Complex shear modulus quantification from acoustic radiation force creep-recovery and shear wave propagation Carolina Amador, Matthew W. Urban, Shigao Chen and James F. Greenleaf. Department of Physiology and Biomedical Engineering Mayo Clinic College of Medicine Rochester, MN, USA
[email protected] Abstract—Quantitative mechanical properties can be measured with shear wave elasticity imaging methods in a modelindependent manner if both shear wave speed and attenuation are known. Typically, only shear wave speed is measured and rheological models are used to solve for the shear viscoelastic complex modulus. This paper presents a method to quantify viscoelastic properties in a model-independent way by estimating the loss tangent over a wide frequency range using timedependent creep-recovery response induced by acoustic radiation force. The shear wave group velocity and the shear wave center frequency in combination with loss tangent are used to estimate the complex modulus so that knowledge of the applied radiation force magnitude is not necessary. Experimental data are obtained in one excised swine kidney.
needs to be maintained for long times, for instance up to 30 ms, which means that the total exposure time to acoustic radiation force in some cases may be too large.
Keywords-creep, recovery, complex shear modulus
I.
INTRODUCTION
Ultrasound-based shear wave elasticity imaging methods have demonstrated that tissue elasticity changes with disease state [1]. Besides shear wave excitation, acoustic radiation force has been proposed to excite the quasi-static response of viscoelastic materials [2, 3]. The majority of current methods, including shear wave and quasi-static excitation, rely on rheological models to estimate mechanical properties such as elasticity and viscosity. Recently, it has been shown that the quasi-static response from acoustic radiation force excitation (creep) can be used to quantify viscoelastic parameter in a model-independent way by estimating the complex shear modulus from creep displacement response [4]. In these acoustic radiation forced induced creep methods, two types of ultrasound beams are used [3]. High intensity pushing beams are interspersed with conventional tracking beams during creep period. These push-tracking beams mimic a temporal stepforce while creep displacements are tracked. Additionally, after the push-track period (creep) additional tracking beams are used to capture the recovery period (as shown in Figure 1). Currently, both creep and recovery are induced and monitored but only creep displacement is used to estimate the model-independent complex shear modulus [4]. A limitation of this model-independent method, is that to study viscoelastic parameters over a wider range of frequencies, the creep period
Figure 1. Acoustic radiation force induced creep and recovery beam sequence.
This paper presents a method to quantify viscoelastic parameter in a model-independent way by estimating the complex shear modulus from recovery response after acoustic radiation force induced creep. Some of the content of this paper is similar to Amador et al.[4]. The method theory is illustrated by Matlab simulation using Kelvin-Voigt model. Experimental data was obtained in one excised swine kidney. II.
MATERIALS AND METHODS
A. Creep-Recovery theory In viscoelasticity theory, creep is a slow, progressive deformation of a material under constant stress. The removal of a constant stress elicits a response just as the imposition of the constant stress does; this response is called recovery. In a shear creep test, the ratio between the unitless measured shear strain response, γ(t), and the applied constant shear stress, τ0 [N/m2], is called the creep compliance, J(t) [m2/N]. The material response to a constant stress is described by the Boltzmann constitutive integral [5]:
t
∫
γ (t ) = J (t − ξ ) 0
∂τ (ξ ) dξ ∂ξ
(1)
where γ is shear strain, τ is shear stress, J is creep compliance and ∂[.]/∂ξ represents the first derivative respect to the independent variable ξ. Evans et al. [6] have reported the analytic solution of (1) by using the Fourier transform and its properties. Briefly, the time-domain compliance is converted to shear complex modulus without fitting any rheological model.
G * (ω ) =
1 FT [τ (t )] = FT [γ (t )] iωFT [J (t )]
(2)
As described in Amador et al.[4], assuming that compliance is proportional to radiation force induced displacement u(t) by a factor β, Evan’s conversion formula (2) gives a relative complex modulus C*(ω) from the displacement u(t).
