Ultrasound in Med. & Biol., Vol. -, No. -, pp. 1–12, 2013Copyright � 2014 World Federation for Ultrasound in Medicine & Biology

Printed in the USA. All rights reserved0301-5629/$ - see front matter

/j.ultrasmedbio.2013.11.010

http://dx.doi.org/10.1016

d Original Contribution

MODELING OF FEMORAL NECK CORTICAL BONE FOR THE NUMERICALSIMULATION OF ULTRASOUND PROPAGATION

QUENTIN GRIMAL,* DANIEL ROHRBACH,y JULIEN GRONDIN,* REINHARD BARKMANN,z

CLAUS-C. GL€UER,z KAY RAUM,y and PASCAL LAUGIER**Sorbonne Universit�es, UPMCUniv Paris 06, UMR 7623, LIP, F-75006, Paris, France; CNRS, UMR 7623, LIP, F-75006, Paris,France; yJulius Wolff Institute and Berlin-Brandenburg School for Regenerative Therapies, Charit�e-Universit€atsmedizin Berlin,Berlin, Germany; and zSektion Biomedizinische Bildgebung, Klinik f€ur Radiologie und Neuroradiologie, Universit€atklinikum

Schleswig-Holstein, Campus Kiel, Germany

(Received 26 April 2013; revised 30 October 2013; in final form 6 November 2013)

AagerieE-mail

Abstract—Quantitative ultrasound assessment of the cortical compartment of the femur neck (FN) is investigatedwith the goal of achieving enhanced fracture risk prediction. Measurements at the FN are influenced by bone size,shape and material properties. The work described here was aimed at determining which FN material propertieshave a significant impact on ultrasound propagation around 0.5 MHz and assessing the relevancy of differentmodels. A methodology for the modeling of ultrasound propagation in the FN, with a focus on the modeling ofbone elastic properties based on scanning acoustic microscopy data, is introduced. It is found that the first-arriving ultrasound signal measured in through-transmission at the FN is not influenced by trabecular bone prop-erties or by the heterogeneities of the cortical bone mineralized matrix. In contrast, the signal is sensitive tovariations in cortical porosity, which can, to a certain extent, be accounted for by effective properties calculatedwith the Mori-Tanaka method. (E-mail: quentin.grimal@upmc.fr) � 2014 World Federation for Ultrasoundin Medicine & Biology.

Key Words: Cortical bone, Elasticity, Fracture risk, Femur, Quantitative ultrasound, Homogenization, Porosity,Acoustic microscopy.

INTRODUCTION

The hip is a site of fracture among elderly people associ-ated with high morbidity and mortality (Cummings andMelton 2002; Keene et al. 1993). Areal bone mineraldensity (aBMD) assessment at the hip by dual X-rayabsorptiometry (DXA) is currently the standard refer-ence method used to characterize fracture risk (Watts2011). However, DXA alone is not sufficient to assessbone strength and often fails to predict fracture (Siriset al. 2004). A significant risk of fracture exists for menand women with an aBMD higher than the osteoporoticrange (Siris et al. 2004; Stone et al. 2003; Wilkin andDevendra 2001). Further, increases in aBMD aftertherapy do not fully explain the observed efficacy ofosteoporosis drug treatments (Delmas and Seeman 2004;Gallacher and Dixon 2010).

ddress correspondence to: Quentin Grimal, Laboratoire d’Im-Biom�edicale, 15 rue de l’�ecole deM�edecine, 75006 Paris, France.: quentin.grimal@upmc.fr

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Although diagnostic bone measurements have fo-cused mainly on trabecular bone, several recent reportshave provided evidence that cortical bone also contrib-utes to a significant amount of the strength of the femoralneck (FN) (Bousson et al. 2006; Manske et al. 2009). Thishas suggested that risk assessment should includeevaluation of cortical bone to improve identification ofindividuals at risk.

Quantitative ultrasound (QUS) assessment of theproximal femur has recently been investigated with thegoal of achieving enhanced fracture risk prediction(Barkmann et al. 2008). In a clinical pilot study, femurQUS allowed good discrimination between women withrecent hip fractures and controls (Barkmann et al. 2010).In an ex vivo study, we found that targeted ultrasoundmea-surement of the FN cortical compartment is possible(Grimal et al. 2013); furthermore, the measurement ishighly correlated with bone strength. In this method, thearrival time of the first ultrasonic signal (FAS) measuredin transmission through the neck is determined. TheFAS is associated with the propagation of waves guidedcircumferentially in the cortical shell (Grimal et al.

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print&

web4C=FPO

Fig. 1. Femoral neck cross-sectional slabs cut perpendicular tothe neck at a thickness of about 10 mm were obtained fromhuman donors and prepared for measurements with scanning

acoustic microscopy (SAM).

