Topology-based orientation analysis of trabecular bone networksBryon R. Gomberg, Punam K. Saha, and Felix W. Wehrlia)

Laboratory for Structural NMR Imaging, Department of Radiology, University of Pennsylvania,Philadelphia, Pennsylvania 19104

�Received 30 August 2002; accepted for publication 2 October 2002�

After bone mineral density, orientation is the major determinant of trabecular bone strength and isthus of significant interest in understanding the clinical implications of osteoporotic bone loss. Themethods used to measure orientation and anisotropy of the trabecular bone have largely relied onderiving global measures along test lines, computing the best-fit ellipsoid, and decomposing toeigenvalue–eigenvector pairs that yield the mean orientation and anisotropy of the region. Thesetechniques ignore the differences between measuring the orientation of trabecular plates versusrods, and do not provide insight into the relationship between local orientation and biomechanicalstresses. Digital topological analysis allows a unique determination of each voxel’s topologicalclass as belonging to a plate, rod, or junction. The digital topology-based orientation analysis�DTA-O� method extracts the voxels belonging to plates and determines the local surface normal byfitting a plane through the local neighborhood BVF map. Modeling regional distributions of thesevectors allows assessment of anisotropy measures, such as mean and variance of the orientationdistribution. High-resolution microcomputed tomography, synthetic, and in vivo images were usedfor a validation of the new method and compare the results with the mean intercept length �MIL�technique. The results indicate that DTA-O is a better measure of trabecular orientation and anisot-ropy than MIL. Applying DTA-O to a recently completed study on the distal radius of 82 subjects�F.W. Wehrli et al., J. Bone Min. Res. 16, 1520 �2001�� shows that the mean orientation andanisotropy at the medial and lateral sides in the distal radius mataphyseal trabecular network areconsistent with the mechanical stresses acting on the radius during common tasks. © 2003 Ameri-can Association of Physicists in Medicine.�DOI: 10.1118/1.1527038�

Key words: structural analysis, trabecular bone structure, trabecular bone characterization,osteoporosis, three-dimensional imaging

I. INTRODUCTION

Osteoporosis is a disease characterized by bone loss and ar-chitectural deterioration1 manifesting in fragility fractures,predominantly of the hip, wrist, and spine—sites that are richin trabecular bone. The diagnosis of osteoporosis is currentlybased on dual-energy x-ray absorptiometry �DXA� arealbone mineral density �BMD� T-scores2,3 and the modality’slow cost and simplicity have rendered DXA the most com-mon screening tool for osteoporosis.3 Yet BMD explainsonly about 60% of the variation in bone strength, with theremainder attributed to architectural changes of the trabecu-lar bone.4 At the other extreme the invasive bone biopsy ofthe iliac crest can determine the microscopic structural ar-rangement of trabecular bone and is used as the gold stan-dard for a pathological diagnosis of metabolic bone disease.Magnetic resonance microscopic imaging ��MRI� of in vivotrabecular bone networks has emerged as a promising alter-native to these two techniques5–7 and is the input modalityfor the method described here.

Wolff’s law8 conveys that trabecular bone responds tomechanical stimuli, leading to the alignment of the trabecu-las with the principal stress directions, explaining the me-chanical anisotropy observed in most bones. As a result, me-chanical anisotropy has been extensively studied by ananalysis of the structural anisotropy, commonly assessed by

the three-dimensional �3-D� mean intercept length-derived�MIL� fabric tensor.9 Odgaard et al.10 found that fabric andcompliance matrix tensors coincide. Zysset et al.11 foundthat the fabric tensor is aligned with the general elastic ten-sor. Others have investigated the structural orientation of tra-becular networks in relation to biomechanical competence,and found the orientation of the trabecular bone network tobe the second largest contributor, after BMD, to bonestrength.12–14

Whitehouse15 first showed that the MIL of trabecular net-works measured along test lines drawn in different direc-tions, has the shape of an ellipse when plotted as polar dia-grams. This finding makes it possible to express thedeparture from isotropy as the ratio of the axes of the ellipse.The technique has later been extended to three dimensions9

by fitting an ellipsoid to MIL measures, and the principalaxes of this ellipsoid describes the trabecular network fabricorientation and anisotropy. While the MIL-derived orienta-tion measurements are commonly used to assess fabric, othermethods for orientation and anisotropy estimation have beenreported as well. Among these are the Sobel operator,5 starlength distribution,16 line fraction deviation methods,17 andautocorrelation analysis.18 Besides the Sobel operator, all ofthese methods rely on statistical averaging requiring by defi-nition sufficient regions of interest for accurate analysis.These same methods also combine the measurements using

1 1Med. Phys. 30 „2…, February 2003 0094-2405Õ2003Õ30„2…Õ1Õ11Õ$20.00 © 2003 Am. Assoc. Phys. Med.

ellipsoid fitting for fabric tensor determination. Finally, allprevious methods for orientation assessment have ignoredthe differences between the orientation of trabecular platesand rods.

