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An ultrasound elastography method to determine the local stiffness of arteries with guided circumferential waves
Journal of Biomechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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An ultrasound elastography method to determine the local stiffness of arteries with guided circumferential waves Guo-Yang Li a,1, Qiong He b,1, Guoqiang Xu a, Lin Jia a, Jianwen Luo b, Yanping Cao a,n a b
Institute of Biomechanics and Medical Engineering, AML, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, PR China Department of Biomedical Engineering, School of Medicine, Tsinghua University, Beijing 100084, PR China
art ic l e i nf o
a b s t r a c t
Article history: Accepted 5 December 2016
Arterial stiffness is highly correlated with the functions of the artery and may serve as an important diagnostic criterion for some cardiovascular diseases. To date, it remains a challenge to quantitatively assess local arterial stiffness in a non-invasive manner. To address this challenge, we investigated the possibility of determining arterial stiffness using the guided circumferential wave (GCW) induced in the arterial wall by a focused acoustic radiation force. The theoretical model for the dispersion analysis of the GCW is presented, and a finite element model has been established to calculate the dispersion curve. Our results show that under described conditions, the dispersion relations of the GCW are basically independent of the curvature of the arterial wall and can be well-described using the Lamb wave (LW) model. Based on this conclusion, an inverse method is proposed to characterize the elastic modulus of artery. Both numerical experiments and phantom experiments had been performed to validate the proposed method. We show that our method can be applied to the cases in which the artery has local stenosis and/ or the geometry of the artery cross-section is irregular; therefore, this method holds great potential for clinical use. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Shear wave elastography Arterial stiffness Guided circumferential wave Finite element analysis Inverse method
1. Introduction Arterial stiffness has a close relationship with some cardiovascular diseases (Cecelja and Chowienczyk, 2012; Shirwany and Zou, 2010), and therefore, non-invasive measurement of arterial stiffness has received considerable attention in recent years. To date, methods based on an evaluation of the pulse wave velocity (Brands et al., 1998; Chubachi et al., 1994; Luo et al., 2012; Meinders et al., 2001; Huang et al., 2016), and the blood pressure induced static-strain within the arterial walls have been proposed by many authors to assess arterial stiffness (Hunter et al., 2010; Khamdaeng et al., 2012; Ribbers et al., 2007). However, in principle, it is difficult to quantitatively determine the local elastic properties of the arterial wall using these methods. Recently, the shear wave elastography method has been investigated for measuring arterial stiffness (Bernal et al., 2011; Couade et al., 2010; Li et al., in press). This method relies on the use of the acoustic radiation force (ARF) (Sarvazyan et al., 1998) to generate broad-band guided axial wave (GAW) in the arterial wall. The dispersion relation of the GAW is then determined and used to n
Corresponding author. Fax: þ 86 10 62781284. E-mail address: [email protected] (Y. Cao). 1 These authors made equal contribution to this study.
deduce the arterial stiffness. This method is promising for instantaneously measuring arterial stiffness because the measurement can be conducted a number of times in each cardiac cycle, e.g., 13 times/cycle (Couade et al., 2010). However, it has been demonstrated that the dispersion relation of the GAW is significantly affected by the curvature of the arterial wall in the low-frequency range (typically 0–1000 Hz) (Li et al., in press; Li and Rose, 2006; Maksuti et al., 2016). In the literature, critical frequency has been suggested beyond which effects of curvature are negligible and the dispersion curve of the guided axial wave can be fitted with the dispersion curve given by the Lamb wave model (Maksuti et al., 2016). In a very recent study, Li et al. (in press) showed that such a critical frequency depends on both elastic properties and geometrical parameters of the arterial wall. Besides, it is difficult to apply the method based on the GAW to the cases in which the wall thickness and/or the inner radius of the artery vary along the axial direction and/or the cross-sectional geometry of the artery is irregular (e.g., non-circular). Considering that these cases are frequently encountered in clinics, the development of a robust method to measure the local arterial stiffness is still urgently needed. Based on the above premise, this study aimed to develop a novel inverse approach based on the guided circumferential wave (GCW) to characterize the elastic properties of the arterial wall. To this end, a finite element (FE) model has been built to study the
http://dx.doi.org/10.1016/j.jbiomech.2016.12.006 0021-9290/& 2016 Elsevier Ltd. All rights reserved.
Please cite this article as: Li, G.-Y., et al., An ultrasound elastography method to determine the local stiffness of arteries with guided circumferential waves. Journal of Biomechanics (2016), http://dx.doi.org/10.1016/j.jbiomech.2016.12.006i
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dispersion relation of the GCW. Our FE results show that the GCW can be well-described with the Lamb wave (LW) model in the frequency range of interest in this study when the ratio of the inner radius of the arterial wall to its thickness is greater than a certain critical value. This finding enables us to develop a simple inverse method to measure the elastic properties of the arterial wall from the dispersion curve of the GCW induced in the artery by the focused ARF. Both numerical experiments and phantom experiments were performed to validate the inverse method. Furthermore, we show that our method is applicable to the cases in which the arterial wall has a local stenosis and/or the geometry of the artery cross-section is irregular, and therefore holds great potential for clinical use.
fluid can be given by € F; ∇ κ ∇ U uF ¼ ρ F u
ð3Þ
where u is the displacement of the fluid and κ and ρ denote the bulk modulus and mass density, respectively. Eq. (3) gives ∇ uF ¼ 0; hence, we can define uF ¼ ∇χ and then Eq. (3) can be rewritten as F
∇2 χ ¼
F
1 χ€ ; c2p
ð4Þ
pffiffiffiffiffiffiffiffiffiffi where cp ¼ κ =ρF is the velocity of the pressure wave in fluid. The pressure p induced by the deformation of the fluid is p ¼ κ ∇ U uF ¼ κ ∇ 2 χ . 2.2. Guided circumferential waves (GCWs)
2. Theoretical model and FE simulations Following previous studies (Bernal et al., 2011; Li et al., in press; Maksuti et al., 2016), the arterial wall was assumed to be a long hollow cylinder surrounded by fluid both inside and outside (see Fig. 1(a)) in the present study, and we generalized our analysis to the case in which the artery has a more general shape (Section 5). The guided wave that is of interest in this study is the so-called GCW (Liu and Qu, 1998a; Rose, 2014). Both the theoretical and the FE models that were used to analyze the GCW are presented below. 2.1. Basic equations for acoustic waves in solid and fluid media The equilibrium equation for the elastic solid free of body force is
€ μ∇2 u þ λ þ μ ∇ð∇ UuÞ ¼ ρu;
ð1Þ
where λ and μ are the Lamé constants, and ρ denotes the mass € denotes ∂2 u=∂t 2 , where t is time. The density of the elastic solid. u constitutive law for the linear elastic solid is σ ¼ μð∇u þu∇Þ þ λð∇ UuÞI. By decomposing the displacement u as u ¼ ∇φ þ∇ ψ, where ∇ U ψ ¼ 0 (Achenbach, 1973), and inserting this expression into Eq. (1), the following wave equation can be obtained 8 2 1 € < ∇ φ ¼ c2 φ l ; ð2Þ € : ∇2 ψ ¼ 12 ψ ct
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi where cl ¼ λ þ 2μ =ρ and ct ¼ μ=ρ denote the velocities for the longitudinal and transverse waves in the elastic solid, respectively. The fluid we considered here is a compressible and inviscid fluid, and the equilibrium equation for the small motion of the
We proceeded to consider the propagation of the GCW, which can be simplified as a plane strain problem. The solution domains are the elastic circular annulus and the surrounding fluid, as shown in Fig. 1(a). Here, we introduce the polar coordinates system in the analysis. The displacements of the elastic solid and the fluid have only two nonzero components, i.e., ur and uθ for the elastic solid and uFr and uFθ for the fluid. At the interfaces between the fluid and elastic solid, the following interfacial conditions (ICs) are specified 8 σ ¼ p > < rr σ rθ ¼ 0 ; at r ¼ R and r ¼ R þ h; ð5Þ > : u ¼ uF r r where R and h denote the inner radius and wall thickness of the circular annulus, respectively. Considering the circumferential wave, which propagates along the θ-direction, we can write (Liu and Qu, 1998a; Rose, 2014) 8 φ ¼ φ0 ðrÞeiðkθ θ ωtÞ > > > > > < ψ ¼ ψ ðr Þeiðkθ θ ωtÞ 0 ð6Þ ; iðkθ θ ωt Þ In In > χ ¼ χ > 0 ðr Þe > > > : χ Out ¼ χ Out ðr Þeiðkθ θ ωtÞ 0
where kθ is the wave number and ω denotes the circular frequency, χ In and χ Out relate to the fluid region inside and outside of the circular annulus, respectively. The phase of the GCW is a constant along the radial direction (Liu and Qu, 1998a), and the phase velocity along a circle with radius r (R r r r R þh) is equal to cðr Þ ¼ r
ω kθ
:
ð7Þ
In the present study, we are interested in the phase velocity at r ¼ R þ h=2, whereas previous studies have centered on the phase
Fig. 1. The schematic diagram of (a) the GCW model and (b) the LW model.
