Theoretical model of guided waves in a bone - mimicking plate coupled with soft - tissue layers

 Vietnam Journal of Mechanics, VAST, Vol.43, No. 1 (2021), pp. 91 – 104
 DOI: https://doi.org/10.15625/0866-7136/15774
 THEORETICAL MODEL OF GUIDED WAVES
 IN A BONE-MIMICKING PLATE COUPLED
 WITH SOFT-TISSUE LAYERS
 Hoai Nguyen1, Ductho Le2, Emmanuel Plan3,4, Son Tung Dang5, Haidang Phan3,6
 1Institute of Physics, Vietnam Academy of Science and Technology, Hanoi, Vietnam
 2Faculty of Mathematics, Mechanics and Informatics, VNU University of Science, Vietnam
 3Institute of Theoretical and Applied Research, Duy Tan University, Hanoi, Vietnam
 4Faculty of Natural Science, Duy Tan University, Da Nang, Vietnam
 5Sintef industry, S. P. Andersens veg 15B, 7031 Trondheim, Norway
 6Faculty of Civil Engineering, Duy Tan University, Da Nang, Vietnam
 ∗E-mail: phanhaidang2@duytan.edu.vn
 Received: 21 December 2020 / Published online: 21 February 2021
 Abstract. Quantitative ultrasound has shown a significant promise in the assessment of
 bone characteristics in the recent reports. However, our understanding of wave interac-
 tion with bone tissues is still far from complete since the propagation of ultrasonic waves
 in bones is a very challenging topic due to their multilayer nature. The aim of the cur-
 rent study is to develop a theoretical model for guided waves in a bone-mimicking plate
 coupled with two soft-tissue layers. Here, the bone plate is modeled as an isotropic solid
 layer while the soft tissues are modeled as fluid layers. Based on the boundary condi-
 tions set for the three-layered structure, a characteristic equation is obtained which results
 in dispersion curves of the phase and group velocities. New expressions for free guided
 waves propagating in the trilayered plate are introduced. The amplitudes of wave modes
 generated by time-harmonic loads applied in the plate are theoretically computed by reci-
 procity consideration. As an example of calculation, the normalized amplitudes of the
 lowest wave modes are presented. The obtained results and equations discussed in this
 study could be, in general, useful for further applications in the area of bone quantitative
 ultrasound.
 Keywords: guided waves; bone plate; trilayered structures; reciprocity; quantitative ultra-
 sound.
 1. INTRODUCTION
 Guided wave propagation in layered structures plays an important role in the study
of quantitative ultrasound (QUS), a method of great potential in the assessment of bone
characteristics. Bone QUS takes advantage of mechanical waves that are more sensitive
 © 2021 Vietnam Academy of Science and Technology
92 Hoai Nguyen, Ductho Le, Emmanuel Plan, Son Tung Dang, Haidang Phan
than conventional X-ray method to the determinants of bone strength [1]. Unlike X-
rays, QUS is safe for newborn babies and pregnant women because it is a non-ionizing
method. Moreover, QUS approach can provide information about the elastic properties
and defects of bones [2]. Numerous studies have been considered to understand how
ultrasound interacts with the bone structure, see, for example, [3,4]. Lowet and Van der
Perre [5] studied simulation of ultrasound wave propagation and the method to mea-
sure velocity in long bones. Numerical simulations of wave propagation and experiment
measurement were used to gain insights into the expected behavior of guided waves in
bones [6]. The velocity dispersion and attenuation in a tri-layered system, which con-
sists of a transversely-isotropic cortical bone plate sandwiched between the soft-tissue
and marrow layers, were computed using a semi-analytical finite element [7]. However,
wave propagation and scattering in bones is a very challenging topic due to the bones’
multi-layer, anisotropic, and viscoelastic nature. The understanding of wave interaction
with bones is, therefore, still quite limited and definitely needs to be expanded [1,3].
 Wave propagation in multi-layered structures is unquestionably one of the most fun-
damental problems of elastodynamics. Study of free guided waves in layered plates can
be found in textbooks [8,9] and research papers [10, 11]. The dispersion equation of
guided waves in fluid-solid bilayered plate is discussed in [12]. The phenomenon of
osculation where two dispersion curves come near to each other was observed and care-
fully studied by [13]. Wave motion generated by a loading is in general solved by the
use of integral transform technique [8, 14–16] and by the reciprocity approach [17–27].
The integral transform approach is usually used for simple half-space problems. How-
ever, it becomes more difficult for anisotropic solids, and impossible for inhomogeneous
solids. The reciprocity approach is therefore suitable for guided wave motions in layered
structures and composites.
 In this work, we present a model for ultrasonic guided waves in a trilayered sys-
