Star length distribution: a volume-based concept for the characterization of structural anisotropy

Star length distribution: a volume-based concept forthe characterization of structural anisotropy

TH. H. SMIT,* E. SCHNEIDER* & A. ODGAARD†*Biomechanics Section, Technical University Hamburg-Harburg, Hamburg, Germany†Biomechanics Laboratory, Orthopaedics Hospital, University of Aarhus, Aarhus, Denmark

Key words. Anisotropy, mean intercept length, porous media, star lengthdistribution, star volume distribution, volume orientation.

Summary

Determination and quantification of anisotropy is of greatinterest in research fields dealing with physical structures orsurface textures. In this paper, a volume-based method ispresented, which essentially determines the mean objectlength in a certain direction for a typical point within astructure or texture. The mean object lengths for allorientations together form the so-called star lengthdistribution (SLD). The validity and the accuracy of theSLD method are investigated, and illustrated by applying itto trabecular bone. By using a line sampling algorithm, therelation with other anisotropy measures could be studiedanalytically. Preliminary tests suggest that with SLD a moreexact description of the mechanical properties of porousstructures may be obtained than with other anisotropymeasures. However, due to possible secondary orientationsthat become apparent with SLD, a fabric tensor must be ofrank higher than two in order to properly describe anorthogonal structure mathematically.

Introduction

Many biological and synthetic materials have a structural ortextural anisotropy that is due to machining, wear or naturalgrowth. Rolled duplex steel, for example, shows differentsurface textures depending on chemical composition,annealing procedure and pressing conditions (Karlsson &Cruz-Orive, 1993), and biological materials like bone have anarchitecture that is determined by their mechanical function(Smit et al., 1997). Also wear patterns, e.g. on artificial kneesurfaces, may show a pronounced orientation, indicating atypical loading pattern during the use of the implant(Wimmer et al., 1998). Analysis of the surface texture orthe structure of materials thus may reveal interesting

information on the anisotropy of material properties(Niessen, 1997; Odgaard et al., 1997), the process of wear(Wimmer et al., 1998) or the mechanical function ofbiological materials (Smit, 1996).

Different methods are available for the characterization ofanisotropy in structures and textures (for a review, seeOdgaard, 1997). To quantify the anisotropy of trabecularbone, Whitehouse (1974) introduced the mean interceptlength (MIL), defined as the average distance between twobone/marrow interfaces (Fig. 1a). Obviously, MIL is afunction of the direction of the line along which it ismeasured. Whitehouse also showed that in the two-dimensional case MIL, plotted in a polar diagram, generatesan ellipse. Based on this, a second rank tensor expressionwas formulated (Harrigan & Mann, 1984) and found to bea proper measure of structural anisotropy. It must beemphasized, however, that MIL determines the orientationof the interface rather than of the material itself. Althoughthese are often well related, this is not necessarily the case;a structure with spherical holes, for example, has no maininterface direction, although the material itself may haveone (Odgaard et al., 1990). As spherical holes are notuncommon in porous media like trabecular bone, MIL maysubstantially underestimate the anisotropy of the material.

Odgaard et al. (1990) introduced the concept of volumeorientation (VO), which describes the anisotropy of thematerial itself and not of the interface. VO is determined bytaking the longest line segment within the volume for apixel within the structure (Fig. 1b). Its direction is called thelocal volume orientation. A statistical sample of localvolume orientations can be collected and analysed byparametric or nonparametric methods (Fisher et al., 1987).Just like MIL, such a curve can be related to an elasticitytensor. However, this is involved with the same problem asfitting an ellipse with MIL: it assumes symmetry within thestructure, and only one maximum can be described. Formany applications these restrictions are not desirable and –as will be shown below – not allowed as well.

Journal of Microscopy, Vol. 191, Pt 3, September 1998, pp. 249–257.Received 24th June 1997; accepted 17 April 1998

249q 1998 The Royal Microscopical Society

Correspondence to: Th. H. Smit. Current address: Department of Clinical Physics

and Engineering, Academic Hospital Vrije Universiteit, PO Box 7057, 1007 MB

Amsterdam, The Netherlands. Tel: þ31 20–444 1245; fax: þ31 20 4444147;

e-mail: [email protected]

Cruz-Orive et al. (1992) introduced the star volumedistribution (SVD), which is closely related to the starvolume measure introduced by Gundersen & Jensen (1985).SVD is defined as the mean volume of an object seenunobscured from a random point within the object,evaluated as a function of each direction in space (Karlsson& Cruz-Orive, 1993). SVD, like VO, thus essentially describesthe distribution of material around a typical point withinthe structure, but – in contrast to VO – all directions aretaken into account (not just the longest), so that possiblesecondary directions get more weight (Fig. 1c). Indeed, SVDis very sensitive for the detection of primary and secondaryorientations, but – due to this sensitivity – also mayoverestimate the degree of structural or textural anisotropy,thereby suppressing secondary orientations within thestructure. This will be discussed in more detail below.

