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Abbreviations: CR

En, apparent elastinCorresponding autE-mail addresses

Research Paper

Evaluation of the structural quality of bone in a caseof progressive osteoporosis complicating a ComplexRegional Pain Syndrome (CRPS) of the upper limb

F. Cosmia,n, G. Mazzolenib

aUniversità degli Studi di Trieste – Dipartimento di Ingegneria e Architettura, Via A. Valerio 10, 34127 Trieste, ItalybUniversità degli Studi di Brescia – Dipartimento di Scienze Cliniche e Sperimentali, viale Europa 11, 25123 Brescia, Italy

a r t i c l e i n f o

Article history:

Received 11 April 2013

Received in revised form

8 July 2013

Accepted 4 August 2013

Available online 5 September 2013

nt matter & 2013 Elsevie.1016/j.jmbbm.2013.08.006

PS, Complex Regional

c modulus; ROI, regionhor. Tel.: þ39 40 558 3431: cosmi@units.it (F. Cosm

a b s t r a c t

Densitometry is considered to be the gold standard in bone quality assessment. However,

since its introduction, the medical community has been aware that mineral density is only

one of the factors that influence the bone risk of fracture, which also depends on the

bone's trabecular arrangement and, in particular, on the trabecular architecture's load

bearing capabilities. At the University of Trieste, in recent years, a test has been developed

that simulates the application of compressive loads on trabecular architecture's recon-

structions extracted from digital radiographs. In this work, the test is described, and the

results obtained by applying the appraisal in a particular case of severe osteoporosis of the

hand, complicating a Complex Regional Pain Syndrome (CRPS) type II, are presented. The

test was able to quantify the pathological alterations of bone micro-architecture by means

of a Structural Index (SI), which was absolutely significant and relevant to the clinical

situation. Important research and clinical opportunities of application of the test include

accurate evaluation of osteoporotic bone diseases, careful clinical follow-up and monitor-

ing of responses to therapeutic approaches, and, prospectively, reliable quantification of

biological damage (forensic field).

& 2013 Elsevier Ltd. All rights reserved.

r Ltd. All rights reserved.

Pain Syndrome; BMD, bone mineral density; CF, content factor; CM, Cell Method;

of interest; SI, Structural Index; fax: þ39 40 568 469.i), giovanna.mazzoleni@med.unibs.it (G. Mazzoleni).

1. Introduction

According to the National Institutes of Health (NIH) defini-tion, osteoporosis is a disease in which the bones becomeweak and are more likely to break (National Institutes ofHealth, 2011). The Dual Energy X-ray Absorptiometry (DEXA)is currently the golden standard for measuring bone density(BMD) and the diagnosis is made based on the number

(T-score) of standard deviations below the young Caucasianfemale adult mean BMD (World Health OrganizationScientific Group, 2008). However, even though a low valueof bone mineral density is considered to increase thefracture risk, the majority of fractures take place in post-menopausal women and elderly men at moderate risk(Pasco et al., 2006; Siris et al., 2004; Sornay-Rendu et al.,2005; Szulc et al., 2005).

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The medical community has long been aware that thereare two factors that increase bone weakness: bone mineraliza-tion loss and micro-architectural deterioration (ConsensusDevelopment Conference, 1993). In effect, without prejudiceto the importance of the contribution to bone resistanceprovided by mineralization, it is well known that the abilityof bone to resist the applied loads also depends on thestructural architecture of the trabeculae, an aspect which thetechniques currently available are not able to take into account(Liu et al., 2009; Yeni et al., 2009; Haiat et al., 2009).

Estimating the bone density alone is not, therefore, aparameter sufficient for a complete evaluation of the loadbearing capability of the bone, and, for a better quantitativeunderstanding of bone quality, also the spatial arrange-ment of the trabecular tissue should be taken intoconsideration.

