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Automated generation of curved planar reformations from MR images of the spine

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2007 Phys. Med. Biol. 52 2865

(http://iopscience.iop.org/0031-9155/52/10/015)

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IOP PUBLISHING PHYSICS IN MEDICINE AND BIOLOGY

Phys. Med. Biol. 52 (2007) 2865–2878 doi:10.1088/0031-9155/52/10/015

Automated generation of curved planar reformationsfrom MR images of the spine

Tomaž Vrtovec1, Sebastien Ourselin2,4, Lavier Gomes3, Boštjan Likar1

and Franjo Pernuš1

1 Faculty of Electrical Engineering, University of Ljubljana, Tržaska 25, SI-1000 Ljubljana,Slovenia2 CSIRO ICT Centre, Autonomous Systems Laboratory, BioMedIA Lab, Locked Bag 17,North Ryde, NSW 2113, Australia3 Department of Radiology, Westmead Hospital, University of Sydney, Hawkesbury Road,Westmead NSW 2145, Australia

E-mail: tomaz.vrtovec@fe.uni-lj.si

Received 20 October 2006, in final form 20 October 2006Published 30 April 2007Online at stacks.iop.org/PMB/52/2865

AbstractA novel method for automated curved planar reformation (CPR) of magneticresonance (MR) images of the spine is presented. The CPR images, generatedby a transformation from image-based to spine-based coordinate system, followthe structural shape of the spine and allow the whole course of the curvedanatomy to be viewed in individual cross-sections. The three-dimensional (3D)spine curve and the axial vertebral rotation, which determine the transformation,are described by polynomial functions. The 3D spine curve passes throughthe centres of vertebral bodies, while the axial vertebral rotation determinesthe rotation of vertebrae around the axis of the spinal column. The optimalpolynomial parameters are obtained by a robust refinement of the initialestimates of the centres of vertebral bodies and axial vertebral rotation. Theoptimization framework is based on the automatic image analysis of MR spineimages that exploits some basic anatomical properties of the spine. The methodwas evaluated on 21 MR images from 12 patients and the results provided agood description of spine anatomy, with mean errors of 2.5 mm and 1.7◦ forthe position of the 3D spine curve and axial rotation of vertebrae, respectively.The generated CPR images are independent of the position of the patient in thescanner while comprising both anatomical and geometrical properties of thespine.

4 Present address: BioMedIA Lab, e-Health Research Centre, Level 20, 300 Adelaide Street, Brisbane, QLD 4000,Australia.

0031-9155/07/102865+14$30.00 © 2007 IOP Publishing Ltd Printed in the UK 2865

2866 T Vrtovec et al

1. Introduction

Images of three-dimensional (3D) anatomical structures, obtained by computed tomography(CT) or magnetic resonance (MR) imaging, are usually presented to clinicians in the form of aseries of two-dimensional (2D) planar cross-sections in the standard axial, sagittal and coronalimage reformation. However, because structures of interest may have curved 3D morphology,they cannot be completely visualized in individual multiplanar or oblique cross-sections. Inthe examination of spine images, for example, the spine may intersect the sagittal or coronalcross-sections due to its natural curved shape, while the axial cross-sections may not alwaysbe positioned at the same level of the vertebral bodies or intervertebral discs. Not all of theimportant details can therefore be shown simultaneously in any planar cross-section, whichmay be even more explicit in the case of significant coronal or sagittal spinal curvature, i.e.scoliosis and kyphosis or lordosis, respectively. To enable an effective clinical evaluation andquantitative analysis of such cases, images should be created in the coordinate system of thecurved 3D anatomical structure, in which cross-sections are either orthogonal or tangent toa curve along the structure. This type of image reformation is referred to as curved planarreformation (CPR), or curved sectioning, and is often used (e.g. in angiography, diagnosticsof colon and pancreas) to overcome the difficulties of conventional representations.

