Parametric Analysis of SV Mode Shear Waves in Transversely Isotropic Materials Using Ultrasonic Rotational 3D SWEI

Abstract

Ultrasonic rotational 3D shear wave elastography imaging (SWEI) has been used to induce and evaluate multiple shear wave modes, including both the shear horizontal (SH) and shear vertical (SV) modes in in vivo muscle. Observations of both the SH and SV modes allow the muscle to be characterized as an elastic, incompressible, transversely isotropic material with three parameters: the longitudinal shear modulus μ_L, the transverse shear modulus μ_T, and the tensile anisotropy χ_E. Measurement of the SV wave is necessary to characterize χ_E, but the factors that influence SV mode generation and characterization with ultrasonic SWEI are complicated. This work uses Green’s function simulations to perform a parametric analysis to determine the optimal interrogation parameters to facilitate visualization and quantification of SV mode shear waves in muscle. We evaluate the impact of five factors: μ_L, μ_T, χ_E, fiber tilt angle θ_tilt, and F-number of the push geometry on SV mode speed, amplitude, and rotational distribution. These analyses demonstrate that: 1) as μ_L increases, SV waves decrease in amplitude so are more difficult to measure in SWEI imaging, 2) as μ_T increases, the SV wave speeds increase, 3) as χ_E increases, the SV waves increase in speed and separate from the SH waves, 4) as fiber tilt angle θ_tilt increases, the measurable SV waves remain approximately the same speed, but change in strength and in rotational distribution, and 5) as the push beam geometry changes with F-number, the measurable SV waves remain approximately the same speed, but change in strength and rotational distribution. While specific SV mode speeds depend on the combinations of all parameters considered, measurable SV waves can be generated and characterized across the range of parameters considered. To maximize measurable SV waves separate from the SH waves, it is recommended to use an F/1 push geometry and θ_tilt = 15°.

I. Introduction

Musculoskeletal shear wave elasticity imaging (SWEI) is a field of wide research, but studies show considerable variability among measurements. For example, in the relaxed vastus lateralis, in vivo longitudinal shear stiffnesses range from 3.3 ± 0.4 kPa [1] to 20.3 ± 2.3 kPa [2], demonstrating a lack of repeatability using current 2D SWEI tools, and limiting clinical adoption [3]–[8]. These studies were fundamentally limited by using a single imaging plane (2D SWEI) and their inherent assumption of an isotropic material, though a few works have highlighted the directional dependence of shear wave speed (SWS, or c) in skeletal muscle [9]–[14] demonstrating the importance of understanding fiber orientation relative to the SWEI imaging plane. There is an ongoing effort in the research community to develop 3D SWEI algorithms capable of estimating the biomechanical properties of muscle and identifying potential relationships between these properties and muscle function to find robust, non-invasive biomarkers of muscle health [1], [2], [12], [15]–[23].

Skeletal muscle is commonly modeled as a transversely isotropic (TI) elastic material due to the alignment of the muscle fibers in the direction of muscle contraction [17], [19], [24]–[26]. If the tissue is also assumed to be incompressible, it can be described with three independent parameters [27]. Here we use the longitudinal shear modulus μ_L, the transverse shear modulus μ_T, and a single parameter modeling the joint effect of the longitudinal and transverse Young’s moduli E_L and E_T called the tensile anisotropy χ_E = (E_L − E_T)/E_T.

In a TI material, there are two shear wave modes with different polarizations that can be excited: the shear horizontal (SH) and shear vertical (SV) [27]. The parameters μ_L and μ_T can be measured solely from observation of the SH mode; however, to quantify tensile anisotropy χ_E, using SWEI, it is necessary to observe the SV mode [28].

Our group has recently observed both wave modes in vivo and demonstrated the ability to estimate all three of these parameters to characterize the vastus lateralis muscle as an incompressible transverse isotropic (ITI) material using rotational 3D SWEI where the transducer rotates around its central axis between 2D SWEI frames to construct a 3D volume of shear wave data [28]. We postulate that tensile anisotropy χ_E, will prove a useful biomarker of muscle health, as in a tensile measurement the displacement is aligned with the direction of biomechanical force generation in contracting skeletal muscle. To assess the use of χ_E as a biomarker of muscle health we must first understand the factors that affect measurement of SV waves using SWEI. Thus the overall goal of this paper is to perform a parametric analysis of the factors affecting SV mode quantification using 3D SWEI.