An example is given to emphasize this point. Fig. 1 showstwo portions of micro-CT slices from a pig humerus, in whichthe trabecular phase has been meshed with triangular cells.Assumed the elastic modulus (E¼1000 MPa) for the trabecularphase in the two images, the stress distributions obtainedby simulating a compression test along the long side of theslices are very dissimilar, as depicted by the Von Mises stressin Fig. 1. The bone fraction BF in both images is the same(BF¼39.9%), but the effective elastic modulus (computed bysimulation) of the top slice is only 25.9% of the bottom one.The images were obtained by phase-contrast micro-CT at theElettra synchrotron radiation facility in Trieste, with a resolu-tion of 14 μm. It must be noted that they represent twoparticular portions from two different slices (not projectedimages) and cannot be used alone to assess the bone tissuefraction in the bone.

In general, the trabecular arrangement influences not onlythe elastic properties of cancellous bone, but also its mechan-ical strength (Kleerekoper et al., 1985; Uchiyama et al., 1999).In effect, the mechanical properties of bone are regulated by

Fig. 1 – Different Von Mises stress distribution (MPa)obtained by simulating a compression test in two micro-CTslices of equal bone fraction from a pig humerus.

the composition and by the structural organization at themicro- and nano-scale (Zysset et al., 1999). Also the failuremechanisms in trabecular bone are related to the tissuecomposition and the microstructure (Morgan, 2008), but, inthis case, the non-linearity of the load-deformation curvemust be taken into account (Linde, 1994). Both elasticmodulus and ultimate strength of cortical and cancellousbone decrease in humans with increasing age (Boukseinet al., 2008) and, even if a direct relation has not beenestablished, inferences can be drawn between the failureand the elastic properties of trabecular bone (Brear et al.,1988) that can be very useful for the purpose of bone qualityranking.

Micro-numerical models applied to 3D micro-CT or micro-MRI reconstructions, based on finite elements or alternativemethods, have been developed to compute the elastic proper-ties of trabecular structures (Cosmi et al., 2009; Niebur et al.,2000; Viceconti et al., 2004, Zysset, 2003) and to performstrength predictions by incorporating the post-yield behaviorof the trabecular bone tissue (Verhulp et al., 2008). Never-theless, a widespread clinical application of 3D methodsseems unlikely in the near future, given the examinationcosts and computational requirements. These considerationsled us to investigate whether the information contained ina 2D digital radiographic image can contribute, along with themethods already in use, to give a clinical indication, namely aranking, of the bone structure's load bearing capabilities.

The approach followed in this work focuses on thecharacterization of the bone structure by numerical simula-tion, obtained from conventional radiographic images ofsuitable anatomical districts. In a few seconds, the test canquantify the possible pathological alterations of bone micro-architecture by means of a Structural Index (SI), which iscalculated from the elastic response of the reconstructedstructure and the normalized sum of gray tones, indicativeof the mineralization in the region under examination(Cosmi, 2008).

For the assessment of bone quality ranking in age-relatedosteoporosis, after the first tests on a small number ofsubjects (Cosmi and Dreossi, 2007a), a wider clinical valida-tion has been recently conducted (Cosmi et al., 2011).The very positive results obtained in these studies, the lowcost of the examination and the wide availability of thenecessary equipment, make the proposed method a highlypromising tool, able to equip the physician with a simple,inexpensive and readily available complementary technique,useful for providing information on the patient's bone tissuequality, thus completing the data supplied by the methodscurrently in use, for a more accurate assessment and also forscreening purposes.

In this paper we present the results obtained by applyingthe test to a particular case of rapidly progressing, limb-confined osteoporosis, which occurred in a case of ComplexRegional Pain Syndrome (CRPS) of the arm.

We chose this particular (and rare) case of CRPS-linked,severe osteoporosis, since, among other bone diseases, thisunique clinical situation provides all the elements necessaryfor proving the validity of the proposed computationalmethod for a wide range of research and clinical applicationsin osteoporotic pathologies (see below).