tem consisting of a bone-mimicking plate coupled with a soft-tissue layer and a marrow
layer. In order to simplify the computation procedure, we ... s set to include only a single wave
mode with amplitude Bn. The negative x-direction state B is written as
 n ˆ n −ikn(x+cnt)
 uˆx = −BnUx (z) e , (45)
 n ˆ n −ikn(x+cnt)
 uˆz = −iBnUz (z) e , (46)
 n n −ikn(x+cnt)
 ux = −BnUx (z) e , (47)
 n n −ikn(x+cnt)
 uz = BnUz (z) e , (48)
 n ˜ n −ikn(x+cnt)
 u˜x = −BnUx (z) e , (49)
 n ˜ n −ikn(x+cnt)
 u˜z = −iBnUz (z) e . (50)
 The next step is to replace the expressions of states A and B into Eq. (44). The left-
hand side of Eq. (44) can be simplified because the loading is applied only at (x0, z0).
Note that the right-hand side of Eq. (44) vanishes when state A and state B are in same
direction. Therefore, there is only contribution from the counter-propagating waves, see
[17,19] for details. Since free boundary conditions are applied on the top and the bottom
of the trilayered plate, there is no contribution of the integration. Moreover, using the
orthogonality condition, derived in Appendix B, Eq. (B.10), the right-hand side of Eq. (44)
cancels out for m 6= n. It should be noted that the time-harmonic load can be arbitrarily
applied at any position in the structure. Without loss of generality, the load is applied
in the solid layer Ω. We finally find, after some manipulation, the amplitude of guided
 Theoretical model of guided waves in a bone-mimicking plate coupled with soft-tissue layers 99
waves in the positive x-direction as
 n −ikn x0
 P+ −iPUz (z0) e
 An =  
, (51)
 2 µˆ Iˆn + λIn + µ˜ I˜n
where
 hˆ
 Z
 ˆ  ˆ n ˆ n 
 In = ikn Txx (z) Ux (z) dz, (52)
 0
 Z0
 n n n n
 In = ikn [Txx (z) Ux (z) + Txz (z) Uz (z)] dz, (53)
 −h
 −h
 Z
 ˜  ˜ n ˜ n 
 In = ikn Txx (z) Ux (z) dz. (54)
 −h+h˜
 Note that Iˆn, In and I˜n are connected to the guided wave of mode n. They are ob-
tained from Iˆmn, Imn and I˜mn expressed in Eqs. (B.7)–(B.9) of Appendix B, respectively, as
m = n.
 If a virtual wave of mode n (state B) in the positive x-direction is chosen, we obtain
 n ikn x0
 P− −iPUz (z0) e
 An =  
. (55)
 2 µˆ Iˆn + λIn + µ˜ I˜n
Similarly, for a horizontal load of the form
 A −ikct
 fx = Qδ (z − z0) δ (x − x0) e , (56)
we find
 n −ikn x0
 Q+ −QUx (z0) e
 An =  
, (57)
 2 µˆ Iˆn + λIn + µ˜ I˜n
 n ikn x0
 Q− QUx (z0) e
 An =  
. (58)
 2 µˆ Iˆn + λIn + µ˜ I˜n
 Eqs. (51) and (55) represent the amplitudes of guided waves of mode n generated
by the application of a vertical time-harmonic load of magnitude P at (x0, z0) obtained
in closed-form solution. Similarly, Eqs. (57) and (58) express the amplitudes due to a
horizontal force of magnitude Q at (x0, z0).
 As an example, we calculate these expressions for a trilayered model which includes
a 3 mm-thick water layer, a 5 mm-thick aluminum layer and a 10 mm-thick water layer.
The material properties are given in Tab.1. In this model, water is used to mimic human
soft-tissue and marrow while the aluminum is used to mimic human cortical bone (see
Fig. 2 of Ref. [7]). For this calculation, the vertical load is applied at the interface of the
upper fluid layer and the solid layer with a magnitude chosen as P = µ/2. Also, low
frequencies ranging from 5 to 40 kHz were used so only the lowest guided wave modes
100 Hoai Nguyen, Ductho Le, Emmanuel Plan, Son Tung Dang, Haidang Phan
A0 and S0 will be generated. The normalized amplitudes of the lowest wave modes at
the interface of the upper fluid and the solid are displayed in Fig.4.
 Fig. 4. Amplitudes of the lowest wave modes due to time-harmonic loading
 4. CONCLUSIONS
 A theoretical approach for guided wave motions in an isotropic solid plate coupled
with two fluid layers has been proposed in this article. We have derived the character-
istic equation and obtained velocity dispersion curves for a three-layered plate. In order
to perform reciprocity application, the expressions of free guided waves have been intro-
duced. It has also analytically computed the amplitudes of guided wave modes subjected
to a time-harmonic load applied in the solid layer. As an example, we have presented the
results for normalized amplitudes of the lowest wave modes. The theoretical predic-
tions obtained in the current work will be beneficial in building models for a cortical
bone-mimicking plate coupled with soft-tissue layers and, in general, useful for further
applications in bone quantitative ultrasound.
 ACKNOWLEDGMENT
 We would like to acknowledge the International Center of Physics for support of this
research (Grant No. ICP.2020.12).
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 Theoretical model of guided waves in a bone-mimicking plate coupled with soft-tissue layers 103
 APPENDIX A. EXPRESSIONS OF GUIDED WAVES IN THE TRILAYERED
 STRUCTURE
 Eight-by-eight matrix
  ˆ ˆ 
 eikαˆ h e−ikαˆ h 0 0 0 0 0 0
  