The strength of trabecular bone networks has been attrib-uted to three major property categories: material, scale, andtopology.19 Material properties refer to the intrinsic proper-ties of the trabecular bone material, such as material strengthand toughness. Scale refers to such properties as density,bone volume fraction �BVF�, trabecular thickness, and meantrabecular plate separation. Topology refers to the propertiesof the bone network that are invariant under elastic deforma-tion and describe the organization of the bone material, suchas whether a particular trabecular network location belongsto a plate-like or rod-like structure. An analysis of trabecularorientation falls under the realm of network topology and hasbeen studied extensively �see Ref. 4 and references citedtherein�. The approach of the present method is to topologi-cally isolate the trabecular plate locations from the skeleton-ized bone network, and analyze surface orientations locallyon a voxel-by-voxel basis. A plate is any two-dimensional�2-D� structure of finite thickness and it can be identified bymeans of digital topological analysis.20 The resulting collec-tions of orientation vectors are aggregated, analyzed, andcompared with conventional methods, testing the hypothesisthat local topology-based measurements more closely reflectchanges in orientation than global MIL statistics.

II. METHODS

A. Digital topological analysis

Central to the method is the classification of each voxel inthe trabecular network by digital topological analysis �DTA�,a method that uniquely defines the topological type of eachnetwork voxel. This method has been previously described21

and thus only a brief summary is given here. The DTA inputis a 3-D BVF image map of the trabecular network, in whicheach voxel signal intensity represents the fraction in percentoccupied by bone.22,23 BVF maps can be derived from boneimages acquired ex vivo by micro computed tomography��CT�12,24 or nuclear magnetic resonance microscopy.25 Invivo, trabecular bone images have been obtained by periph-eral computed tomography26 and �MRI.5–7

The �MRI images, typically having resolution on the or-der of the structure size, must be suitably preprocessed toyield BVF probability maps.22,23 The BVF maps are thenbinarized using a connectivity-preserving threshold20 andskeletonized.27 The result is a surface skeleton representationof the trabecular network that contains only one- and two-dimensional elements �surfaces and curves�. Digital topo-logical classification28 determines the topological class ofeach voxel, such as surface interior �S-type�, surface edge�SE-type�, surface–surface junction, curve interior, curveedge, surface–curve junction, etc. The digital topology-basedorientation analysis method �henceforth referred to asDTA-O� extracts voxels that belong to surface interiors oredges, calculates the local surface normal of each voxel, ana-lyzes the network orientation, and displays the results.

B. Extracting measurement locations

Deciding at which locations to measure local orientationwas important for obtaining accurate representations fromthe trabecular network. The locations were evenly distributedalong the surfaces, and not biased by trabecular thickness.Using the DTA voxel classification maps, the surface nor-mals were computed at all locations identified as belongingto a surface interior �S-type� or surface edge �SE-type�, afterremoval of junctions and their immediate neighborhoods toavoid mixing the surface and junction orientations. Sincethese maps have previously been skeletonized, the influenceof trabecular thickness was assumed to be negligible.

C. Local surface normal measurement

The local surface normals at each location were obtainedby fitting a plane through the neighboring surface voxelsfound in the BVF map. Plane orientations defined in spheri-cal coordinates were related to the surface normal vectorcoefficients a, b, and c using a�sin(�)cos(�), b�sin(�)sin(�), and c�cos(�). Here � is the angle from thez axis �typically aligned with the anatomic axis—Fig. 1� tothe surface normal �polar angle� and � is the angle from thex axis to the surface normal vector’s x-y plane projection�azimuthal angle�. The azimuthal orientation distributionswere computed as the sum of orientation vectors, weightedby each vector’s transverse component. In this manner thecontributions of surface normals that have a more transverseorientation are increased and contribution of surface normalsthat are oriented near the poles are lowered. The image orlaboratory coordinate system z axis was chosen on the basisof anatomical landmarks, such as parallel to the shaft of longbones, and the x and y axis chosen usually on the basis ofpatient position comfort. Thus, the z axis of the image coor-dinate system �direction perpendicular to the axial imageplane� and tissue fabric axis were not necessarily aligned,though the degree of misalignment should, in practice, besmall �Fig. 1�.

The formula for a plane passing through the voxel loca-tion �taken as the origin� was expressed by ax�by�cz

FIG. 1. Projection image of 9 mm trabecular bone plug portion with ananatomical axis slightly askew of the image coordinate frame (x,y,z).When the angle between the anatomical axis and coordinate system z-axis issmall, the azimuthal orientations in the image coordinate system can be usedto represent the orientations transverse to an anatomical axis.