II. Background and Motivation

A. Incompressible, Transversely Isotropic (ITI) Materials

Ultrasonic SWEI measures the mechanical properties of biological tissues by first creating a localized acoustic radiation force (ARF) impulse in tissue at the transducer focus that generates outwardly propagating shear waves. These waves can be tracked by subsequent ultrasound B-mode frames. In an infinite, homogeneous, isotropic, linear elastic material, the SWS is related to the shear modulus μ and density ρ of the material as μ = ρc² [29], [30]. However, in many biological tissues, including skeletal muscle, the relationship between SWS and μ is not as simple because the material is not isotropic. Skeletal muscle is commonly modeled as a linear, elastic, TI material due to the alignment of the muscle fibers in the direction of muscle contraction, with an axis of symmetry along the muscle fiber direction $\widehat{A}$. A TI material is characterized by 5 independent elements in the elasticity tensor c_ijkl relating the stress tensor σ and strain tensor ϵ using a generalized Hooke’s law relation: σ_ij = c_ijklϵ_kl [26], [31]. These 5 elements are commonly expressed in terms of 6 engineering constants and one constraint: longitudinal (L) and transverse (T) Young’s moduli (E_L, E_T), two shear moduli (μ_L, μ_T), two Poisson’s ratios (ν_LT, ν_TT) and the constraint μ_T = E_T/(2(1 + ν_TT)) [26], [31].

When the incompressible limit is taken, the two Poisson’s ratios have specific values: ν_LT = 1/2, and ${\nu }_{TT}=1-\left(\frac{E_T}{2E_L}\right)$ [26], [27]. These relations mean that in an elastic incompressible transversely isotropic (ITI) material the shear wave propagation can be described by three independent material parameters. In this work we use μ_L, μ_T, and a term combining the two Young’s moduli, called tensile anisotropy ${\chi }_E=\frac{E_L-E_T}{E_T}$ [27]. We can also define a shear anisotropy as ${\chi }_{\mu }=\frac{{\mu }_L-{\mu }_T}{{\mu }_T}$.

B. Wave Propagation in ITI Materials

As seen in Fig. 1, in an elastic ITI material, both the SH and SV shear wave polarizations can be excited. This setup diagram shows the shear wave polarizations generated during SWEI imaging: the ARF push $\overrightarrow{F}$, in the axial Z direction, the muscle symmetry axis $\widehat{A}$, the direction of shear wave propagation, $\widehat{n}$, the angle ϕ_rot between $\widehat{A}$ and $\widehat{n}$ in the XY plane, the angle θ_tilt between $\widehat{A}$ and the XY plane, and ψ the angle between the vectors $\widehat{A}$ and $\widehat{n}$. The SH mode, shown in pink, has particle motion always perpendicular to the $\widehat{A}-\widehat{n}$ plane, and the SV mode, shown in yellow, has particle motion in the $\widehat{A}-\widehat{n}$ plane. Given that acoustic radiation force (ARF) methods preferentially push and track displacements in the axial (Z) dimension, it is expected that when the fibers are normal to the ARF push (θ_tilt = 0°) only the SH mode is tracked, as the corresponding SV mode has minimal displacement in the Z direction. When the fiber direction $\widehat{A}$ is tilted relative to the transducer face, as seen in Fig. 1, both the SH and SV modes have displacement components that can be measured in the Z direction.

Fig. 1.

Two shear wave modes with different polarizations (pink = SH; yellow = SV) can be excited in TI materials. The polarizations are defined relative to the plane created by the muscle symmetry axis $\widehat{A}$ and the propagation direction $\widehat{n}$ being ultrasonically tracked. With the push $\overrightarrow{F}$ tilted at an angle θ_tilt from normal incidence with $\widehat{A}$, both the SH mode (pink plane polarization) and the SV mode (yellow plane polarization) generate Z direction displacement which can be quantified using conventional ultrasonic motion tracking methods. Figure adapted from Knight et al. [28]

The phase velocity v_SH for the SH mode is given by [25]

$$\rho v_{SH}^2={\mu }_{L}co{s}^2\psi +{\mu }_{T}si{n}^2\psi ,$$ (1)

where ρ is the material density (treated as a constant), ψ is the angle between the fiber orientation $\widehat{A}$ and the tracked propagation direction $\widehat{n}$. The corresponding group velocity V_SH for the SH propagation mode is given by [25]

$$\rho V_{SH}^2=\frac{{\mu }_L{\mu }_T}{{\mu }_{L}si{n}^2\psi +{\mu }_{T}co{s}^2\psi },$$ (2)

which shows that group velocity depends on the same factors as the SH mode phase velocity. These relations indicate that the SH wave speed depends on μ_L and μ_T and the fiber orientation ψ, as it relates to $\widehat{A}$, and is not dependent on tensile anisotropy [26], [27].