G * (ω ) =
1 1 = ⋅ C * (ω ) β ⋅ iωFT [u (t )] β
(3)
The relative complex modulus C*(ω) is proportional to the actual complex modulus G*(ω) by the proportionality constant β:
C (ω ) = β (Gs (ω ) + iGl (ω )) *
(4)
where Gs(ω) and Gl(ω) are the real (storage modulus) and imaginary (loss modulus) parts of the complex shear modulus G*(ω). Because the magnitude of the acoustic radiation force, F = 2αI/c, is proportional to the absorption coefficient of the media, α, and the temporal average intensity of the acoustic beam at a given spatial location, I [7], in a homogenous material, the magnitude of the extracted relative complex modulus C*(ω) will vary as a function of material absorption and acoustic beam intensity, therefore, the extracted relative complex modulus C*(ω) would not be a very useful measure. To overcome this problem, a widely used property of viscoelastic materials called loss tangent or tan(δ), defined as the ratio between the loss modulus and the storage modulus, is used [5]:
tan (δ ) =
Gl (ω ) β ⋅ C l (ω ) C l (ω ) = = G s (ω ) β ⋅ C s (ω ) C s (ω )
(5)
where Cs(ω) and Cl(ω) are the real and imaginary part of the relative complex modulus C*(ω). The previous creep theory can be applied to the recovery period as follow. The recovery period or the response to the removal of the step-stress may be expressed as the application at t = t’ of another step of equal magnitude but opposite sign to the step applied at t = 0. This excitation can be expressed in terms of the unit step function as:
τ (t ) = τ 0u (t ) − τ 0u (t − t ')
(6)
The first term of (6) represents the imposition of a step-stress of magnitude τ0 at t = 0, while the second term represents the removal of the stress, expressed at the imposition of a step-
stress of equal magnitude but opposite direction delayed by the amount t’. The response of such stimulus is illustrated in Figure 2(a), where the creep or the response to the step-stress is shown with asterisk and the recovery or the response to an opposite step-stress is shown with circles and triangles. The creep and recovery curves can be brought into superposition by reflecting either one around the time axis and then shifting them along both axes. Then, creep and recovery contain the same information on the time-dependent behavior of a linear viscoelastic material. Moreover, for the recovery period, (1) remains the same but with negative magnitude from the negative step-stress and it is defined from t’ to t. B. Shear wave excitation Because a shear wave is generated from the creep excitation; the shear wave group velocity can be estimated and the center frequency of the wave can be used in combination with the estimated loss tangent to find the model-independent shear complex modulus. The wavenumber, k, and the shear elastic modulus, G, are linked through the shear wave propagation equation. In an elastic medium, they are related by:
G=ρ
(7)
ω2 k2
where ρ is density of the medium and ω the angular frequency. In the case of a linear viscoelastic medium the wave number, k, and shear elastic modulus, G, are complex, written as k* = kr – iki and G*(ω)=Gs(ω) + iGl(ω) [8]. Then for a viscoelastic medium, (7) can be written as:
Gs (ω ) = ρω 2
k r2 − k i2
(k
2 r
Gl (ω ) = −2 ρω 2
+ k i2
(k
)
k r ki 2 r
+ ki2
(8)
)
(9)
where kr is the real part of the wave number (defined as kr = ω/cs, where cs is the shear wave speed), ki is the imaginary part of the wave number (defined as ki = αs, where αs is shear wave attenuation) [9]. The loss tangent or tan(δ) written in terms of the complex wave number is [9]:
tan (δ ) =
(10)
2k r k i k r2 − k i2
If both tan(δ) and kr are known, the negative root for ki in (10) is [9]: ⎡ ⎛ 1 ⎞ ⎟− k i = k r ⎢⎜⎜ ⎢⎝ tan (δ ) ⎟⎠ ⎢⎣
⎛ ⎛ 1 ⎞2 ⎞ ⎤ ⎜1 − ⎜ ⎟ ⎟⎥ ⎜ ⎜⎝ tan (δ ) ⎟⎠ ⎟ ⎥ ⎝ ⎠ ⎥⎦
(11)
Finally, by knowing kr and ki, the shear storage (Gs) and loss (Gl) moduli are obtained from (8) and (9). We propose to use shear wave group velocity, cg, and its center frequency, ωc,
to calibrate the relative complex modulus. The shear wave group velocity is calculated by evaluating the time shifts in the shear wave versus position and using (12)
cg = Δx/Δt
where cg is the group velocity and Δt is the time shift measured over a distance Δr. The center frequency is defined as the frequency at which the magnitude spectrum of particle velocity is highest. Tan(δ) at the center frequency is used in equation (11) to calculate ki(ωc), then both ki(ωc) and kr(ωc) are used in (8) and (9) to calculated the storage and loss modulus at the center frequency, Gs(ωc) and Gl(ωc). Finally, the proportionality constant is estimated by: β=
(G s (ωc ) + iGl (ωc )) C * (ω c )
(13)
and it is used in combination with the relative complex modulus C*(ω) to calculate the model-independent complex shear modulus. C. Simulation and experiments The theoretical Kelvin-Voigt model is used to demonstrate that creep (14) and recovery (15) contain the same information on the time-dependent behavior of a linear viscoelastic material. 1β u creep (t ) = G
u re cov ery (t ) =
G ⎛ − t⎞ ⎜ η ⎟ ⎜⎜1 − e ⎟⎟, 0 < t < t ' ⎝ ⎠
1β G
starting at 5 ms and circles (o) for recovery starting at 10 ms. The model-independent loss tangent or tan(δ) calculated using (3) and (5) on simulated creep and recovery displacements are shown in Figure 2b; by inspection, the creep and recovery processes contain the same loss tangent information on the time-dependent behavior of a linear Kelvin-Voigt material.