For a study in which the primary and secondaryorientations in cubes of vertebral trabecular bone had tobe found, a method was developed that essentiallydetermined the mean bone length, and consequently wascalled MBL (Smit, 1996; Smit et al., 1997). The original ideaof MBL was to combine the multidirectionality of MIL withthe sensitivity of the VO method. In order to do that, onlythe bone intersections with a line grid were considered;these were then weighted by the length of each intersectionitself. The result appeared to be mathematically the same asthe so-called star length distribution (SLD), a method thatwas used later by Odgaard et al. (1997) as a minormodification of SVD, and which was based on a point-sampling procedure. We are not aware of any publication,however, in which the properties of SLD are studied in moredetail and compared with other anisotropy measures.

This paper presents the line-sampling procedure for thedetermination of MBL, along with its implementation ontwo- and three-dimensional pixel data. However, in order toavoid confusion with the previously used SLD and tounderline the close relationship of MBL to SVD, the methodwill be called star length distribution (SLD). The method isvalidated on several artificial 2D and 3D structures, andillustrated by applying it to trabecular bone specimens. Therelation between the geometrical and material properties ofa porous structure is investigated using a 2D finite elementmodel of an idealised structure. In the discussion, SLD ismathematically compared with MIL and SVD, which resultsin a concise overview of the different methods andalgorithms to characterize structural anisotropy.

The star length distribution

SLD(J) is defined as the length L of an object (or set ofobjects) measured along a line l with direction J for atypical point within that (set of ) object(s):

SLDðfÞ ¼ EðLðfÞÞ; 08 # f1808; ð1Þ

Fig. 1. Principles for determining structural anisotropy. (a) Meanintercept length. A grid of test lines at an angle J is placed onthe structure. The total length of the grid lines is divided by thenumber of intersections with the bone/marrow interfaces to findMIL(J). (b) Volume orientation. A point grid is laid on the struc-ture. For each point within the structure, the local volume orienta-tion is determined. (c) Star volume distribution: a point grid is laidon the structure. For each point within the structure, the interceptlength through the point is determined in several directions.

250 TH . H. SMIT ET AL.

q 1998 The Royal Microscopical Society, Journal of Microscopy, 191, 249–257

in which E() denotes the expectation or mean value overuniform random positions of the point within the object(s).Using a line grid to define the intersections (as in MIL; seeFig. 1a), it can be shown that the expectation of a pointbeing a part of intersection i is proportional to the length ofthat section Li, whereas the total number of intersections nis proportional to the total length of the intersections, sothat Eq. (1) can be written as:

SLDðfÞ ¼

Pni¼1ðLiðfÞÞ2Pni¼1ðLiðfÞÞ

; 08<¼ f <¼ 1808: ð2Þ

Note that – in contrast to MIL – the length of eachintersection is weighted by the length itself. The algorithmof SLD is illustrated for the two-dimensional case in Fig. 2,showing a matrix of pixels (grey) representing the crossingof a vertical and a horizontal strut. A series of equallyspaced grid lines is laid over the structure under an angle J.The intersection i of a grid line with the bone structure haslength Li, which is related to the number of (quadratic)pixels in this intersection npixi, the pixel size (or pictureresolution) dpix, and the angle J:

Li ¼npixi·dpix

maxðcosðfÞj; sinðfÞÞ; 08<¼ f <¼ 1808: ð3Þ

Note that effectively the longest projection of Li on thecoordinate axes is measured, and that the length issubsequently calculated by correcting for the angle.Obviously, the intersecting line may contain only one pixelin the direction of the projection, because otherwise Li

would be overestimated for most angles J. The grid lines

thus must be defined as a function of the largest coordinate,which is x for J # 458 and J $ 1358, and y for458 # J # 1358. Intersection i thus contains npixi pixels,all of which can be used as a sample point for thedetermination of SLD(J):