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�0, where the surface normal to this plane is �a,b,c� andthe x, y, and z distances were referenced to the central voxel,p. For the rest of this paper, a, b, and c are considered nor-malized to a unit length vector. To find the best-fitting planethrough voxel p, a cost function is used that takes the sum ofweighted squared distances from all neighboring BVF vox-els, computed as

cost�p�� �i �N�p�

wi�axi�byi�czi �2. �1�

In other words, the cost of fitting the plane represented by�a,b,c� to the voxel location p is the sum across all voxels i,belonging to neighborhood N(p) of the weighted squareddistance from each point �xi ,yi ,zi � to the plane. The BVFvalues are used as weights, wi , thus allowing voxel locationswith more bone to give more weight to the cost function.N(p) is the n3, 3-D neighborhood around p, where n is anodd integer �3, 5, 7, 9, etc.�, and for this study all analyseswere performed using n�5 (53 neighborhood�, but studiesat higher resolutions might want to use a larger region. Thebone voxel locations used in the fitting are determined byfirst removing junctions from the skeleton representation ofN(p), followed by region growing the surface structurestarting from p. This operation enables isolating the voxels ofthe specific trabecula to which the point p belongs, withoutincluding voxels from adjacent trabeculas that might be inthe same neighborhood. The resulting set of locations wasthen morphologically dilated using a 33 kernel to locate allBVF voxels belonging to the surface structure being mea-sured in the neighborhood of p. This approach rests on theassumption that the skeleton locations are within this ker-nel’s shape of the trabecula’s midline. The plane that mini-mizes the cost function is the estimated local surface orien-tation at this network point.

Rearranging Eq. �1� allows an efficient computation of thecost function:

cost�p��As•D

As��a2,b2,c2,2ab,2ac,2bc� ,

D�� �i

wixi2,�

iwiyi

2,�i

wizi2,�

iwixiyi ,

�i

wixizi ,�i

wiyizi � .

Here the cost function has been separated into two six-element vectors that represent the plane’s orientation, As ,and the weighted data, D . In order to efficiently computeeach surface voxel’s orientation from the BVF data As isprecomputed for all possible orientations. The minimum costorientation is then determined using the Newton–Raphsonsearch technique.29

D. Anisotropy analysis

As opposed to the mechanical and structural anisotropies,the anisotropy of local trabecular orientation distributions

was assessed using a model-based approach. Such distribu-tions may be modeled in many ways, but following others,5 aGaussian model was chosen here. The Gaussian model canbe used for both one-dimensional �1-D�, or 2-D distributionsin regions that have monomodal histograms, hypothesized tobe produced by monoaxially stresses, such as those acting onlong bones. After determining the transverse component ofsurface normal orientations, DTA-O anisotropy was deter-mined by fitting a Gaussian distribution. The best-fitting 1-Dand 2-D Gaussian distributions were computed by theGAUSSFIT and GAUSS2DFIT functions in IDL version 5.4 �Re-search Systems, Inc., Boulder, CO�. For 1-D data, the meanorientation was evaluated as the Gaussian mean �peak loca-tion� and the anisotropy by the Gaussian height, width, andbaseline height. For 2-D orientation distribution data, theangle of orientation of the 2-D distribution is used as a mea-sure of mean orientation.

E. Display strategies

Among the possible ways of displaying 3-D vector fieldsrepresenting the orientation of trabecular bone, 1-D or 2-Dhistograms, orientation distribution cartographic maps, andregional anisotropy maps are particularly useful. One-dimensional histograms can be displayed by separating thesurface normals to polar and azimuthal histogram distribu-tions. These have the advantage of the easy visual identifi-cation of mean orientations and anisotropy, while sacrificingthe 3-D aspect of the orientation vector distributions. Theresults for a typical in vivo histogram are shown in Fig. 2�a�.Projecting the 2-D distributions of surface normal orienta-tions onto a plane has been addressed extensively by cartog-raphers, for example, by means of the Hammer–Aitoff equalarea projection method30 used in Fig. 2�b�. In these orienta-tion distribution cartographic maps the horizontal axis repre-sents the azimuthal angle and the vertical axis the polarangle. This method of viewing the orientation data gives avisual indication of the tilt between the anatomical and im-age coordinate systems and shows clusters of predominanttrabecular orientations. Figure 2�c� shows a rose plot for theazimuthal orientation distributions, but the bimodal shapebell-shaped plots are not as obvious. This visual inspectionof the orientation data is important in making decisions abouthow to anatomically segment the different biomechanicalsupport regions for a more detailed analysis, such as in howto divide the distal radius into medial and lateral sides.