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In most cases, CRPS syndrome is a limb-confined chronicand progressive diseases (de Mos et al., 2007). Formerlyknown as “reflex sympathetic dystrophy” (RSD) and “causal-gia” (now classified as CRPS type I and CRPS type II, respec-tively) (Stanton-Hicks et al., 1995), this syndrome, whichinvolves both the central and peripheral nervous systems,is a highly heterogeneous pathological condition, whoseclinical diagnosis is still underestimated and pathophysiol-ogy poorly understood (de Mos et al., 2008; Marinus et al.,2011). Often arising after triggering events (e.g. traumaticlesions, fractures or elective surgery of the limbs) (490% ofcases) (de Mos et al., 2007), this severe and painful disease isassociated with a particularly poor quality of life and largehealth-care and societal costs (Goebel, 2011). Due to thecomplexity of the disorder, and to the lack of scientificallyvalidated studies, the correct diagnosis of CRPS syndrome isoften delayed (even for years after the disease onset), and thetreatment, which is multidisciplinary and very complex, isnot yet standardized (Harden et al., 2013).

Besides intense pain, the heterogeneous clinical expressionof long-standing CRPS is characterized by multiple systemdysfunctions, which may include, in combination with sensoryabnormalities, autonomic disturbances of vasomotor origin,trophic skin and muscle changes, also a rapidly progressivedemineralization of the bones (“patchy osteoporosis”) (Park,2012). Since bone loss in CRPS occurs regionally, with prevalentloss of the trabecular bone and marked bone demineralizationpredominant at the epiphyseal regions (Doury, 1988), and therecovery of lost mineral bone is very slow (even several years) –thus predisposing the patient to future fractures after minorinjuries (Sarangi et al., 1993) – the severe clinical case of CRPSobject of our study appeared particularly suitable for ourinvestigation purpose.

The results obtained show how the developed tool may beable to open important prospective applications for the studyof bone pathologies (namely osteoporotic ones), not only inregards to the assessment of the illness course and theefficacy of potential therapies, but also with the aim to clarifythe pathophysiological aspects of those diseases. Moreover,interesting possibilities of application exist also in the med-ical forensic field, where this new method can help thephysician in the quantification of the biological damage.

2. Materials and methods

2.1. Image elaboration and structural analysis

As already mentioned in the Introduction, the test simulatescompression tests on a structure obtained by processingradiographic images, allowing for a quantitative assessmentof trabecular bone quality, which can be usefully employed inclinical practice.

The structural numerical model is based on the CellMethod, a recently introduced discrete method (Tonti, 2001),which is particularly interesting from the point of view ofcalculation time and memory requirements, without com-promising the accuracy of the results.

Without going into the details of the formulation, which canbe found in the relevant literature mentioned in Section 2.1

here below, suffice it to say that the application of thismethod is particularly appropriate in the presence of hetero-geneities. In effect, the Cell Method is based on a directdiscrete formulation of field laws, so that the characteristicdimension of the mesh can be of the same order of magni-tude as that of the heterogeneities, without any constraintsimposed by mathematical differentiability.

The details of the Cell Method formulation for elastostaticsand elastodynamics are discussed in detail in Cosmi (2001,2005) and Tonti and Zarantonello (2009, 2010). The method hasalready been successfully applied for the estimation of theelastic properties of sintered materials, short fiber reinforcedpolyamide composites, and for the analysis of three-dimensional models of the structure of the trabecular bone(Cosmi, 2011a, b, c, 2004, 2003; Cosmi and Dreossi, 2007b;Cosmi and Di Marino, 2001; Taddei et al., 2008). The websitehttp://discretephysics.dic.units.it/ also collects work on the CellMethod by several authors.

The anatomical regions chosen as the most appropriatefor the analyses are the first phalanx of the second, thirdand fourth finger (proximal epiphyses). In these anatomicregions, in fact, it is possible to identify the outline of thetrabecular structure even in a plain radiograph, since thepattern develops in layers with a certain regularity, despitethe irregularities of the bone shape.

The operator interface with the radiographic image of thehand, a zoom of the second finger proximal epiphysis, andthe region of interest, ROI, selected for the structural analysis,is shown in Fig. 2. The size of the ROI can be adjusted by theoperator in each finger, so that it covers the largest possiblesquare trabecular region within the cortical boundaries.

The structure used for the numerical analysis is obtainedfrom the digital X-ray image through a number of steps:

1.

A nonlinear sub-threshold erosion filter is applied to theROI image in order to remove the smaller elements, notconnected with the tissue. Each pixel of the selected imagehas a gray tone value between 0 and 255, Fig. 3(a).