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2013; Grondin et al. 2010; Nauleau et al. 2012). Asopposed to X-rays, ultrasound is intrinsically sensitiveto mechanical properties, which contribute to boneintegrity. We assumed that enhanced risk predictionmay be achieved through ultrasonic assessment of theproximal femur cortex. This assumption motivated theresearch reported here on analysis of the propagation ofultrasound in the cortical shell of the FN.

Ultrasound transmission through the FN is complex.Measured signals are influenced by bone size, geometryand material properties. The respective effects of eachof these factors must be thoroughly understood to designefficient signal processing methods. At frequencies usedin clinical devices (0.5–1 MHz), the wavelength in boneis a few millimeters, much larger than the typical micro-structural heterogeneities (50–200 mm for the diameter ofvascular pores) (Grimal et al. 2011a). Hence ultrasound issensitive to effective properties that are determined by theelastic properties of the mineralized matrix and by thepervading porous network. Matrix properties (Bergotet al. 2009; Bousson et al. 2011; Roschger et al. 2008)and porosity (Bousson et al. 2004; Zebaze et al. 2010)change with age, pathological status and use of ther-apeutic agents. It is thus important to assess the impactof such changes on ultrasonic signals, for example, byusing numerical simulations.

The advent of powerful desktop computers and theavailability of finite-difference and finite-element soft-ware dedicated to elastic wave propagation have encour-aged rapid expansion of the numerical simulation ofultrasound propagation in bone. During the last decadethe research methodology in QUS has shifted from essen-tially experimental research to research in part guided bythe results of simulations. The approach has been vali-dated in some cases in which the main propagation char-acteristics observed experimentally could be accuratelypredicted by numerical simulations (Bossy and Grimal2010; Bossy et al. 2004a, 2005; Luo et al. 1999).

The objective of the work described here was todetermine which bone properties have a significantimpact on ultrasound propagation in FN cortical boneand to assess the relevancy of some models differing incomplexity. For these purposes, we introduce amethodol-ogy for the modeling of ultrasound propagation, with afocus on the modeling of elastic properties. We proposethat the microscopic features of bone tissue (pores, inho-mogeneities of mineralization or other material proper-ties) do not have to be accounted for explicitly. Instead,they may be represented by effective elastic properties.

Scanning acoustic microscopy (SAM) was used toobtain maps of FN material properties and microstruc-ture, which served as a basis for the modeling of elasticproperties. SAM is the state-of-the-art method forproviding sample-specific (individualized) quantitative

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assessment of the microstructure and elastic propertiesof mineralized tissues (Malo et al. 2013; Raum 2008;Raum et al. 2007; Rohrbach et al. 2012; Sa€ıed et al.2008).

METHODS

Sample preparation and scanning acoustic microscopyNine FN cross-sectional slabs about 10 mm thick

were obtained from seven human donors (age: 64.4 68.8 years) who received a hip implant. The experimentalprotocol was approved by the ethics commission of theMartin Luther University, and informed consent for thestudy was obtained from all subjects. Cross sectionswere cut approximately perpendicular to the FN axis be-tween the femoral head and the central region of the FN(Fig. 1). After fixation and dehydration in ethanol, thesamples were embedded in polymethylmethacrylate,ground and polished (Lakshmanan et al. 2007). An ortho-normal frame and a system of coordinates (x1, x2, x3) wereassociated with the FN based on anatomical landmarks:direction 3 was attached to the neck axis, and directions1 and 2 to the antero-posterior and supero-inferior direc-tions, respectively.

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Numerical simulation of US propagation d Q. GRIMAL et al. 3

A scanning acoustic microscope (SAM200Ex, Q-BAM, Halle, Germany) equipped with a spherical focus-ing transducer with 50-MHz center frequency (V605/60�,Valpey Fisher, Hopkinton, MA, USA) was used to mea-sure reflected signals (C-scan mode), in direction 3, onthe surface of the slabs that contained the central regionof the FN. Distilled water, thermostated at 25�C, wasused as coupling liquid. This technique provides images,in the plane (x1, x2), of the acoustical impedance (Z) with aspatial resolution of 23 mm (Raum et al. 2008). The pixelsize was set to 16 mm. A binary map of the acoustical im-ages was defined where pixels with value 1 and 0 are asso-ciated with areas occupied by pores (Haversian channelsand resorption lacunae) and the bone matrix, respectively(Lakshmanan et al. 2007). The binary map is used to allo-cate different properties to the coordinate points corre-sponding to pores or matrix. Hereinafter, the ‘‘porositymap’’ refers to pixels of value 1.