Several approaches to reformation of CT images of the spine have already been presentedand reported to be useful for evaluation of spinal deformities. Rothman et al (1984)demonstrated that curved images of the coronal spine region, obtained by connecting manuallyselected points into a continuous curve, are useful in evaluation of anatomical relationships.After reformation, structures such as nerve roots, facet joints and spinal cord could be observedin a single 2D image. By generating oblique sagittal images, Rabassa et al (1993) showedthat visualization of the vertebral facets improved, while oblique axial images allowed viewsparallel to the intervertebral discs in the case of scoliosis or an increased lumbosacral curvatureof the spine. Although the reformation was limited to oblique cross-sections, the authorsconcluded that in certain clinical situations, such as in evaluation of neural foraminal stenosisor localization of spinal lesions, the reformatted images could supplement the acquired CTimages. Congenital spinal deformities were examined by Newton et al (2002), who manuallyoutlined the boundaries of the spine in standard reformation and created CPR images thatimproved the identification and interpretation of abnormalities. The benefit of CPR images,in comparison with multiplanar or oblique cross-sections, was most valuable in the case ofsignificant sagittal or coronal curvature of the spine, as CPR images might help spine surgeonsto achieve a more complete understanding of spinal deformity. Roberts et al (2003) reformattedthe images orthogonally to the long axis of both left and right neural foraminae of the cervicalregion of the spine. By oblique reformation, they improved the consistency in the interpretationof neural foraminal stenosis between observers. They also suggested that such an approachshould be considered in routine evaluation. In order to improve the definition of congenitalabnormalities of the spine, Menten et al (2005) described a planospheric reformation methodthat was based on a reconstruction from a cylindrical plane, defined around the approximateboundary of the spinal canal within an axial CT cross-section. As a result, the anterior andposterior elements of the spine were visualized in the same plane. Manual determination of thecurve points was required in all of the studies reported above. A semi-automated method waspresented by Kaminsky et al (2004), who performed segmentation of the spine on reformattedCT images in order to overcome the problems of orientation in the standard reformation. Thetransformation axis was determined by a 3D spline, obtained either manually by delineatingcentrelines in sagittal and coronal planes or automatically by dropping spheres of maximumpossible radius through the vertebral bodies or spinal canal. Recently, an automated CPR

Automated generation of CPRs from MR images of the spine 2867

method for 3D CT spine images was presented by Vrtovec et al (2005), which required manualdetermination of just two points at both ends of the spinal section of interest. By introducingthe notion of the spine-based coordinate system, they represented the spinal curvature and therotation of vertebrae as polynomial functions, which formed the transformation axes for thereformation procedure. However, the method was developed for CT images and is thereforemodality dependent.

Over the past few years, MR has become a more dominant modality in spine imaging,providing high-quality 3D images of soft tissues and bone structures of the spine, includingthe spinal cord, by a correct selection of imaging parameters. The poor resolution of earlyMR scanners has been improved by dedicated multichannel spine coils with better signal-to-noise ratio (SNR). Visualization of abnormalities, injuries and diseases in the spinal region isoften superior in MR imaging than in other imaging methods, such as CT or myelography.Moreover, as there is no exposure to ionizing radiation, MR is considered to be the modalityof choice for follow-up examinations and longitudinal studies. Image reformation has alreadybeen identified as a valuable technique in MR imaging of the spine. Apicella and Mirowitz(1995) reported that image reformation could compensate for the apparent asymmetry of 3Danatomical structures, caused by improper patient positioning or patient motion during imageacquisition, and that reformatting can be applied to different anatomical structures, includingthe spine, where it can be used to improve the visualization of the spinal canal and neuralforaminae. Liljenqvist et al (2002) focused their study on vertebral morphology, which is ofsignificant importance in pedicle-screw placement for the treatment of scoliosis. In order toobtain true values of the pedicle width, length and angle, the MR spine images were manuallyreformatted so that the measurements could be performed in cross-sections perpendicular tothe vertebral bodies. Very recently, in a study of automated survey of MR spine images (Weisset al 2006), it was reported that automated reformation of 3D spine images along the truesagittal, coronal or axial axis of the vertebral bodies and discs may potentially facilitate theradiologist’s image interpretation. On the other hand, dedicated commercial software andpackages provided by CT/MR scanner vendors enable generation of multiplanar and curvedcross-sections. However, the points that are connected into a continuous curve also haveto be manually selected by the user, which requires navigation through complex 3D (spine)anatomy. Although the MR scanners allow acquisition of images in arbitrary planes (i.e. ina rotated standard reformation), they are selected by the machine operator and dependent onthe position of the patient in the scanner. The possibility of obtaining a curved slice from theMR scanner (Jochimsen and Norris 2002, Bornert 2003) was achieved by the application ofspatially selective 2D radio-frequency (RF) pulses, thus allowing the imaged slice to be curvedonly in one dimension. Poor spatial resolution in initial experiments was improved; however,the image quality was still not satisfying due to intensity modulation artefacts.

