebralbody line is represented by a parametric polynomial function. It is the geometric propertyof a polynomial function that the number of its flexion points is equal to the polynomialdegree, decreased by 1. As three distinctive flexion points exist in normal spinal curvature,i.e. the thoracolumbar junction (TJ), the maximal thoracic kyphosis (TK) and the maximallumbar lordosis (LL), polynomial functions of the fourth degree were chosen to describenormal spine anatomy in the sagittal and coronal image planes (functions x(p) and y(p)

in (1), respectively). Accordingly, the 3D vertebral body line of the studied scoliotic spinewas described by polynomial functions of the fifth degree. Along the axial image planes,polynomial functions of the first degree (i.e. straight lines) were always used to represent thenatural cranial–caudal length of the spine (function z(p) in (1)).

2.3. Geometric curvature and curvature angle

The geometric properties of a spine can be described by 3D descriptors that capture the 3Dspinal curvature in the coronal and sagittal planes as functions of the position on the spine. Asa result, the strength of the spinal curvature is given quantitatively for a specific position onthe spine. GC and CA are the 3D descriptors with such properties, moreover, their evaluationis straightforward for a given parametric 3D vertebral body line. GC is defined for an arbitraryposition on the spine as the reciprocal to the radius R of the osculating circle in 3D at thatposition (i.e. the circle that best approximates the 3D vertebral body line at an arbitrary positionp on the spine) and represents the amount by which the 3D vertebral body line deviates frombeing flat (× denotes the vector cross-product):

GC(p) =∣∣ dC(p)

dp× d2C(p)

dp2

∣∣∣∣ dC(p)

dp

∣∣3 = 1

R(p). (3)

Hence, the smaller the radius of the osculating circle, the higher is the GC. Since GC isan inherent property of the 3D vertebral body line, it is independent of the spatial orientationof spine anatomy and is therefore neither affected by the position of the patient in the scannernor by the orientation of image acquisition planes.

The descriptor CA is defined as the angular magnitude of GC on an arbitrary spinesection. CA can be considered as the angular representation of GC, normalized to the specified

Quantitative analysis of spinal curvature in 3D 1899

spine section. The curvature angle (CA, in radians) of the parametrized geometric curvature,

CA(p1, p2)[rad] = L(p1, p2)

R(p12)= L(p1, p2) · GC(p12), (4)

is determined as the angle that spans two given positions p1 and p2 on the spine with thearc L of the osculating circle, which is defined at the mean position p12 = 1

2 (p2 − p1) onthe spine (figure 1(b)). In the case that the specified section is relatively small, the arc of theosculating circle that is used for computation of CA can be approximated with the length ofthe parameterized 3D vertebral body line on that section:

L(p1, p2) =∫ p2

p1

∣∣∣∣dC(p)

dp

∣∣∣∣ dp. (5)

For example, vertebra centroids of two adjacent vertebrae may represent a section wheresuch an approximation can be applied (figure 1(b)). Therefore, besides inheriting all theproperties of GC, the normalization to a specified section makes CA independent of the sizeof the spine. The representation of CA in degrees is given by:

CA[degrees] = 180◦

πCA[rad]. (6)

3. Results

In order to obtain the GC and CA values, the 3D vertebral body line in the form of a parametricpolynomial function in 3D had to be extracted first. We applied two methods, the LSF andEDO methods, to the studied 30 images of normal spine and to the scoliotic spine image.The small distances between the obtained 3D vertebral body lines and manually identifiedvertebra centroids indicate that the selection of the degree of the polynomial functions wasappropriate (figure 2). The EDO method is in general less accurate and more variable than theLSF method (table 1), and the obtained distances between the 3D vertebral body line and thevertebra centroids for the EDO method are on average approximately twice the magnitude ofthe corresponding distances for the LSF method. However, the LSF method required manualidentification of all vertebra centroids, while the EDO method required the approximateidentification of two vertebra centroids only.