As can be seen from Fig. 2, the distribution of trabecularbone orientations over a region may be nonuniform or mul-timodal, having more than one principal direction. It wasfound that when such multimodal regions are made smaller,monomodal distributions appear, indicating a more uniformregion. A regional anisotropy mapping technique was devel-oped to visualize the monomodal components of local trans-verse anisotropies across a cross-sectional 3-D virtual bonebiopsy �VBB� dataset. The surface normal vector locationswere projected onto the x-y plane projection and dividedinto smaller subregions by uniformly dividing the x directioninto n equal columns and then along y within each column

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into m equal count boxes. The number and shape of subre-gions generated, n�m, was modified to match the transverseanatomical shape. The subregion boundaries were displayedas a series of grid lines on the middle anatomical image

to visually assess the surface density within each region.Each subregion’s 1-D azimuthal surface normal orienta-

tion histogram was analyzed by phase shifting the distribu-tion’s minimum to the first value and then padding eitherside for another length before performing the Gaussian fit.This helped isolate and center the peak in the field of view.The resulting values were displayed as vectors overlaid ontothe central axial image. To qualitatively assess the alignmentof surface normal vector orientations along the trabecularbone, the vectors were rotated 90° to lie parallel to the bonestructures and the vector length was set proportional to eachsubregion’s anisotropy.

F. Mean intercept length analysis

Mean intercept length �MIL�15 in three dimensions wascomputed by first extracting intensity profiles along test linesthrough the region of interest, followed by locating bone tomarrow interfaces along each profile. The voxel locations ofone profile were determined first, and then translated to getthe series of parallel profiles. The histogram of interceptlengths was calculated and the histogram average used as theMIL along each test line. An ellipsoid was then fit to thehemispheric MIL data by rearranging the (x,y,z) data intosecond-order polynomials, and solving for z2 using multipleregression. In a method similar to Chung et al.,25 the result-ing coefficients were used to derive eigenvalues and eigen-vectors. The MIL-derived anisotropy was expressed as theratio of major to minor axes of the MIL ellipsoid fit, asdetermined by the maximum and minimum ellipsoid eigen-values.

G. Experiments

The methods described above were applied in a series ofexperiments designed to investigate the response of theDTA-based method to changes in orientation and anisotropy.The experiments included tests of sensitivity to polar rota-tion, azimuthal rotation, and tests of trabecular plate spacingand varying degrees of anisotropy. These experiments wereperformed using high-resolution �CT images from trabecu-lar bone specimens resampled to resolutions mimicking invivo acquisition, in addition to synthetically generated data.MIL analysis was also performed on the validation data tocompare the accuracy, precision, and performance. Finally,DTA-O was performed on in vivo images for a group of 82subjects on the medial and lateral sides of the distal radius,comparing the anisotropy and orientation of these two re-gions and showing a sample application of the method.

1. Polar and azimuthal rotation experiments—Goldstandard images (�-CT)

The accuracy in the measured orientation was assessedwith the aid of �CT acquired at 23�23�23 �m3 voxel size�Scanco Medical AG, Bassersdorf, Switzerland� from a 9mm diameter and 9 mm long cylindrical core of distal radiusmetaphyseal trabecular bone from a 78 year-old male ca-daver. To simulate rotation of the object in the image, eachx-y or y-z plane was numerically rotated within the image

FIG. 2. Typical in vivo surface orientation data from the distal radius dis-played in three formats: �a� separate polar and azimuthal angle histograms.Notice the bimodal distribution of the transverse orientations; �b� 2-D ori-entation histogram displayed in Hammer–Aitoff cartographic format; �c�azimuthal and polar ‘‘rose’’ plot. The bimodal distribution of the 2-D orien-tations is clearly seen at around 40° and 145°, except in the difficult tointerpret rose plot, as well as the tilt of the polar orientations, indicating thatthe bone anatomical axis is slightly off the image z axis.

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grid, resulting in a polar or azimuthal rotation of the object.Lower resolution images were created by voxel averaging bya factor of 4 in each dimension, simulating images at near invivo resolution. The angles spanned the range from 5°–175°,rotated in increments of 5°, and DTA-O analysis as describedabove was performed on each dataset. The 2-D histogramsfor the polar rotation images were fit to a 2-D Gaussian, andthe angle of rotation for that Gaussian fit was plotted againstthe actual rotation angle. To measure the azimuthal rotationsin each image, the histogram of each rotated image wassmoothed by a five-element window, incrementally shiftedand correlated with the nonrotated dataset. The shift thatminimized this correlation is the measured rotation of thisimage, and all rotations plotted against the known rotationangles. The rotated datasets were also analyzed by MILanalysis and the measured rotation angle determined fromthe fit ellipsoid.

2. Waviness and spacing experiments—Syntheticimages

A method for creating synthetic images of trabecularbone, conceived as part of a previous study,20 was modifiedto allow for variations in plate spacing and orientations.First, grid points, regularly spaced at ten voxels on a plane ofspecified orientation were calculated. These grid points weresubsequently displaced by known distances along the surfacenormals. The displacements were chosen from a random nor-mal distribution of mean zero and a specific standard devia-tion, resulting in a ‘‘wavy’’ surface of known deviation fromplanarity. A smooth, single-voxel thick surface was generatedusing a cubic spline technique �IDL, Research Systems, Inc.,Boulder CO� and this surface was multiply replicated andtranslated to obtain a ‘‘wavy’’ plate of constant thickness.Combining multiple parallel plates at three orthogonal orien-tations into a single object resulted in a closed-cell, trabecu-lar plate model with known thickness, orientation, and vari-ability. Resampling the binary images to a lower resolutioncreated images of synthetic plates that simulate the partialvolume effect seen in trabecular bone images at in vivo reso-lutions.