2.

A grid of nodes is laid on the image, Fig. 3(b). The sameinitial spacing has been used throughout this work.The nodes are then automatically connected to form amesh of triangular cells, with larger cells over regions withsame gray level. Consequently, the number of cells canchange with the ROI structure. In this work, the number ofcells in the simulations ranged roughly from 3200 to 3500,depending on the bone dimensions and the local pattern.

3.

The elastic modulus of each cell is scaled between 0 and1000 MPa on the basis of the average gray level in 7 pointsof the cell (barycenter, vertexes and sides middle points).A linear elastic constitutive law has been assumed, with aPoisson's ratio of 0.3. The cells with average gray level zerodo not have mechanical characteristics. Fig. 3(c) graphi-cally depicts the elastic modulus distribution in the cells.

By means of the described procedure, the image has beentransformed into a structural model suitable for simulatingthe application of compressive loads along the orthogonalaxes. The average value of the apparent elastic moduli inthe two directions, En, can be then computed. The sum of the

Fig. 2 – Snapshot of the operator interface. The radiographic image of the hand (left), a zoom of the second finger's proximalepiphysis (top right), and the region of interest, ROI, selected for the structural analysis (bottom right) are shown.

Fig. 3 – (a) ROI original image; (b) mesh of triangular cells; and (c) graphical representation of the elastic modulus distributionin the cells.

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gray tone values in the ROI, normalized to 100, is the CFand provides an indication of the local level of mineraliza-tion, somehow related to the matter content in the ROI(Cosmi, 2008).

These two values are then combined in a Structure Index,SI. In order to explain the meaning of SI, it can be usefulto point out that the simulation results in terms of apparentelastic modulus incorporate both aspects of bone strength:the matter content and its architectural organization.Since the purpose of the proposed approach is to highlightthe contribution to bone quality of the structural arrange-ment, as a complement to the well-established bone mineraldensity measurements, the SI has been introduced with

the aim of removing the effect linked to the mineralizationlevel from the apparent elastic moduli in the ROI. Themathematical expression for computing the Structure Indexis as follows:

SI¼ a1½b1En�b2CF�

where a1 is a normalization factor computed from theradiographic acquisition parameters, and b1 and b2 arepositive constants. The complied code running in theIDL Virtual Machine platform (http://www.exelisvis.com/accessed January 28, 2013) gives the results in less than1 min on a normal personal computer, including both imageprocessing and numerical model solution.

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In application to the evaluation of the risk of fracture inosteopenic and osteoporotic patients, the first trials performedare described in Cosmi and Dreossi (2007a) and Cosmi et al.(2011). These studies confirmed that Structural Index is able toseparate the clinically positive subjects from the healthy ones,despite a possible information loss due to the use of planarradiographic images.

This last aspect is detailed in Section 4, where it is shownthat, even if a 2D image based simulation cannot offer anaccurate prediction for the 3D structure mechanical properties,the SI can still provide an accurate tool for bone quality ranking,being able to preserve, and even amplify, the differences amongtrabecular patterns detected in appropriate anatomical regions.

2.2. Application of the method in a case of ComplexRegional Pain Syndrome of the upper limb

The object of the present study is a case of Complex RegionalPain Syndrome (CRPS), type II (ICD-10 GM 2010, G56.4) (Albazazet al., 2008), which was diagnosed in a woman of 50 years,following a lacerated major blunt trauma of the right hand,with partial nerve injury (second finger) and fracture of theproximal epiphysis of the first phalanx of the third finger. Thediagnosis was performed according to the Budapest criteriaendorsed by the International Association for the Study of Pain(Budapest/IASP CRPS criteria), recently validated for clinicaland research use (Harden et al., 2013).

The case was particularly interesting, as complicated bysevere osteoporosis, which, during the six months followingthe trauma, extended in a distal to proximal direction,affecting, progressively, all the bone structures of the hand,radius and ulna, mostly involving, as typically occurs in thisparticular syndrome, the trabecular bone of the epiphysealportions of the interested bones (Doury, 1988).