In this paper, we present a novel method for automated CPR of MR images of the spine.The reformation framework is based on the transformation from the standard image-basedto the spine-based coordinate system, which we proposed for automated CPR of CT images(Vrtovec et al 2005). The main contribution of this work is the automated extraction ofthe 3D spine curve and axial vertebral rotation from MR images. The method is based onautomated detection of spine bone structures from MR images, which is far more difficultthan the detection of these structures from CT images. For this purpose, we exploit some ofthe basic anatomical features of the spine, i.e. the mid-sagittal symmetry and circularity ofthe vertebral column. Besides being diagnostically valuable, the resulting CPR images areindependent of the position of the patient in the scanner, as they are referenced to the locationof the vertebral bodies and the relative orientation of the vertebrae while comprising bothanatomical and geometrical properties of the spine.

2868 T Vrtovec et al

(a) (b) (c)

Figure 1. In the image-based coordinate system, the spine intersects sagittal and/or coronal cross-sections (a). The transformation from image-based to spine-based coordinate system is determinedby the 3D spine curve c(n) and axial vertebral rotation ϕ(n) (b). In the spine-based coordinatesystem, the whole course of the spine can be observed in sagittal and coronal cross-sections (c).

2. Problem description

In order to generate CPR images of the spine, information on individual vertebrae andintervertebral discs and their spatial relationship has to be properly extracted from MR imagesof the spine. The standard image-based reformation is represented by the Cartesian coordinatesystem, in which the x, y and z axes represent the standard sagittal, coronal and axial directions,respectively. With the introduction of the spine-based coordinate system (Vrtovec et al 2005),the spine domain �n is mapped from the image-based to the spine-based coordinate systemby applying the transformation T:

T : (x, y, z)�n−→ (u, v,w), (1)

where u, v and w are the axes of the spine-based coordinate system, defined by continuousparametric functions c(n) and ϕ(n), which are specific for the spine. The 3D spine curvec(n) is a curve that passes through the centres of vertebral bodies and intervertebral discs andrepresents the central axis of the spinal column. The axial vertebral rotation ϕ(n) representsthe rotation of the vertebral spinous processes around the axis of the spinal column. Thecontinuous variable n parameterizes the spine domain �n. For an arbitrary position on thespine, the axes of the spine-based coordinate system (figure 1) can be therefore defined asfollows:

• The axis w is tangent to the 3D spine curve c(n) and points in the direction of the axis ofthe spinal column.

• The axis v is orthogonal to axis w and points in the direction of the vertebral spinousprocesses. The rotation of the axis v around the axis w is determined by the angleϕ(n) = � (v, y⊥), which is defined as the angle between the axis v and the projection y⊥of the Cartesian axis y onto the plane P⊥(n), in which the axis v lies.

• The axis u is orthogonal to both axes w and v.

The key problem in automated generation of CPR is the determination of the 3D spinecurve c(n) and axial vertebral rotation ϕ(n), which is especially demanding for MR spine

Automated generation of CPRs from MR images of the spine 2869

images in which bone structures, needed to define c(n) and ϕ(n), are difficult to extract. Themajor contribution of this paper is a novel solution to this challenging problem, which isaddressed in the following subsections.

3. Methods

In order to automatically detect the centres of vertebral bodies and intervertebral discs, whichwill serve to define the 3D spine curve c(n) and axial vertebral rotation ϕ(n), we exploitsome of the basic anatomical properties of the spine. First, when observed in axial cross-sections, the vertebral bodies and intervertebral discs are nearly circular in shape and thevertebrae are symmetric over the lines that pass through the centres of vertebral bodies (orintervertebral discs) and vertebral spinous processes. These anatomical properties are usedfor initial estimation of the centres of vertebral bodies and axial vertebral rotation in each 2Daxial cross-section of the original 3D MR spine volume. Second, assuming that the 3D spinecurve and axial vertebral rotation are smooth functions, the initial estimates are refined in 3Dby robust nonlinear regression.