Please cite this article as: Li, G.-Y., et al., An ultrasound elastography method to determine the local stiffness of arteries with guided circumferential waves. Journal of Biomechanics (2016), http://dx.doi.org/10.1016/j.jbiomech.2016.12.006i
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velocity at the outer boundary of the circular annulus (Liu and Qu, 1998a, b; Valle et al., 2001; Yeh and Yang, 2011), which is convenient to be measured in the non-destructive testing of pipeline. Inserting Eq. (6) into Eqs. (2) and (4), the resulting equations together with the ICs given in Eq. (5) forms the eigenvalue problem which determines the dispersion relations of the GCW. Liu and Qu (1998a) studied the GCW within a circular annulus in vacuum and numerically solved the dispersion equation. Fong (2005) performed a systematic investigation of the GCW and addressed the effect of the curvature. It is noted that when the inner radius of the elastic circular annulus is much greater than the wall thickness, i.e., R=h- þ 1, the GCW model reduces to the LW model as shown in Fig. 1(b). In this case, two types of guided waves namely the anti-symmetric and symmetric modes exist. The dispersion equation for the anti-symmetric mode is (Osborne and Hart, 1945) 2 2 2 2 sin h ðαhÞ cosh βh 4k αβ cosh ðαhÞ sin h β h 2k kt ð8Þ ρF αk4 ¼ ραp t cosh ðαhÞ cosh β h ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 2 where α ¼ k kl , β ¼ k kt and αp ¼ k kp . k denotes the wave number of the LW, kl ¼ ω=cl , kt ¼ ω=ct and kp ¼ ω=cp . Eq. (8) is a transcendental equation, which can only be solved numerically. When R=h is finite, Fong (2005) found that the curvature effect on the GCW could be negligible for the issues he investigated. This interesting finding indicates that even when R=h is finite, Eq. (8) may be used to approximate the dispersion relation of the GCW. To confirm this important finding and develop an inverse approach to determine arterial stiffness using the GCW, a
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finite element (FE) model was built in this study to calculate the dispersion relation of the GCW in the arterial wall. 2.3. A FE model to study the dispersion relation of the GCW in the arterial wall The dispersion relation of the GCW in the low-frequency range were studied with the FE analysis. Details of the FE model can be found in the Supporting Information (SI-1). In Fig. 2(a) and (b), the typical dispersion curves of the GCW with different R=h and E are plotted. For comparison, the dispersion curves of the LW with the same wall thickness and physical properties are also presented by solving Eq. (8). Our FE results show that in the frequency range of 0–2500 Hz, the dispersion curves of the GCW are very close to those of the LW, in particular when R=h is large. This relationship can be understood as follows. In the low-frequency range, the wave length of the GCW is greater than the wall thickness h. In this case, the elastic annulus can be approximated as a curved beam along the circumferential direction. The vibration frequency of the beam is 3 determined by its bending stiffness, which is proportional to Eh , and the effect of the curvature 1=R is insignificant in this case (Fong, 2005). In Fig. 2(c), variation of the dimensionless phase velocity, which is defined as the ratio of the phase velocity of GCW to the phase velocity of the LW at 1000 Hz, with R=h is plotted. It can be seen that when R=h is small (e.g., R=h ¼1) effects of curvature come into play even in the low-frequency range. Whereas, when R=h 4 1, the dimensionless phase velocity is larger than 0.95, indicating that the relative error caused by using Eq. (8) to predict the phase velocity of GCW is no more than 5%.
Fig. 2. The dispersion curves of the GCWs in the elastic circular annulus with different R=h and elastic moduli. For different (a) R=h and (b) elastic moduli, the dispersion curves of the GCWs are very close to those of the LWs in the frequency range from 0 to 2500 Hz when R=h 41. (c) Plot of the dimensionless phase velocity, which is defined as the ratio of the phase velocity of GCW to the phase velocity of the LW at 1000 Hz. The shaded area in the figure indicates the dimensionless phase velocity greater than 0.95. The other parameters used in the FE simulations were h ¼ 1:0 mm, κ ¼ 2:2 106 kPa and ρ ¼ ρF ¼ 1000 kg/m3.
Please cite this article as: Li, G.-Y., et al., An ultrasound elastography method to determine the local stiffness of arteries with guided circumferential waves. Journal of Biomechanics (2016), http://dx.doi.org/10.1016/j.jbiomech.2016.12.006i
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3. An inverse approach to determine arterial stiffness based on the GCW
4. Validation of the inverse method 4.1. Numerical experiments
The finding and conclusion in Section 2.3 enable the development of an inverse approach to determine arterial stiffness by approximating the dispersion curve of the GCW in the arterial wall using that given by Eq. (8). The procedures are shown as follows. (1) Utilizing the ARF, broad-band shear waves can be generated in the arterial wall in experiments. The velocity field vr r; θ in the arterial wall can be expressed as (Alleyne and Cawley, 1991) vr r; θ; t ¼
Z ω
Aðr; ωÞeiðkθ θ ωt Þ dω;
ð9Þ
where Aðr; ωÞ is the frequency-dependent amplitude. Eq. (9) can be regarded as the superposition of GCWs with different frequencies. (2) For a specified radius r, e.g., r ¼ R þ h=2 which is adopted in the present adopting a two-dimensional Fourier transform study, to vr r; θ; t gives Z H kθ ; ω ¼
þ1 1
Z
þ1 1
vr R þ h=2; θ; t e iðkθ θ þ ωtÞ dθdt :
ð10Þ
The implementation of Eq. (10) can be realized by means of a two-dimensional fast Fourier transformation (2DFFT). (3) At each frequency ω, the maximum value of H kθ ; ω was identified, from which the wave number kθ ðωÞ was determined. The phase velocity therefore can be calculated based on Eq. (7). Once the dispersion curve is obtained from experiments, Eq. (8) can be used to fit the experimental data and infer the elastic modulus of the arterial wall.