SLDðfÞ ¼

Pni¼1ððnpixi·LiÞðfÞÞPn

i¼1 npixi;08<¼ f <¼ 1808: ð4Þ

In practice, SLD(J) is determined with a pixel countingalgorithm, using Eqs. (3) and (4) according to

SLDðfÞ ¼

Pni¼1ðnpixiÞ

2

maxðjcosðfÞj; sinðfÞÞ:Pn

i¼1 npixidpix;

08<¼ f ¼ 1808: ð5Þ

Note that, using Eqs. (3) and (4), SLD(J) also may beexpressed in Li, which reduces Eq. (5) to Eq. (2).

For the three-dimensional case, the algorithm is essentiallythe same as for the two-dimensional case, except for thesampling of the test line orientations. While in 2D the angle J

can be sampled uniformly (i.e. increased with a constantangle step), doing the same with v and J would lead to aconcentration of test line orientations near the pole of thehemisphere, which would favour this orientation when fabrictensors are derived from the SLD results. Further, it would bemore accurate in finding the maximum or minimum SLDorientation near the pole of the hemisphere than near theequator. It does not, however, affect the accuracy of the SLDmeasurements themselves, as these are independentlydetermined for each direction. Nonetheless, for the reasonsgiven above, a cosine-weighted sampling algorithm was used,leading to a true isotropic random test line orientation (for adetailed description, see Odgaard et al., 1997).

Validation of SLD

Performance

Four structures with idealized geometries were tested andcompared with analytical solutions of SLD. The first one wasa solid sphere, which obviously must show the same SLD inall directions. Using

x2 þ y2 þ z2 ¼ r2 ð6Þ

and Eq. (2), the analytical solution for SLD as a function ofthe sphere radius r, can be found with

SLD ¼

�x

�yð2zÞ2dydx�

x

�yð2zÞdydx

¼16·

� r0

� ��r2¹x2

p0 ðr2 ¹ x2 ¹ y2Þdydx

8·� r

0

� ��r2¹x2

p0

��ðr2 ¹ x2 ¹ y2Þ

pdydx

¼2p·r4

43 p·r3 ¼

32

·r: ð7Þ


Fig. 2. The procedure of the star length distribution (SLD). A grid oftest lines (black) at an angle J is placed on a cancellous bone speci-men (grey). For each pixel of these lines that lies within the bonevolume, the bone length in the direction J is determined. Thebone lengths in this direction are averaged over all pixels to getSLD(J).

SLD: A CHARACT ER IZAT ION OF STRU CTURAL ANISOT ROP Y 251

A solid sphere with a diameter of 151 pixels wasexamined. The mean value for SLD(v, J) was 112·98pixels, which is close to the analytical solution of 113·25.The standard deviation is 0·04 (0·035%), which meansthat the method is not very sensitive in the direction ofscanning.

As the goal of SLD was to find the degree of structural ortextural anisotropy, a test object with known degree ofanisotropy, an ellipsoid with the longest axis twice as longas the other axes, was analysed. A fabric tensor wasgenerated, and the ratio of the eigenvalues of the fittedtensor then defines the degree of anisotropy. Using theequation of an ellipsoid and substituting in Eq. (7), it can beshown that SLD along the main ellipsoid axis is 3r. As SLDfor the minor axes remains the same, the theoretical degreeof anisotropy DA ¼ 2; this was also found numerically(DA ¼ 2·00). Interestingly, DA ¼ 2·0 is also found with MIL,but SVD gives DA ¼ 8·0.

The third case studied is a series of vertical cylinders witha length of 200 pixels and diameter of 15 pixels. SLD in thehorizontal direction must be the same as for a solid circle,which can be calculated analytically with

SLD ¼

�xð2yÞ2dx�xð2yÞdx

¼8·� r

0ðr2 ¹ x2Þdx

4·� r

0

��ðr2 ¹ x2Þ

pdx

¼163 ·r3

p·r2 ¼163p

·r: ð8Þ

The SLD plots are shown for the horizontal and thevertical plane (Fig. 3). The maximum length (200·00 pixels)was found in the vertical direction, and a minimum of13·62 was found in the horizontal direction. The analytical

value for the circular cross-section of these beams is 12·73(Eq. 8); this difference is due to the inaccurate description ofthe cylindrical beams by cubic voxels.