3.1. Initial estimation of centres and rotations of vertebrae

In each axial cross-section Pax(z = zj ), j = 1, 2, . . . , Z, of the 3D image, an in-plane lineof symmetry yj (x)⊂ Pax(zj ) that passes through the centre of vertebral body and spinousprocess is defined. The line yj (x), which splits the corresponding vertebral body into twohalves, is obtained by maximizing the similarity function sim(·) between the two half-images(figure 2):

{γj , λj } = arg max{γj ,λj }

sim

(Pax(zj ); y < yj (x)

Pax(zj ); y > yj (x)

), (2)

where tan(90◦ − γj ) is the slope and λj is the intersection of the optimal in-plane line ofsymmetry yj (x) = tan(90◦ − γj )(x − λj ) with the axis x.

Next, the centre of the vertebral body is searched for along the obtained optimal in-planeline of symmetry yj (x) by an operator, sensitive to the circular structure of the vertebral bodyin the axial cross-section. For a circular structure, a certain intensity variation along any radialdirection is always present, while the intensity variation in the direction perpendicular tothe radial direction should be relatively small. To estimate these properties of a circularstructure, the proposed operator is made of concentric rings. Intensity variation in thedirection perpendicular to the radial direction is estimated by the sum of entropies of pixelintensities in individual concentric rings. On the other hand, to estimate the intensity variationalong radial directions and to penalize the homogeneous regions, the entropy of the entireoperator is also computed. The operator �, which consists of M concentric rings of radiirm, rm < rm+1,m = 0, 1, . . . ,M − 1, is defined as

� =∑M−1

m=0 wmHm

H∑M−1

m=0 wm

; wm = e− 12 (

mM

·S)2

, (3)

where Hm,Hm = −∑Qq=1 pq,m log pq,m, is the entropy defined by the probability distribution

pq,m of intensities in the mth ring; H,H = −∑Qq=1 pq log pq , is the entropy defined by the

probability distribution pq of intensities within the entire operator and Q is the number ofbins used for probability estimation. The ring weights wm are chosen to be within S standarddeviations of the Gaussian distribution (3), so that the inner rings have a relatively largerimpact on the operator response than the outer ones. The centre (xj , yj (xj )) = (xj , yj ) of the

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(b)(a) (c)

Figure 2. Detection of the centres of vertebral bodies in cross-sections of the cervical (a), thoracic(b) and lumbar (c) spinal regions. The response of the operator � has a (local) minimum in thecentre of the vertebral body.

vertebral body is found by minimizing the response of the entropy-based operator � along theline of symmetry yj (x) in plane Pax(zj ):

xj = arg minx

(Pax(zj )|�(x, yj (x), zj )). (4)

Initial estimates of the centres of vertebral bodies {c} = {cj = (xj , yj , zj ); j =1, 2, . . . , Z} and axial vertebral rotations {γ } = {γj ; j = 1, 2, . . . , Z} along the spine areobtained by applying the above procedure (2), (4) to all axial cross-sections Pax(z = zj ), j =1, 2, . . . , Z, of the original 3D spine image.

3.2. Robust refinement of centres and rotations of vertebrae

The initial estimates of the centres of vertebral bodies and axial vertebral rotations, obtainedby the procedure above, are refined in 3D by robust nonlinear regression. For this purpose,we introduce the continuous 3D spine curve c(n) = (x(n), y(n), z(n)) and axial vertebralrotation ϕ(n) in the form of polynomial functions

{x, y, z, ϕ}(n) =K{x,y,z,ϕ}∑

k=0

b{x,y,z,ϕ},knk

bk

, (5)

where Kx,Ky,Kz and Kϕ are the degrees and bx = {bx,k}, by = {by,k}, bz = {bz,k} andbϕ = {bϕ,k} are the parameters of polynomials x(n), y(n), z(n) and ϕ(n), respectively. Theparameters are normalized over the spine domain �n:

bk =∫

�n

|nk| dn, (6)

so that the modification of each parameter has the same impact on the absolute variation ofthe corresponding polynomial term. For the purpose of implementation, the continuous spine

Automated generation of CPRs from MR images of the spine 2871

domain �n is discretized into N samples, {�n} = {ni; i = 1, 2, . . . , N}. The discretizationyields a discrete 3D spine curve c(ni) = (x(ni), y(ni), z(ni)), i = 1, 2, . . . , N , and a discreteaxial vertebral rotation ϕ(ni), i = 1, 2, . . . , N .