Additional experiments were performed in order to test the robustness of the EDO method.The vertebra centroids at T1 and L5, which served as initialization points, were displaced fora randomly defined direction and amplitude of up to 5 mm, thus simulating coarse manualidentification of the initialization points in a 10 mm area. The EDO method was then appliedfor 100 randomly defined initialization points to each of the 30 images of normal spine andto the scoliotic spine. Average distances between the obtained 3D vertebral body lines andthe vertebra centroids were computed, and the results show that the EDO method is relativelyrobust, as the resulting 3D vertebral body line is relatively independent of the initializationpoints (figure 3).

We first present a detailed analysis for two illustrative cases, chosen from among the30 images of normal spine. For image 4, the average distance between the 3D vertebral bodyline and vertebra centroids is approximately equal to the average along all 30 images for boththe LSF and EDO methods. On the other hand, the average distance for image 25 is above thisaverage, again for both the LSF and EDO methods (figure 2, table 1). In order to not influencethe interpretation of the results by presenting detailed analysis of images with relatively lowaverage distances, images 4 and 25 were chosen for the illustrative cases (cases 1 and 2,respectively).

1900 T Vrtovec et al

Figure 2. Distribution of distances between the 3D vertebral body line, obtained by the least-squares fitting (LSF, top) and edge distance optimization (EDO, bottom) methods, and vertebracentroids for each of the 30 images of normal spine and for the scoliotic (SC) spine image. Foreach image, the results are presented as a box-whiskers diagram, which shows the median, firstand third quartile, minimal and maximal distance and outliers (values that are more than 1.5 timesthe interquartile range away from the first or third quartile).

Table 1. Distance d to the 3D vertebral body line (VL), and the geometric curvature (GC) andcurvature angle (CA) at the thoracic kyphosis (TK), thoracolumbar junction (TJ) and lumbarlordosis (LL), obtained by applying the least-squares fitting (LSF) and edge distance optimization(EDO) methods to illustrative cases 1 and 2, and to all the 30 images of normal spine used in thisstudy.

LSF method EDO method

Mean d (±std) [1] 1.3 mm (±0.7 mm) 2.4 mm (±1.7 mm)VL Mean d (±std) [2] 1.5 mm (±0.9 mm) 3.5 mm (±1.2 mm)

Mean d (±std) [all] 1.1 mm (±0.6 mm) 2.1 mm (±1.4 mm)

Minimal GC (R) [1] 0.6 m−1 (1.67 m) [∼T11] 0.6 m−1 (1.67 m) [∼T11]TJ Minimal GC (R) [2] 0.03 m−1 (33.3 m) [∼T12] 0.14 m−1 (7.14 m) [∼T12]

Mean CA (±std) [all] 1.2◦ (±1.0◦) [T12/L1] 1.0◦ (±0.8◦) [T12/L1]

Maximal GC (R) [1] 2.9 m−1 (0.35 m) [∼T3] 3.4 m−1 (0.29 m) [∼T1]TK Maximal GC (R) [2] 5.6 m−1 (0.18 m) [∼T5/T6] 5.1 m−1 (0.19 m) [∼T5/T6]

Mean CA (±std) [all] 3.4◦ (±1.3◦) [T3/T4] 3.5◦ (±1.2◦) [T3/T4]

Maximal GC (R) [1] 5.7 m−1 (0.17 m) [∼L5] 4.6 m−1 (0.22 m) [∼L5]LL Maximal GC (R) [2] 4.3 m−1 (0.23 m) [∼L5] 5.6 m−1 (0.18 m) [∼L5]

Mean CA (±std) [all] 6.5◦ (±4.4◦) [L4/L5] 5.0◦ (±3.4◦) [L4/L5]

Clinically relevant features of the normal spine that are located at flexion points canbe extracted from the course of GC. The flexion point where the course of GC reaches its

Quantitative analysis of spinal curvature in 3D 1901

Figure 3. Distribution of average distances between 100 3D vertebral body lines, obtained by theedge distance optimization (EDO) method with randomly defined initialization points, and vertebracentroids for each of the 30 images of normal spine and for the scoliotic (SC) spine image.