We performed numerical experiments based on FE analysis (FEA) (see SI-2 for details). According to the FEA, the variation of vr in the arterial wall with time can be obtained, and Fig. 3 (a) illustrates the distribution of vr in a typical time period. To calculate the dispersion curve, we concentrated on the vr along the circle of r ¼ R þ h=2, as shown in Fig. 3(b). Then, following the procedures described in Section 3, the dispersion curve of the GCW can be obtained (see Fig. 3(b)–(d)). For the typical example given in Fig. 3(d), the dispersion curve is fitted with Eq. (8). The 0 best fitting gives E 72 kPa, which is slightly smaller than the value of E ¼ 75 kPa input in the FE model. More numerical experiments have been conducted, and the results are shown in Table 1. The relative differences between the identified results E0 using the present inverse approach and the Young’s moduli E input in the FE models show that the larger the R=h is, the smaller the relative error will be. For the critical case of R=h¼ 1, the relative error can be up to 17%. Whereas, when R=h Z 2, the error in E0 will be smaller than 10%. 4.2. Phantom experiments Phantom experiments was further carried out to validate the present inverse approach. Two polyvinyl alcohol (PVA) cryogel vessel-mimicking phantoms, whose inner radii and wall thicknesses were 4 mm and 2 mm, and 3 mm and 1.5 mm, respectively, were prepared. The elastic moduli of the phantoms were identified with the present inverse approach (see Supporting Information, SI-3, for the details of experiments). The phantom mold and the experimental setup are shown in Fig. 4(a) and (b). For one of the phantoms (inner radius¼4 mm, wall thickness ¼ 2 mm), the radial motion at about 0.6 ms after
Fig. 3. The validation of the inverse approach by FEA. (a) The FE model used to study the propagation of the GCW in the arterial wall and the map of vr in a typical time period. The path at r ¼ R þ h=2, vr was recorded as shown in (b). (c) The map of Hðkθ ; ωÞ. (d) The dispersion curve of the GCW obtained by FEA and the fitting curve with the LW model. The parameters used in this model were E ¼ 75 kPa, h ¼ 1:0 mm, R ¼ 2:0 mm, κ ¼ 2:2 106 kPa and ρ ¼ ρF ¼ 1000 kg/m3.
Please cite this article as: Li, G.-Y., et al., An ultrasound elastography method to determine the local stiffness of arteries with guided circumferential waves. Journal of Biomechanics (2016), http://dx.doi.org/10.1016/j.jbiomech.2016.12.006i
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applying of the ARF is shown in Fig. 4(c) for illustration with the velocity overlaid on the B-mode image (with a dynamic range of 60 dB). The spatio-temporal imaging of the shear wave propagating along a path of interest (red circle in Fig. 4(d)) is shown in Fig. 4(e). By the inverse approach suggested above, the dispersion curve was further obtained as shown in Fig. 4(f). Fig. 5 gives the dispersion relations of the two phantoms. The discrete points in the figures were obtained from the experiments. It can be seen that, when the frequency is higher than about 1500 Hz, some high-order modes and noises may emerge. This phenomenon has also been observed for the GAW in previous studies (Bernal et al., 2011). In our analysis, only the experimental data of the lowest mode was fitted with Eq. (8). The best fittings gave the Young’s moduli of the vessel-mimicking phantoms were 90 kPa and 95 kPa, respectively. The deviation might stem from
Table 1 A comparison of the elastic moduli E input in the FEA with those (E0 ) extracted using the present inverse approach. Other parameters used in the simulations were h ¼ 1:0 mm, κ ¼ 2:2 106 kPa and ρ ¼ ρF ¼ 1000 kg/m3. R=h
E (kPa)
E0 (kPa)
Relative error
1.0
50 100 150
48 90 125
4.0% 10% 16.7%
2.0
50 100 150
49 96 139
2.0% 4.0% 7.3%
3.0
50 100 150
50 100 148
0 0 1.3%
5
the different geometries of the two phantoms which could have some effects on the mechanical properties of the phantoms. The experimental results obtained here match those reported by Maksuti et al. (2016) very well. In their study, they tested the phantoms prepared in the same procedure as ours. Their elastography experiments and tensile tests found that the shear modulus of the phantom was approximately 30.5 kPa, i.e., the Young’s modulus was about 91.5 kPa. The relative error between our experimental results and theirs is less than 4%. Clearly, the phantom experiments performed here together with the numerical experiments in Section 4.1 demonstrate that the present inverse approach is effective and can be used in practical measurements.
5. Discussion In the analysis above, the artery was assumed to be a hollow cylinder surrounded by fluid. However, the cross-section of a practical artery may have more complex geometries instead of a circular annulus. Moreover, local stenosis may exist in the artery. Here, we explore the applicability of the present method to these critical cases. Besides, some limitations in this study are discussed. 5.1. Effect of the arterial cross-section shape The aim of the present study was to develop an inverse approach to identify arterial stiffness in vivo, which is of great importance for the diagnosis of cardiovascular diseases. In this case, it is necessary to explore whether the present method is effective when the cross-section of the artery is not a circular annulus, which is frequently encountered in practice. A real artery as illustrated in Fig. 6(a) may be modeled as an elastic tube of which the thickness is h and the shape of the crosssection is determined by a function of Γ ðx; yÞ ¼ 0, as shown in Fig. 6
Fig. 4. (a) The phantom mold and the vessel-mimicking phantom. (b) Schematic diagram of the experimental setup (c) The velocity field of a vessel-mimicking phantom overlaid on the B-mode imaging at about 0.612 ms after applying the ARF. (d) The B-mode imaging of the phantom in which the path of interest (the red circle) is plotted. (e) The spatio-temporal imaging of the shear wave along the path of interest. (f) The dispersion curve obtained from the spatio-temporal imaging by the present inverse approach. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Please cite this article as: Li, G.-Y., et al., An ultrasound elastography method to determine the local stiffness of arteries with guided circumferential waves. Journal of Biomechanics (2016), http://dx.doi.org/10.1016/j.jbiomech.2016.12.006i
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Fig. 5. Experimental results for vessel-mimicking phantoms. (a) R¼ 4 mm, h ¼2 mm; (b) R ¼ 3 mm, h ¼1.5 mm. When the frequency is higher than about 1500 Hz, other high-order modes and noises (the red points) which are not used to conduct the inverse analysis emerges.
Fig. 6. (a) Cross-section of a human artery (from Wikipedia). (b) The schematic of the artery wall with variable curvature radius. (c) The propagation and (d) the dispersion curve of the GCW in the artery wall with an elliptic cross-section.
(b). In this case, the velocity component Ralong the normal direction n ¼ ∇Γ =ð∇Γ Þj can be expressed as vn ω AðωÞeiðks ωt Þ dω, where s is arc length as shown in Fig. 6(b). Then, the 2DFFT is conducted to vn to obtain the dispersion curve. For illustration, we consider an artery with an elliptic crosssection. The propagation of the elastic wave in the artery is simulated with the FEA, and the typical result is shown in Fig. 6(c). The dispersion curve is calculated based on the FEA results. As shown in Fig. 6(d), the dispersion curve can be well described by Eq. (8), indicating that the inverse method developed in this paper is applicable to more general cases. It should be noted that the effects of curvature may come into play when R=h is small. According to our computational results, for an artery with non-circular cross-section, curvature effects can be negligible when the ratio of the local curvature radius R to the wall
thickness h is greater than 1. For the example shown in Fig. 6, R=h Z 1:3. 5.2. Applicability of the method to the characterization of an arterial wall with local stenosis Local stenosis may occur in the human artery, and it is of great importance to evaluate the stiffness of the stenosis region. The present method has the potential to be used for such a purpose. To illustrate this point, an arterial wall with local stenosis was studied by a three-dimensional FE model as shown in Fig. 7(a). The propagating of the guided wave in the cross section of the stenosis region was monitored (see Fig. 7(b)) and used in the inverse analysis. Fig. 7(c) gives the dispersion curve provided by the FEA, which was fitted using Eq. (8) to infer the elastic modulus of the
Please cite this article as: Li, G.-Y., et al., An ultrasound elastography method to determine the local stiffness of arteries with guided circumferential waves. Journal of Biomechanics (2016), http://dx.doi.org/10.1016/j.jbiomech.2016.12.006i
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Fig. 7. Mechanical characterization of the local stenosis region of an artery. (a) The model and geometrical parameters used in the FEA, L ¼ 4 mm, t 1 ¼ 1 mm, t 2 ¼ 2 mm, R1 ¼ 4 mm, and R2 ¼ 3 mm. (b) The normalized radial velocity field in the arterial wall. (c) The dispersion curve given by the FEA and the fitting curve with Eq. (8). The best fitting gave an elastic modulus of 70 kPa, which was 6.7% smaller than the input value.