To evaluate the effect of the ‘‘waviness’’ of each plate, anexperiment was performed using 11 images of wavy plates.The displacements came from normal distributions with stan-dard deviations incrementally varied from 0 through 10 vox-els. Three orthogonal sets of 1.2 voxel thick wavy plateswere spaced 5, 10, and 15 voxels apart within an image sizeof 853 voxels. In these images the Gaussian peak shouldbecome wider with increasing waviness, indicating a moreuniform distribution of orientations. These images were ana-lyzed using DTA-O and MIL methods, and the anisotropyplotted as a function of the standard deviations of the dis-placements. To test the case where DTA-O is expected to failin detecting a change in structure, a plate ‘‘spacing’’ experi-ment was designed to show the effect of varying plate sepa-ration on DTA-O and MIL anisotropy analysis. Seven syn-thetic images were created with three orthogonal platesystems, where two of the three orthogonal plate series were

spaced 10 voxels apart, and the spacing of the third seriesranged from 10 through 40 voxels in 5 voxel increments. Theresulting anisotropy measurements were plotted versus thespacing increments of the third series of plates.

3. Orientation of trabecular structure in the distalRadius—In vivo

Data from a previously reported study involving high-resolution �MRI images of the distal radius �79 women and3 men, aged 22–76 years 54�14�31 were reanalyzed usingDTA-O and MIL. Images of the right distal radius metaphy-sis had been acquired and preprocessed to produce 68�68�87 �m3 BVF maps. The structural orientation of thetrabeculas was compared between the medial and lateral sideof the radius using anatomic landmarks using each method.Toward this goal the radio-ulnar joint and Lister’s tuberclewere located on each subject, and the bisecting line betweenthese two points partitioned the distal radius into medial andlateral sides �Fig. 3�. This bisecting line also served as thereference for angle measurements. The transverse anisotropyassessed by DTA-O and MIL of the medial side of eachsubject was plotted against the corresponding values on thelateral side. All anisotropy parameters were compared withthe subject’s age, the DTA-O azimuthal histograms of allsubjects were summed for each side, and the histograms ana-lyzed for orientation and anisotropy.

H. Processing and statistics

All orientation processing, analysis, and visualizationwere done using automated programs written in IDL 5.4 �Re-search Systems, Inc., Boulder, CO�. Preprocessing and DTAwere implemented in the C programming language and in-terfaced to the automated processing system.32 The onlymanual part of the process was tracing 3-D regions of inter-est for analysis, necessary to locate the trabecular boneboundary and exclude artifacts if present in the �MRI im-

FIG. 3. Example of the bisection method for in vivo data. The left arrowindicates the radio-ulnar joint and the right one indicates Lister’s tubercle.The dashed line is the connecting line between the points that is used as thereference orientation for the transverse angles, and the solid line indicatesthe bisecting line between the points that separate the distal radius into themedial and lateral sides.

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ages. All statistics were performed with JMP-IN �SAS Insti-tute, Cary, NC� and Excel 2000 �Microsoft Corp., Redmond,WA�. Pearson correlation coefficients were computed for therotation, spacing, and waviness experiments assessed by bothDTA-O and MIL. Gaussian fitting, both one- and two-dimensional, was done using the IDL routines GAUSSFIT andGAUSS2DFIT. The Fisher z transformation was used to deter-mine if the DTA-O and MIL correlation coefficients weresignificantly different. In vivo data was also analyzed usingPearson correlation coefficients for medial versus lateral pa-rameters, and for correlations with age.

III. RESULTS

Some results of the rotation experiments are shown in Fig.4. There was a strong correlation between actual and DTA-Omeasured polar rotation angles (r 2�0.997). The correlation(r 2�0.978) with MIL-derived measurements was signifi-cantly weaker (p�0.0001), indicating higher precision ofthe derived local orientations for the new method. The slopesof the linear correlations were 1.08 for DTA-O and 0.86 forMIL, suggesting higher accuracy of the former. The azi-muthal rotations provided significantly different (p�0.0006) correlations of 0.995 and 0.975, and slopes of 1.04and 0.93 for DTA-O and MIL, respectively, again suggestinghigher accuracy and precision for DTA-O. It should be notedthat DTA-O correlation intercepts were close to zero, unlikethe polar angle MIL plots where the trend line had a y inter-cept of 28°.

