Comparison of phase velocity in trabecular bone mimicking-phantoms by time domain numerical (EFIT) and analytical multiple scattering approaches

Ultrasonics 52 (2012) 809–814

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Short Communication

Comparison of phase velocity in trabecular bone mimicking-phantoms by time domain numerical (EFIT) and analytical multiple scattering approaches M. Molero ⇑, L. Medina Departamento de Fı´sica, Facultad de Ciencias, UNAM, Ciudad Universitaria, CP 04150, Mexico

a r t i c l e

i n f o

Article history: Received 5 November 2011 Received in revised form 20 April 2012 Accepted 24 April 2012 Available online 11 May 2012 Keywords: Multiple scattering Waterman–Truell model EFIT

a b s t r a c t The corrected Waterman–Truell model and the Elastodynamic Finite Integration Technique were used to analyze the ultrasonic wave dispersion in trabecular bones mimicking phantoms. A simple two-phase model of the trabecular bone is assumed; the trabeculae structure and the bone marrow. The phase velocity for frequencies within the range from 400 kHz to 800 kHz were computed for different scatterer arrays varying their dimensions and number. The theoretical and numerical results were compared to experimental published data, obtained from a mimicking phantom composed by a periodic array of nylon shreds (trabeculae array) immersed in a water tank. Our results showed an excellent consistency when compared to experimental data. The negative dispersions of 8.48 m/s/MHz and 9.16 m/s/MHz were computed by the multiple scattering method and the numerical approach, respectively, where the latter is closer to the experimental dispersion of 12.09 m/s/MHz. Similar result has been reported in the literature, where the dispersion predicted by the Generalized Self-Consistent Method [J. Acoust. Soc. Am. 124 (2008) 4047] is 9.96 m/s/MHz.  2012 Elsevier B.V. All rights reserved.

1. Introduction Ultrasonic techniques based on the speed of sound (SOS) and the broadband ultrasonic attenuation (BUA) measurements have been widely used for trabecular bone characterization and fracture risk assessment (QUS) [1]. The analysis of QUS data is very complex due to the structural complexity of the bone. There are numerous studies focused on understanding the interaction between acoustic waves and trabecular bone, related to the phase velocity and attenuation of the incident ultrasonic wave [2–4]. Several theories of wave propagation in viscoelastic media have been applied to the trabecular bone considered as a two-phase composite. Some of them predict the dependency of backscattering coefficient with frequency enabling the assessment of density and micro-structural characteristics [5–12]. Other investigations are based on the effective acoustic properties computation for low concentration of scatterers taking into account multiple-scattering phenomena [13–16]. Also the ultrasonic wave propagation in fluidsaturated cancellous bone has been investigated in terms of Biot’s model [17–21], and an alternative theoretical approach in which the bone is considered as multilayer composite was introduced in [22,23]. Time domain numerical approaches have also been proposed to simulate the complex propagation phenomena in

⇑ Corresponding author. E-mail address: [email protected] (M. Molero). 0041-624X/$ - see front matter  2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ultras.2012.04.007

cancellous bone [24–28]. Up to now the interaction between ultrasonic energy and trabecular bone has not fully understood. The main discussion is on the behavior of the phase velocity, as function of frequency, that leads to a negative dispersion value. The aim of this work is to study the phase velocity of the ultrasonic waves interacting with trabecular bone using theoretical and numerical approaches. The first one corresponds to the analytical approximation of Waterman and Truell’s (WT2D) corrected model [29]. The second one deals with the Elastodynamic Finite Integration Technique (EFIT) to numerically solve the linear elastodynamic equations in heterogeneous media [30]. The advantages and drawbacks of both schemes are discussed. Experimental ultrasonic data of bone-mimicking phantoms reported by Wear [31,32] was used to validate our theoretical work.

2. Multiple scattering methods The elastic wave propagation in composite materials consists of a complex mixture of multiple mode conversion and multiple scattering which results in a ‘‘diffusive’’ energy transport. The effect of inhomogeneity manifests itself in slowing down the propagation and dispersion of a source wave, and it is represented as change in the phase velocity and attenuation of the incident acoustic waves. Several methods have been developed to describe these effects, among them are the analytical dynamic homogenization (e.g. WT2D) and the time-domain numerical (e.g. EFIT) approaches. The WT2D is a very

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accurate technique based on incident normal waves into randomly distributed scatterers with regular geometries, while EFIT allows to built a more realistic experimental setup for precise scatterers position and scanning procedures.