stenosis region. A comparison of the identified elastic modulus with the actual value input in the FEA shows that the relative error was 6.7%, indicating that indeed the present method may be used to determine the mechanical properties of the local stenosis region of an arterial wall, which appears to be impossible to access with other methods. 5.3. Limitations in this study This study has some limitations that deserve further efforts. First, real arterial walls have layered structures as shown in Fig. 6. Usually two layers can be observed with the ordinary ultrasound imaging method. In this case, the present inverse approach can provide the effective elastic modulus of the arterial wall as shown in the Supporting Information (SI-4). Second, variation of the arterial wall thickness along the circumferential direction has not been considered in this study. However, this issue is of practical interest in some cases and deserves further efforts. Third, real artery tissues are anisotropic with distributed collagen fibers (Gasser et al., 2006). Although recent studies have demonstrated that it is possible to determine the anisotropic and hyperelastic properties of a bulk anisotropic soft tissue (Li et al., 2016b; Li et al., 2016c), assessing the anisotropic parameters of arterial walls using the shear wave elastography method represents a more challenging issue because the wave in the arterial wall is guided. To characterize the anisotropic properties of the arteries, both the GCW and the GAW may need to be measured; this important issue is beyond the scope of this study and deserves further investigations. Fourth, here the arterial wall is assumed as timeindependent material, however, a real artery may exhibit viscoelastic deformation. For viscoelastic soft tissues, the wavenumber k is no longer a real number, but could be a complex number, which stands for the dissipation of the wave. Previous studies have
shown that the dispersion relation of the lowest mode, i.e., the mode adopted in our inverse analysis, is basically unaffected by the viscosity of the media (Chan and Cawley, 1998; Nguyen et al., 2011). Fifth, in this study, the outer region of the arterial wall was simplified as inviscid fluid to derive the theoretical solution. In this case, the phase velocity of the GCW is much smaller than that of the sound wave in the fluid; therefore, the GCW is basically nonleaky, i.e., the guided waves are trapped within the arterial wall. Whereas for a real artery, the perivascular tissues may cause the leakage of the GCW and affect the dispersion relation of the GCW. When perivascular tissues are much softer than the arterial wall, their effects on the dispersion relation of the GCW may be negligible. To quantitatively address their effects in general cases, a FE model including the perivascular tissues can be developed and this work is underway and the results may be reported in the coming future. Finally, effects of blood pressure will come into play in the in vivo experiments. When interpreting the in vivo data using the present method, in principle, it is possible to determine both the elastic modulus of the artery and the variation of blood pressure. This interesting issue requires further efforts.
6. Concluding remarks Quantitatively assessing local arterial stiffness in a non-invasive manner is of great importance and remains a challenging issue. Here, we addressed this challenge by exploring the use of the GCW induced in the arterial wall by the focused ARF to determine the arterial stiffness. In summary, the following key results were achieved. The dispersion properties of the GCW have been investigated using a finite element model. Our results show that when the ratio of the inner radius of the artery to its thickness is greater than a
Please cite this article as: Li, G.-Y., et al., An ultrasound elastography method to determine the local stiffness of arteries with guided circumferential waves. Journal of Biomechanics (2016), http://dx.doi.org/10.1016/j.jbiomech.2016.12.006i
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certain critical value, the dispersion properties of the GCW are basically independent of the curvature of the arterial wall and can be well-described using the LW model. Based on the FE results, an inverse method is proposed to characterize the elastic modulus of an artery. Both numerical experiments and phantom experiments had been performed to validate the proposed method. We show that our method can be applied to the cases in which the arterial wall has a local stenosis and/or the geometry of the arterial cross-section is irregular, thereby it has great potential to be used in clinics.
Conflict of interest statement Authors have no financial and personal relationships that could inappropriately influence or bias this work.
Acknowledgement We acknowledge the support from the National Natural Science Foundation of China (Grant nos. 11572179, 11172155, 11432008, and 81561168023). English through the whole text has been edited using the NPG (Nature Publishing Group) Language Editing service.
Appendix A. Supplementary material Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.jbiomech.2016.12. 006.
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biomechanical properties using shear wave imaging. Ultrasound Med. Biol. 36, 1662–1676. Fong, K.L.J., 2005. A Study of Curvature Effects on Guided Elastic Waves (Ph.D. thesis). Imperial College London. Gasser, T.C., Ogden, R.W., Holzapfel, G.A., 2006. Hyperelastic modelling of arterial layers with distributed collagen fibre orientations. J. R. Soc. Interface 3, 15–35. Huang, C., Guo, D., Lan, F., Zhang, H., Luo, J., 2016. Noninvasive measurement of regional pulse wave velocity in human ascending aorta with ultrasound imaging: an in-vivo feasibility study. J. Hypertens 34, 2016–2037. Hunter, K.S., Albietz, J.A., Lee, P.-F., Lanning, C.J., Lammers, S.R., Hofmeister, S.H., Kao, P.H., Qi, H.J., Stenmark, K.R., Shandas, R., 2010. In vivo measurement of proximal pulmonary artery elastic modulus in the neonatal calf model of pulmonary hypertension: development and ex vivo validation. J. Appl. Physiol. 108, 968–975. Khamdaeng, T., Luo, J., Vappou, J., Terdtoon, P., Konofagou, E.E., 2012. Arterial stiffness identification of the human carotid artery using the stress–strain relationship in vivo. Ultrasonics 52, 402–411. Li, G.-Y., He, Q., Jia, L., He, P., Luo, J., Cao, Y., 2016a. An inverse method to determine the arterial stiffness with guided axial waves. Ultrasound Med. Biol. (in press) Li, G.-Y., He, Q., Qian, L.-X., Geng, H., Liu, Y., Yang, X.-Y., Luo, J., Cao, Y., 2016b. Elastic Cherenkov effects in transversely isotropic soft materials-II: ex vivo and in vivo experiments. J. Mech. Phys. Solids 94, 181–190. Li, G.-Y., Zheng, Y., Liu, Y., Destrade, M., Cao, Y., 2016c. Elastic Cherenkov effects in transversely isotropic soft materials-I: theoretical analysis, simulations and inverse method. J. Mech. Phys. Solids 96, 388–410. Li, J., Rose, J.L., 2006. Natural beam focusing of non-axisymmetric guided waves in large-diameter pipes. Ultrasonics 44, 35–45. Liu, G., Qu, J., 1998a. Guided circumferential waves in a circular annulus. J. Appl. Mech.