In contrast, the phase velocity v_SV for the SV propagation mode is given by [26], [27]

$$\rho {\upsilon }_{SV}^2={\mu }_L+4\left(\left({\chi }_E+1\right){\mu }_T-{\mu }_L\right)si{n}^2\psi co{s}^2\psi .$$ (3)

Equation 3 indicates that SV mode phase velocity is affected by μ_L, μ_T, ψ and χ_E [26], [27]. Thus, while μ_L and μ_T can be measured solely using the SH mode it is necessary to measure the SV mode to be able to determine the third independent material parameter, tensile anisotropy χ_E, needed for complete characterization of muscle as an ITI material.

Furthermore, while the SH wave mode has a closed form solution for the group shear wave speed (Eq. 2), the SV wave mode has no such simple solution [27]. In previous work, we quantified χ_E using an iterative Green’s function (GF) approach for a specific experimental geometry [28]. We quantified χ_E by measuring the group speed of the SH and SV mode experimentally, then calculated the GF speeds for a wide variety of χ_E values and minimized the difference to estimate χ_E [28]. This procedure allows for in vivo measurement of all three parameters of an ITI material with a single 3D SWEI acquisition. Fig. 2 shows an example acquisition and corresponding matched GF simulations and experimental SWS obtained in an ongoing study.

Fig. 2.

a) Following the methods of Knight et al., a healthy 37 year old female volunteer was consented under an approved Duke Institutional Review Board protocol and strapped into a BioDex System 4 Pro™ to ensure a hip flexion 90° and knee flexion of 60°. The transducer was rotated around its central axis in 5° increments. Repeated 3D SWEI rotational data sets were acquired with small movements (< 1 cm) between acquisitions. b) For each 2D frame (36 frames in 5° increments per 3D acquisition), the Radon Sum approach was used to find the SH and SV speeds [28], [32] as shown in blue marks and green circles respectively. χ_E was determined by simulating shear wave signals with Green’s functions (GF) with matched parameters to the experimental data using a range of values for χ_E and minimizing the difference between simulated and experimental SV mode speeds [27]. Simulated shear wave speeds and strength are shown in red to black circles. Red being the strongest shear wave, primarily the SH ellipse, and black being the weaker SV waves, including open circles where the strength of the wave is below −30 dB. Across 8 measurements, χ_E was found to be 10.9 ± 1.4 (corresponding to E_T/E_L of 0.085 ± 0.009).

C. Parameter Space: Factors Affecting SV Shear Wave Speed Measurement

A more complete understanding of the potential for χ_E to serve as a biomarker of disease requires a more robust understanding of the factors affecting χ_E, as measured by factors affecting SV mode waves. In reports from studies of healthy volunteer’s muscles in literature and as our group has observed empirically, μ_L and μ_T can vary based on a variety of factors including knee flexion angle, active stretching, and contraction, though μ_L varies more than μ_T and is the parameter more often measured [1], [4], [8], [12], [22], [23], [32]–[35]. Additionally, χ_E has been measured in vivo in limited studies [16], [20], [21], [28], [32], from which we have established a range of feasible χ_E values to investigate in simulations.

Also affecting χ_E measurement and SV wave speed are the factors controlled by the experimental imaging configuration and fiber orientation during imaging. The tilt angle between the transducer and the fibers as they appear on B-mode can be adjusted by the experimental setup. Empirically, our group has observed that SV waves can be measured in healthy volunteers, even at low fiber tilt angles (θ_tilt < 10°). The innate fiber tilt θ_tilt, measured in the vastus lateralis of healthy volunteers when imaging transcutaneously with ultrasound on the leg has ranged from 0° to 20° [36], [37], and small adjustements with the transducer and gel path can change the observed θ_tilt.

Additionally, as we have varied the shape of the excitation push, as characterized by the lateral F-number between F/1 and F/2, we have noticed changes in the amplitude of the SV waves and transducer rotation positions (ϕ_rot) where the SV waves are observed. Thus F-numbers of 1 and 2 were also included as parameters to investigate the effect of push geometry (F-number) on SV wave speed measurement. By elucidating the effects of these parameters on SV waves, which directly relate to the measurability of χ_E, we aim to more fully understand the potential of quantifying χ_E as a biomarker for muscle health.

III. Methods

The five factors affecting SV shear waves studied in this paper can be grouped into three material parameters: μ_L, μ_T, χ_E, and two ARF push parameters: θ_tilt and F-number. As seen in Table I, ranges and step sizes were chosen for each investigated parameter, based upon 2D SWEI values reported in literature [1], [4], [8], [12], [16], [20]–[23], [33]–[35], our group’s 3D SWEI measurements [28], [32], and feasible interrogation geometries [36], [37]. The Z displacement component from the focal plane XY (transverse) plane centered at the axial focus (20 mm) was calculated following the methods of Rouze et al. and Knight et al. for each of these 18,120 parameter combinations [27], [28]. In computationally efficient GF calculations as proposed by Rouze et al., the parameter space representing tensile anisotropy ${\chi }_E=\frac{E_L-E_T}{E_T}$ is implemented as $\frac{E_L{\mu }_T}{E_T{\mu }_L}$ (see Equations 32–36 of Rouze et al.), so this is used as the varied parameter herein [27]. $\frac{E_L{\mu }_T}{E_T{\mu }_L}$ can be related to χ_E using the equations in Section II. A.