Fig. 4 shows a detail of the specific alterations observedunder X-rays in the case object of study, exhibiting the typical“patchy” osteoporosis pattern, with a predominant involve-ment of the trabecular bone of the epiphyses.

3. Results

As already discussed in Section 2, the software computes:

1.

Figradthe5 m

the average value of the apparent elastic moduli along theaxes of the structure obtained from the ROI image elabora-tion, En,

. 4 – “Patchy” osteoporosis pattern in the magnifiediographic image of the first phalanx of the third finger ofright hand in the case under study (the image was takenonths after the onset of the CRPS syndrome).

2.

the CF, that gives an indication of the local level ofmineralization, and combines these values in a SI.

In a first series of evaluations, the data were obtained fromthe radiographs of the right hand, carried out during theusual clinical examination procedures provided in thesecases, i.e. immediately following the trauma, and after 2and 5 months from the onset of the syndrome. In particular,the radiography performed at the time of the trauma corre-sponds to a situation of healthy trabecular structure, notyet altered by the CPRS-induced abnormalities. The damageinduced by the osteoporotic disorder, however, is well detect-able in the successive radiographs. An interesting visualcomparison is shown in Fig. 5, where the trabecular regionin the proximal epiphyses of the first phalanx (middle finger)is depicted. The progressive damage to the trabecular struc-ture (bone resorption) appears to be easily observable,but cannot be quantified by the visual examination of theradiographs alone.

In this particular case of CRPS, that involved all thestructures of the hand, it was possible to perform a morespecific assessment of the clinical situation by following intothe detail the time evolution of each finger. For a quantitativeassessment of the overall situation of the bone's features, thevalues of the Structural Index (SI), of the apparent elasticmodulus (En), and of the content factor (CF) were computed ineach finger (thumb excluded). Given the non-optimal imageresolution, the average of three tests was always consideredin each finger. The results are shown in Fig. 6.

The SI, as well as being able to quantify the bone's damageand of following its evolution over time, appears to be moresensitive to the alterations of the bone structure rather thanthe average modulus of elasticity and the parameters relatedto mineralization. For example, the percentage of changebetween the initial (healthy) situation and the conditionfound after 5 months for SI is 48%, while it is 27% for En,and only 20% if CF is considered. These indications reflectthose obtained from the analysis of the simplified modelsdiscussed in Section 4. It is noteworthy the strict correspon-dence found between the time-course of SI values (wholehand and single fingers) and the corresponding clinicalsituation (i.e. severity of pain, local swelling, tropic condition,articular mobility). Images of the affected hand after 2 (a) and5 (b) months from the onset of the syndrome are shownin Fig. 7. Consequently, En and CF were not considered in thesuccessive clinical assessments, and only SI, the parameterwith the greatest ability to discriminate differences, wasrecorded.

Given the encouraging results obtained, in the followingradiologic controls it was decided to acquire higher resolutionradiographs by a digital mammograph, with an appropriatesetting of the parameters and the adoption of a Rh–Rh filter,similarly to that described in Cosmi and Dreossi (2007a). Inthis case, since no reference-data were available for the righthand in good physical shape (before or immediately after thetrauma), the left hand not affected by the CPRS syndromewas used as internal healthy control, in order to assess thecharacteristic values of an unaltered trabecular structure. Theimages of the trabecular region analyzed offer again aninteresting visual comparison, and allow to appreciate the

Fig. 5 – Trabecular region in the proximal epiphyses of the first phalanx of the third finger of the right hand (above), and thecorresponding enlarged ROI (below), used for the analyses of the radiograms taken immediately following the trauma (a), 2 (b),and 5 (c) months later.

Fig. 6 – Detail of single finger's evolution of SI (a), and average values in the right hand (thumb excluded) of the SI (b), of theapparent average elastic modulus En (c), and of the CF (d).

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improvement in the definition of the trabecular structureobtained by the mammograph (Fig. 8).