(a) (b) (c)

Figure 4. Normal spine, case 1. (a) The 3D vertebral body line (white line), obtained by the least-squares fitting (LSF) method and (b) by the edge distance optimization (EDO) method. Blackdiamonds represent the vertebra centroids. (c) Geometric curvature (GC) and curvature angle(CA), evaluated along the spine by applying the LSF (squares, dotted lines) and EDO (circles,solid lines) methods.

minimal value represents the position where the spine is most straight in 3D, i.e. the positionof TJ. For the first case (figure 4(c)), the TJ occurs approximately at T11 (table 1). Since theminimal value of GC is not equal to zero, the spine at TJ is not completely straight. It can beobserved in figures 4(a) and (b) that although the 3D vertebral body line is straight at TJ in thesagittal, it is curved in the coronal image plane. The true nature of the observed spine wouldtherefore be overlooked if observed in the sagittal image plane only, i.e. in 2D. Similarly,the TJ can be identified and evaluated for the second illustrative case (figure 5(c), table 1),where it occurs approximately at T12. Since GC at that point is almost equal to zero, the TJis almost completely straight in 3D, which can be observed from the 3D vertebral body lines(figures 5(a) and (b)). The flexion point where the course of GC reaches its maximal valuetowards the cranial and caudal directions represents the position of maximal TK and maximal

1902 T Vrtovec et al

(a) (b) (c)

Figure 5. Normal spine, case 2. (a) The 3D vertebral body line (white line), obtained by the least-squares fitting (LSF) method and (b) by the edge distance optimization (EDO) method. Blackdiamonds represent the vertebra centroids. (c) Geometric curvature (GC) and curvature angle(CA), evaluated along the spine by applying the LSF (squares, dotted lines) and EDO (circles,solid lines) methods.

LL, respectively. For the two illustrative cases, the maximal TK occurs approximately at T3in the first and approximately at T5/T6 in the second case, while the maximal LL occurs atL5 in both cases (table 1).

The GC descriptor can be evaluated continuously along the whole spine, on the otherhand, the CA descriptor can be evaluated only on specified spine sections (figures 4(c)and 5(c)). The exact positions of TJ, maximal TK and maximal LL can therefore not bedetermined using CA. Moreover, CA directly depends on GC (4) and can be therefore viewedupon as an alternative descriptor to GC. For these reasons, CA is less appropriate for theanalysis of spinal curvature in a single image. On the other hand, CA is more suitable for theevaluation of spinal curvature along a series of images where the spine may differ in size, astwo spines with equal spinal curvatures but different sizes may have different GC but equalCA values. We evaluated the distribution of CA along the 30 images of normal spine, whichwas obtained by grouping the CA values on spine sections from T1/T2 to L4/L5 (figure 7).The CA is on average the lowest at section T12/L1, which means that generally, TJ will occurat this section. The maximal TK occurs on average at section T3/T4 and the maximal LL atsection L4/L5 (table 1). A number of other features that are characteristic for the normal spinecan also be observed. The course of normal TK is represented by a gradual increase of CAin the cranial direction from section T12/L1 to T3/T4, followed by a slight decrease towardssection T1/T2, while the course of normal LL is represented by a gradual increase of CA inthe caudal direction from section T12/L1 to L4/L5. Furthermore, the obtained average CAvalues along the spine (figure 7) are comparable to the normal sagittal angulations measuredin 2D radiographic images acquired in standing position (Bernhardt and Bridwell 1989).