-T ASME 65, 424–430. Liu, G., Qu, J., 1998b. Transient wave propagation in a circular annulus subjected to transient excitation on its outer surface. J Acoust. Soc. Am. 104, 1210–1220. Luo, J., Li, R.X., Konofagou, E.E., 2012. Pulse wave imaging of the human carotid artery: an in vivo feasibility study. IEEE Trans. Ultrason Ferroelectr. Freq. Control 59, 174–181. Maksuti, E., Widman, E., Larsson, D., Urban, M.W., Larsson, M., Bjällmark, A., 2016. Arterial stiffness estimation by shear wave elastography: validation in phantoms with mechanical testing. Ultrasound Med Biol. 42, 308–321. Meinders, J.M., Kornet, L., Brands, P.J., Hoeks, A.P., 2001. Assessment of local pulse wave velocity in arteries using 2D distension waveforms. Ultrason. Imaging 23, 199–215. Nguyen, T.M., Couade, M., Bercoff, J., Tanter, M., 2011. Assessment of viscous and elastic properties of sub-wavelength layered soft tissues using shear wave spectroscopy: theoretical framework and in vitro experimental validation. IEEE Trans. Ultrason Ferroelectr. Freq. Control 58, 2305–2315. Osborne, M.F.M., Hart, S.D., 1945. Transmission, reflection, and guiding of an exponential pulse by a steel plate in water. I. theory. J. Acoust. Soc. Am. 17, 1–18. Ribbers, H., Lopata, R.G.P., Holewijn, S., Pasterkamp, G., Blankensteijn, J.D., de Korte, C.L., 2007. Noninvasive two-dimensional strain imaging of arteries: validation in phantoms and preliminary experience in carotid arteries in vivo. Ultrasound Med Biol. 33, 530–540. Rose, J.L., 2014. Ultrasonic Guided Waves in Solid Media. Cambridge University Press, New York. Sarvazyan, A.P., Rudenko, O.V., Swanson, S.D., Fowlkes, J.B., Emelianov, S.Y., 1998. Shear wave elasticity imaging: a new ultrasonic technology of medical diagnostics. Ultrasound Med. Biol. 24, 1419–1435. Shirwany, N.A., Zou, M.-h, 2010. Arterial stiffness: a brief review. Acta Pharmacol. Sin. 31, 1267–1276. Valle, C., Niethammer, M., Qu, J., Jacobs, L., 2001. Crack characterization using guided circumferential waves. J. Acoust. Soc. Am. 110, 1282–1290. Yeh, C.-H., Yang, C.-H., 2011. Characterization of mechanical and geometrical properties of a tube with axial and circumferential guided waves. Ultrasonics 51, 472–479.
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An ultrasound elastography method to determine the local stiffness of arteries with guided circumferential waves Guo-Yang Li a,1, Qiong He b,1, Guoqiang Xu a, Lin Jia a, Jianwen Luo b, Yanping Cao a,n a b
Institute of Biomechanics and Medical Engineering, AML, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, PR China Department of Biomedical Engineering, School of Medicine, Tsinghua University, Beijing 100084, PR China
art ic l e i nf o
a b s t r a c t
Article history: Accepted 5 December 2016
Arterial stiffness is highly correlated with the functions of the artery and may serve as an important diagnostic criterion for some cardiovascular diseases. To date, it remains a challenge to quantitatively assess local arterial stiffness in a non-invasive manner. To address this challenge, we investigated the possibility of determining arterial stiffness using the guided circumferential wave (GCW) induced in the arterial wall by a focused acoustic radiation force. The theoretical model for the dispersion analysis of the GCW is presented, and a finite element model has been established to calculate the dispersion curve. Our results show that under described conditions, the dispersion relations of the GCW are basically independent of the curvature of the arterial wall and can be well-described using the Lamb wave (LW) model. Based on this conclusion, an inverse method is proposed to characterize the elastic modulus of artery. Both numerical experiments and phantom experiments had been performed to validate the proposed method. We show that our method can be applied to the cases in which the artery has local stenosis and/ or the geometry of the artery cross-section is irregular; therefore, this method holds great potential for clinical use. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Shear wave elastography Arterial stiffness Guided circumferential wave Finite element analysis Inverse method
1. Introduction Arterial stiffness has a close relationship with some cardiovascular diseases (Cecelja and Chowienczyk, 2012; Shirwany and Zou, 2010), and therefore, non-invasive measurement of arterial stiffness has received considerable attention in recent years. To date, methods based on an evaluation of the pulse wave velocity (Brands et al., 1998; Chubachi et al., 1994; Luo et al., 2012; Meinders et al., 2001; Huang et al., 2016), and the blood pressure induced static-strain within the arterial walls have been proposed by many authors to assess arterial stiffness (Hunter et al., 2010; Khamdaeng et al., 2012; Ribbers et al., 2007). However, in principle, it is difficult to quantitatively determine the local elastic properties of the arterial wall using these methods. Recently, the shear wave elastography method has been investigated for measuring arterial stiffness (Bernal et al., 2011; Couade et al., 2010; Li et al., in press). This method relies on the use of the acoustic radiation force (ARF) (Sarvazyan et al., 1998) to generate broad-band guided axial wave (GAW) in the arterial wall. The dispersion relation of the GAW is then determined and used to n
Corresponding author. Fax: þ 86 10 62781284. E-mail address: [email protected] (Y. Cao). 1 These authors made equal contribution to this study.
deduce the arterial stiffness. This method is promising for instantaneously measuring arterial stiffness because the measurement can be conducted a number of times in each cardiac cycle, e.g., 13 times/cycle (Couade et al., 2010). However, it has been demonstrated that the dispersion relation of the GAW is significantly affected by the curvature of the arterial wall in the low-frequency range (typically 0–1000 Hz) (Li et al., in press; Li and Rose, 2006; Maksuti et al., 2016). In the literature, critical frequency has been suggested beyond which effects of curvature are negligible and the dispersion curve of the guided axial wave can be fitted with the dispersion curve given by the Lamb wave model (Maksuti et al., 2016). In a very recent study, Li et al. (in press) showed that such a critical frequency depends on both elastic properties and geometrical parameters of the arterial wall. Besides, it is difficult to apply the method based on the GAW to the cases in which the wall thickness and/or the inner radius of the artery vary along the axial direction and/or the cross-sectional geometry of the artery is irregular (e.g., non-circular). Considering that these cases are frequently encountered in clinics, the development of a robust method to measure the local arterial stiffness is still urgently needed. Based on the above premise, this study aimed to develop a novel inverse approach based on the guided circumferential wave (GCW) to characterize the elastic properties of the arterial wall. To this end, a finite element (FE) model has been built to study the
http://dx.doi.org/10.1016/j.jbiomech.2016.12.006 0021-9290/& 2016 Elsevier Ltd. All rights reserved.