TABLE I.

Parameter Space Used in GF simulations

Parameter	Values
CL (relates to μL by ${\mu }_L=\rho c_L^2$)	2, 3, 4, 5 m/s
cT (relates to μT by ${\mu }_T=\rho c_T^2$)	0.75, 1.00, 1.25 m/s
θtilt	0°, 5°, 10°, 15°, 20°
Lateral F-number	1, 2
$\frac{E_L{\mu }_T}{E_T{\mu }_L}$ (relates to χE)	0.250:0.025:4.000
χE	min 1.64 max 178.8

For all simulations: push frequency = 4 MHz attenuation α = 0.5 dB/cm/MHz, axial push focus = 20 mm

The force profiles of Z displacements were calculated with Field II [38] and used to model the ARFI excitation force geometry of an L7–4 linear array with a fixed elevation focus of 25 mm as shown in Fig. 3. Fig. 3 shows the lateral-axial and elevational-axial profiles of the two push geometries: one with a lateral F/1 push and one with a lateral F/2 push, both with a 4 MHz push frequency, 20 mm push focal depth, a 200 μs ARF push, and assuming a tissue attenuation α = 0.5 dB/cm/MHz.

Fig. 3.

Force profiles with different F-numbers: On the left is the XZ (lateral-axial) profile of normalized forces calculated by Field II [38] with an F/2 push focused at an axial depth of 20 mm, assuming a frequency of 4 MHz, α = 0.5 dB/cm/MHz. Next is the YZ (elevational-axial) profile of the same force as the leftmost plot. The third plot is the XZ (lateral-axial) profile of normalized forces calculated by Field II [38] with an F/1 push focused at an axial depth of 20 mm, assuming a frequency of 4 MHz, α = 0.5 dB/cm/MHz. The fourth plot is the YZ (elevational-axial) profile of the F/1 push. In all four images, the red dot represents the ARF push focus, and the black asterisk represents the axial position of peak displacement. In the F/2 push images, the maximum displacement is 1.5 mm shallow to the axial push focus (20 mm). In the F/1 images, the maximum displacement is 0.25 mm shallow to the axial push focus.

GF simulations modeled shear wave propagation for 10 ms, by calculating axial (Z) displacement in a single transverse (XY) plane at the push focus (20 mm). The GF simulations account for simulated rotation of the transducer such that the lateral profile aligns with radius at each position of ϕ_rot.

For each shear wave data set, the displacement through time at each position was differentiated in time to calculate particle velocity, matching experimental processing. Shear wave speeds were identified using a Radon sum approach from 8 to 16 mm radially (laterally outward), similar to the lateral range used in experimental data [28], [30], [32], using an automated algorithm that limits the identified speed between 0.5 and 12 m/s [39]. SH wave speeds were identified by finding the peak in the Radon sum closest to the expected speed given the ellipse shape of the SH group speed equation (Eq. 2) [28]. The algorithm then identified SV waves (up to two per rotational angle) as the next largest Radon sum peak present. Fig. 4 shows a sample GF plane where c_L = 2 m/s, c_T = 1 m/s, θ_tilt = 10°, χ_E = 10, with F/2 push. Both the SH and SV speeds identified are shown in the polar plot in the center of Fig. 4, and the identified shear wave trajectories can be seen as white trajectories on the space-time plots around the circle.

Fig. 4.

Example GF Plane with c_L = 2 m/s, c_T = 1 m/s, θ_tilt = 10°, χ_E = 10, and a F/2 push. The center polar coordinates show the SWS, from both SH (elliptical shape) and SV waves. The color of the points represents the strength of the wave as the average particle velocity along the radial path used to estimate SWS. Solid points are measurable shear wave speeds (greater than −30 dB cutoff), while open circles are below the measurable cutoff. in The space time plots around the edges show the particle velocity (calculated from normalized particle displacement) at each rotational angle (ϕ_rot step size = 15° for space time plots, while for data collection and speeds shown in center are ϕ_rot step size 5°). An example of labeled axis is in the top left. It is important to note that the color axis is greatly zoomed in so as to show the weaker amplitude SV waves in addition to the SH waves. The white lines represent trajectories identified by the Radon sum speed finding algorithm, and the slope of each white line corresponds to a shear wave speed point on the center plot.