Finally, a 3D structure was analysed with struts in thevertical direction, and within the horizontal plane atangles of 208 and 958. These directions were more or lessfound (Fig. 4). Quantitatively, however, the SLD resultsare no longer trivial: in the vertical direction, forexample, the struts had a length of 252 pixels, whileSLD gives only 112·24. This is due to the horizontalstruts, which obviously have a very small object length inthe vertical direction (12 pixels) and thus decrease SLDfor the whole structure. The smallest SLD, on the otherhand, is 15·95, which is more than the strut thickness of12 pixels. Although for special cases the absolute value ofSLD may still have a meaning, this seems to be not thecase for irregular structures like that of trabecular bone.SLD therefore can be used only to find the degree ofanisotropy, and one must be careful with interpretingabsolute values.

An important parameter in the accuracy of SLD is thedensity of the line grid, because this determines the numberof points that are actually used to quantify the bonestructure: too many lines leads to unacceptable computingtime, while too few lines may affect the accuracy. Todetermine a proper grid line density, an ellipsoid withDA ¼ 2·0 and a cubic vertebral trabecular bone specimenoriginating from the centre of the vertebral body wereanalysed with line distances ranging from 1 to 30 pixels.This means that SLD was determined for 100% – 0·11% ofall voxels within the specimen, the first one obviously givingthe exact solution for a given picture resolution. In Fig. 5,

Fig. 3. Perspective (left) and top view (right) of the beams alongwith the SLD plots.

Fig. 4. Artificial 3D structure with three main directions. SLD isgiven for the frontal (top) and cranial (below) view.



the degree of anisotropy of both specimens is given as afunction of the line density. Apparently, the line distance canbe set at 10 pixels, which corresponds to a grid line densityand a percentile voxel sample of 1%. That the solutionactually does converge for finer line grids supports thenotion that the way of sampling the structure, i.e. by linesampling or by point sampling, does not affect themeasurement of SLD itself: a line distance of one meansthat all points are used for the determination of SLD, whichis the same as taking the maximum possible sample size in apoint sampling procedure.

The accuracy of finding the structural orientations andthe maximum SLD in 3D depends on the number ofdirections (‘partitions’) in which a hemisphere is analysed.For a unit hemisphere and with the uniform isotropicscanning procedure, each partition has an area of 2p/n, inwhich n is the total number of directions. For n ¼ 1000, theangle between two sample lines is approximately 5·28, sothat the maximum error is approximately 2·68. If there areno sharp protuberances on the structure (as for instance ona sea urchin), this can be refined by fitting of an ellipsoid:the eigenvectors give the main orientations of the structure,and the eigenvalues the SLD in these directions, the largestbeing the maximum SLD. A second rank tensor (like anellipsoid), however, can only detect the primary orientationin an orthogonal structure because the secondary orienta-tions are veiled by this procedure. This is discussed furtherbelow.

The application: trabecular bone

Trabecular bone has an architecture that reflects itsmechanical function: bone struts and plates are alignedalong the lines of principal stress (trajectories), an observa-tion which is often referred to as Wolff ’s law (Wolff, 1892).The trajectorial structure of trabecular bone is due to theprocess of functional adaptation, which states that

unloaded bone becomes resorbed, and that mechanicalstimulation induces bone formation (Roux, 1881). Thearchitecture of trabecular bone thus contains informationon how it is actually loaded, which can be very helpful forthe study of functional anatomy or for the clinical diagnosisof mechanical malfunctions (Schluter, 1965). A quantita-tive description of the architecture of this porous andanisotropic material is a first and necessary step for suchanalyses. Architectural information is also important for thedetermination of the mechanical properties of trabecularbone, which is relevant for the estimation of fracture risk inpatients with osteoporosis (see e.g. Linde et al., 1990).

The architecture of trabecular bone can be described bythe amount of bone and its structure. Usual measures forthe amount of bone are ash weight, bone mineral content,the ratio of bone volume and total volume (BV/TV) orvolume fraction, and apparent density (Goldstein, 1987).The trabecular bone structure is essentially described by itsanisotropy, usually quantified by MIL or – more recently –SVD. Application of SLD to a slice of vertebral bone readilyshows the advantage of a volume-based method over aninterface-based method (Fig. 6): the main orientation of thestructure is more pronounced, for SVD even more than forSLD. Further, SLD and SVD detect the second peak in thenear-horizontal direction, which is visible from the struc-ture ‘at first sight’, but was not found with MIL. This showsthat surface anisotropy is essentially different from volume


Fig. 5. The degree of anisotropy as a function of the line distancefor a trabecular bone specimen and an ellipsoid with DA ¼ 2·0.DA converges for a line distance of 10.