Please cite this article as: Li, G.-Y., et al., An ultrasound elastography method to determine the local stiffness of arteries with guided circumferential waves. Journal of Biomechanics (2016), http://dx.doi.org/10.1016/j.jbiomech.2016.12.006i
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dispersion relation of the GCW. Our FE results show that the GCW can be well-described with the Lamb wave (LW) model in the frequency range of interest in this study when the ratio of the inner radius of the arterial wall to its thickness is greater than a certain critical value. This finding enables us to develop a simple inverse method to measure the elastic properties of the arterial wall from the dispersion curve of the GCW induced in the artery by the focused ARF. Both numerical experiments and phantom experiments were performed to validate the inverse method. Furthermore, we show that our method is applicable to the cases in which the arterial wall has a local stenosis and/or the geometry of the artery cross-section is irregular, and therefore holds great potential for clinical use.
fluid can be given by € F; ∇ κ ∇ U uF ¼ ρ F u
ð3Þ
where u is the displacement of the fluid and κ and ρ denote the bulk modulus and mass density, respectively. Eq. (3) gives ∇ uF ¼ 0; hence, we can define uF ¼ ∇χ and then Eq. (3) can be rewritten as F
∇2 χ ¼
F
1 χ€ ; c2p
ð4Þ
pffiffiffiffiffiffiffiffiffiffi where cp ¼ κ =ρF is the velocity of the pressure wave in fluid. The pressure p induced by the deformation of the fluid is p ¼ κ ∇ U uF ¼ κ ∇ 2 χ . 2.2. Guided circumferential waves (GCWs)
2. Theoretical model and FE simulations Following previous studies (Bernal et al., 2011; Li et al., in press; Maksuti et al., 2016), the arterial wall was assumed to be a long hollow cylinder surrounded by fluid both inside and outside (see Fig. 1(a)) in the present study, and we generalized our analysis to the case in which the artery has a more general shape (Section 5). The guided wave that is of interest in this study is the so-called GCW (Liu and Qu, 1998a; Rose, 2014). Both the theoretical and the FE models that were used to analyze the GCW are presented below. 2.1. Basic equations for acoustic waves in solid and fluid media The equilibrium equation for the elastic solid free of body force is
€ μ∇2 u þ λ þ μ ∇ð∇ UuÞ ¼ ρu;
ð1Þ
where λ and μ are the Lamé constants, and ρ denotes the mass € denotes ∂2 u=∂t 2 , where t is time. The density of the elastic solid. u constitutive law for the linear elastic solid is σ ¼ μð∇u þu∇Þ þ λð∇ UuÞI. By decomposing the displacement u as u ¼ ∇φ þ∇ ψ, where ∇ U ψ ¼ 0 (Achenbach, 1973), and inserting this expression into Eq. (1), the following wave equation can be obtained 8 2 1 € < ∇ φ ¼ c2 φ l ; ð2Þ € : ∇2 ψ ¼ 12 ψ ct
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi where cl ¼ λ þ 2μ =ρ and ct ¼ μ=ρ denote the velocities for the longitudinal and transverse waves in the elastic solid, respectively. The fluid we considered here is a compressible and inviscid fluid, and the equilibrium equation for the small motion of the
We proceeded to consider the propagation of the GCW, which can be simplified as a plane strain problem. The solution domains are the elastic circular annulus and the surrounding fluid, as shown in Fig. 1(a). Here, we introduce the polar coordinates system in the analysis. The displacements of the elastic solid and the fluid have only two nonzero components, i.e., ur and uθ for the elastic solid and uFr and uFθ for the fluid. At the interfaces between the fluid and elastic solid, the following interfacial conditions (ICs) are specified 8 σ ¼ p > < rr σ rθ ¼ 0 ; at r ¼ R and r ¼ R þ h; ð5Þ > : u ¼ uF r r where R and h denote the inner radius and wall thickness of the circular annulus, respectively. Considering the circumferential wave, which propagates along the θ-direction, we can write (Liu and Qu, 1998a; Rose, 2014) 8 φ ¼ φ0 ðrÞeiðkθ θ ωtÞ > > > > > < ψ ¼ ψ ðr Þeiðkθ θ ωtÞ 0 ð6Þ ; iðkθ θ ωt Þ In In > χ ¼ χ > 0 ðr Þe > > > : χ Out ¼ χ Out ðr Þeiðkθ θ ωtÞ 0
where kθ is the wave number and ω denotes the circular frequency, χ In and χ Out relate to the fluid region inside and outside of the circular annulus, respectively. The phase of the GCW is a constant along the radial direction (Liu and Qu, 1998a), and the phase velocity along a circle with radius r (R r r r R þh) is equal to cðr Þ ¼ r
ω kθ
:
ð7Þ
In the present study, we are interested in the phase velocity at r ¼ R þ h=2, whereas previous studies have centered on the phase
Fig. 1. The schematic diagram of (a) the GCW model and (b) the LW model.
Please cite this article as: Li, G.-Y., et al., An ultrasound elastography method to determine the local stiffness of arteries with guided circumferential waves. Journal of Biomechanics (2016), http://dx.doi.org/10.1016/j.jbiomech.2016.12.006i
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velocity at the outer boundary of the circular annulus (Liu and Qu, 1998a, b; Valle et al., 2001; Yeh and Yang, 2011), which is convenient to be measured in the non-destructive testing of pipeline. Inserting Eq. (6) into Eqs. (2) and (4), the resulting equations together with the ICs given in Eq. (5) forms the eigenvalue problem which determines the dispersion relations of the GCW. Liu and Qu (1998a) studied the GCW within a circular annulus in vacuum and numerically solved the dispersion equation. Fong (2005) performed a systematic investigation of the GCW and addressed the effect of the curvature. It is noted that when the inner radius of the elastic circular annulus is much greater than the wall thickness, i.e., R=h- þ 1, the GCW model reduces to the LW model as shown in Fig. 1(b). In this case, two types of guided waves namely the anti-symmetric and symmetric modes exist. The dispersion equation for the anti-symmetric mode is (Osborne and Hart, 1945) 2 2 2 2 sin h ðαhÞ cosh βh 4k αβ cosh ðαhÞ sin h β h 2k kt ð8Þ ρF αk4 ¼ ραp t cosh ðαhÞ cosh β h ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 2 where α ¼ k kl , β ¼ k kt and αp ¼ k kp . k denotes the wave number of the LW, kl ¼ ω=cl , kt ¼ ω=ct and kp ¼ ω=cp . Eq. (8) is a transcendental equation, which can only be solved numerically. When R=h is finite, Fong (2005) found that the curvature effect on the GCW could be negligible for the issues he investigated. This interesting finding indicates that even when R=h is finite, Eq. (8) may be used to approximate the dispersion relation of the GCW. To confirm this important finding and develop an inverse approach to determine arterial stiffness using the GCW, a
3
finite element (FE) model was built in this study to calculate the dispersion relation of the GCW in the arterial wall. 2.3. A FE model to study the dispersion relation of the GCW in the arterial wall The dispersion relation of the GCW in the low-frequency range were studied with the FE analysis. Details of the FE model can be found in the Supporting Information (SI-1). In Fig. 2(a) and (b), the typical dispersion curves of the GCW with different R=h and E are plotted. For comparison, the dispersion curves of the LW with the same wall thickness and physical properties are also presented by solving Eq. (8). Our FE results show that in the frequency range of 0–2500 Hz, the dispersion curves of the GCW are very close to those of the LW, in particular when R=h is large. This relationship can be understood as follows. In the low-frequency range, the wave length of the GCW is greater than the wall thickness h. In this case, the elastic annulus can be approximated as a curved beam along the circumferential direction. The vibration frequency of the beam is 3 determined by its bending stiffness, which is proportional to Eh , and the effect of the curvature 1=R is insignificant in this case (Fong, 2005). In Fig. 2(c), variation of the dimensionless phase velocity, which is defined as the ratio of the phase velocity of GCW to the phase velocity of the LW at 1000 Hz, with R=h is plotted. It can be seen that when R=h is small (e.g., R=h ¼1) effects of curvature come into play even in the low-frequency range. Whereas, when R=h 4 1, the dimensionless phase velocity is larger than 0.95, indicating that the relative error caused by using Eq. (8) to predict the phase velocity of GCW is no more than 5%.