For each shear wave, we identified a strength metric, or peak velocity amplitude metric, calculated by measuring the Radon sum of the particle velocity normalized by path length. This metric can also be described as the average particle velocity under the trajectory path, representing the amplitude of the shear wave measured relative to other shear waves (both SH and SV) in the same GF simulation plane. As seen in Fig. 4, when the strength of a given SWS estimate at a given rotation angle ϕ_rot was 30 dB down from the maximum seen in the GF simulation plane, it was considered unmeasurable, as represented by open circles, as opposed to the filled circles which could be feasibly measured by ultrasonic tracking. This threshold was selected based upon our experience with in vivo experimental data. Thus use of a 30 dB threshold in shear wave strength, based on typical in vivo data as shown in Fig. 2b, increases confidence that although the GF simulations are noiseless, results from GF simulations are still applicable in vivo.

Fig. 4 also shows the GF data are much less noisy than similar in vivo data (for example, see Knight et al. 2022 Fig. 6 [28]) and have narrower trajectories, allowing for as many as three distinct wave fronts, one SH and two SV, to be found at a single ϕ_rot (multiple SV mode waves at a single ϕ_rot with different SWS are possible for certain combinations of material parameters [27]). In the cases where three shear wave fronts are found, generally one of the SV waves is too low in amplitude to be feasibly measured using ultrasonic SWEI (for example see ϕ_rot = 30° in Fig. 4).

IV. Results

Fig. 5 shows polar plots of SH and SV speeds for simulations with c_L varying from 2 m/s to 5 m/s and θ_tilt varying from 0° to 20°, with fixed values of c_T = 1 m/s, χ_E = 10, and an F/2 push. The top row of Fig. 5 shows that at θ_tilt = 0° there are no measurable SV waves, and even the unmeasurable waves (hollow black circles) disappear as c_L increases. The middle row shows at a tilt of θ_tilt = 10° and shows some measurable SV waves at lower c_L speeds, but as c_L increases with all other factors remaining the same, the number of angles with measurable SV wave speeds decreases. The bottom row shows that at a tilt of θ_tilt = 20° even more SV speeds are measurable, and while the number of angles of ϕ_rot with measurable SV waves remains the same (almost all angles), the strength of the SV waves on average decreases as c_L increases. Overall, as θ_tilt increases, there are more measurable SV waves at all c_L values studied herein.

Fig. 5.

Varying c_L and θ_tilt: All 12 examples have c_T = 1 m/s, χ_E = 10, and an F/2 push. In the top row θ_tilt = 0°, the middle row θ_tilt = 10°, and the bottom row θ_tilt = 20°. For each subplot the maximum speed is 6 m/s, and ϕ_rot angles are as seen in Fig. 4. Solid points are measurable shear wave speeds (greater than −30 dB cutoff), while open circles are below the measurable cutoff.

Fig. 6 shows results obtained using values of c_T ranging from 0.75 m/s to 1.25 m/s, with fixed values of c_L = 2 m/s, θ_tilt = 10°, χ_E = 10, and F/2 push. As c_T increases with all other factors remaining the same, the speed of the measurable (solid points) SV waves increases. In this example when c_T = 0.75 m/s, the strongest SV wave speed is 2 m/s to 2.5 m/s, when c_T = 1 m/s the strongest SV wave speed is 3.5 m/s to 4 m/s, and when c_T = 1.25 m/s, the strongest SV wave speed increases to 4 m/s to 5 m/s. Additionally, the number of ϕ_rot rotational angles where SV waves can be measured increases as c_T increases.

Fig. 6.

Varying c_T : All 3 examples have c_L = 2 m/s, θ_tilt = 10°, χ_E = 10 and an F/2 push. For each subplot the maximum speed is 6 m/s, and the ϕ_rot angles are as seen in Fig. 4. Solid points are measurable shear wave speeds (greater than −30 dB cutoff), while open circles are below the measurable cutoff.

Fig. 7 shows results obtained using χ_E varying from 5.0 to 15.0, with fixed values of c_L = 2 m/s, c_T = 1 m/s, θ_tilt = 10°, and an F/2 push. As χ_E increases with all other factors remaining the same, the measurable SV waves increase in speed and separate from the SH waves. Additionally, the number of ϕ_rot rotational angles where SV waves can be measured increases as χ_E increases.

Fig. 7.

Varying χ_E: All 5 examples have c_L = 2 m/s, c_T = 1 m/s, θ_tilt = 10°, an an F/2 push. For each subplot the maximum speed is 6 m/s, and the ϕ_rot angles are as seen in Fig. 4. Solid points are measurable shear wave speeds (greater than −30 dB cutoff), while open circles are below the measurable cutoff.