(k, l) and the mass density (q). A discretised version of Eqs. (3)–(7) respect to the n-cell and regular grid Dx = Dy is [41]:

v nx;i;j ¼ v n1 x;i;j þ

 Dt  n1=2 n1=2 n1=2 Bx sxx;iþ1;j  sn1=2 xx;i;j; þ sxy;i;j;  sxy;i;j1 þ DtBx F x;i;j Dx ð8Þ

v ny;i;j ¼ v n1 y;i;j þ

 Dt  n1=2 n1=2 n1=2 þ DtBy F y;i;j By sxy;i;j  sn1=2 xy;i1;j þ syy;i;jþ1  syy;i;j Dx ð9Þ

2.1. Theoretical method: dynamic homogenization The multiple scattering model formulated by Waterman and Truell has been extensively used for predicting effective properties in composite materials, such as the longitudinal and shear wavenumbers which are related to the phase velocity and attenuation coefficient as function of frequency [33]. This formulation based on the Foldy’s approximation [34] expresses the far-field scattered from any particular scatterer as a multipole expansion. For spherical scatterer geometry see Eq. (3.8) in [33] and cylindrical scatterers see Eq. (2) in [35]. The effective wavenumber for multiple scattering of scalar waves by random distribution of isotropic scatterers is given as 2

2

keff ¼ km  ð4n0 f0 Þi  Q ðxÞ where Q ðxÞ ¼

4n20 k2m



ð1Þ

8n20 2 km

p

Z p 0

cotðh=2Þ

d 2 f dh dh h

ð2Þ

where fh is the far field pattern and h is the polar angle of a cylindrical scatter. From Eqs. (1) and (2), the phase velocity is calculated as v / ¼ Refkx g. eff

2.2. Numerical method: Elastodynamic Finite Integration Technique A reliable way to numerically emulate the wave propagation in heterogeneous media is the Elastodynamic Finite Integration Technique (EFIT). This method is able to compute N-phase composite material scenarios and to carry out ultrasonic scanning procedures similar to the experimental setup. The time-domain numerical scheme EFIT is used to simulate wave propagation in heterogeneous materials [30,37–40]. The field component to be calculated, is placed at the centre of an appropriate control volume and the integration of the corresponding equation is calculated over this cell. The linear elastodynamics equations in terms of the longitudinal particle velocity (vx, vy) and stress (sxx, syy, sxy) components are:

@ sxx @ sxy þ þ Fx @x @y @s @s qv_ y ¼ xy þ yy þ F y @x @y @v @v s_ xx ¼ ðk þ 2lÞ x þ k y þ Gxx @x @y @v y @v x s_ yy ¼ ðk þ 2lÞ þk þ Gyy @y @x   @v x @v y s_ xy ¼ l þ þ Gxy @y @x

qv_ x ¼

h i h io Dt n ðk þ 2lÞ v ny;i;j  v ny;i;j1 þ k v nx;i;j  v nx;i1;j Dx þ DtGyy;i;j ð11Þ

n1=2 snþ1=2 yy;i;j ¼ syy;i;j þ

n1=2 snþ1=2 xy;i;j ¼ sxy;i;j þ

 f02  fp2 ; keff and km are the wave numbers associ-

ated with the effective medium and the matrix, n0 refers to the P n number of scatterers per unit area. f0 ¼ 1 and n ðiÞ An P1 n fp ¼ n ðiÞ An , are the forward and backward scattering amplitudes and An are unknown constants (see Appendix A), respectively. However, this expression is valid only for far field approximations or normal incidence. A derivation with no dependency on the incidence angle, which provides a good prediction of phase velocity at low scatterers concentration [36], was proposed by Linton and Martin [29]:

Q ðxÞ ¼ 

h i h io Dt n ðk þ 2lÞ v nx;i;j  v nx;i1;j þ k v ny;i;j  v ny;i;j1 Dx þ DtGxx;i;j ð10Þ

n1=2 snþ1=2 xx;i;j ¼sxx;i;j þ

ð3Þ ð4Þ ð5Þ ð6Þ

h i Dt lxy v nx;i;jþ1  v nx;i;j þ v ny;iþ1;j  v ny;i;j þ DtGxy;i;j Dx ð12Þ

where Bx, By are effective buoyancies, and lxy is the effective rigidity defined as:

Bx ¼

By ¼

2

qiþ1;j þ qi;j 2

qi;jþ1 þ qi;j

lxy ¼

4 1 1 1 þ þ li;j liþ1;j li;jþ1 þ liþ1;jþ1 1

ð13Þ

ð14Þ

ð15Þ

To guarantee numerical stability of the EFIT code, the grid size in x and y directions and the time step must be Dx 6 Cmin/(10Fmax) pffiffiffi and Dt 6 ð1= 2ÞDx=C max , respectively, where Cmin and Cmax correspond to the lowest and highest velocities in the heterogeneous medium, and Fmax is the highest frequency component in the signal respect to the operating frequency Fc (e.g. Fmax = 2Fc). To implement the stress-free boundary conditions, a vacuum formulation (VCF) is used, where the elasticity moduli and Bx and By tend to zero [42]. 3. Two-phase wave propagation models The trabecular bones are highly dispersive acoustic media composed by randomly shaped trabeculae (considered as scatterers) surrounded by bone marrow. The trabecular bone can be represented as fluid-like medium (bone marrow) containing a low volume fraction (5–20%) of solid structure of interconnected trabecular elements (mineralized collagen or bone). A simple two-phase phantom of the trabecular bone microstructure has been developed by Wear [31,32]. It consists of two-dimensional regular array of parallel nylon cylinders (trabeculae) immerse in a water tank (bone marrow), where nylon–water composite has been chosen due to its similar frequency-dependent scattering effects behavior as the trabecular bone microstructure. Experimental acoustic phase velocity results were published for different cylinder diameters and inter-element distances, which varied from 152 to 305 lm and 700 to 1000 lm, respectively. The two-phase multiple scattering trabecular models were simulated, as follows:

ð7Þ

where F(~) and G(~) are the force and the stress sources components, and the elastic medium properties are given by the Lamé constants

 The Waterman–Truell model (WT2D) and its corrected version (WT2DLM) models were used for the analytic computation of the phase velocity for different phantoms arrays. Both models

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assume the composite material as an array of identical cylindrical scatterers randomly distributed into a host medium (water). For each phantom array the forward and backward scattering amplitudes are computed by solving the equation system (see Eq. (21) in Appendix A) for a given boundary conditions (see Eq. (20) in Appendix A).  The EFIT is able to handle numerical simulation of ultrasonic wave propagation in geometrical scenarios, as the trabecular bone mimicking phantoms and the ultrasonic experimental setup developed by Wear. Fig. 1 shows two examples of geometrical scenarios used in our numerical simulations composed by an array of parallel cylinders uniformly distributed across a 55  76.2 mm2 water tank. The through-transmission ultrasonic inspection is emulated as follows: (i) two ultrasonic transducers, one acting as a transmitter (T) and the other as a receiver (R), are modeled as line sources of 19.05 mm length, (ii) it is assumed that the incident energy behaves as a longitudinal ultrasonic pulse defined by p(t) = (1  cos(pFct))cos(2pFct)" t < 2/Fc where Fc is the operating frequency, (iii) the acoustic propagation is computed under this specific conditions, and (iv) the received energy is the contribution of each point related to the receiver line sources. The phase velocity of each received signal is calculated by

v ðFÞ / ¼

ds ts  tw þ Cdws

ð16Þ

where ds is the bone phantom length (12.7 mm), Cw is the speed of sound in water, ts and tw are the time taken by the wave to travel ds with and without the phantom, respectively. The difference in transit times is computed by a zero-crossing algorithm. This time domain algorithm consists on finding the specific time at both signals are in phase; one coming from the water (reference signal) and other coming from two-phase composite. The simulation of scenarios with no phantom were carried out by the Acoustic Finite Integration Technique (AFIT) [43]. The acoustic parameters used in the simulations are: (i) longitudinal velocities 2600 m/s (Cn, nylon) and 1480 m/s (Cw, water), (ii) nylon shear velocity 1300 m/s, and (iii) mass densities 1100 kg/m3 (qn, nylon) and 1000 kg/m3 (qw, water). 4. Analytical and numerical results Phase velocity as function of frequency computed by WT2D and WT2DLM are shown in Fig. 2. In this figure, the phase velocity was computed for different phantom scenarios: the nylon cylinders

