Full Characterization of in vivo Muscle as an Elastic, Incompressible, Transversely Isotropic Material using Ultrasonic Rotational 3D Shear Wave Elasticity Imaging

Abstract

Using a 3D rotational shear wave elasticity imaging (SWEI) setup, 3D shear wave data were acquired in the vastus lateralis of a healthy volunteer. The innate tilt between the transducer face and the muscle fibers results in the excitation of multiple shear wave modes, allowing for more complete characterization of muscle as an elastic, incompressible, transversely isotropic (ITI) material. The ability to measure both the shear vertical (SV) and shear horizontal (SH) wave speed allows for measurement of three independent parameters needed for full ITI material characterization: the longitudinal shear modulus μ_L, the transverse shear modulus μ_T, and the tensile anisotropy χ_E. Herein we develop and validate methodology to estimate these parameters and measure them in vivo, with μ_L = 5.77 ± 1.00 kPa, μ_T = 1.93 ± 0.41 kPa (giving shear anisotropy χ_μ = 2.11 ± 0.92), and χ_E = 4.67 ± 1.40 in a relaxed vastus lateralis muscle. We also demonstrate that 3D SWEI can be used to more accurately characterize muscle mechanical properties as compared to 2D SWEI.

I. Introduction

SHEAR wave elasticity imaging (SWEI) and other acoustic radiation force impulse (ARFI) methods have been used to measure mechanical properties in a wide variety of tissues [1]–[3]. ARFI and SWEI methods employ a conventional clinical ultrasound transducer to create a small amplitude (< 50 μm) localized displacement in the tissue, inducing shear waves that spread outward radially from the push location and are tracked by subsequent ultrasound frames [4]. Under the assumption of a linear, isotropic, and elastic material [4]–[6], the shear modulus μ can be calculated from the measured shear wave speed c and tissue density ρ as

$$\mu =\rho c^2.$$ (1)

Musculoskeletal SWEI is a wide field of research. Studies have indicated differences in shear wave speeds related to sex [7]–[9], age [8], [10], [11], contractile state [12]–[14] and muscle related diseases such as myopathies [15], [16], dystrophies [17], [18], and muscle spasticities [19]–[22]. The existing research shows uncertainty and lack of repeatability among SWEI measurements in skeletal muscle using current tools [23]–[28], which limits clinical adoption. However, the volume of research to date demonstrates an interest by the orthopedic and neurology communities in noninvasively assessing muscle material properties.

Skeletal muscle is commonly modeled as a transversely isotropic (TI) material due to the alignment of muscle fibers along an axis of symmetry, typically oriented along the direction of muscle contraction. A linear, elastic TI material can be characterized by five parameters [29]–[34]. Furthermore, if the material is assumed to be incompressible, these five parameters are reduced to three [29], [30]. In this study we model skeletal muscle as an incompressible TI (ITI) material characterized by three parameters: 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. In previous work we have reported this third parameter as the ratio E_T/E_L [29], [30]; herein we report tensile anisotropy χ_E [35], defined as

$${\mathrm{\chi }}_E=\frac{E_L-E_T}{E_T}=\frac{1}{E_T/E_L}-1.$$ (2)

This allows for easy comparison to shear anisotropy χ_μ ^[35], defined as

$${\mathrm{\chi }}_{\mu }=\frac{{\mu }_L-{\mu }_T}{{\mu }_T}.$$ (3)

Many studies to date have measured the directional dependence of shear wave propagation in muscle, reporting a wide range of speeds along and across the fiber axis [16], [24], [36]–[50]. These reports used an experimental setup similar to that shown in Fig. 1(a), where the fiber symmetry axis is in a plane parallel to the transducer face, allowing for the measurement of shear wave speeds along and across the muscle fiber axis. The shear moduli μ_L and μ_T that characterize shear wave propagation along and across the muscle fiber axis can be calculated from these speeds using (1). However tensile anisotropy χ_E cannot be determined using (1).

Fig. 1.