The parametric form of the 3D spine curve c(n), determined by the 3D spine curveparameters bc = bx ∪ by ∪ bz, is obtained by fitting a polynomial curve (5) to the set {c} ofthe centres of vertebral bodies in the discrete spine domain {�n}. Nonlinear least trimmedsquares (LTS) regression method (Rousseeuw and Leroy 2003) is used to determine the optimalparameters bc:

bc = arg minbc

hc∑i=1

r2c,[i](bc), (7)

where r2c,[i](bc), r

2c,[1](bc) � r2

c,[2](bc) � · · · � r2c,[N](bc), represent the ordered squared

residuals r2c,i (bc) = (ci − c(ni))

2 and hc is the trimming constant that satisfies the condition0.5 < hc

N� 1 and determines the number of ordered residuals that are used in the computation.

The initial estimates of rotations {γ } = {γj ; j = 1, 2, . . . , Z} of the vertebral spinousprocesses, which are obtained from the lines of symmetry yj (x), serve to initialize thecomputation of the axial vertebral rotations {ϕ} = {ϕj ; j = 1, 2, . . . , Z}. The rotations{ϕ} are recomputed in planes P⊥(n) orthogonal to the 3D spine curve c(n) (figure 1). Theoptimal parameters bϕ that determine the parametric form of the axial vertebral rotation ϕ(n)

(5) are obtained by applying the nonlinear LTS regression method to the set {ϕ} in the discretespine domain {�n}:

bϕ = arg minbϕ

hϕ∑i=1

r2ϕ,[i](bϕ). (8)

Similarly as for the 3D spine curve (7), the trimming constant hϕ, 0.5 <hϕ

N� 1,

determines the number of ordered squared residuals r2ϕ,[i](bϕ), r2

ϕ,i(bϕ) = (ϕi − ϕ(ni))2, that

are used in the computation.

3.3. Curved planar reformation

Curved surfaces are obtained from original MR images of the spine by following the computed3D spine curve c(n) and axial vertebral rotation ϕ(n), which determine the transformation T

from image-based to spine-based coordinate system (1). The curved transformation does notpreserve distances. However, by applying the inverse transformation T

−1, distances can bemeasured in the image-based coordinate system. By folding the obtained curved surfaces ontoa plane, CPR images of the spine are generated. Similarly as the axial, sagittal and coronalcross-sections determine the position of the anatomy relative to imaging planes in the image-based coordinate system (x, y, z), cross-sections that determine the position of the anatomyrelative to the spine can be determined in the spine-based coordinate system (u, v,w) (figure 1).A spine-based axial, sagittal and coronal cross-section is any plane which is parallel to theaxes u and v, v and w, and u and w of the spine-based coordinate system, respectively.

4. Experiments and results

4.1. MR spine images

The proposed method was tested on 21 axial MR scans of the spine from 12 patients, acquiredby a spine array coil. T1-weighted (average repetition time TR = 550 ms, average echo time

2872 T Vrtovec et al

TE = 15 ms) and T2-weighted (average repetition time TR = 4560 ms, average echo timeTE = 110 ms) images of the lumbar and thoracic spinal regions from nine patients wereobtained by a 1.5 T (tesla) MR scanner (General Electric Signa Excite). The matrix sizewas X × Y = 512 × 512 pixels, the average voxel size Sx × Sy = 0.398 × 0.398 mm2

(field of view FOV = 200 × 200 mm2), the cross-section thickness Sz = 3–6 mm and thenumber of axial cross-sections Z = 23–31. In addition, three whole-length T2-weighted spineimages were acquired (Sz = 3 mm, Z = 208–230, one on a 1.5 T and two on a 3 T GESigna Excite scanner) using the same imaging protocol in order to test the performance of thealgorithm on the whole course of the spine. The whole-length spine images were acquired inthree separate acquisitions and joined into one image by using the information in the headerof the DICOM files. Landmarks, manually placed in 3D at the centres of vertebral bodiesand corresponding tips of vertebral spinous processes in each original MR image (i.e. in theimage-based coordinate system), served as ground truth data. Ground truth data were used toquantitatively evaluate the performance of the proposed automated CPR method.