The numerical results are shown in Table 1. By 8 monthsafter the disease onset, a significant recovery of the bone qualitywas evident, and, specifically, it was noteworthy at the level of

the third and the fourth fingers, which, in the analyzed region,showed SI values similar to those observed in the correspondingregions of healthy left hand's fingers. Interestingly, when eachsingle finger was considered, a close correlation between theestimated recovery of bone loss and the objective osteo-articular

Fig. 7 – Representative photographs of the CRPS-affected hand, taken after 2 (a), 5 (b), 8 (c and c1) and 12 (d and d1) monthsfollowing the onset of the disease. It is possible to appreciate the clinical evolution of the syndrome. In particular, (c1) and (d1)refer to soft tissues appearance and articular mobility after 8 (c) and 12 (d) months after the beginning of the disease.

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Fig. 8 – Trabecular region in the proximal epiphyses of the first phalanx of the third finger of the left (a) and the right hand (band c) (above), and the corresponding enlarged ROI (below), used for the analyses of the radiograms. Images in (b) and (c) weretaken after 8 and 12 months from the onset of the CRPS syndrome, respectively.

Table 1 – Values of SI in each finger, and average values in the healthy left hand and in the CRPS-affected right hand thumbexcluded after 8 and 12 months from the disease's onset. SI values were computed from mammographic images.

SI First finger Second finger Third finger Fourth finger Hand average

Left hand 314 163 160 179 2048 months 86 106 122 144 11512 months 105 127 161 176 142

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improvement of function (motility) and soft tissues' trophy wasobserved, thus confirming the potential clinical value of theadopted method for a correct estimation of the severity andcourse of CRPS-linked osteoporotic bone diseases.

Pictures of the CRPS-affected hand taken 8 (c and c1) and12 (d and d1) after the onset of the syndrome are shown inFig. 7. In particular, in Fig. 7 (c1) and (d1), which refer to softtissues appearance and articular mobility after 8 (c) and 12 (d)months from the onset of the disease, it is possible toappreciate the parallel between bone quality (expressed bythe value of the calculated SI) (Fig. 6 and Table 1) and theclinical situation.

4. Discussion

In this work we decided to use planar radiographies to inves-tigate the mechanical properties of the bone during the course

of a severe CRPS-induced “patchy osteoporosis” of the upperlimb. The proposed method is based on the in silico mechanicalcharacterization of bone trabecular structures obtained byconventional radiographic images.

In our study, information from the measurement of bonedensitometry obtained by conventional Dual Energy X-rayAbsorptiometry (DEXA) scans used for the analysis of general-ized age-related osteoporosis was not employed for crossedassessments with the SI. Provided that our main objective wasnot to compare the accuracy of the results obtained with ourmethod with those from DEXA scans (or from other imagingtechniques, that, except for radionuclide bone imaging, arenot clinically accepted for providing objective and relativelyspecific evidence of CRPS disease), the internal controls pro-vided by the Rx images taken from the right hand immediatelyafter the trauma and from the healthy left hand, the informa-tion obtained from the CF (that gives an indication of the locallevel of bone mineralization), and the typical images obtained

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from three-phase radionuclide bone scans, used for confirmingthe diagnosis of CRPS-related bone alterations (Mackinnon andHolder, 1984) (data not shown), made unnecessary to performother tests such as DEXA.

By definition, it is not possible to resolve a 3D structure by asingle radiographic image. An example is here discussed toshow that, even if a 2D image-based simulation cannot providean accurate prediction for the 3D structure mechanical proper-ties, the SI can still provide an accurate tool for bone qualityranking, being able to preserve, and even to amplify, thedifferences among trabecular patterns in the particular anato-mical region used for the analysis.

Quite obviously, since this study regards a living patient,it was not possible to conduct any direct mechanical tests.More pointedly, the Structural Index could be, in theory,obtained from cadaver's hand bones, which could be testedfor mechanical-property data, such as elastic modulus orbone strength, and if this possibility arises, the authors fullyintend to follow this path. Nevertheless, since the mechanicalproperties of a composite material strictly depend on itsstructure and composition, even if the direct mechanical testcould be performed on a cadaver's hand bones, the resultsobtained could obviously differ from those of a living, blood-perfused bone.