(a) 3D SWEI orientation with ARFI push excitation $\widehat{F}$ normal to muscle fiber direction (i.e. muscle symmetry axis, $\widehat{A}$). $\widehat{n}$ represents the direction of wave propagation along which the shear wave speed is computed. ϕ_rot is the angle between $\widehat{A}$ and $\widehat{n}$ in the XY plane. In a TI material, two shear wave modes with different polarizations can be excited: the shear horizontal (SH) mode, shown in pink, which is always perpendicular to the $\widehat{A}-\widehat{n}$ plane and the shear vertical (SV) mode, shown in yellow, which is in the $\widehat{A}-\widehat{n}$ plane. With $\widehat{F}$ and $\widehat{A}$ at normal incidence, as in (a), the SH mode is the primary mode excited in this configuration. (b) 3D SWEI orientation with ARFI push excitation $\widehat{F}$ at an angle θ_tilt away from normal incidence to the muscle symmetry axis ($\widehat{A}$). With a nonzero θ_tilt both the SH and SV modes are excited and have displacement components in the Z (axial) imaging direction in the material. The angle ψ between A and $\widehat{n}$ is shown in orange in (b), while in (a) ψ = ϕ_rot.

Recently, our group has demonstrated that it is possible to excite and observe multiple shear wave propagation modes in a uniaxially stretched polyvinyl alcohol phantom in which the stretch axis represents the material symmetry axis $\widehat{A}$ [51]. This experimental setup is similar to that shown in Fig. 1(b). The key difference compared to the typical setup shown in Fig. 1(a) is that the material symmetry axis $\widehat{A}$ is tilted with respect to the ultrasonic acoustic radiation force push beam $\widehat{F}(\widehat{A}\perp \widehat{F})$. With this setup, both the SH and SV propagation modes are excited by an ARFI force directed along the Z axis. Measurement of both the SH and SV wave modes enables characterization not only of μ_L, μ_T, and χ_μ from the SH mode, as measured by a typical setup (Fig. 1(a)), but also of χ_E, which is dependent upon the SV mode. Herein, we present a method to exploit this geometry and estimate μ_L, μ_T, χ_μ and χ_E using Green’s function (GF) simulations. We validated our method using finite element method (FEM) simulations, and we also apply this approach to estimate these parameters in in vivo muscle.

II. Background

A. Linear, Elastic, Incompressible, Transversely Isotropic (ITI) Material Models

A linear, elastic material’s stress-strain relationship can be written in terms of a generalized Hooke’s law between the stress tensor σ and strain tensor ϵ as

$${\sigma }_{ij}=c_{ijkl}{ϵ}_{kl}$$ (4)

where c_ijkl are the elements of the stiffness tensor that characterize the material. For a TI material, rotational and reflective symmetries of the material about its symmetry axis $\widehat{A}$ (Fig. 1) reduce the stiffness tensor to five independent quantities [30]–[33]. These five quantities can be expressed in terms of six engineering constants and one constraint: two Young’s moduli E_L and E_T along the longitudinal (L) and transverse (T) directions, two shear moduli μ_L and μ_T, two Poisson’s ratios, ν_LT and ν_TT, and the constraint

$${\mu }_T=\frac{E_T}{2(1+{\nu }_{TT})}.$$ (5)

In SWEI methods, soft tissues are often treated as quasi-incompressible materials with an acoustic wave (compressional) speed of 1540 m/s, and the shear wave propagation is modeled using incompressible material models. For the case of an ITI material, the Poisson’s ratios, ν_LT and ν_TT have specific values [29], [30]

$${\nu }_{LT}=1/2$$ (6)

and

$${\nu }_{TT}=1-\frac{1}{2}\frac{E_T}{E_L}.$$ (7)

These relations, plus the constraint in equation (5), imply that shear wave propagation in an ITI material can be described by three independent material parameters. Here, we use the parameters μ_T, μ_L, and χ_E . Note that we have previously used E_T/E_L [29], [30] which is related to χ_E by (2).

B. Wave Propagation in ITI Materials

Plane wave propagation in a TI material can be described in terms of three propagation modes: the pressure or compressional mode (P), in which the polarization is aligned with the propagation direction $\widehat{n}$ (Fig. 1), and the shear horizontal (SH) and shear vertical (SV) wave modes, in which the polarization is perpendicular to the propagation direction $\widehat{n}$. Particle motion in the SH wave mode is in the plane perpendicular to the plane formed by the material symmetry axis $\widehat{A}$ and propagation direction $\widehat{n}$ (the pink planes in Fig. 1), while particle motion in the SV wave mode is in the $\widehat{A}-\widehat{n}$ plane (yellow planes in Fig. 1).