The proposed 3D descriptors can also be used to detect and quantify the pathological(e.g. scoliotic) spinal curvatures. In order to demonstrate the difference between normal andscoliotic spinal curvatures, we applied the proposed methods to the image of a scoliotic spine

Quantitative analysis of spinal curvature in 3D 1903

(a) (b) (c)

Figure 6. Scoliotic spine. (a) The 3D vertebral body line (white line), obtained by the least-squaresfitting (LSF) method and (b) by the edge distance optimization (EDO) method. Black diamondsrepresent the vertebra centroids. (c) Geometric curvature (GC) and curvature angle (CA), evaluatedalong the spine by applying the LSF (squares, dotted lines) and EDO (circles, solid lines) methods.

Table 2. Distance d to the 3D vertebral body line (VL) and the geometric curvature (GC) at thesuperior end (SE), inferior end (IE) and apical (AP) vertebra, obtained by applying the least-squaresfitting (LSF) and edge distance optimization (EDO) methods to the scoliotic spine image.

LSF method EDO method

VL Mean d (±std) 0.9 mm (±0.4 mm) 1.8 mm (±1.0 mm)

AP Maximal GC (R) 5.6 m−1 (0.18 m) [∼T9] 4.9 m−1 (0.20 m) [∼T9]SE Minimal GC (R) 1.1 m−1 (0.91 m) [∼T5] 0.8 m−1 (1.25 m) [∼T5]IE Minimal GC (R) 0.8 m−1 (1.25 m) [∼T12] 0.7 m−1 (1.43 m) [∼T12]

(figures 6(a) and (b)). Similarly as for the normal spines, the flexion points of the course of GCrepresent clinically relevant features in the case of the scoliotic spine (figure 6(c)). The flexionpoint where the GC reaches its maximal value represents the apex of the scoliotic deformity(i.e. the apical vertebra), which occurs approximately at T9 (table 2). The flexion point wherethe course of GC reaches its minimal value towards the cranial and caudal directions representsthe position of the superior and inferior end vertebra, respectively. The superior end vertebraoccurs at approximately T5, while the inferior end vertebra occurs at approximately T12(table 2). By comparing the values of the CA for the scoliotic spine to the distribution ofthe CA along the 30 images of normal spine (figure 7), the location and extent of a scolioticdeformity can be determined. The CA for the scoliotic spine is significantly higher thanthe maximal distribution of CA for normal spines in the region around the apical vertebra,however, around both superior and inferior end vertebrae the CA values for the scoliotic spineare comparable to the CA values for normal spines.

We also studied the influence of the variation in the initialization points of the EDOmethod to the resulting GC and CA values. The results are shown for the two illustrative cases

1904 T Vrtovec et al

Figure 7. Distribution of the curvature angle (CA) for different spine segments, obtained byapplying the least-squares fitting (LSF, top) and edge distance optimization (EDO, bottom) methodsto 30 CT images of normal spine.

and for the image of the scoliotic spine in figure 8. There is some variation in the results, butthe basic shape and tendency of the courses of the GC and CA remained relatively unchanged.The identification of distinctive flexion points, required for the evaluation of the positions ofTJ, maximal TK and maximal LL, is therefore not substantially affected by the variation inthe initialization points of the EDO method.

4. Discussion

The purpose of this study is to present a framework for quantitative analysis of spinal curvaturein 3D. Spinal deformities may occur in the axial, sagittal and coronal image planes, or in acombination of any of these planes. In order to study the properties of such complex 3Dstructures, descriptors that capture the characteristics of the spine in 3D should be applied.Although the Cobb angle (Cobb 1948) is the established method for quantifying spinaldeformities, its measurement depends on various factors. Among the most significant ones

Quantitative analysis of spinal curvature in 3D 1905

(a) (b)

Figure 8. Distribution of (a) the geometric curvature (GC) and (b) the curvature angle (CA) for thecase 1 (top), case 2 (middle) and for the scoliotic spine (bottom), computed from 100 3D vertebralbody lines that were obtained by the edge distance optimization (EDO) method with randomlydefined initialization points.