In principle, a 3D trabecular architecture can be reconstructedby CT scans and successively imported in a numerical model, e.g. based on the Finite Element Method, to calculate its apparentelastic properties. Unfortunately, in the case of the present studya comparison between data obtained from 3D direct numericalmodels and from planar radiographic images in the sameanatomical region was not possible, since 3D scans at the levelof resolution required for this purpose were not available at thefacility where this study was conducted.

Fig. 9 – 3D_A and 2D_A, respectively, 3D and 2D models of simpanatomical region; 3D_B and 2D_B, 3D_C and 2D_C, analogous mareas are shown darker for clarity of representation).

To overcome this limitations, we considered a very sim-plified, out-of-scale model of the three-dimensional trabecu-lar structure in the examined region (the proximal epiphysesof the first phalanx of the second, third and fourth finger),which, as already mentioned, develops in a planar, regularfashion, very different from the complex 3D micro-architecture, as found, for example, in the femoral neck orin the calcaneus. The simplified model consists of twoorthogonal square-section rods, lying in the xy plane andjoined by a frame (see 3D_A, Fig. 9 top left). Frame side size is11 mm, bar thickness in the xy plane is 3 mm, and 1 mmalong z. The bars are separated along the z-axis, and thedistance between the two orthogonal rods is 1 mm. In an X-ray image (X-ray path following z-axis), the structure appearswith areas of increased absorption at the greatest thickness,as in the model 2D_A shown in Fig. 9 (bottom left). In asimilar manner, it is possible to obtain the out-of-scalemodels of an altered trabecular structure (as occurs, forexample, during osteoporosis) by changing the rods thicknessin the yx plane. The frame side size in the corresponding3D_B (Fig. 9 top center) and 2D_ B (Fig. 9 bottom center)models is the same, but the rod thickness in the xy plane isnow changed to 1 mm. A third structure is obtained bychanging the frame side size to 7 mm and by keeping thetrabecular thickness value 1 mm in the yx plane. The corre-sponding 3D_C and 2D_C models are shown in Fig. 9 top rightand Fig. 9 bottom right, respectively.

In the three-dimensional models, an arbitrary value ofE¼1000MPa is assumed as the elastic modulus of the bars.In the planar models, by proceeding in a manner similar to thecalculations described in the previous Section 2.1, a value ofE¼1000MPa is assumed for the bars elastic modulus, except inthe central area, where the difference in X-ray absorption can

lified, out-of-scale-trabecular structure in the examinedodels of two differently altered structures (higher absorption

Table 2 – Ratios of the apparent elastic moduli En, CF and SI computed for the different structures.

EnA/EnB EnC/EnB CFA/CFB CFC/CFB SIA/SIB SIC/SIB

3D 1.4 1.6 1.3 1.52D 1.3 1.6 1.3 1.5 1.5 1.7

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be taken into account by posing E¼2000 MPa, and in the frame,where for the same reason E¼3000 MPa.

Compression tests along the x and y-axes can be simulated,and the apparent elastic moduli En of the six models can beevaluated, together with the CF defined in Section 2.1 and the SIof the 2Dmodels (the index is only defined for plane structures).

Obviously, since a 2D simulation cannot accurately predictthe behavior of a 3D structure, the apparent elastic moduliand the content factor in the 2D_A, 2D_B, and 2D_C modelsare quite different from those computed for the correspond-ing 3D_A, 3D_B, and 3D_C models, as expected, but the ratiobetween the apparent elastic moduli in the 2D and in the 3Dstructures is approximately the same (and equal to 3) for allthe structures A, B and C. Nevertheless, it must be pointedout that, for this application, the most important element tobe conserved in the 3D to 2D simplification is the ability todiscriminate between the structures, rather than the exactvalue of the considered parameter, i.e. the apparent elasticmoduli or the content factor.

A simple indicator of the changes between the two struc-tures is represented by the ratio between the parameterscomputed for the different structures, as summarized inTable 2. It can be appreciated that the 3D and the 2D resultsare practically equivalent for the purpose of structure ranking,since there is no difference between the ratios computed in the2D and in the 3D structures. It can be therefore affirmed that, inthis regard, the 2D and 3D models are equivalent.