In the limit of an ITI material, the phase velocity of the P-wave becomes infinite, and we will ignore this propagation mode. The phase velocities v_SH and v_SV of the SH and SV modes are given in terms of the angle ψ between $\widehat{A}$ and $\widehat{n}$ [29], [30] as

$$\rho v_{SH}^2={\mu }_L{\cos }^2\psi +{\mu }_T{\sin }^2\psi ,$$ (8)

and

$$\rho v_{SV}^2={\mu }_L+4\left(({\mathrm{\chi }}_E+1){\mu }_T-{\mu }_L\right){\sin }^2\psi {\cos }^2\psi .$$ (9)

Equation (8) shows that the SH mode depends only on the observation direction, μ_L, and μ_T, but is not affected by the tensile anisotropy, χ_E. For this mode, the group velocity V_SH(ψ) is given by an elliptical shape [30], [32], [34], [52] as

$$\rho V_{SH}^2(\psi )=\frac{{\mu }_L{\mu }_T}{{\mu }_L{\sin }^2\psi +{\mu }_T{\cos }^2\psi }.$$ (10)

The group velocity for the SV propagation mode does not reduce to a simple expression like (10) as the SV mode can support multiple shear wave speeds for a particular propagation direction, depending on the specific material parameters considered (see Fig. 1 of Rouze et al. [29]). However, we have recently reported tractable calculations using Green’s functions that describe both the SV and SH modes in an elastic, ITI material, which we employ herein to characterize χ_E [30].

C. Practical Implementation in Ultrasound

In each of the two configurations seen in Fig. 1, the transducer rotates around its central axis. As such, each 2D imaging frame contains the axial imaging dimension Z, perpendicular to the transducer face, and a lateral (radial) dimension, parallel to the transducer face, corresponding to $\widehat{n}$ in Fig. 1 as determined by the rotational angle ϕ_rot of the transducer.

SWEI methods preferentially push and track material displacements in the axial (Z) dimension. As such, when the excitation $\widehat{F}$ from the transducer is normal to the fiber direction $\widehat{A}$ (Fig. 1(a), $\widehat{F}\perp \widehat{A}$), we only track the SH mode of propagation. In this configuration, the SV mode does not have an appreciable component in the Z (axial imaging) direction, and thus would not be detected with our experimental setup. However when the fiber direction $\widehat{A}$ is tilted relative to the transducer face, as seen in Fig. 1(b), the planes containing the SV mode and SH modes are tilted such that both modes have components in the Z dimension, and thus can be excited and monitored with our experimental setup. In both Figs. 1(a) and (b) the transducer rotates around its axis to allow us to measure the shear wave propagation in 3D (see Supplemental Animation 1). Similarly, the tilt angle θ_tilt between the fibers $\widehat{A}$ and the lateral imaging axis at each propagation direction $\widehat{n}$ can change depending on the probe positioning (ϕ_rot) and muscle architecture because different muscles have different fiber orientations relative to the skin surface. This affects the magnitude of the tissue displacement in the axial imaging orientation (Z) for both the SV and SH modes (see Supplemental Animation 2).

Typically, experiments using 2D SWEI determine μ_L and μ_T from measurements of the group shear wave speed (SWS) along and across the muscle fibers using the configuration shown in Fig. 1(a). Because the tensile anisotropy χ_E appears only in the equation describing the SV wave mode (9), it is necessary to measure both the SV and SH waves to fully characterize an ITI material. This is achieved herein by leveraging the tilted fiber geometry inherent to the vastus lateralis during in vivo imaging.

III. Methods

A. Experimental Acquisitions in in vivo Muscle

A volumetric rotational 3D-SWEI acquisition system was obtained using a rotating linear array probe [53]. A Philips ATL L7-4 array (Philips®, Andover, MA) was attached to a Verasonics Vantage 256 (Verasonics®, Kirkland, WA) ultrasound system. The transducer was mounted in a custom holder attached to a rotation stage (XPS-Q8, Newport®, Irvine, CA) to allow rotation about the transducer’s vertical axis.

A healthy, 53 year old female volunteer with no known muscular diseases or injuries was selected and written informed consent was obtained for participation in a Duke University Institutional Review Board-approved study. The subject was placed in a lateral position with a rolled towel between the knees. A water bath with Tegaderm bottom was used to ensure a consistent water path between the transducer and the muscle throughout the full rotation of the probe. Ultrasound gel was applied between the leg and the Tegaderm box, and any air bubbles were manually removed by smoothing the Tegaderm face against the leg. The water bath and transducer were positioned so the center of rotation was in the main body of the vastus lateralis muscle, approximately halfway between the upper trochanter and the quadriceps tendon (Fig. 2a).