are the correct identification of the end vertebrae of the spinal curvature and the determinationof vertebral endplate inclination (Chockalingam et al 2002), both of which may be affectedby the lower visibility (separability) of anatomical structures in the thoracic region and by thevariable shape of vertebral endplates (Polly et al 1996). Moreover, since the measurementis based solely on the inclination of vertebral endplates, the method is unable to describethe global spine geometry (Stokes et al 1987, Harrison et al 2000, Pinel-Giroux et al 2006).The described limitations induce a high variability in the measurement of the Cobb angle,which stimulated the development of new techniques for the evaluation of spinal curvature.Descriptors such as posterior tangent lines (Harrison et al 2000), best-fit ellipses (Harrisonet al 2002), mean radius of curvature (Goh et al 2000), geometric torsion (Poncet et al 2001) ortangent circles (Pinel-Giroux et al 2006) are either associated with a large number of clinicallyunintuitive parameters or require a relatively high amount of user interaction (e.g. preciseoutlining or drawing tangent lines to vertebral bodies). As complicated models do not helpsurgeons and clinicians in understanding the spinal deformity, such techniques are too complexfor routine use in clinical environment. Moreover, all techniques focus on spinal curvatureseparately in either the coronal or the sagittal image plane, i.e. in 2D, although they can occurin an arbitrary plane. This requires the images to be uniform in orientation and size, whichcan be achieved by a standardized image acquisition process. On the other hand, descriptorsthat measure the spinal curvature in 3D and are independent of the orientation and size of thespine may allow a more general (e.g. using images of different modalities) and more objective(e.g. measuring the curvature in multiple image planes) evaluation of spinal deformities.

The presented GC and CA descriptors of spinal curvature in 3D are evaluatedstraightforwardly from the 3D vertebral body line. The LSF method for the determination

1906 T Vrtovec et al

of 3D vertebral body line is based solely on the spatial coordinates of vertebra centroids,which are clinically intuitive and can be unambiguously defined on images of differentsize, dimensionality (e.g. 2D or 3D images) and modality (e.g. radiographs, CT or magneticresonance (MR) images). Although a polynomial function that would pass exactly throughvertebra centroids could be determined by adequately higher polynomial degrees, smallerdegrees help in compensating the variability in the identification of vertebra centroids.However, the identification of vertebra centroids is a process that requires clinical experienceand a relatively high amount of user interaction. The EDO method is an example of howcomputer-assisted image analysis techniques can be used to overcome these problems. Theamount of user interaction is limited to the approximate determination of the two end points ofthe 3D vertebral body line. The identification of vertebra centroids from T1 to L5 in 3D equals51 spatial coordinates, i.e. one coordinate in each of the three image planes (axial, sagittal andcoronal) for each of the 17 vertebrae. This number represents the number of input parametersfor the LSF method, while for the EDO method, just two initialization points are required,therefore the number of input parameters reduces to six. While the LSF method is designed forapplication to an arbitrary form of spinal curvature and imaging modality, the presented EDOmethod was developed for CT images. In the case of other imaging modalities, however, theproposed framework for the quantitative analysis of spinal curvature may remain unchanged,except for the automated extraction of 3D vertebral lines, which should be modality-dependent,e.g. adapted for MR images (Vrtovec et al 2007).