For each sample, the average of the mineralized ma-trix impedance (points outside the porosity map), denotedbZ , was calculated. Within each region of interest, the en-docortical boundary between cortical and trabecular bonewas determined by an experienced histomorphometrist.The cortical bone was demarcated by the following

Fig. 2. Illustration, for one of the nine femoral neck cross sectiof scanning acoustic microscopy (SAM) data. See the descriptirepresent impedance values that were converted to stiffness valu

to large and small impedanc

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criteria (Malo et al. 2013): (i) Cortical bone includesHaversian canals. (ii) If the pore is less than twice theaverage size of pores in the region, then the pore isincluded in the cortex. (iii) For larger pores, the endocort-ical boundary is set to split the pore. (iv) If the diameter ofthe pore is smaller than the distance from the pore to theendosteal region, then the pore is included in the cortex.In most cases, cortical bone could easily be distinguished.The result of the segmentation of the cortical shell is illus-trated in Figure 2.

Trabecular (Tb.Ar) and cortical (Ct.Ar) areas weredefined as the areas of the trabecular and cortical com-partments, respectively. Cortical porosity (Ct.Po) wasdefined as the ratio of the total area of pores in the cortexto Ct.Ar. Cortical thickness (Ct.Th) was defined as themean distance between the endosteal and periostealinterfaces.

SimulationsThe quantitative ultrasound transverse transmission

technique at the hip was simulated according toGrondin et al. (2010) in the framework of linear elasticityand assuming a lossless propagation medium. The equa-tion of motion is given by.

ons, of the five bone model types constructed on the basison of the models in Table 1. The gray levels in S0 and S1es as described in the text; light and dark gray corresponde values, respectively.

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Fig. 3. Configuration of the 2-D finite-difference time-domainsimulations.

4 Ultrasound in Medicine and Biology Volume -, Number -, 2013

rðxÞ vvivt

ðx; tÞ5 vsij

vxjðx; tÞ (1)

and the constitutive law by

sij 5 cijklðxÞ˛klðxÞ (2)

where the Einstein summation convention on repeatedindices applies. r is mass density, vi are the componentsof particle velocity, sij and ∊ij are the components ofthe stress and strain tensors and cijkl are the componentsof the fourth-order stiffness tensor. Using the Voigt matrixnotation, in what follows we denoteCij the components ofthe 2-D stiffness matrix. We set a 2-D numerical experi-ment in which a plane wave emitted by a virtual mono-element transducer is incident on a 2-D bone model(Fig. 3). The direction of propagation, which was approx-imately the antero-posterior direction, was chosen consis-tent with accessibility of the neck to ultrasound asdemonstrated by Barkmann et al. (2010). On the posteriorside, a plane receiver was placed aligned face to face withthe emitter. Emitter and receiver were separated by a dis-tance of 50 mm, and their height was 50 mm, which isgreater than the largest diameter of the FN for all femurs.

Fig. 4. Emitted signal used

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A finite-difference time-domain (FDTD) simula-tion code (Bossy 2012) was used to solve the wave prop-agation problem. The simulation box was surrounded byperfectly matched layers to avoid any spurious reflec-tions. At the onset of each simulation, a short pulsewas emitted (0.5-MHz central frequency, 0.4-MHzbandwidth [–6 dB]) (Fig. 4) and the simulated signalat the receiver was processed to determine the time-of-flight of the first arriving signal, denoted T. The latterwas defined as the arrival time of the maximum of thefirst oscillation of the signal (Grondin et al. 2010).The values of the acoustic field variables in the wholesimulation domain (‘‘snapshots’’) are recorded at severaltime points.

The FN SAM images were directly imported into theFDTD code, which uses a structured mesh with an iden-tical grid step in directions xl and x2. It follows that thegrid step is identical to pixel size, which is small enoughto ensure the stability of the numerical solution (Bossyand Grimal 2010).

The model includes a complete description of theelastic behavior and mass density of the bone. The fluidsand soft tissues filling the pores (porosity map) and themedullar cavity were allocated elastic properties and den-sity close to those of water (Kubo et al. 2011); that is, thebulkmodulus, shear modulus and density are respectively2.25 GPa, zero and 1000 kg/m3.

Bone tissue elastic propertiesThe bone matrix was assumed to be transversely

isotropic, with the plane (x1–x2) as isotropy plane. Thematrix elastic behavior is fully described at each coordi-nate point (outside of the porosity map) by five indepen-dent stiffness coefficients, which are components of thestiffness tensor Cijðx1; x2Þ. The coefficients were derivedfrom the measured impedance using a method introducedpreviously (Grimal et al. 2008), the main steps of whichare reported below. In a first step, the impedance valuesZðx1; x2Þ were converted to elastic coefficientsC33ðx1; x2Þ using an empirical relationship (Raum et al.2006), which was derived from the site-matched

for the simulations.

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Fig. 5. Representation of the material properties of the bone matrix used for simulations. The images display heteroge-neous properties corresponding to model S0. Left: r (kg/m3), middle: C11 (GPa), right: C44 (GPa).