In Fig. 5�a� the width of the Gaussian fit was plottedversus waviness �expressed as the standard deviation of thedisplacement from planarity� in synthetic images derivedfrom a 3-D array of mutually orthogonal plates as describedin the Methods section. The correlations yielded r 2�0.963for DTA-O and r 2�0.829 for MIL, but the significance isinconclusive due to a low number of datasets. In this experi-ment a narrower Gaussian corresponds to a higher anisot-ropy. The slope of the DTA-O trend line was 2.6, whichindicates the DTA-O method’s sensitivity to variations insurface orientations.

In Fig. 5�c� the width of the fitted Gaussian was correlatedwith plate spacing in a model of orthogonal plates of mod-erate waviness. Here, two of the three orthogonal plate series

maintained a constant spacing of ten voxels while the spac-ing for the third series of orthogonal plates was varied. In thespacing experiment, the data indicate no correlation betweenDTA-O-measured anisotropy with plate spacing, as would bepredicted since plate orientation has not changed. The fluc-tuations seen in the DTA-O anisotropy measurements of Fig.5�c� represent the analysis error. In contrast, a strong positivecorrelation was obtained between anisotropy measured bythe MIL method and plate spacing (r 2�0.933). This result,of course, is expected since MIL depends on the plate spac-ing along each test line direction. Notice that the change inMIL anisotropy with spacing and waviness are of a similarmagnitude, potentially confounding the results.

Figure 6 shows the DTA-O data derived from high-resolution �MRI images of the distal radius for several sub-jects. These examples show typical variations in the bell-shaped histograms that we have modeled as Gaussiandistributions in this article. Combined medial and lateralsides indicate a bimodal distribution, while medial and lat-eral histograms indicate different regional orientations. Smallwrist outlines indicate the localization of the radio-ulnar jointand Lister’s tubercle, as well as the resulting bisecting planeof the volume. The data of individual medial–lateral orien-tation distributions shows that despite the different condi-tions of the subjects, there are similarities in trabecular ori-entation on the regional maps. The orientation histogramsindicate alignment of the trabeculas on the medial side to-ward the radio-ulnar joint, and on the lateral side towardLister’s tubercle. Also notice the marked differences in rela-tive anisotropies in the different subregions

In our study of 82 subjects, the typical total number oforientation vectors in a trabecular bone VBB scan of thewrist is on the order of 30,000. The results of plotting thelateral versus medial anisotropy separately for all subjectsare shown in Fig. 7�a�. The slope of 0.45 (r 2�0.43) forDTA-O indicates that structural anisotropy is greater on themedial than on the lateral side. The MIL results substantiatethis, but with a significantly different correlation coefficientof r 2�0.23. Since the coordinate system is defined with thez axis along the anatomic axis, the polar angle distributiondescribes the deviation of plates from this axis and the azi-muthal angle describes the transverse distribution of trabecu-

FIG. 4. The results of the rotation experiments: �a� Azi-muth rotation analyzed by DTA-O. �b� Azimuth rotationanalyzed by MIL. A comparison of the trend line fitsbetween DTA-O and MIL analysis shows that DTA-Ois much more precise than MIL. Systematic variationsfrom linearity are possibly due to interpolation effectsin the numerical estimation of rotation. Overlaid on �a�are a few images of the �CT data rotated and resampledto model datasets of the same object acquired in vivo atvarious rotations. Shown are axial views of azimuthrotations at 45° increments increasing from left to right.

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lar orientation, which should reflect the off-axis mechanicalloads at the anatomic location analyzed. The anisotropy pa-rameter with highest correlation with age was lateral Gauss-ian height �Fig. 7�c�� at r 2�0.155 (p�0.0003), and medialGaussian height was not far behind with r 2�0.088 (p�0.007). MIL anisotropy on the other hand had a lower,though not significantly, correlation coefficient of r 2

�0.0544 (p�0.035). For both medial and lateral sides theGaussian width, baseline, and peak location were not signifi-cantly correlated with age. In Fig. 8 the summed histogramsfor medial and lateral sides in all subjects are plotted and theanisotropy parameters are given in Table I. The data of thecombined distributions shows the same orientations as mostindividuals, where the medial plates are oriented toward theradio-ulnar joint, and the lateral ones toward Lister’s tu-bercle.

IV. DISCUSSION

The results show that the DTA-O derived local orienta-tions give better precision and accuracy than measurementsbased on the MIL method. The rotation experiment indicatesthe significantly increased precision of the DTA-O methodby the results of a linear regression analysis. The wavinessand spacing experiments indicate that the DTA-O and MILmethods measure distinctly separate properties of the trabe-cular network. It should be noted that the cavities in thesynthetic image model, a topological feature not present intrabecular bone, were assumed not to influence the DTA

measurements since the scale of these cavities is much largerthan the local neighborhood size used for DTA. The DTA-Otrend line has a high slope and good correlation, indicatingthe sensitivity of DTA-O to changes in local orientations dueto the increased waviness.