Fig. 8 shows results obtained using θ_tilt varying from 0° to 20° with c_L = 2 m/s, c_T = 1 m/s, χ_E = 10, and an F/2 push. As θ_tilt increases with all other factors remaining the same, the SV waves remain approximately the same speed, but change in strength, starting as unmeasurable (below −30 dB) then having more ϕ_rot angles with measurable SV waves as θ_tilt increases.

Fig. 8.

Varying θ_tilt: All 5 examples have c_L = 2 m/s, c_T = 1 m/s, χ_E = 10, and an F/2 push. For each subplot the maximum speed is 6 m/s, and the ϕ_rot angles are as seen in Fig. 4. Solid points are measurable shear wave speeds (greater than −30 dB cutoff), while open circles are below the measurable cutoff.

Fig. 9 shows results obtained using an F/1 push and an F/2 push, both with c_L = 2 m/s, c_T = 1 m/s, θ_tilt = 10°, and χ_E = 10. As F-number changes from 1 to 2, with all other factors remaining the same, the measurable SV waves remain approximately the same speed and strength, but change in rotational distribution in that with the F/2 push there are measurable SV waves near the 0° – 180° axis that are close in speed to the corresponding SH waves.

Fig. 9.

Varying F number: Both examples have c_L =2 m/s, c_T = 1 m/s, θ_tilt = 10°, and χ_E = 10. The left plot has an F/1 push while the right plot has an F/2 push. For each subplot the maximum speed is 6 m/s, and the ϕ_rot angles are as seen in Fig. 4. Solid points are measurable shear wave speeds (greater than −30 dB cutoff), while open circles are below the measurable cutoff.

V. Discussion

3D rotational SWEI using a tilted excitation geometry allows for measurement of both SH and SV mode propagation. Observing the SV mode waves requires use of a tilted geometry and allows for estimation of χ_E, the third parameter necessary for complete characterization of an ITI material. By looking at factors that affect the SV waves, we can understand factors that affect χ_E, with an eye towards potential use of χ_E as a biomarker of in vivo muscle health.

Evaluating the SV wave speeds, SV wave strength, and SV wave rotational angle distribution across the simulation space of five parameters led to the identification of some major trends, as discussed below.

As seen in the top row of Fig. 5, when c_L increases while c_T, χ_E, and F-number remain the same, and θ_tilt = 0°, there are no SV wave speeds that are above the measurable cutoff of −30 dB relative to the SH mode, though there are some very small amplitude SV waves that would not be measurable, shown in open black circles. As c_L increases, even unmeasurable SV waves are not seen. At a θ_tilt of 10°, as seen in the middle row of Fig. 5, at a c_L of 2 m/s there are measurable SV waves in all four quadrants (ϕ_rot = 0° → 90°, 90° → 180°, 180° → −90°, −90° → 0°). As c_L increases to 3 m/s, the rotational distribution of measurable SV waves changes: there are measurable SV waves in all four quadrants but at fewer angles overall. At c_L = 4 m/s and then c_L = 5 m/s, θ_tilt = 10°, there are almost no measurable SV waves. When θ_tilt = 20°, as seen in the bottom row of Fig. 5, the SV wave speeds are approximately the same as the row above where θ_tilt = 10°, but the strength of the SV waves has changed. At almost all rotation angles for all values of c_L with θ_tilt = 20° there is at least one, if not two measurable SV speeds. When c_L = 2 m/s, and c_L = 3 m/s, many of the SV wave speeds are in red, meaning they are nearly the same strength as the SH wave, however as c_L increases the strength of the SV waves decreases quickly.

As seen in Fig. 6, when c_T increases while all other factors remain the same, the SV speeds change in multiple ways: the speed of the SV waves increases and the rotational distribution changes. In this example, when c_T = 0.75 m/s, the largest amplitude SV wave speed is 2 m/s to 3.5 m/s and the SV waves are measurable from ϕ_rot = 15° → 45°, 135° → 165°, 195° → 225°, and −45° → −15°. When c_T = 1 m/s the strongest SV wave speed is 3.5 m/s to 4 m/s and the SV waves are measurable from −60° → −30°, 30° → 60°, 120° → 155°, and 205° → 240°. In addition to these points, there are a few measurable SV waves very close to the SH ellipse in each of the four quadrants. These SV waves demonstrate how at rotation angles close to the major axis of the SH ellipse, SV waves and SH waves can appear very close in speed, which can lead to merged waves appearing particularly in noisier in vivo scenarios [28]. At c_T = 1.25 m/s, the largest amplitude SV wave speed is 4 m/s to 5 m/s and the SV waves are measurable in all four quadrants. As c_T increases the SV speed increases, and the number of ϕ_rot angles with measurable SV waves also increase.