G G ⎛ − t ' ⎞ − (t −t ' ) ⎜ η ⎟ η 1 e e , t > t' − ⎟⎟ ⎜⎜ ⎠ ⎝
Figure 2. Kelvin-Voigt model (a) creep and recovery displacement (b) loss tangent from (a).
Figure 3 shows the (a) experimental recovery displacement. (b) model-independent relative complex shear modulus and (c) loss tangent in a region of 3x3 mm2 in the renal cortex of an excised swine kidney.
(14) (15)
The shear modulus G and viscosity η used in simulations were 3 kPa and 4 Pa·s, respectively. The creep displacement, ucreep(t), response was simulated up to 10 ms and the recovery displacement, urecovery(t), was simulated for 5 and 10 ms. The sampling frequency for all simulations was 6.25 kHz. Evan’s conversion formula (3) was used to calculate the loss tangent (5) using simulated creep and recovery displacements from (14) and (15). Experimental data was obtained from one excised swine kidney. A Verasonics ultrasound system (Verasonics, Redmond, WA) equipped with L7-4 linear array transducer (Philips Healthcare, Andover, MA) was used in the experiment. Excised kidney was placed in a saline bath at room temperature. Acoustic radiation force was used to induce creep by using 32 short push beams of 2 μs length with a focused push beam of 5 MHz center frequency. The short pushes where 80 ms apart. Flash imaging was used to detect recovery at a 12.5 kHz frame rate for 20 ms. A 2D autocorrelation method was used to estimate tissue displacement [10]. III.
RESULTS
Figure 2a illustrates the normalized Kelvin-Voigt model creep displacement, ucreep(t), with asterisk (*) and recovery displacement, urecovery(t), with triangles (Δ) for recovery
Figure 3. Experimental results in excised swine kidney, (a) Recovery displacement, (b) relative storage and loss moduli, (c) tan(δ).
Figure 4 illustrates the (a) time to peak regression to calculate shear wave group velocity and (b) magnitude spectrum of velocity to calculate center frequency. The group velocity was 1.37 m/s at 71 Hz center frequency.
In this paper, we described and validated a method to fully quantify regional viscoelastic properties in a manner independent of models. Previous work in this area involved the use of rheological models, but the need for such models affects the viscoelastic parameter estimation as well as the fitting process. Future work will include evaluating method variability and effects of tissue density, as well as in vivo applications. V. Figure 4. Group velocity estimation and analysis, (a) Time to peak regression, (b) magnitude spectrum of velocity.
The shear complex modulus is shown in Figure 5. The average storage modulus and the slope of the loss modulus were 2.11 kPa and 3.05 Pa.s between 50 and 1000 Hz.
CONCLUSION
The proposed method is a novel approach to overcome limitations encountered with model-dependent methods. Moreover, the short pulses could be used with conventional ultrasound scaneers. ACKNOWLEDGMENT The project described was supported by grants R01EB002167, R01EB002640 and DK082408 from the National Institute of Health. Disclosure: Mayo and the authors have financial interest in the technology described here. REFERENCES
Figure 5. Shear complex modulus of excised swine kidney.
IV.
DISCUSSION
Very few methods have been proposed to characterize tissue mechanical properties in a model-independent manner. The method presented in this paper is a novel approach to overcome difficulties encountered with rheological model and fitting approaches. Kelvin-Voigt model simulations illustrate that creep and recovery contain the same information. Additionally, it is demonstrated that the steady state does not need to be reached on creep when using the recovery response (refer to Figure 2), which means that there is no need to push for long time. Another advantage of using the recovery signal is that it can be monitored as long as needed without using high intensity beams. As a consequence of the multiple short pushes, a shear wave is generated. The group velocity and center frequency were measured and are useful to calculate the modelindependent complex modulus, which provides a more complete characterization of viscoelastic properties.
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