Fig. 6. Comparison between SLD, MIL and SVD for a slice of verteb-ral trabecular bone near the endplate. Note that the second boneorientation, which is clearly visible with SVD, is not detected byMIL and is suppressed by SVD.


anisotropy, and that surface anisotropy may fail to detectstructural information that is relevant for the characteriza-tion of anisotropy or for the mechanical evaluation oftrabecular bone specimens. It is interesting to see that thesecond orientation is more pronounced for SLD than forSVD, resulting in degrees of anisotropy of 1·75 and 4·27,respectively.

SLD is illustrated in 3D with a trabecular bone cubeoriginating from a location near the centre of the endplateof a lumbar vertebral body (Fig. 7). Again, a secondmaximum is found in the horizontal direction, while withinthe horizontal plane the anisotropy is marginal. The degreeof anisotropy of the specimen was 1·33. This is lower thanthat of the specimen originating from the centre of thevertebral body, which was used in the convergence testdescribed above (DA ¼ 2·07). Using MIL, however, thespecimen was found to be only slightly anisotropic(DA ¼ 1·02), while SVD gave DA ¼ 2·35. (For comparison,in the specimen from the centre of the vertebral body, thedegree of anisotropy was 1·13 and 13·5 for MIL and SVD,

respectively.) This means that MIL suggests that bothspecimens are more or less isotropic, while SVD suggeststhat the specimen in the centre of the vertebral body is farmore anisotropic than the specimen near the endplate. Alldegrees of anisotropy lie well within the ranges given byOdgaard et al. (1997) for a large series of trabecular bonespecimens.

Fig. 7. A 3D-reconstructed trabecular bone specimen from a sitenear the lower endplate of a human lumbar vertebra, along withthe SLD results in a spherical and a Cartesian coordinate system.The plots give a right lateral and a cranial view on the ‘cloud’ ofSLD points presented in a 3D co-ordinate system.

Fig. 8. Synthetic 2D structure with an oval hole pattern at anangle J. The inverted maximum von Mises stress as well as thestructure stiffness E are well approximated by SLD. MIL underesti-mates this anisotropy and fails to detect the secondary orientationof the structure; SVD overestimates the anisotropy of the structureand veils the secondary orientation.



The relation between SLD and material properties

Structural anisotropy is a fundamental quantity in thedetermination of the mechanical properties of trabecularbone (e.g. Linde et al., 1990). Relations between MIL and thestiffness tensor of trabecular bone have been formulatedmathematically (Cowin, 1985; Van Rietbergen et al., 1995;Zysset & Curnier, 1995), and this has improved the estimationof mechanical properties from the trabecular bone structure(Snyder & Hayes, 1990; Turner et al., 1990; Goulet et al.,1994). However, while MIL as a surface anisotropy measuremay fail to detect secondary orientations, SLD and SVD mayreveal the structural information that is necessary for aproper material characterization.

This hypothesis was studied with an artificial 2D structure,analysed with the finite element method (Fig. 8). This plate,with a regular pattern of oval holes, is loaded under auniformly divided load at a certain angle J. The loading of thematerial can then be quantified by the maximum equivalentvon Mises stress found anywhere in the structure. Thereliability of the structure is then proportional to the inverse ofthe maximum von Mises stress. When this parameter isplotted against the varying angle J of the hole pattern,together with SLD, SVD and MIL of the structure, thesuperiority of SLD over MIL and SVD is evident: SLD is more orless proportional to the inverted maximum von Mises stressand the stiffness of the structure, while MIL gives only a slightidea of anisotropy. SVD, on the other hand, overestimates theprincipal orientation over the secondary ones. The same trendwas found by Odgaard et al. (1997) for a series of trabecularbone volumes: SLD gives a degree of anisotropy close to themechanical anisotropy of the structures (DA ¼ 1·70 and1·99, respectively), while SVD underestimates the secondaryorientations (DA ¼ 7·12). MIL, on the other hand, only showsminor degrees of anisotropy (1·36). The structure character-ized by SLD thus gives a more exact estimation of themechanical properties of anisotropic structures.