The main advantage of the 3D descriptors of GC and CA is that the measurementsare independent of the orientation and size of the spine. The acquisition of images istherefore not required to be standardized, thus allowing a comparison between the imageswith different properties (e.g. different scanners, acquisition parameters, patient orientation,spinal deformities, clinical environments). The measures also do not depend on vertebralbody or intervertebral disc shape, or vertebral endplate tilt, but solely on the global propertiesof the obtained 3D vertebral body line. The parametric description of the 3D vertebralbody line may be useful in different applications and studies, for example, it can be appliedstraightforwardly to measurements of spine torsion (Poncet et al 2001) or to the automatedspine survey technique (Weiss et al 2006). Moreover, the analysis can be reduced to 2Dwithout losing the independence of the orientation and size by observing the 3D vertebralbody line separately in the sagittal and coronal image planes (1). If the exact location ofTJ was automatically identified, the automated evaluation of total TK (CA between T1 andTJ) and total LL (CA between TJ and L5) may be possible. Normal TK and normal LL areof significant importance in the maintenance of an adequate sagittal spinal balance and theevaluation of the low back pain (Berthonnaud et al 2005, Vedantam et al 1998). By comparingan arbitrary normal spine anatomy to the mean GC and CA values over a healthy population,the presence and exact location of hyperkyphosis and/or hyperlordosis may be identifiedand more objectively evaluated. The automated detection of characteristic spine regions (i.e.flexion points of the 3D vertebral body line) may be further used to classify spinal deformities(King et al 1983, Lenke et al 2001, Roussouly et al 2005). To conclude, the application of theproposed framework to the analysis of spinal curvatures in 3D may improve the understandingof spine anatomy and aid in clinical quantitative evaluation of spinal deformities.

Acknowledgments

This work has been supported by the Ministry of Higher Education, Science and Technology,Slovenia, under grants P2-0232, L2-7381 and L2-9758. The authors would like to thank the

Quantitative analysis of spinal curvature in 3D 1907

Images Sciences Institute, Medical University Utrecht, The Netherlands, for providing the CTimages used in this study.

References

Adam C, Izatt M, Harvey J and Askin G 2005 Variability in Cobb angle measurements using reformatted computerizedtomography scans Spine 30 1664–9

Asazuma T, Nakamura M, Matsumoto M, Chibo K and Toyama Y 2004 Postoperative changes of spinal curvatureand range of motion in adult patients with cervical spinal cord tumors: analysis of 51 cases and review of theliterature J. Spinal Disord. 17 178–82

Bernhardt M and Bridwell K 1989 Segmental analysis of the sagittal plane alignment of the normal thoracic andlumbar spines and thoracolumbar junction Spine 14 717–21

Berthonnaud E, Dimnet J, Roussouly P and Labelle H 2005 Analysis of the sagittal balance of the spine and pelvisusing shape and orientation parameters J. Spinal Disord. 18 40–7

Bjorck A 1996 Numerical Methods for Least Squares Problems (Philadelphia, PA: SIAM)Cheung J, Wever D, Veldhuizen A, Klein J, Verdonck B, Nijlunsing R, Cool J and Van Horn J 2002 The reliability of

quantitative analysis on digital images of the scoliotic spine Eur. Spine J. 11 535–42Chockalingam N, Dangerfield P, Giakas G, Cochrane T and Dorgan J 2002 Computer-assisted Cobb measurement of

scoliosis Eur. Spine J. 11 353–7Cobb J 1948 Outline for the study of scoliosis Am. Acad. Orthop. Surg. Instr. Course Lect. 5 261–75Duke K, Aubin C, Dansereau J and Labelle H 2005 Biomechanical simulations of scoliotic spine correction due to

prone position and anaesthesia prior to surgical instrumentation Clin. Biomech. 20 923–31Duong L, Cheriet F and Labelle H 2006 Three-dimensional classification of spinal deformities using fuzzy clustering

Spine 31 923–30Goh S, Price R, Leedman P and Singer K 2000 A comparison of three methods for measuring thoracic kyphosis:

implications for clinical studies Rheumatology 39 310–5Harrison D E, Harrison D D, Cailliet R, Troyanovich S, Janik T and Holland B 2000 Cobb method or Harrison

posterior tangent method: which to choose for lateral cervical radiographic analysis Spine 25 2072–8Harrison D E, Janik T, Harrison D E, Cailliet R and Harmon S 2002 Can the thoracic kyphosis be modeled with

a simple geometric shape? The results of circular and elliptical modeling in 80 asymptomatic patientsJ. Spinal Disord. 15 213–20