Table 1. Model types that served as a basis for thesimulation of ultrasound propagation in individualized

femoral neck models*

Model type Description

S0 Reference model: detailed description of microstructureand elastic heterogeneity with resolution 23 mmconstructed after the SAM image. This modeldescribes: cortical and trabecular bone; microstructurein the cortical shell (vascular porosity); stiffnessheterogeneity of the mineralized matrix

S1 Identical to S0 with the trabecular tissue removedS2 Identical to S1 with the mineralized matrix properties

homogenized (averaged)S3 Identical to S2, but the pores are also allocated the

averaged matrix propertiesS4 Mechanical properties are homogeneous in the cortical

shell; these properties are obtained throughhomogenization of pores and matrix properties withthe Mori-Tanaka method

* Nine individualized models of bone samples pertaining to eachmodel type were constructed from acoustic microscopy data. SeeFigure 2.

Numerical simulation of US propagation d Q. GRIMAL et al. 5

assessment of mineral density and impedance of severalbone samples:

C33 5 1:3110:075Z10:5Z2 (3)

Stiffness measured in the trabecular and cortical compart-ments are denoted Tb.C33 and Ct.C33, respectively. In asecond step, the remaining stiffness coefficients werederived using a micro-mechanical model proposed byHellmich et al. (2004) (referred to as ‘‘mineral foam ma-trix with collagen inclusions’’), which estimates all theterms of the stiffness tensor as a function of the volumefractions of the three elementary constituents: water(fH2O), collagen (fcol) and hydroxyapatite (fHA). Usingdata collected by Broz et al. (1995), Raum et al. (2006)found that the ratio fcol/fH2O is highly correlated withvariation in fHA (R . 99). Making use of the propertythat the sum of the volume fractions is necessarily one,the Hellmich model fH can be used to calculate all theterms of the stiffness tensor:

Cij 5FHðfHAÞ (4)

The value of the parameter fHA for each coordinate pointðx1; x2Þ is derived by solving C33 5 FH (fHA) using thevalue of C33 derived from impedance (eqn [3]). All termsof the stiffness tensor can be obtained with eqn (4), butbecause the simulation is 2-D and the waves are polarizedin the isotropy plane ðx1; x2Þ, only two independent coef-ficients are required for the simulation. The micro-mechanical model described above was chosen becauseit yields sufficiently realistic values of elasticity for thepurposes of the work, as discussed in Grimal et al.(2008). The mass density of the bone matrix is obtainedusing the definition of the impedance for longitudinalwaves propagating in direction x3:

r5 Z2=C33

Examples of bone property maps used for the simulationare provided in Figure 5.

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Bone models of decreasing complexityThe model of each FN cross section (geometry, elas-

ticity, density) built as described above is referred to as aS0-type model. This type accounts for the sample’s over-all shape, microstructure, cortical porosity and heteroge-neous matrix elasticity and density. The numericalsimulations of wave propagation based on S0-typemodels are as realistic as possible, being based on state-of-the-art techniques for quantitative imaging of boneproperties. To determine which bone characteristicshave a significant impact on the transmitted ultrasonicsignal, we defined four other model types that wereobtained by progressive simplification of the initial S0-type model (Fig. 2 and Table 1).

Model type S1 was derived from S0 to assess specif-ically the impact of the trabecular compartment. S1 wasobtained by removing the trabecular bone and replacingit with water (Fig. 2b), all cortical characteristics remain-ing identical for S0 and S1. The aim with model type S2

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Fig. 6. Variation in effective elastic constants, Cij, with porosityfor a set of given mineralized matrix stiffness constants, ascalculated with the Mori-Tanaka method. The method wasused to build model type S4. The numerical code used to calcu-late these curves was provided as supplementary material inGrimal et al. (2011b). C11 5 C22 and C33 are longitudinal stiff-ness coefficients; C44 5 C55 and C66 are shear stiffness

coefficients.

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was to assess the effect of the heterogeneous spatial dis-tribution of the tissue matrix elasticity. S2 was identical toS1 except that the matrix impedance was no longer het-erogeneous: the average impedance bZ replaced thepoint-dependent impedance Zðx1; x2Þ (Fig. 2c).