The variation in plate waviness is not expected to affectthe MIL along any direction, but MIL also detected a changein anisotropy in this experiment. The slight change in MILanisotropy with plate waviness is most likely a consequenceof the increased surface area and decreased spacing as wavi-ness increases, thus raising the probability that a test line willintersect one of the orthogonal planes. The different effectsof these experiments on structural anisotropy obtained withthe two methods are a consequence of the different defini-tions of anisotropy, which should be taken into account whenanalyzing in vivo measurements. The systematic variation inthe rotation results seen in both the DTA-O and MIL fits aremost likely caused by the inaccuracies of the numerical in-terpolation method used to rotate the object in the imagegrid. This might be avoided with a better interpolationscheme that takes into account the two-phase nature of tra-becular bone.

The distal radius is one of the skeletal locations amenableto in vivo microstructural analysis by �MRI since it allowsthe use of a tightly coupled radio-frequency coil �antenna�needed to achieve sufficient SNR and thus imageresolution.7,33 Further, the distal radius is a site of os-teoporotic fractures and has been shown to be useful as a

FIG. 5. Results from the waviness �a�,�b� and spacing�c�,�d� experiments for DTA-O �a�,�c� and MIL �b�,�d�analysis. Overlaid on �a� and �c� are sample imagesfrom each experiment, showing �a� off-plane wavinessexamples of 0, 6, and 9 voxels and �c� plate spacing of10, 20, 30, and 40 voxels. Trend-line correlations andslopes indicate that there DTA-O anisotropy variesstrongly with plate orientation but is not dependent ontrabecular spacing. MIL analysis is moderately depen-dent in both experiments but in opposite manners, po-tentially confounding the MIL evaluation of complextrabecular changes.

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FIG. 6. Examples of in vivo orientation analysis on three female subjects: �a�,�b� 62 year old emphysematic �c�,�d� 47 year old control �e�,�f� 68 year old severeosteopenic showing across variation of the orientation distributions. �a�,�c�,�e� Gray scale images are axial images from the center of the volume, grid linesdivide the volume into x-y projection subregions, and vectors indicate mean projected transverse orientation and anisotropy within each region. Notice thesimilarities in the regions to the left side �medial� of each image that point towards the radio-ulnar joint and on the right side �lateral� towards Lister’s tubercle.�b�,�d�,�f� show projected transverse orientation histograms for the same subjects.

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surrogate site for assessing fractures of the vertebras,31,34,35

and thus may give an indication of the systemic causes ofbone structural degeneration. The mean orientations seen oneach side of the distal radius are consistent with the stressesoccurring during typical uses of the forearm such as pushingthe fist against some horizontal support, producing stresses inthe ulna that are transmitted to the radius through the radio-ulnar joint. Trabecular orientation on the lateral side is con-

sistent with stresses such as those resulting from pushing upwith flat palms, which produces compressive forces on thedorsal side of the distal radius that concentrate at Lister’stubercle. Finally, it is well known that age-related bone lossis not uniform, for example, recent data in the tibia suggestthat the transverse trabeculae are lost preferentially.36 Thisobservation suggests that remodeling during aging occurs soas to increase structural alignment with the principal direc-tion of loading. In the present case of regional orientation inthe radius, the opposite behavior appears to be present, i.e., adecrease in anisotropy with age. These in vivo data suggestthat the DTA-O-based orientation method might serve as atool for studying remodeling changes in vivo.

The Gaussian model for fitting orientation histograms canbe justified theoretically with the argument based on mostlong bones having a preferential biomechanical direction.Thus, it is reasonable to assume that the forces acting at alocal neighborhood of the trabeular network will have a prin-cipal longitudinal component, convolved with a broadeningfunction that was assumed to be Gaussian in DTA-O anisot-ropy analysis. Such a model was prompted by the observa-tion that transverse orientation distributions in smaller re-

FIG. 7. �a�,�b� Lateral versus medial anisotropy in thedistal radius for 82 subjects, for DTA-O �a� and MIL�b�. The good correlation indicates that the anisotropymeasurements are consistent within subjects, yet thevariation could be due to differences in physical activity�biomechanics�. �c�,�d� Anisotropy versus age showssignificant trends, but not significantly different be-tween the two methods.

FIG. 8. Results of medial and lateral accumulated transverse orientationhistograms for all subjects in the distal radius. Gaussian fitting of the histo-grams indicate that trabeculas on the medial side of the distal radius areorientated toward the radio-ulnar joint and in the lateral side toward Lister’stubercle. The results also indicate that the medial side is more anisotropicthan the lateral �bigger difference in baseline to peak difference�, but alsohas more variation in orientations as indicated by the wider peak.

TABLE I. Parameters for Gaussian fits to the accumulated plate orientationsof the medial and lateral sides of the distal radius in 82 subjects.