(a)

randomly distributed into a water matrix are varied in size (254 lm and 305 lm) and number (7.9% and 15.8%). At low frequencies (100–800 kHz) both methods have similar results; the dispersion, defined as the slope of a curve Fc vs v/, increases as the volume fraction of scatterers increases. The maximum frequency, before the transition zone starts (oscillation occurs), depends on the size of the scatterers; the first maximum frequency (1.9 MHz) correspond to scatterers of size d = 305 lm, where the phase velocity is smaller at higher volume fraction, the second maximum frequency is at 2.25 MHz corresponding to smaller scatterers (d = 254 lm). However, the ratio between scatterers-diameter/wavelength at the global minima is almost 0.4 for all cases. Phase velocity results of a bone-phantom composed by 254 lm diameter and 7.9% volume fraction of cylindrical nylon scatterers for a different operating frequencies are shown in Fig. 3. The negative dispersion values as reported in [31,32] are 12.09 m/s/MHz (Experimental), 8.48 m/s/MHz (WT2DLM) and 9.16 m/s/MHz (EFIT). A different behavior is found when the operating frequency and volume fraction are fixed to 500 KHz and 7.9% respectively (see Fig. 4), the phase velocity increases while the scatterers diameters increases. When the scatterers diameter is twice as the initial value (152 lm) the velocity increases 3%, it is expected that a larger the diameter the velocity will tend to nylon velocity. The phase velocity error with respect to the experimental data is less than 0.15%, where the minimum error is obtained by the WT2DLM. Fig. 5 shows a linear increment of the phase velocity while the volume fraction is increased from 1.8% to 11.4% of scatterers of 254 lm diameter and 500 kHz of operating frequency. The relative errors are in the range of 0.05–0.06%.

5. Concluding remarks The theoretical models of Waterman–Truell and its corrected version have been compared, at low frequencies both methods shows similar results for a specific two-phase models with low volume fraction of scatterers (7.9–15.8%). For frequencies greater than 800 kHz the difference between both approaches becomes notorious, mainly at higher volume fraction of scatterers. It seems that the main contribution of the corrected version by Linton and Martin becomes important for larger number of scatterers and high frequency analysis. Even though that the experimental data is in the lower frequency range the corrected version was chosen to be validated, since its formulation is closer to an experimental

(b)

Fig. 1. Two trabecular bone phantom scenarios: (a) 1.8% volume fraction of 152 lm diameter scatterers and (b) 305 lm diameter scatterers at 11.4% of volume fraction.

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M. Molero, L. Medina / Ultrasonics 52 (2012) 809–814

Fig. 2. Phase velocity predictions as a function of frequency obtained by the WT2D and WT2DLM for a different phantom array.

1525

EFIT

1520

1510 1505 1500 1495

1515 1510 1505 1500 1495

1490

1490 0.4

0.5

0.6

0.7

0.8

Freq [MHz]

1525 EFIT WT2DLM

1520

Experimental

1515 1510 1505 1500 1495 1490

152

203

1485

1.82.2 2.8

3.7

5.1

7.9

11.4

scatterer volume fraction [%]

Fig. 3. Phase velocity vs frequency. Experimental data from [32].

1485

WT2DLM Experimental

1515

Phase velocity [m/s]

Phase veocity [m/s]

1520

Phase velocity [m/s]

1525

EFIT WT2DLM Experimental

254

305

scatterer diameter [µm] Fig. 4. Phase velocity vs scatterer diameter. Experimental data from [32].