The effect of porosity on time-of-flight, T, was inves-tigated with model types S3 and S4 in which the poreswere not modeled explicitly (Fig. 2d). In S3, the corticalshell was considered as being a non-porous homogeneousmaterial with properties identical to those of the corticaltissue defined in S2; that is, all the points of the corticalshell (inside and outside the porosity map) had thesame stiffness and density derived from impedance bZ .In contrast, the cortical shell in S4 was made of a homo-geneous material the properties of which were ‘‘effec-tive’’ properties defined to account for the presence ofpores. Cortical bone was regarded as a two-phase com-posite material made of a mineralized matrix (trans-versely isotropic as described above) pervaded byinfinitely long cylindrical pores. The effective elasticitytensor was obtained by using theMori-Tanaka homogeni-zation method (Zaoui 2002). Effective elasticity wascalculated for each bone using the corresponding porosityand elasticity derived from the SAM measurement. Thetypical variations in effective elasticity with porosity, ascalculated with the Mori-Tanaka method, are plotted inFigure 6 for the purpose of illustration. The effective den-sity in model-type S4 was calculated as the arithmeticaverage of matrix and water densities.

Two sets of simulations were performed (for all sim-ulations, the time-of-flight of the first-arriving signal, T,was evaluated): (1) A total of 9 3 4 simulations for thenine cross sections and the four different model typeswere conducted to assess the influence of model typeon T (see Methods). (2) For three of the nine cross sec-tions, simulations were conducted to evaluate T whenthe porosity or matrix stiffness was modified within aphysiological range (see Methods). The three chosencross sections were representative of the investigateddata set.

Analysis of simulation results for model types S0–S4Post-processing of simulation output yields nine

time-of-flights (cross sections i 5 1$$$9), Tk(i), foreach model type (k 5 0$$$4). Several quantities werecalculated to gain insight into the relationships betweenthe model characteristics and the acoustic response, Tk(i):

� For each cross section, the difference between thereference time-of-flight T0 obtained for the S0-typemodel, and Tk obtained for another model type wascalculated:

dTkðiÞ5 TkðiÞ-T0ðiÞ; i5 1$$$9; k5 1$$$4

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� Student t-tests were performed to test whether themean of each dTk (k5 1$$$4) is significantly differentfrom zero.

� The statistical relationships between T0ðiÞ and theoutput of one of the simplified models are summarizedin the equation

dTk 5 dMk6ICk; k5 1$$$4 (5)

where

dMk 51

9

X9

i5 1

dTkðiÞ

is group-averaged bias and ICk is the 95% confidence in-terval, which represents random fluctuation caused bydifferences in the geometry and distribution of materialproperties in the cross sections (inter-sample variability).

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Table 2. Geometrical parameters and tissue mechanicalproperties of the femoral neck cross sections derived

Numerical simulation of US propagation d Q. GRIMAL et al. 7

� Linear regressions between T0 and Tk (k5 1$$$4) werecalculated.

from scanning acoustic microscopy measurements(N 5 9)

Mineralized matrix elasticity Cortical bone, Ct.C33 38.6 6 3.2 GPaMineralized matrix elasticity Trabecular bone,

Tb.C33

32:96 2:1 GPa

Cortical porosity, Ct.Po 18:16 3:5 %Cortical thickness, Ct.Th 2:56 0:7 mmCortical area/trabecular area 58:76 21:6 %

Assessment of sensitivity to variation in materialproperties

Another set of FDTD simulations were performedfor three cross sections (S2-type models) to assess thesensitivity of T to physiological variations in porosity(Ct.Po) and mineralized matrix elasticity (Ct.C33)(Bousson et al. 2004; Fratzl-Zelman et al. 2009),independently of variations in cross-sectional sampleshape. Only independent variations of Ct.Po (‘‘porosity’’data set) and Ct.C33 (‘‘elasticity’’ data set) were consid-ered. To obtain the ‘‘porosity’’ data set, Ct.Po wasincreased and decreased stepwise by dilation and erosionof pixels starting from the measured porosity map,whereas Ct.C33 was set to its measured value at each pointof the cross sections. Erosion was combined with logicaloperations to avoid the perforation of bone inner or outersurfaces. Accordingly, bone thickness remained un-changed. To obtain the ‘‘elasticity’’ data set, Ct.C33 wasincreased and decreased stepwise starting from themeasured value, while the porosity map remained un-changed. Numerical simulations were performed foreach cross section of the ‘‘porosity’’ and ‘‘elasticity’’data sets, and arrival time Twas calculated. For each crosssection, the variations of time-of-flight (DT), with respectto the non-modified model, associated with variations inporosity (DCt.Po) and matrix stiffness (DCt.C33) was as-sessed. The influence of porosity variations (respectively,matrix stiffness) on acoustic response was quantified withlinear regressions of DT versus DCt.Po (respectively,DCt.C33) using the ‘‘porosity’’ data set (respectively,‘‘elasticity’’ data set).

All statistics were performed using the MATLABstatistics toolbox (The MathWorks, Natick, MA, USA).p-Values,0.05 were considered to indicate significance.

RESULTS

Bone characteristics assessed with SAM exhibitednoticeable variability, indicating the diversity of theselected FN cross sections (Table 2).