Fig. 2. The dispersion curves of the GCWs in the elastic circular annulus with different R=h and elastic moduli. For different (a) R=h and (b) elastic moduli, the dispersion curves of the GCWs are very close to those of the LWs in the frequency range from 0 to 2500 Hz when R=h 41. (c) Plot of the dimensionless phase velocity, which is defined as the ratio of the phase velocity of GCW to the phase velocity of the LW at 1000 Hz. The shaded area in the figure indicates the dimensionless phase velocity greater than 0.95. The other parameters used in the FE simulations were h ¼ 1:0 mm, κ ¼ 2:2 106 kPa and ρ ¼ ρF ¼ 1000 kg/m3.
Please cite this article as: Li, G.-Y., et al., An ultrasound elastography method to determine the local stiffness of arteries with guided circumferential waves. Journal of Biomechanics (2016), http://dx.doi.org/10.1016/j.jbiomech.2016.12.006i
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3. An inverse approach to determine arterial stiffness based on the GCW
4. Validation of the inverse method 4.1. Numerical experiments
The finding and conclusion in Section 2.3 enable the development of an inverse approach to determine arterial stiffness by approximating the dispersion curve of the GCW in the arterial wall using that given by Eq. (8). The procedures are shown as follows. (1) Utilizing the ARF, broad-band shear waves can be generated in the arterial wall in experiments. The velocity field vr r; θ in the arterial wall can be expressed as (Alleyne and Cawley, 1991) vr r; θ; t ¼
Z ω
Aðr; ωÞeiðkθ θ ωt Þ dω;
ð9Þ
where Aðr; ωÞ is the frequency-dependent amplitude. Eq. (9) can be regarded as the superposition of GCWs with different frequencies. (2) For a specified radius r, e.g., r ¼ R þ h=2 which is adopted in the present adopting a two-dimensional Fourier transform study, to vr r; θ; t gives Z H kθ ; ω ¼
þ1 1
Z
þ1 1
vr R þ h=2; θ; t e iðkθ θ þ ωtÞ dθdt :
ð10Þ
The implementation of Eq. (10) can be realized by means of a two-dimensional fast Fourier transformation (2DFFT). (3) At each frequency ω, the maximum value of H kθ ; ω was identified, from which the wave number kθ ðωÞ was determined. The phase velocity therefore can be calculated based on Eq. (7). Once the dispersion curve is obtained from experiments, Eq. (8) can be used to fit the experimental data and infer the elastic modulus of the arterial wall.
We performed numerical experiments based on FE analysis (FEA) (see SI-2 for details). According to the FEA, the variation of vr in the arterial wall with time can be obtained, and Fig. 3 (a) illustrates the distribution of vr in a typical time period. To calculate the dispersion curve, we concentrated on the vr along the circle of r ¼ R þ h=2, as shown in Fig. 3(b). Then, following the procedures described in Section 3, the dispersion curve of the GCW can be obtained (see Fig. 3(b)–(d)). For the typical example given in Fig. 3(d), the dispersion curve is fitted with Eq. (8). The 0 best fitting gives E 72 kPa, which is slightly smaller than the value of E ¼ 75 kPa input in the FE model. More numerical experiments have been conducted, and the results are shown in Table 1. The relative differences between the identified results E0 using the present inverse approach and the Young’s moduli E input in the FE models show that the larger the R=h is, the smaller the relative error will be. For the critical case of R=h¼ 1, the relative error can be up to 17%. Whereas, when R=h Z 2, the error in E0 will be smaller than 10%. 4.2. Phantom experiments Phantom experiments was further carried out to validate the present inverse approach. Two polyvinyl alcohol (PVA) cryogel vessel-mimicking phantoms, whose inner radii and wall thicknesses were 4 mm and 2 mm, and 3 mm and 1.5 mm, respectively, were prepared. The elastic moduli of the phantoms were identified with the present inverse approach (see Supporting Information, SI-3, for the details of experiments). The phantom mold and the experimental setup are shown in Fig. 4(a) and (b). For one of the phantoms (inner radius¼4 mm, wall thickness ¼ 2 mm), the radial motion at about 0.6 ms after
Fig. 3. The validation of the inverse approach by FEA. (a) The FE model used to study the propagation of the GCW in the arterial wall and the map of vr in a typical time period. The path at r ¼ R þ h=2, vr was recorded as shown in (b). (c) The map of Hðkθ ; ωÞ. (d) The dispersion curve of the GCW obtained by FEA and the fitting curve with the LW model. The parameters used in this model were E ¼ 75 kPa, h ¼ 1:0 mm, R ¼ 2:0 mm, κ ¼ 2:2 106 kPa and ρ ¼ ρF ¼ 1000 kg/m3.
Please cite this article as: Li, G.-Y., et al., An ultrasound elastography method to determine the local stiffness of arteries with guided circumferential waves. Journal of Biomechanics (2016), http://dx.doi.org/10.1016/j.jbiomech.2016.12.006i
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applying of the ARF is shown in Fig. 4(c) for illustration with the velocity overlaid on the B-mode image (with a dynamic range of 60 dB). The spatio-temporal imaging of the shear wave propagating along a path of interest (red circle in Fig. 4(d)) is shown in Fig. 4(e). By the inverse approach suggested above, the dispersion curve was further obtained as shown in Fig. 4(f). Fig. 5 gives the dispersion relations of the two phantoms. The discrete points in the figures were obtained from the experiments. It can be seen that, when the frequency is higher than about 1500 Hz, some high-order modes and noises may emerge. This phenomenon has also been observed for the GAW in previous studies (Bernal et al., 2011). In our analysis, only the experimental data of the lowest mode was fitted with Eq. (8). The best fittings gave the Young’s moduli of the vessel-mimicking phantoms were 90 kPa and 95 kPa, respectively. The deviation might stem from
Table 1 A comparison of the elastic moduli E input in the FEA with those (E0 ) extracted using the present inverse approach. Other parameters used in the simulations were h ¼ 1:0 mm, κ ¼ 2:2 106 kPa and ρ ¼ ρF ¼ 1000 kg/m3. R=h
E (kPa)
E0 (kPa)
Relative error
1.0
50 100 150
48 90 125
4.0% 10% 16.7%
2.0
50 100 150
49 96 139
2.0% 4.0% 7.3%
3.0
50 100 150
50 100 148
0 0 1.3%
5
the different geometries of the two phantoms which could have some effects on the mechanical properties of the phantoms. The experimental results obtained here match those reported by Maksuti et al. (2016) very well. In their study, they tested the phantoms prepared in the same procedure as ours. Their elastography experiments and tensile tests found that the shear modulus of the phantom was approximately 30.5 kPa, i.e., the Young’s modulus was about 91.5 kPa. The relative error between our experimental results and theirs is less than 4%. Clearly, the phantom experiments performed here together with the numerical experiments in Section 4.1 demonstrate that the present inverse approach is effective and can be used in practical measurements.
5. Discussion In the analysis above, the artery was assumed to be a hollow cylinder surrounded by fluid. However, the cross-section of a practical artery may have more complex geometries instead of a circular annulus. Moreover, local stenosis may exist in the artery. Here, we explore the applicability of the present method to these critical cases. Besides, some limitations in this study are discussed. 5.1. Effect of the arterial cross-section shape The aim of the present study was to develop an inverse approach to identify arterial stiffness in vivo, which is of great importance for the diagnosis of cardiovascular diseases. In this case, it is necessary to explore whether the present method is effective when the cross-section of the artery is not a circular annulus, which is frequently encountered in practice. A real artery as illustrated in Fig. 6(a) may be modeled as an elastic tube of which the thickness is h and the shape of the crosssection is determined by a function of Γ ðx; yÞ ¼ 0, as shown in Fig. 6
Fig. 4. (a) The phantom mold and the vessel-mimicking phantom. (b) Schematic diagram of the experimental setup (c) The velocity field of a vessel-mimicking phantom overlaid on the B-mode imaging at about 0.612 ms after applying the ARF. (d) The B-mode imaging of the phantom in which the path of interest (the red circle) is plotted. (e) The spatio-temporal imaging of the shear wave along the path of interest. (f) The dispersion curve obtained from the spatio-temporal imaging by the present inverse approach. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Please cite this article as: Li, G.-Y., et al., An ultrasound elastography method to determine the local stiffness of arteries with guided circumferential waves. Journal of Biomechanics (2016), http://dx.doi.org/10.1016/j.jbiomech.2016.12.006i
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Fig. 5. Experimental results for vessel-mimicking phantoms. (a) R¼ 4 mm, h ¼2 mm; (b) R ¼ 3 mm, h ¼1.5 mm. When the frequency is higher than about 1500 Hz, other high-order modes and noises (the red points) which are not used to conduct the inverse analysis emerges.