4.2. Implementation details

User interaction, required by the automated CPR method, was limited to pinpointing theapproximate centre of the vertebral body in only one axial cross-section of each MR spineimage in either cervical, thoracic or lumbar region. The centre of the vertebral body servedfor initializing the search of the sets {c} (2) and {ϕ} (4), while the radius of the entropy-basedoperator � (3) was automatically adjusted from M = 15 rings in the cervical region, toM = 20 rings in the thoracic region, and to M = 30 rings in the lumbar region of the spine(the ring size was �m = 1 mm

ring , the weights wm were within S = 2 standard deviations ofthe Gaussian distribution, the probability distributions were computed using Q = 16 bins).Standard mutual information (Weisstein 1999) was used as the similarity measure (functionsim(·) in (2)) and the simplex method in multidimensions (Press et al 2002) was used for theoptimization procedure (2), (4). The number of samples N in the discrete spine domain {�n}was set to the number of axial cross-sections Z, N = Z. For describing the 3D spine curvesand axial vertebral rotations (5), polynomials of degree K{x,y,ϕ} = 4 were used in images ofspine segments and polynomials of degree K{x,y,ϕ} = 6 in whole-length spine images. Thedegree Kz of the polynomials z(n) (5) was fixed to 1. The trimming constants of the nonlinearLTS regression method (Rousseeuw and Leroy 2003) were set to hc = hϕ = 2N

3 (7), (8). Thesummary of parameter values used in the experiments is presented in table 1.

4.3. Results

The whole course of the spine cannot be observed in the image-based coordinate system, asthe spine intersects with sagittal and coronal cross-sections. This shortcoming is overcome byCPR images, which allow the inspection of the whole course of the spine in single spine-basedcross-sections (figure 1). By following the course of the computed 3D spine curve c(n) andthe course of the computed axial vertebral rotation ϕ(n) in the spine domain �n (5), and byfolding the obtained curved surfaces onto a plane, CPR images were successfully generatedfrom images of spine segments (figure 3) and from whole-length spine images (figure 4).

The courses of the sagittal spine curve x(n) and coronal spine curve y(n) are shown infigure 5. Since the images used in this study represented normal spine anatomy, the coronalspine curves show the natural coronal curvatures of the spine, while the sagittal spine curvesreflect the relatively small natural sagittal curvatures. Such results were expected and alsoconfirmed by the close match to the ground truth data. On the other hand, the courses of

Automated generation of CPRs from MR images of the spine 2873

(a) (b)

Figure 3. CPR T1-weighted images (top row) and a T2-weighted (bottom row) images of a spinesegment, created by folding curved surfaces (a) onto a plane (b).

Table 1. Summary of parameter values used in the experiments.

Entropy-based operator �

Number of rings M = 15–20–30a

Ring size �m = 1 mmring

Standard deviation of weights S = 2Number of bins Q = 16

Spine domain �n

Number of samples N = Z

3D spine curve c(n)

Polynomial degree K{x,y} = 4–6b, Kz = 1LTS trimming constant hc = 2N

3

Axial vertebral rotation ϕ(n)

Polynomial degree Kϕ = 4–6b

LTS trimming constant hϕ = 2N3

a Cervical region–thoracic region–lumbar region.b Spine segment–whole-length spine.

axial vertebral rotation ϕ(n) (figure 5) indicate that there may be no common course of axialvertebral rotation in normal spines but only a slight natural fluctuation that is specific for theobserved spine anatomy.

Since axial vertebral rotation is computed in planes orthogonal to the 3D spine curve,the estimation of the axial rotation parameters depends on the previous estimation of the3D spine curve parameters. This makes the procedure inconsistent with the Serret–Frenetreference frame in differential geometry (Weisstein 1999), i.e. the axis v of the spine-basedcoordinate system does not equal the Frenet normal vector, but is oriented in the direction ofthe vertebral spinous processes. However, such an approach assures that besides geometricalalso the anatomical properties of the spine are incorporated in the resulting CPR images. Thecurvature, which is defined in differential geometry as an intrinsic property of the curve, cannevertheless be observed. For a parameterized curve c(n) it is defined in 3D as (the sign ×

2874 T Vrtovec et al

(a) (b) (c)

Figure 4. The sagittal and coronal CPR images of the three whole-length images of the spine((a)–(c), respectively) obtained by folding the sagittal and coronal curved surfaces (top row) ontoa plane (bottom row).

denotes vector cross product)

κ(n) =∣∣ dc(n)

dn× d2c(n)

dn2

∣∣∣∣ dc(n)

dn

∣∣3 . (9)

The spinal curvature κ(n) at an arbitrary position n on the spine has a magnitude equal tothe reciprocal value of the radius of an osculating circle to the 3D spine curve at that position.As a result, the 3D spine curve can be presented by a one-dimensional curve that capturesthe characteristics of the spine in 3D. The computed courses of the spinal curvature for thewhole-length spine images are shown in figure 5.