The differences in acoustic propagation for thedifferent model types are illustrated in Figure 7. Thereceived signals for the different model types had similarprofiles, although the amplitude of the ‘‘direct wave’’(which travels along a straight line through the trabecularcompartment) was smaller for S0 because of the scat-tering by the trabeculae. In particular, the wave frontsand received signals for S4 were very similar to thosefor S1, although porosity is not modeled explicitly.

Inspection of the snapshots of the simulated acousticfields indicated that the FAS time-of-flight, T,was always

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associated with the wave guided by the cortical shell. Thedirect wave does not contribute to the FAS. This is consis-tent with previous simulations and experimental results(Grondin et al. 2010; Grimal et al. 2013).

The Lilliefors test indicated that none of the sets ofdTk followed a non-normal distribution. Model types S1and S2 did not significantly differ from S0 as evidencedby the outcome of the t-test and the small confidence in-tervals (Table 3, Fig. 8). As expected, times-of-flight forS3 models, which are stiffer models because they do notaccount for porosity, were significantly smaller than thosefor S0 (negative bias dM).

The effective model S4 was found to be statisticallydifferent from the reference case S0, but with a moderatebias. Times-of-flight obtained with S0 and S4 were highlycorrelated (R2 5 0.96), and the slope of the linear regres-sion was equal to one.

Regarding the sensitivity of T to variations inporosity and matrix elasticity, we found a linear relation-ship between DT (ms) and DCt.Po (%) (DT 5 0.12 3DCt.Po 1 0.03, R2 5 0.94, p , 5 3 1023) (Fig. 9)and between DT (ms) and DCt.C33 (GPa) (DT 5–0.08 3 DCt.C33 1 0.04, R2 5 0.95, p , 5 3 1023)(Fig. 10).

DISCUSSION

Different models of FN cross sections with sample-specific geometry and mechanical properties were testedto assess the impact of modeling assumptions for thesimulation of through-transmission ultrasound measure-ments in the megahertz range. Scanning acoustic micro-scopy, here with a resolution of 23 mm, not onlyprovided access to the overall shape of the cross sections,but also allowed mapping of the vascular porosity, trabec-ular microstructure and elastic properties of the matrixafter conversion of impedance to elasticity.

We have focused our interest on the time-of-flight, T,of the FAS of the wave transmitted through the FN, aparameter that can be measured with a clinical prototype(Barkmann et al. 2010). Furthermore, T had been found tobe highly correlated with femur strength in an ex vivostudy (Grimal et al. 2013).

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Fig. 7. Snapshots of the acoustic displacement field in one of the femoral neck cross sections for three model-types: left:S0,middle: S1, right: S4. The circumferential wave propagating in the cortical shell is visible in advance of the direct wave(in the medullar cavity), particularly in the inferior part of the femoral neck. The acoustic signal integrated on the wholesurface of the receiver is shown together with a magnification of the early-arriving signal where the time-of-flight, T, is

retrieved (time indicated by vertical line).

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The most comprehensive model type (S0) appearedto be unnecessarily complex for the purposes of modelingthe propagation of the waves guided by the cortical shelland determining T. First, T did not change when trabec-ular bone was removed and replaced with water. Thiswas a consequence of the weak coupling of the guidedwaves in the cortical shell to the low-density trabecularbone. This strongly suggests that it is not necessary tomodel the trabecular compartment to simulate FAS mea-surements. Second, the heterogeneity of the mineralizedmatrix did not have an effect on T, which remained un-changed in type S2 as compared with type S0 models.

Table 3. First-arriving signal time-of-flight (T) obtained with eachmodel (S

Model type Bias, dM (ms) Confidence interval, IC

S1 –1.2 3 10–3 4.4 3 10–2

S2 6.1 3 10–2 0.28S3 –1.6 0.9S4 –0.24 0.57

* See eqn (5). The fourth column gives the result of the Student t-test testingmeans the difference in results between model S0 and the tested model are sta

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The latter heterogeneity is due to local variations in thedegree of mineralization and elasticity of osteons andinterstitial tissue (Currey 2002). Third, it seems that itis not necessary to know the detailed structure of thecortical porosity. The results obtained with S4-typemodels are close to those obtained with the reference S0models, as evidenced by the linear regressions, althoughthere was a small but significant difference in the meanvalues (t-test). This means that the representation of thepore structure in the cross section may be replaced by ahomogeneous material with effective properties esti-mated solely from the overall cross-section porosity and

model type compared with that obtained with the reference

0)*

k (ms) p (t-test) Regression equation, R2

0.88 T1 5 0:99 T010:32 ð1Þ0.22 T2 5 0:96 T011:14 ð0:99Þ5 3 10–6 T3 5 1:06 T023:53 ð0:92Þ3.5 3 10–2 T4 5 T020:31 ð0:96Þ

whether a model gives different T values on average: a p-value, 0.05tistically significant.