  
  αˆ −αˆ 1/α −α −1/α α 0 0 
  1 2 1 2 
  
  1 1 
  0 0 α1 − 2α2 − α1 −2α2 0 0 
  αˆ 1 α1 
          
  ˆ 2 ˆ 2 2 2 
  λ 1 + αˆ λ 1 + αˆ 2µ 1 − α1 µ 2µ 1 − α1 µ 0 0 
 =  
D  −1 1  .
  −ikα1h −ikα2h ikα1h − ikα2h − −ikα˜ h ikα˜ h 
  0 0 e α2e e α2e α˜ e α˜ e 
  α1 α1 
      
  1 −ikα h −ikα h 1 ikα h ikα h 
  0 0 α1 − e 1 2α2e 2 −α1 e 1 −2α2e 2 0 0 
  α1 α1 
          
  −ikα1h 2 −ikα2h ikα1h 2 ikα2h ˜ 2 −ikα˜ h ˜ 2 ikα˜ h 
  0 0 2µe µ 1−α1 e 2µe µ 1−α1 e λ 1+α˜ e λ 1 + α˜ e 
  
 ˜ ˜
 0 0 0 0 0 0 e−ikα˜ (h+h) eikα˜ (h+h)
 (A.1)
 Dimensionless quantities
 "       #T
 h iT D∗ D∗ ∗ ∗ ∗ ∗ D∗
 ˆ ˆ ˜ ˜  1ˆ   2ˆ  |D | |D | |D | |D |  1˜ 
 d = d1 d2 d1 d2 d3 d4 d1 d2 = 1 2 3 4 1 ,
 |D∗| |D∗| |D∗| |D∗| |D∗| |D∗| |D∗|
 (A.2)
where
  ˆ ˆ 
 eikαˆ h e−ikαˆ h 0 0 0 0 0
  
  
  αˆ −αˆ 1/α1 −α2 −1/α1 α2 0 
  
  1 1 
  0 0 α1 − 2α2 − α1 −2α2 0 
  αˆ 1 α1 
  
 ∗  λˆ 1 + αˆ 2
 λˆ 1 + αˆ 2
 2µ 1 − α2
 µ 2µ 1 − α2
 µ 0 
D =  1 1 
  −1 1 
  −ikα1h −ikα2h ikα1h ikα2h −ikα˜ h 
  0 0 e α2e e −α2e −α˜ e 
  α1 α1 
   1   1  
  − e−ikα1h e−ikα2h − eikα1h − eikα2h 
  0 0 α1 2α2 α1 2α2 0 
  α1 α1 
 −ikα1h  2
 −ikα2h ikα1h  2
 ikα2h ˜  2
 −ikα˜ h
 0 0 2µe µ 1 − α1 e 2µe µ 1 − α1 e λ 1 + α˜ e
  