As seen in Fig. 7, as tensile anisotropy χ_E increases, with all other factors remaining the same, the SV waves increase in speed and begin separating from the SH wave ellipse. This behavior means that in vivo it will be easier to measure higher χ_E values than low χ_E values as when the SV and SH speeds are close to each other, particularly along the main fiber axis (ϕ_rot = 0° – 180°), the SV and SH waves can appear merged in vivo and become difficult to separate [28]. As χ_E increases, the rotational distribution changes as well: the absolute value of angle between the ϕ_rot values where the SV waves are measurable increases, meaning that at higher χ_E values the SV waves are measurable at a wider range of ϕ_rot angles.

We now turn to discussion of variables that can be controlled by the experimental setup. When θ_tilt = 0°, the fiber direction $\widehat{A}$ is in the XY plane (see Fig. 1) and parallel to the transducer face, only the SH mode is measurable (seen Fig. 8 for example). As θ_tilt increases, the SV waves are first measurable only on the the right side, from ϕ_rot = −60° → −45°, and 45° → 60° then as tilt increases to θ_tilt = 10° and 15° SV waves are measurable at multiple ϕ_rot angles in all four quadrants. At θ_tilt = 20° SV waves are measurable at almost all ϕ_rot, and at some rotation angles there are two measurable SV waves. At a tilt of 5°, 6 of 72 ϕ_rot angles have measurable SV waves. This ratio increases as θ_tilt increases until at θ_tilt = 20° 65 of 72 ϕ_rot angles have measurable SV waves, representing a ten-fold increase from θ_tilt = 5° to 20°. This result matches our expectation that as θ_tilt increases, more rotation angles will have measurable SV waves. What is most notable about Fig. 8 is that the speeds of the measurable SV speeds do not change appreciably as tilt increases, even while the strength and rotational distribution of the SV waves change.

As seen in Fig. 9, as the F-number changes from 1 to 2 with all other factors remaining the same, the measurable SV wave speeds and strength remain approximately the same, but change in ϕ_rot distribution. When F/2 is used, there is more energy solely in the axial (Z) direction. Notably, in the F/2 case, there are measurable SV speeds on the left side of the image, meaning more regions where SV speeds can be measured than with F/1. In the F/1 case, 32 out of 72 rotation angles have measurable SV waves, while in the F/2 case 38 of 72 rotational angles have measurable SV waves, representing a 19% increase in angles with measurable SV angles.

The percent increase in ϕ_rot angles with measurable SV speeds for both F-number and θ_tilt would seem to indicate maximizing θ_tilt and using an F/2 push would maximize the number of SV waves seen and increase opportunities for measuring SV wave speeds in vivo. However, it is critical to account for the fact that in vivo SWEI imaging is noisier than GF simulations. Thus, the presence of measurable amplitude SV waves close in speed to the SH speeds, as seen in the F/2 case and at high values of θ_tilt, would likely lead to merged waves in the estimation of the SH ellipse and lead to more variability in estimation of the SH ellipse. Given that estimation of μ_L and μ_T from the SH ellipse is crucial for all steps of identifying in vivo ITI parameters, it is important to avoid combinations of F-number and θ_tilt where many of the measurable SV waves are close to the SH ellipse, as these waves will likely appear as merged and add variability to the measurements. We conclude that the optimal ARF parameters for 3D rotational SWEI in in vivo muscle are an F/1 push and a 10–20° θ_tilt to increase likelihood of successfully quantifying SV wave propagation.

The minor asymmetry between the left and right side of many of the plots (for example Fig. 5, first column middle and bottom row) is likely attributable the F-number of the ARF push and to the wave guiding effect of tilt in a TI material. As seen in Fig. 3, the locations of the peak displacement on axis and the push focus are not the same due to material attenuation (α = 0.5 dB/cm/MHz). As has been previously demonstrated for ARFI excitations [29], the peak displacement occurs shallow to the push focal depth, with the distance between them increasing for greater tissue attenuations. This axial asymmetry of the ARF push (distance between peak displacement and push focus) means that the relative strengths of the SV and SH waves will vary spatially when the direction of fibers $\widehat{A}$ tilts upward towards the transducer than when the direction of fibers $\widehat{A}$ tilts away from the transducer.

As seen in Fig. 3, there is a larger axial asymmetry in the F/2 push than in the F/1 push. This axial asymmetry is dependent on the specific attenuation α simulated. As α increases, the axial asymmetry in F/2 pushes increases more than in F/1 pushes. Thus, an F/1 push reduces the dependence of SV mode characterization on attenuation.