Fig. 5. Phase velocity vs scatterer volume fraction. Experimental data from [32].

setup respect to the incident ultrasonic waves. The chosen theoretical method WT2DLM and EFIT were compared to experimental data, which uses several phantom arrays. For all different scenarios analytical and numerical models behave very close to the experimental data, where the analytical technique gives a closer results with an error less than 0.2%, for all cases, where the error is defined as the linear difference between experimental and computed dataHowever, the phase velocities computed by WT2DLM were closer to the experimental data, even though that the numerical EFIT computations can emulate a closer experimental setup and similar geometry arrangements while WT2DLM is based on a random distribution of scatterers. If the nodes in EFIT computation increases, it is expected that the phase velocity values will be closer than the analytical one, however the computational requirements increase. Since the relative errors between the calculated phase velocities and experimental data are less than 0.16%, cannot justify the increment on computational requirements. On the other hand negative dispersions computed by the Multiple Scattering Method (WT2DLM) and Time Domain Elastodynamic Technique (EFIT) are lower than the one published by Generalized Self-Consistent Method (GSCM) of 9.96 m/s/MHz

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M. Molero, L. Medina / Ultrasonics 52 (2012) 809–814

[14] and higher to the Independent Scattering (IS) of 6.95 m/s/ MHz [12], where GSCM and EFIT results are closer to the experimental dispersion of 12.09 m/s/MHz. From these theoretical results, it seems that the trabecular bone mimicking phantom can be approximated as a two-phase model surrounded by an artificial synthetic medium.

!

ð2Þ

Q 22 ¼  n2  n 

ðiÞ

Q 11 ¼  and

Acknowledgements

ðj2 aÞ2 J n ðj2 aÞ  ðj2 aÞJ nþ1 ðj2 aÞ 2

1 q1 ðj2 aÞ2 J n ðk1 aÞ 2 q2

n ¼



1; 2;

ð30Þ

ð31Þ

n¼0 . nP1

References The authors would like to acknowledge PAPIIT (IN118811-3) and CONACyT (131937) for their financial support. M.M. thanks to the National Council of Science and Technology of Mexico (CONACyT) for the postdoctoral stipend and the Spanish Science and Innovation Ministry (PN I+D+I BIA 2009-14395-C04-01).

Appendix A The longitudinal and shear acoustic potentials, for cylinders (2) of radio a randomly distributed into a fluid medium (1) are defined as:

/1 ðr >a ; #; tÞ ¼

1 X An Hn ðk1 rÞein# eixt

ð17Þ

n¼0

/2 ðr
1 X Bn J n ðk2 rÞein# eixt

ð18Þ

n¼0

w2 ðr
1 X C n J n ðj2 rÞein# eixt

ð19Þ

n¼0

where k~ and j~ are the longitudinal and shear wavenumbers, respectively. An, Bn and Cn are the unknown constants that can be determined for specific boundary conditions, such as: ð1Þ

ð2Þ

ur ¼ ur

ð2Þ sð1Þ rr ¼ srr ð2Þ 0 ¼ srh

ð20Þ

where u and s define the displacement and stress at the boundary r = a. The linear equation system to find the unknowns constants is

2

ð1Þ

P 6 11 6 Q ð1Þ 4 11 0

32 3 2 ðiÞ 3 An P11 7 7 n 6 ðiÞ 7 ð2Þ ð2Þ 6 ¼  B  i 4 4 5 Q 11  Q 12 7 n n Q 11 5 5 ð2Þ ð2Þ C n 0 Q Q ð2Þ

ð2Þ

P 11  P 12

21

ð21Þ

22

where ð1Þ

ð22Þ

ð2Þ

ð23Þ

ð2Þ

ð24Þ

ðiÞ

ð25Þ

P11 ¼ nH1n ðk1 aÞ  ðk1 aÞH1nþ1 ðk1 aÞ P11 ¼ nJ n ðk2 aÞ  ðk2 aÞJ nþ1 ðk2 aÞ P12 ¼ nJ n ðj2 aÞ P11 ¼ nJ n ðk1 aÞ  ðk1 aÞJ nþ1 ðk1 aÞ ð1Þ

Q 11 ¼ 

ð2Þ Q 11

¼

1 q1 ðj2 aÞ2 H1n ðk1 aÞ 2 q2 ! ðj2 aÞ2 J n ðk2 aÞ þ ðk2 aÞJ nþ1 ðk2 aÞ n n 2 2

ð26Þ

ð27Þ

ð2Þ

ð28Þ

ð2Þ

ð29Þ

Q 12 ¼ nðn  1ÞJ n ðj2 aÞ  ðj2 aÞJ nþ1 ðj2 aÞ Q 21 ¼ nðn  1ÞJ n ðk2 aÞ  ðk2 aÞJ nþ1 ðk2 aÞ

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