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Fig. 8. Box plot of the time-of-flight, T, for model types S0–S4,all nine cross-sections pooled. The 25th and 75th percentiles aregiven by the lower and upper bounds of the box; the medianvalue is indicated by the line inside the box; and extreme values

are indicated by the ends of the vertical dashed bar.

Fig. 10. Relative variations in time-of-flight, T, versus stepwiserelative variations in cortical stiffness, Ct.C33, for three samplesin the configuration of model type S2. Each marker type corre-sponds to one sample. The straight line is the linear regression

for the pooled data.

Numerical simulation of US propagation d Q. GRIMAL et al. 9

the mean value of the matrix elasticity. The Mori-Tanakamethod was chosen to derive effective properties becauseit is known to model correctly the decrease in effectivebone stiffness with increasing porosity. More precisely,cortical bone effective stiffness coefficients calculatedwith the Mori-Tanaka method and the method of asymp-totic homogenization (Grimal and Parnell 2011; Parnelland Abrahams 2006; Parnell and Grimal 2009) arevirtually identical (Parnell et al. 2012), and the latterwas validated by comparison with experiments (Grankeet al. 2011).

Fig. 9. Relative variations in time-of-flight, T, versus stepwiserelative variations in cortical porosity, Ct.Po, for three samplesin the configuration of model type S2. Each marker type corre-sponds to one sample. The straight line is the linear regressionfor the pooled data.Models S3 and S4 yielded significantly lower

values compared with model S0 (see Table 3).

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Two explanations are possible for the small bias be-tween T values obtained for the S4 and S0model types: (1)The homogenization method may not give a sufficientlyaccurate estimation of the apparent elasticity sensed bythe propagating wave. Parnell et al. (2012) found thatthe Mori-Tanaka solution is close to the upper Hashin-Rosen bound. Therefore, the Mori-Tanaka method mayover-estimate the apparent elastic properties sensed bythe wave. (2) The wave propagation paths may be slightlymodified in the homogenized cross sections as comparedwith the S0 case, because of a heterogeneous distributionof pores and matrix properties in the cross section. Themeasured value of T is related to a certain wave path,likely through the inferior part of the cortex. Comparedwith the case of the full model S0, the material propertiesof the cross section in the simplified model S4 are iden-tical at all points. In particular, if a difference exists be-tween properties in the superior and inferior parts of theFN, this difference is lost in S4, for which the materialproperties lie between the inferior and superior propertiesof model S0. In our samples, however, the small differ-ences in material properties observed between upperand lower FN are not likely the cause of the negativebias observed between the results for model types S0and S4. Nevertheless, it might prove useful to incorporateinto future modeling possible differences in effective ma-terial properties, in particular, between the inferior andsuperior FN (Malo et al. 2013). This can be done withoutadditional numerical cost with FDTD methods.

The absence of noticeable effects of bone matrix het-erogeneity and the possibility of replacing pore structurewith effective properties is related to the fact that thewave-length (�8 mm at the central frequency) was much largerthan the characteristic dimension of inhomogeneities.

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10 Ultrasound in Medicine and Biology Volume -, Number -, 2013

The possibility of replacing pore structure witheffective properties is an important finding because itallows considerable simplification of sample-specificmodels for simulation of the FAS, as well as for simula-tion of other femur QUS techniques (e.g., guided waves).The actual porosity and matrix elasticity of one samplecan be accounted for without the need to model the poresexplicitly. Rather, they can be taken into account in an im-plicit manner by homogenization. Therefore, the FDTDspatial grid step for numerical simulations can be chosenlarger than it would be to model explicitly the microstruc-ture, leading to significant reductions in computationalcosts. In additional computations (not presented here),we found that the computation time is reduced by a factorof 7 by changing the FDTD grid step from 16 to 200 mm,without compromising the stability of the numerical solu-tion. The size and complexity of the simulation input dataare also reduced by resorting to homogenization.

The models described in this article allow a system-atic investigation of the influence of the variations in boneproperties like porosity and elasticity that have a possibleimpact on strength. Bone pathologies and therapeuticagents for osteoporosis have been reported to alter theporosity and mineral content or elasticity of the corticalbone matrix (Boivin and Meunier 2002; Borah et al.2010; Brennan et al. 2009). Simply disregarding thepores (model S3) leads to a large decrease in T, meaningthat although the porosity is on average ,20%, itcontributes to a large part of the variations in T. Also,the sensitivity of QUS to variations in cortical porosityhas been reported in experimental studies (Bossy et al.2004b; Grondin et al. 2012; Raum et al. 2005).Simulations illustrated here (Figs. 9 and 10) indicate thatvariations in the porosity and elasticity of the mineralizedmatrix, within a physiological range, both have an impacton T. Note that an increase in porosity may imply adecrease in cortical shell thickness because of endocorti-cal trabecularization. Coupled and independent effectsof porosity and thickness variations on QUS propagationat the FN have been investigated by Rohde et al. (inpress). Variations in T are likely to be measurable in anexperiment given that the precision of the estimation ofT is typically well below 1 ms (Grimal et al. 2013). Therate of variation in Twith varying porosity andC33 slightlydepends on the sample (Figs. 9 and 10), that is, on theshape of each cross section. Therefore, it seems important,in future studies, to performmultivariate analyses on largecollections of samples to investigate separately the effectsof matrix material properties, porosity, thickness andshape of the sample.