King H, Moe J, Bradford D and Winter R 1983 The selection of fusion levels in thoracic idiopathic scoliosis J. BoneJoint Surg. Am. 65 1302–13

Korovessis P, Stamatakis M and Baikousis A 1998 Reciprocal angulation of vertebral bodies in the sagittal plane inan asymptomatic Greek population Spine 23 700–4

Kouwenhoven J-W, Vincken K, Bartels L and Castelein R 2006 Analysis of preexistent vertebral rotation in thenormal spine Spine 31 1467–72

Lenke L, Betz R, Harms J, Bridwell K, Clements D, Lowe T and Blanke K 2001 Adolescent idiopathic scoliosis: anew classification to determine extent of spinal arthrodesis J. Bone Joint Surg. Am. 83-A 1169–81

Mac-Thiong J, Labelle H, Vandal S and Aubin C 2000 Intra-operative tracking of the trunk during surgical correctionof scoliosis: a feasibility study Comput. Aided Surg. 5 333–42

Morrissy R, Goldsmith G, Hall E, Kehl D and Cowie G 1990 Measurement of the Cobb angle on radiographs ofpatients who have scoliosis: evaluation of intrinsic error J. Bone Joint Surg. Am. 72 320–7

Pinel-Giroux F-M, Mac-Thiong J, de Guise J, Berthonnaud E and Labelle H 2006 Computerized assessment of sagittalcurvatures of the spine: comparison between Cobb and tangent circles techniques J. Spinal Disord. 19 507–12

Polly D, Kilkelly F, McHale K, Asplund L, Mulligan M and Chang A 1996 Measurement of lumbar lordosis:evaluation of intraobserver, interobserver, and technique variability Spine 21 1530–5

Poncet P, Dansereau J and Labelle H 2001 Geometric torsion in idiopathic scoliosis: three-dimensional analysis andproposal for a new classification Spine 26 2235–43

Roussouly P, Gollogly S, Berthonnaud E and Dimnet J 2005 Classification of the normal variation in the sagittalalignment of the human lumbar spine and pelvis in the standing position Spine 30 346–53

Shea K, Stevens P, Nelson M, Smith J, Masters K and Yandow S 1998 A comparison of manual versus computer-assisted radiographic measurement: intraobserver measurement variability for Cobb angles Spine 23 551–5

Stokes I 1994 Three-dimensional terminology of spinal deformity: a report presented to the Scoliosis ResearchSociety by the Scoliosis Research Society Working Group on 3-D terminology of spinal deformity Spine 19236–48

Stokes I and Aronsson D 2006 Computer-assisted algorithms improve reliability of King classification and Cobbangle measurement of scoliosis Spine 31 665–70

1908 T Vrtovec et al

Stokes I, Bigalow L and Moreland M 1987 Three-dimensional spinal curvature in idiopathic scoliosisJ. Orthop. Res. 5 102–13

Vedantam R, Lenke L, Keeney J and Bridwell K 1998 Comparison of standing sagittal spinal alignment inasymptomatic adolescents and adults Spine 23 211–5

Villemure I, Aubin C, Grimard G, Dansereau J and Labelle H 2001 Progression of vertebral and spinal three-dimensional deformities in adolescent idiopathic scoliosis: a longitudinal study Spine 26 2244–50

Vrtovec T, Likar B and Pernuš F 2005 Automated curved planar reformation of 3D spine images Phys. Med.Biol. 50 4527–40

Vrtovec T, Ourselin S, Lavier G, Likar B and Pernuš F 2007 Automated generation of curved planar reformationsfrom MR images of the spine Phys. Med. Biol. 52 2865–78

Weiss K, Storrs J and Banto R 2006 Automated spine survey iterative scan technique Radiology 239 255–62