As seen in Fig. 8 and Fig. 9, the measured speed of the SV waves is not greatly affected by F-number and θ_tilt, though the strength and rotational distribution of SV waves does change with F-number and θ_tilt. All measurable SV speeds seen across Fig. 8 and Fig. 9 are approximately 3.5 m/s to 4 m/s, and appear strongest at the same ϕ_rot (45°, 135°, 225°, −45°) relative to other SV waves in the same GF plane. This independence from setup geometry indicates that in future work, a look up table approach could be utilized to estimate the material parameter χ_E for a given μ_L, μ_T, and measured SV waves speeds at all rotational angles ϕ_rot. The results shown herein also predict that this look up table could be reasonably independent of the push geometry (F-number and θ_tilt) used to excite the shear waves.

This work shows that SV waves of a strength that could be feasibly measured with ultrasound can occur in most of the clinically likely combinations of the material parameters (μ_L, μ_T, and χ_E) and ARF push parameters (θ_tilt and F-number) and thus their quantification is feasible. The speed of the measurable SV wave is affected most by μ_T and χ_E. The strength and rotational distribution of the SV wave is affected by all five parameters investigated.

As seen in Fig. 5, if c_L increases, the SV waves remain approximately the same speed, however as seen in Fig. 6, as c_T increases, the SV waves increase in speed. If both c_L and c_T increase, and the χ_E values remain approximately the same between skeletal muscles, we would still expect measurable SV waves in the range of speeds for c_L and c_T seen in other muscles such as the biceps brachii. If anything, the SV waves would be faster and more separated from the SH waves, as seen in the 3rd panel of Fig. 6. As such, it is unlikely that the optimal acquisition characteristics would change. An advisable acquisition change would be an increase in tracking frame pulse repetition frequency (PRF) to better capture faster moving SV waves, but the tracking PRF used in this study was already at the implementable limit for the Verasonics scanner. Additionally, increasing the tracking PRF is not always possible when imaging at depth, thus measurements in stiffer materials may not be possible if SV speeds are faster than some maximum value likely around 10 m/s.

The methods presented herein, specifically for identification of χ_E are limited to incompressible, transversely isotropic materials. A TI model is likely appropriate for unipennate muscles such as the vastus lateralis. Its limitations when applied to bipennate, circumpennate, or multipennate muscles become a question of if the fibers in the region of excitation appear locally unipennate enough that the material could be reasonably assumed to be ITI. However, this remains an open question that will be pursued in future work.

Herein, we have assumed that muscle is incompressible, elastic, and TI [27]. In contrast, some groups have assumed that muscle is compressible, viscoelastic, and TI. Using these assumptions, those groups have used measurement of the acoustic wave in addition to both the SH and SV waves in conjunction with Field II and GF simulations in TI materials to reconstruct viscoelastic properties of tissue [40], [41]. Under these assumptions, none of the measured parameters are sensitive to changes in χ_E, while this is a critical aspect of our model. Since tensile anisotropy is one of the key focuses of this work and of Knight et al. [28], the viscoelastic model was set aside in favor of ability to measure tensile anisotropy. The question as to whether viscosity or χ_E provide useful diagnostic information remains to be determined.

While the sequence used in Knight et al. [28] required approximately 3 minutes per rotational 3D-SWEI acquisition with 5° spacing between planes, recent improvements in our system have reduced this time to 11 seconds for a full rotational 3D-SWEI acquisition with 5° spacing. This facilitates measurements during rest, isometric stretching, and during moderate levels of contraction which can be held for the required 11 seconds.

Similarly, the time of acquisition could be decreased further if greater rotational spacing is utilized, which can be readily accomplished if only SH wave characterization is desired. However, we have found empirically that if characterization of χ_E, and thus of SV waves, is desired then using a rotational spacing of 5° increases the likelihood of visualizing SV waves.

Two of the largest limitations of this work are that only the Z displacement components are measured, and only in a single coronal XY plane at the push focus. Investigating only the Z displacement component, while most realistic for ultrasonic tracking applications, limits overall understanding of energy propagation in 3D. Only calculating displacements in a single coronal plane limits understanding of what is happening to the spatial distribution of SH and SV waves above or below the studied plane, particularly what is happening primarily along the muscle fibers, where the shear wave energy is expected to be greatest. Investigating 3D propagation, including exploration of 3D cross correlation methods [42], is the focus of future work.

VI. Conclusion

This work investigates how tissue properties (μ_L, μ_T, and χ_E) and imaging configurations (θ_tilt and F-number) affect the SV mode speed, amplitude and spatial distribution in a variety of cases likely to occur in 3D rotational SWEI in vivo imaging of skeletal muscle. While the specific SV mode speeds depend on the combinations of all parameters considered, measurable SV waves occur in most of the likely biologically occurring in vivo muscle SWEI imaging combinations of the material parameters and ARF push parameters. To maximize the number of rotation angles ϕ_rot where measurable SV waves are found, an F/1 push and θ_tilt = 15° is recommended. This work demonstrates that SV wave speeds are important to understand in the context of muscle SWEI, and worthy of more study.