Ground truth data were used to quantitatively evaluate the performance of the proposedautomated CPR method. The 3D spine curve error was estimated as the Euclidean distancebetween the computed 3D spine curve and the ground truth landmarks. The axial vertebralrotation error was estimated as the difference between the computed values and the rotationsfrom ground truth data, obtained as the angles between the lines through the centres of vertebral

Automated generation of CPRs from MR images of the spine 2875

(a) (b) (c) (d)

Figure 5. Course of the sagittal spine curve x(n) (a), coronal spine curve y(n) (b), axial vertebralrotation ϕ(n) (c) and spinal curvature κ(n) (d) of the three (top to bottom row, respectively)whole-length images of the spine. The vertical lines represent the position of the ground truthlandmarks.

Table 2. Quantitative comparison between the computed 3D spine curve and axial vertebralrotation and ground truth data. The results are shown separately for T1- and T2-weighted images.Overall mean values and the corresponding standard deviations are shown in the bottom rows.

Spine segments Whole spines

Images 1 2 3 4 5 6 7 8 9 10 11a 12a Mean Std.

Mean 3D spine curve error �c(n) (mm)T1-weighted 2.7 4.1 1.0 1.5 4.3 3.7 2.7 2.8 1.9 – – – 2.8 1.1T2-weighted 3.2 3.9 1.2 1.1 1.7 1.7 1.4 1.5 2.0 2.6 3.6 3.7 2.3 1.0

Mean axial vertebral rotation error �ϕ(n) (◦)T1-weighted 3.1 1.3 1.5 0.9 2.6 0.8 0.9 2.2 1.6 – – – 1.7 0.8T2-weighted 3.9 2.5 0.9 1.3 2.6 1.7 0.8 2.0 0.9 1.4 2.1 1.2 1.8 0.9

3D spine curve error �c(n)� (mm) 2.5 1.1Axial vertebral rotation error �ϕ(n)� (◦) 1.7 0.9

a Images acquired on a 3T MR scanner.

bodies and the corresponding tips of vertebral spinous processes, and the sagittal referenceplane. The results are presented in table 2 and indicate that the method performed well bothfor T1- and T2-weighted images. For all images used in this study, the mean 3D spine curveerror was �c(n)� = 2.5 mm (standard deviation σ = 1.1 mm) and the mean axial vertebralrotation error was �ϕ(n)� = 1.7◦ (standard deviation σ = 0.9◦).

5. Discussion and conclusion

The proposed automated method for generating CPR images from MR images of the spineallows the visualization and inspection of images in the coordinate system of the spine. Mostof the existing reformation techniques for 3D spine images require manual determination of the

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spinal curvature and/or vertebral rotations and are usually determined in 2D. With minimaluser interaction (i.e. by identifying the centre of only one vertebral body), the proposedmethod automatically extracts the 3D spine curve and axial vertebral rotation, which determinethe transformation from image-based to spine-based coordinate system (1). The measuredquantities are described continuously along the whole spinal length in a low-parametric formby polynomials of degree 4–6. The proposed method was qualitatively and quantitativelyevaluated on 18 MR images of spine segments (nine T1- and nine T2-weighted) and threeT2-weighted whole-length MR images of normal spines. The computed 3D spine curvesand axial vertebral rotations are consistent with the manually defined ground truth data andcomparable to the results obtained on CT images (Vrtovec et al 2005).

Quantitative assessment of spinal curvature and axial vertebral rotation is important forsurgical planning, analysis of surgical results and monitoring of the progression of spinaldeformities (Aronsson et al 1996, Birchall et al 1997). Describing spinal curvatures withmathematical functions is not a new concept, and functions such as sinusoids (Drerup andHierHolzer 1996), splines (Kaminsky et al 2004, Verdonck et al 1998) and polynomials(Vrtovec et al 2005, Peng et al 2005) were used for that purpose. For measuring the axialvertebral rotation, CT is the most accurate imaging modality (Krismer et al 1996, Kuklo et al2005). Different techniques for CT were developed (Aaro and Dahlborn 1981, Ho et al1993, Gocen et al 1999) and later improved by taking into account sagittal and coronalinclinations of vertebrae (Skalli et al 1995, Krismer et al 1996, Hecquet et al 1998). Similartechniques were also applied to MR images of the spine. Birchall et al (1997) computedthe axial vertebral rotation from the position of landmarks that were manually placed in eachoblique axial cross-section, defined through superior and inferior end-plates of each vertebrain the acquired axial MR images. In order to assess intervertebral rotations, Rogers et al(2002) registered pairs of axial cross-sections that captured most of the transverse, spinousand superior articular processes of each vertebra. However, the initial centre of rotation wasmanually defined at the dorsal edge of the vertebral body on each cross-section.