Fig. 6. (a) Cross-section of a human artery (from Wikipedia). (b) The schematic of the artery wall with variable curvature radius. (c) The propagation and (d) the dispersion curve of the GCW in the artery wall with an elliptic cross-section.
(b). In this case, the velocity component Ralong the normal direction n ¼ ∇Γ =ð∇Γ Þj can be expressed as vn ω AðωÞeiðks ωt Þ dω, where s is arc length as shown in Fig. 6(b). Then, the 2DFFT is conducted to vn to obtain the dispersion curve. For illustration, we consider an artery with an elliptic crosssection. The propagation of the elastic wave in the artery is simulated with the FEA, and the typical result is shown in Fig. 6(c). The dispersion curve is calculated based on the FEA results. As shown in Fig. 6(d), the dispersion curve can be well described by Eq. (8), indicating that the inverse method developed in this paper is applicable to more general cases. It should be noted that the effects of curvature may come into play when R=h is small. According to our computational results, for an artery with non-circular cross-section, curvature effects can be negligible when the ratio of the local curvature radius R to the wall
thickness h is greater than 1. For the example shown in Fig. 6, R=h Z 1:3. 5.2. Applicability of the method to the characterization of an arterial wall with local stenosis Local stenosis may occur in the human artery, and it is of great importance to evaluate the stiffness of the stenosis region. The present method has the potential to be used for such a purpose. To illustrate this point, an arterial wall with local stenosis was studied by a three-dimensional FE model as shown in Fig. 7(a). The propagating of the guided wave in the cross section of the stenosis region was monitored (see Fig. 7(b)) and used in the inverse analysis. Fig. 7(c) gives the dispersion curve provided by the FEA, which was fitted using Eq. (8) to infer the elastic modulus of the
Please cite this article as: Li, G.-Y., et al., An ultrasound elastography method to determine the local stiffness of arteries with guided circumferential waves. Journal of Biomechanics (2016), http://dx.doi.org/10.1016/j.jbiomech.2016.12.006i
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Fig. 7. Mechanical characterization of the local stenosis region of an artery. (a) The model and geometrical parameters used in the FEA, L ¼ 4 mm, t 1 ¼ 1 mm, t 2 ¼ 2 mm, R1 ¼ 4 mm, and R2 ¼ 3 mm. (b) The normalized radial velocity field in the arterial wall. (c) The dispersion curve given by the FEA and the fitting curve with Eq. (8). The best fitting gave an elastic modulus of 70 kPa, which was 6.7% smaller than the input value.
stenosis region. A comparison of the identified elastic modulus with the actual value input in the FEA shows that the relative error was 6.7%, indicating that indeed the present method may be used to determine the mechanical properties of the local stenosis region of an arterial wall, which appears to be impossible to access with other methods. 5.3. Limitations in this study This study has some limitations that deserve further efforts. First, real arterial walls have layered structures as shown in Fig. 6. Usually two layers can be observed with the ordinary ultrasound imaging method. In this case, the present inverse approach can provide the effective elastic modulus of the arterial wall as shown in the Supporting Information (SI-4). Second, variation of the arterial wall thickness along the circumferential direction has not been considered in this study. However, this issue is of practical interest in some cases and deserves further efforts. Third, real artery tissues are anisotropic with distributed collagen fibers (Gasser et al., 2006). Although recent studies have demonstrated that it is possible to determine the anisotropic and hyperelastic properties of a bulk anisotropic soft tissue (Li et al., 2016b; Li et al., 2016c), assessing the anisotropic parameters of arterial walls using the shear wave elastography method represents a more challenging issue because the wave in the arterial wall is guided. To characterize the anisotropic properties of the arteries, both the GCW and the GAW may need to be measured; this important issue is beyond the scope of this study and deserves further investigations. Fourth, here the arterial wall is assumed as timeindependent material, however, a real artery may exhibit viscoelastic deformation. For viscoelastic soft tissues, the wavenumber k is no longer a real number, but could be a complex number, which stands for the dissipation of the wave. Previous studies have
shown that the dispersion relation of the lowest mode, i.e., the mode adopted in our inverse analysis, is basically unaffected by the viscosity of the media (Chan and Cawley, 1998; Nguyen et al., 2011). Fifth, in this study, the outer region of the arterial wall was simplified as inviscid fluid to derive the theoretical solution. In this case, the phase velocity of the GCW is much smaller than that of the sound wave in the fluid; therefore, the GCW is basically nonleaky, i.e., the guided waves are trapped within the arterial wall. Whereas for a real artery, the perivascular tissues may cause the leakage of the GCW and affect the dispersion relation of the GCW. When perivascular tissues are much softer than the arterial wall, their effects on the dispersion relation of the GCW may be negligible. To quantitatively address their effects in general cases, a FE model including the perivascular tissues can be developed and this work is underway and the results may be reported in the coming future. Finally, effects of blood pressure will come into play in the in vivo experiments. When interpreting the in vivo data using the present method, in principle, it is possible to determine both the elastic modulus of the artery and the variation of blood pressure. This interesting issue requires further efforts.
6. Concluding remarks Quantitatively assessing local arterial stiffness in a non-invasive manner is of great importance and remains a challenging issue. Here, we addressed this challenge by exploring the use of the GCW induced in the arterial wall by the focused ARF to determine the arterial stiffness. In summary, the following key results were achieved. The dispersion properties of the GCW have been investigated using a finite element model. Our results show that when the ratio of the inner radius of the artery to its thickness is greater than a
Please cite this article as: Li, G.-Y., et al., An ultrasound elastography method to determine the local stiffness of arteries with guided circumferential waves. Journal of Biomechanics (2016), http://dx.doi.org/10.1016/j.jbiomech.2016.12.006i
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certain critical value, the dispersion properties of the GCW are basically independent of the curvature of the arterial wall and can be well-described using the LW model. Based on the FE results, an inverse method is proposed to characterize the elastic modulus of an artery. Both numerical experiments and phantom experiments had been performed to validate the proposed method. We show that our method can be applied to the cases in which the arterial wall has a local stenosis and/or the geometry of the arterial cross-section is irregular, thereby it has great potential to be used in clinics.
Conflict of interest statement Authors have no financial and personal relationships that could inappropriately influence or bias this work.
Acknowledgement We acknowledge the support from the National Natural Science Foundation of China (Grant nos. 11572179, 11172155, 11432008, and 81561168023). English through the whole text has been edited using the NPG (Nature Publishing Group) Language Editing service.
Appendix A. Supplementary material Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.jbiomech.2016.12. 006.
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Please cite this article as: Li, G.-Y., et al., An ultrasound elastography method to determine the local stiffness of arteries with guided circumferential waves. Journal of Biomechanics (2016), http://dx.doi.org/10.1016/j.jbiomech.2016.12.006i