 0
  
  
  0 
  
  
  0 
  
  
and k =  0  ,
  
  ikα˜ h 
  −α˜ e 
  
  
  0 
  
 −λ˜ 1 + α˜ 2
 eikα˜ h
 (A.3)
 ∗ ∗
with Di formed by replacing the column i of matrix D by k.
 APPENDIX B. ORTHOGONALITY CONDITION
 We derive here an orthogonality condition for counter-propagating guided waves
in the trilayered plate using the reciprocity relation given in Eq. (44). The condition is
related to two free guided waves of mode m with wavenumber km (state A) and mode n
with wavenumber kn (state B). The domain is defined by a ≤ x ≤ b, −(h + h˜) ≤ z ≤ hˆ
as demonstrated in Fig.3. With the absence of a force term, the left-hand side of Eq. (44)
104 Hoai Nguyen, Ductho Le, Emmanuel Plan, Son Tung Dang, Haidang Phan
vanishes. This leads to
 Z   Z   Z  
 B A A B ˆ B A A B B A A B ˜
 τˆij uˆj − τˆij uˆj nˆ idS + τij uj − τij uj nidS + τ˜ij u˜j − τ˜ij u˜j n˜ idS = 0. (B.1)
 Sˆ S S˜
Applying free boundary conditions on the top and the bottom of the trilayered plate,
Eq. (B.1) becomes
 hˆ 0 −h hˆ 0 −h
 Z Z Z Z Z Z
 ˆmn + mn + ˜mn = ˆmn + mn + ˜mn
 FAB x=a dz FAB |x=a dz FAB x=a dz FAB x=b dz FAB |x=b dz FAB x=b dz,
 0 −h −(h+h˜) 0 −h −(h+h˜)
 (B.2)
 ˆmn mn ˜mn
where FAB , FAB and FAB are expressed as
 ˆmn Bn Am Am Bn
 FAB = τˆxx uˆx − τˆxx uˆx , (B.3)
 mn Bn Am Bn Am Am Bn Am Bn
 FAB = τxx ux + τxz uz − τxx ux − τxz uz , (B.4)
 ˜mn Bn Am Am Bn
 FAB = τ˜xx ux − τ˜xx u˜x . (B.5)
Using the displacement and stress expressions of the states A and B into Eq. (B.2), after
some manipulation, yields
  
 i(km−kn)a i(km−kn)b  

 e − e µˆ Iˆmn + µImn + µ˜ I˜mn = 0, (B.6)
where
 hˆ
 1 Z
 Iˆ = i k Tˆ n (z)Uˆ m(z + k Tˆ m (z)Uˆ n(z) dz, (B.7)
 mn 2 n xx x m xx x
 0
 0
 1 Z
 I = i [k (Tn (z)Um(z) − Tn (z)Um(z)) + k (Tm (z)Un(z) − Tm (z)Un(z))] dz, (B.8)
 mn 2 n xx x xz z m xx x xz z
 −h
 −h
 1 Z
 I˜ = i k T˜ n (z)U˜ m(z) + k T˜ m (z)U˜ n(z) dz. (B.9)
 mn 2 n xx x m xx x
 −(h+h˜)
Note that Eq. (B.6) must be satisfied for arbitrary values of a and b. Clearly, it is satisfied
if m = n. When m 6= n, it can be satisfied only if µˆ Iˆmn + µImn + µ˜ I˜mn = 0. For m 6= n,
thus, we obtain the orthogonality condition of guided waves in a trilayered plate as
 µˆ Iˆmn + µImn + µ˜ I˜mn = 0. (B.10)
 It should be noted that as m = n, quantities Iˆn, In and I˜n expressed in Eqs. (52)–(54)
are Iˆmn, Imn and I˜mn given in Eqs. (B.7)–(B.9).