In the model, the effective elasticity of cortical bonewas derived from estimations of porosity and elasticproperties (C33) of the mineralized matrix obtained byacoustic microscopy. The latter was estimated from

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acoustic impedance according to the method describedin Raum et al. (2006). The effective stiffness in the direc-tion of the femoral neck axis, which is strongly related tothe macroscopic Young modulus in this direction, is animportant determinant of the behavior of the femoralneck in axial loading (traction-compression) as well asin bending (Cowin 2001). The purpose of using QUS isto obtain non-invasively a proxy of strength. Therefore,the question of the relationship between the latter andthe elasticity of the mineralized matrix, or corticalporosity, warrants consideration. Porosity is a well-established determinant of strength (Wachter et al.2002; Zioupos 2001) and is likely risk factor forfracture (Zebaze et al. 2010), but the relationship betweenstrength and C33 has not been investigated, to the best ofour knowledge. Note, however, that some authors havefound a positive correlation between the matrix Young’smodulus assessed by nano-indentation and strength(Zioupos 2001).

The work described here has several limitations.Only 2-D simulations in a non-dissipative propagationmedium were considered. In practice, 3-D effects dueto the complex FN shape and attenuation are expectedto play a role, which needs to be assessed with more com-plex simulations. Nevertheless, the negligible effect oftrabecular bone on the FAS and the possibility of replac-ing, to some extent, the detailed cortical bone microstruc-ture with homogenized properties for simulation are alsolikely to be true in 3-D configurations and when theintrinsic absorption of the tissue is accounted for. In areal experiment (3-D), the wave measured via the FASmay not be strictly confined in the plane considered inthe simulation because of the slight asymmetry of thefemoral neck. We proposed in Grimal et al. (2013) a tech-nique to measure the femur neck with a pair of cylindricaltransducers (center frequency5 0.5 MHz), which probesa cross-sectional slab of the femur neck approximately6 mm thick. Because the effective material properties ofthe femur neck are not expected to vary significantlywithin the thickness of the slab, the simulation based ona single cross section can be considered a reasonablemodel of the physical problem. Moreover, the effect ofbone intrinsic absorption is to diminish the signal ampli-tude; however, it is known that the low-amplitude signalof the circumferential wave can actually be measured(Grimal et al. 2013).

Because samples were embedded in polymethylme-thacrylate, the mineralized matrix stiffness probed bySAM is slightly over-estimated compared with nativestiffness. The conclusions of the study should neverthe-less be the same for native bones because here we inves-tigated relative differences between different models.

The time-of-flight of the FAS was found to be influ-enced by the stiffness of the mineralized matrix and by

17 January 2014 � 6:56 pm � ce

Numerical simulation of US propagation d Q. GRIMAL et al. 11

the porosity. In a previous work, Grondin et al. (2010)found that the time-of-flight of the FAS is also sensitiveto the geometric parameters of the FN, such as its mo-ments of inertia and cross-sectional area. Because T issensitive to several bone properties of the femur neck—material and geometric properties—it may prove difficultin a clinical configuration to interpret variations in T interms of a single property among several. Instead, severalQUS parameters should be measured. Of interest is therecent shift in the axial transmission technique frommea-surement of the FAS to a multi-modal analysis of guidedmodes (Talmant et al. 2010). Such an approach could beextended to the circumferential propagation studied here.The emerging new generation of QUS methods are aimedat recovering different bone characteristics (e.g., porosity,stiffness, geometric parameters)–some of which are notmeasurable with X-ray approaches, to enhance fracturerisk prediction. Elaborated ultrasound measurement andsignal processing techniques are under development toderive several independent variables from an experiment(Minonzio et al. 2010; Nauleau et al. 2012). Themeasured variables can then be used to solve an inverseproblem to determine elementary bone characteristics.The inverse problem will be based on a physical modelsuch as those investigated in the present work.

Acknowledgments—This work has been conducted within the EuropeanAssociated Laboratory ‘‘Ultrasound Based Assessment of Bone’’(ULAB), a cooperation of centers in Paris (France) and Kiel and Berlin(Germany), funded by CNRS (France). We thank Hendrick de Graaf forhis contribution to the statistical analysis.

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