It is worthwhile emphasizing here that SLD apparentlymay not be described by a second rank tensor, althoughthe procedure is common practice and comfortable for thedescription of the trabecular bone architectures and theirmechanical properties (e.g. Van Rietbergen, 1996). The firstreason is that the main bone directions in a vertebra maynot be assumed to be orthogonal (e.g. Kuo & Carter, 1991),although the bone architecture generally is supposed to bean optimum structure. This, however, is only the case whenthe bone is subjected to one major load case, as for examplein the human radius. For a bone like the lumbar vertebra,however, this is much less clear, so that the possibility of anonoptimum structure (or rather, a structure optimized totwo or more loads) has to be left open.

The second reason is that when a structure has two orthree distinctive bone directions, SLD of the smallestmaximum becomes the global minimum when SLD isapproximated by an ellipsoid. This is best illustrated for the2D case (Fig. 9). The vertebral trabecular bone slice shownon the left has a maximum both in the horizontal (1788)and in the vertical direction (878). When SLD for thisstructure is approximated by an ellipse, however, the formbecomes almost a circle, in which no main directions can bedetermined. The orthogonal graph on the right shows theseverely changed description of the bone architecture evenmore clearly: instead of two bone directions there remainsonly one small maximum at 878, while the other has beenturned into a global minimum at 1778. This description isobviously too much of a simplification of the structure, andveils important information. Apparently, higher ranktensors are necessary to describe such trabecular bonestructures properly.

Discussion

SLD was found to be a more sensible method for thedetection of bone orientations than MIL. This is a


Fig. 9. The problem of the SLD approximation by an ellipse: (a) a slice of vertebral trabecular bone from near the endplate; (b) the resultingpolar plot of SLD (thin line) and the approximation by an ellipse (thick line); (c) as for (b) in an xy-graph.


consequence of the SLD definition, in which the interceptsare squared before division by the total number of points:

SLDðfÞ ¼zPn

i¼1ðLiðfÞÞ2Pni¼1ðLiðfÞÞ

; 08<¼ f <¼ 1808; ð2Þ

which shows that long bone lengths are weighted more inSLD than short ones, and inaccuracies at the pixel levelhave less influence. In contrast, the mean intercept length(MIL) includes long and short intercepts with equal weight:

MIL ðfÞ ¼ C·

Pni¼1ðLiÞ

n; 08<¼ f <¼ 1808 ð9Þ

with n the number of intersections and C ¼ 2·VF; VF is thebone volume fraction.

With SVD, on the other hand, the longer intercepts areweighted more heavily than with SLD, and SVD thereforeoverestimates the degree of anisotropy. If SVD weredetermined using grid lines, the definition would be

SVDðfÞ ¼p

3·

Pni¼1ðLiÞ

4Pni¼1ðLiÞ

; 08<¼ f <¼ 1808: ð10Þ

The exponent of 4 in Eq. (10) surely makes the methodmore sensitive for the detection of the main bone orienta-tion, but it also suppresses eventual secondary orientationswith respect to the primary one. Figure 8 suggests that SLDprovides an almost linear relationship with the materialproperties of the 2D structure, which is a nice feature. It isconceivable, however, that another exponent – close to 2 –

may give better prognoses of the global material propertiesof a trabecular bone structure in 3D.

Although SLD clearly has a physical meaning, itsmagnitude can probably be interpreted only in the contextof the structure itself. This is due to the simple fact thatperpendicularly orientated struts in a structure reduce themagnitude of SLD (Fig. 10a,b). A lower value for SLD thusdoes not mean that there is less bone, nor that the structurewould be less stiff or strong. In fact, it is possible to have lessbone and yet high SLD values; in osteoporotic bone, forexample, severe loss of the bone structure may becompensated by thickening of the remaining trabeculaeand thus the same value of SLD can be found (Fig. 10c,d).SLD thus serves to determine only the anisotropy of astructure, although it is conceivable that, in combinationwith some density measure, it also allows a prediction of themechanical properties of the whole structure. The potentialof SLD for this purpose, though preliminary, has beenindicated above.

In conclusion, SLD is a good method for the detection ofstructural or textural anisotropy of trabecular bone, and –more than MIL or SVD – considers the secondaryorientations within the structure or texture. It thus appearsto be essential that long intercepts are weighted moreheavily than short ones, but if the exponent is too high, thedegree of anisotropy is overestimated.

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Star length distribution: a volume-based concept for the characterization of structural anisotropy