Side Gaussian height Mean �degrees� Width �degrees� Baseline height

Lateral 0.003 46 15.5 24.6 0.004 28Medial 0.006 61 155.8 36.0 0.002 35

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gions displayed bell-shaped curves. The division of the distalradius into medial and lateral halves were the largest possibleregions that displayed the monomodal shape needed for theGaussian modeling. At smaller region sizes, the distributionswere monomodal, and the Gaussian fitting gives reasonableestimates of regional trabecular plate orientations. The re-gional anisotropy maps were developed to visualize thesetransverse anisotropies overlaid on an anatomical cross sec-tion image. Future studies might investigate the suitability ofother models, such as fractals but the literature is controver-sial in their application. Clearly, a detailed continuum me-chanics investigation of trabecular network biomechanicsmight provide insight into the nature of biomechanical andorientational distributions.

The combination of reduced precision by statistical blur-ring in determining a MIL along each test direction, and theuncertainty in ellipsoid fitting are proposed as the reasons forthe reduced performance of MIL results. In MIL anisotropyanalysis, and other secant-based methods, the ratio of maxi-mum to minimum fit ellipsoid axes represent a measure ofanisotropy, and the major axis orientation is the primary net-work orientation. Ellipsoid fitting assumes orthotropic me-chanical symmetry, which might exist at some trabecularbone locations such as in the spine,9 but perhaps not at morecomplicated skeletal locations such as the proximal femurand calcaneus. Thus, in general, MIL-derived second-rankfabric tensors might not fully reflect the local structural an-isotropy of all anatomical trabecular locations.37

Another potential source of error in the MIL methodmight be the confounding effects from measurements ofplates and rods. While the trabecular network has been de-scribed morphologically as a mixture of plates and rods,38

supporting the biomechanical forces with the minimumamount of material, methods for analysis of orientation haveignored the differences between these two trabecular types.The orientation of trabecular plates is defined by a surfacenormal and the orientation of rods is defined by a curvedirection. Therefore, the orientation vectors for the formerare perpendicular to the structure whereas in the latter casethey are aligned with the structure. Thus, it is clear fromthese definitions that previous methods of orientation analy-sis are unable to distinguish between these two topologicaltypes.

While the method presented here shows a higher precisionthan the MIL method, it has limitations. The MIL methodwas developed for analysis of optical images at very highresolution, that are difficult to obtain in vivo, and thus theapplication of MIL at in vivo resolutions should be furtherinvestigated. By contrast, a texture-based method such asLFD is not subject to such constraints. Further, each prepro-cessing and processing step has the potential to introduceerror in the data, thus making the results susceptible to inac-curacies due to these accumulated errors. The modeling oforientation histograms with Gaussian distributions is appli-cable in many cases, but not all patient data in the radiusfollows this model as accurately. Once a deeper understand-ing of the orientation distributions can be achieved, a morecomplicated model can be developed.

The current method is not applicable to the measurementof curve orientation, but it can be modified to follow thesame approach for them. Difficulties arise because of thesmaller structure size of trabecular rods, and the resultinghigher partial volume effect. Acquisition and preprocessingimprovements would help separate this partial volume effectfrom noise, and enable an analysis of rod orientations. Futureexpansion would analyze rod orientations and possible otherssuch as profiles and surface–surface junctions, all of whichcan be modeled as curves. The method also assumes that theanatomical axis is close to the z axis of the reference frame,but to completely remove off-axis bias, a technique for 3-Drotation of the vector field will be necessary to ensure com-plete alignment. For example, in the calcaneus, there is noclear anatomical axis due to the complex shape and loadingpoints.

In conclusion, the ability to measure standard orientationwith high precision opens numerous possibilities for futureresearch, notably in vivo the modeling and remodeling ef-fects on the orientation of trabecular bone. Monitoring thetemporal change of in vivo network orientation followingacute changes in biomechanical loading should be feasible.Examples include the study of the structural implications ofchanges in exercise regimes, or monitoring the effects oflong-term exposure to microgravity. The observation of tra-becular orientation changes in response to biomechanicalloading would represent the first in vivo confirmation ofWolff’s law.

ACKNOWLEDGMENTS

National Institutes of Health �NIH� Grants No. RO141443 and No. T32 CA 74781. Dr. Ruegsegger and Dr. Laibfrom the Institute for Biomedical Engineering, Departmentof Information Technology and Electrical Engineering, SwissFederal Institute of Technology, Zurich, Switzerland, for pro-viding the microcomputed tomography scans of our speci-mens used in the rotation experiment.

a�Corresponding author Dr. Felix W. Wehrli, Laboratory for StructuralNMR Imaging, Department of Radiology, 3400 Spruce Street, 1 Founders,Philadelphia, Pennsylvania 19104. Telephone: 215-662-7951; fax: 215-349-5925; electronic mail: wehrli@oasis.rad.upenn.edu

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