The spinal curvature κ(n), which is relative to the osculating circle to the 3D spine curve,and the axial vertebral rotation ϕ(n), which is the rotation of vertebrae around the 3D spinecurve, are inherent properties of the spine (i.e. spine-specific) and therefore not affected byrigid body transformations. The generated CPR images are independent of the position of thepatient in the MR scanner and of the orientation of the imaging planes. Moreover, healthyanatomy is represented in the same coordinate system as abnormal anatomy, which may allowa more objective evaluation and diagnosis of the abnormalities. Displaying the whole courseof the spinal column within a single 2D image may be of significant importance in the caseof increased coronal (i.e. scoliosis) or sagittal (i.e. kyphosis, lordosis) spinal curvature. Thevertebral rotation is determined by taking into consideration sagittal and coronal inclinationsof the vertebrae, which are known to significantly influence the measurements. However, thevertebral rotation cannot be always uniquely defined, as it is affected by the natural bending ofspinous and transverse vertebral processes. The same observation can be made for vertebralrotations, obtained by manually defined landmarks. The major limitation of the presentedmethod is that the determination of axial vertebral rotation depends on prior estimation of the3D spine curve, as the rotation is measured in planes that are centred in the computed spinecurve samples.

Among the most significant parameters that may assist an orthopaedic surgeon inevaluating spinal deformities, is the length of the spinal axis, the Cobb angle, the locationsof the centres of vertebral bodies and vertebral rotation angles, i.e. axial rotation, sagittal andcoronal inclinations (Stokes 1994). Besides direct automated localization of the centres ofvertebral bodies, i.e. the 3D spine curve, and measurement of axial vertebral rotation (5), the

Automated generation of CPRs from MR images of the spine 2877

proposed method implicitly allows automated measurement of the remaining parameters. Thelength of the spinal axis can be computed from the arc length, which is a geometrical propertyof the polynomial function that represents the 3D spine curve. In the case of a scoliotic spinaldeformity, the location of the end vertebrae (i.e. the vertebrae with maximal slant towards theconcavity of the curve above and below its apex) could be extracted from the course ofcurvature (9) and axial vertebral rotation, allowing the measurement of the Cobb angle(Cassar-Pullicino and Eisenstein 2002). The sagittal and coronal vertebral inclinations can beassociated with the inclination of the planes, orthogonal to the 3D spine curve. Moreover, theidentification of the superior and inferior end-plate on individual vertebral bodies would allowthe determination of inter-segmental (i.e. between neighbouring vertebrae) and intra-segmental(i.e. within individual vertebrae) rotation angles (Birchall et al 1997). These are of significantimportance, since only the inter-segmental rotations can be corrected by surgical intervention(Cassar-Pullicino and Eisenstein 2002). Automated measurement of the significant spine-specific parameters may therefore provide a complete quantitative representation of the spinein 3D, which may improve preoperative planning and postoperative evaluation.

The main purpose of the proposed automated CPR method is, however, to reduce thestructural complexity in favour of an improved feature perception of the spine and to provideclinically relevant quantitative analysis of the 3D spine anatomy. However, the knowledgeof the location and orientation of the spine in 3D can be exploited by other image analysistechniques and applied in a clinical environment, e.g. for the identification and measurementof dimensions of the spinal canal and the spinal cord. The notion of the spine-based coordinatesystem is modality independent and can therefore be used for data fusion, i.e. merging of CTand MR images of the same patient. We will focus our future research on some of these topics.

Acknowledgments

This work has been supported by the Slovenian Ministry of Higher Education, Science andTechnology under grant P2-0232 and by a six-month fellowship from the CommonwealthScientific and Industrial Research Organisation (CSIRO), Australia.

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