If the discriminatory power of parameters such as contentfactor (CF) and apparent elastic modulus (En)are considered, theCF ratio, linked to the differences in mineralization, has adiscriminatory capability that is comparable to that of the 2Delastic moduli in the case of the structures A and B, and a littlesmaller than the En ratio in all the other cases, that is the 2Dmodels of structures A and C and all the 3D models. But, if forthe planar models the Structural Index (SI) ratio is consideredinstead of the elastic moduli, it can be seen that SI is able toamplify the differences among the structures with respect to theother parameters. In the present case of very simplified bonearchitectures, this amplification effect is small, but it becomesmore relevant when more dense patterns, in which the trabe-culae are regularly organized, are considered, like in the proximalside of the first phalanges of hand fingers, used for the clinicalevaluation of the bone structures. This is a definitely positiveeffect for the purposes of clinical classification.

Therefore, from the methodological point of view, it can beconcluded that the planar model obtained from a digitalradiography with the procedure described in Section 2.1,appears to be able to give results that can be useful forclassifying the quality of bone for the purpose of a researchapplication or a clinical evaluation, even if it is not possible touse a 2D model for predicting the exact value of the elasticmodulus of the 3D bone structure.

5. Conclusions

The present paper illustrates the results derived from theanalysis of a particular case of rapidly progressing, severeosteoporosis of the upper limb, associated with a ComplexRegional Pain Syndrome (CRPS) type II, and obtained by apply-ing a new computational method, developed in order to permitto “classify” bone's quality in osteoporotic syndromes.

Based on the Cell Method, the approach followed in thiswork focuses on the characterization of bone strength, obtainedby numerical simulations of load application to a numericalstructure derived from a conventional (2D) digital radiographicimage of the hand. The proposed test can quantify thepathological alterations of bone micro-architecture by meansof a Structural Index (SI), which is calculated from the elasticresponse of the reconstructed structure and the normalizedsum of gray tones, indicative of the mineralization in the regionof interest. In the particular (and not frequent) CRPS casechosen for our study, the calculated Structural Index wasabsolutely significant and relevant to the clinical situation.

The test, originally developed as a complementary tool forthe evaluation of the risk of fracture in osteopenic andosteoporotic patients, is very fast (the result is given within1 min), can be performed at low cost, and can be easilyexecuted and implemented within a traditional unit of Radi-ology, since it requires only a conventional 2D digital radio-graph of the hand.

The results obtained in the present study demonstrate, intheir complex, as the developed method may have wide andimportant prospects of application in the medical context,that include, besides specific research objective directedto add new insight into the pathogenesis of complex bonealterations (such as CRPS-linked osteoporosis is), also thepossibility of a more accurate clinical evaluation of the degreeand course of osteoporotic bone diseases, a careful monitor-ing of their responses to specific therapeutic approaches(e.g. functional, pharmacologic or other interventional proce-dures), and, prospectively, a more precise and reliable quan-tification of the biological damage (forensic field).

Acknowledgments

This study has been performed on a CRPS patient followed bythe clinical team of the “Casa di Cura San Francesco” (Bergamo,Italy). The authors are particularly thankful to Dr. D. Malgrati forhis generous enthusiasm and motivation, which made possiblethe realization of this work, to Dr. G. Ronzoni, for providing theX-ray images used in this study, and to Mr. R. Pozzoni, for hisexpert technical assistance in their acquisition. Ing. S. Scozzese'scontribute to the simulations of the simplified model of the

j o u r n a l o f t h e m e c h a n i c a l b e h a v i o r o f b i o m e d i c a l m a t e r i a l s 2 9 ( 2 0 1 4 ) 5 1 7 – 5 2 8 527

trabecular structure is also acknowledged. G.M. is particularlygrateful to Mr. P. Panichi, whose professional competence,human sensibility and insight permitted to achieve the clinicalresults shown in this study. The authors are also indebted to Dr.N. Steimberg, Dr. J. Boniotti and Dr. M. Attanasio for theirconstant encouragement and assistance all during the prepara-tion of this paper. This work was supported by local funds of theUniversity of Trieste (to F.C.) and of the University of Brescia (toG.M.).

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