Fig. 2.

Data acquisition setup. Fig. a shows an L7-4 transducer placed in the rotation stage set up, Tegaderm box with gel against the vastus lateralis muscle, to be rotated in 5 degree steps. Fig. b shows an example B-mode image obtained from one of the rotational imaging planes (ϕ_rot = 15°) The red arrow shows the axial depth of the SWEI push focus. The yellow lines are approximations of the proximal and distal vastus lateralis aponeuroses

Thirty-six frames were acquired in 5 degree rotation (ϕ_fiber) increments with a total of 3 seconds between each frame, including time for the transducer to rotate. As each frame included a left moving and right moving wave, the transducer only had to be rotated 180 degrees to acquire a full 360 degree volume. The sequence for each rotation position consisted of a 4 MHz push focused at 20 mm axially, centered under the transducer laterally, followed by a repeating pattern of 5.2 MHz tracking plane waves at 10 kHz pulse repetition frequency angled at −3°, 0°, and 3° consecutively for ~ 40 ms. The tracking planes were coherently compounded with a rolling window so that each resulting track plane included one of the three original angles as seen in Table I, while still allowing for full temporal sampling.

TABLE I.

Coherently Compounded Track Configuration

Sub-Frame 1	−3°	→	0°	→	3°
Sub-Frame 2	0°	→	3°	→	−3°
Sub-Frame 3	3°	→	−3°	→	0°

For each track pulse, a 4x4 cm (axial x lateral imaging plane) region of interest was reconstructed with a pixel resolution of 400 μm axially and 200 μm laterally. This process of acquiring a full 360 degree rotation acquisition (36 frames) was performed a total of 4 times on two different days in the same subject.

Within each rotational plane, axial displacements were calculated using Kasai’s algorithm with a 1.5λ kernel [?] and separated into the left and right moving shear waves. The displacement data were temporally low pass filtered at 600 Hz, and differentiated into particle velocity data. The transverse plane (X-Y imaging plane in Fig. 1) was extracted for further analysis by averaging over a 1 mm depth axially centered around the push focal depth of 20 mm. A sample B-mode image is shown in Fig. 2b for rotation position ϕ_rot = 15° and shows the tilted fiber orientation in the lateralis muscle. The fiber tilt angle θ_tilt was determined by 2DFT of the B-mode image, as described in Section III-C.

B. Shear Wave Speed Estimation

Shear wave speeds were calculated for each space time trajectory using the Radon sum algorithm [56]. Typically, a lateral range of 6 to 16 mm was used in the estimate, however, this range was adjusted for each space-time trajectory based on the range over which a distinct trajectory could be visually identified. The white lines overlaid on the space time signals in Fig. 3 show the lateral range used in the speed estimate. If a single wave mode was present, it was assumed to be SH mode, if two trajectories were present, the faster was assumed to be the SV mode. For cases with two shear wave trajectories, such as Fig. 3b, individual speeds were estimated by masking the Radon sum images (see Fig. 2 of Rouze et al. 2010 [56]) to identify the trajectory for each propagation mode. All trajectories were verified visually by a single reader.

Fig. 3.

Example space time plots from experimental acquisitions. The top row (Fig. a) shows an example space time plot of a transducer angle (ϕ_rot = 120°) where there is only one trajectory and one wave mode present, which is the SH mode. Fig. b shows an example of an angle (40°) from the same acquisition as Fig. a where there are two shear wave speeds clearly present on the space time plot. The faster wave is the SV mode (3.49 m/s) and the slower the SH mode (2 m/s). The bottom row, Fig. c, shows an example of an angle (180°) from the same overall acquisition as Fig. a and Fig. b where the SV mode and the SH mode waves are merged. Although a speed is identified on this image, visual assessment indicates that it is neither the SV nor the SH trajectory, and thus the speed estimate is excluded from the SH mode ellipse fitting. All of these data in Fig. a, b, and c, are also shown in Fig. 6

For some trajectories, such as Fig. 3,c it was not possible to assign the estimated speed to either the SH or SV mode as the waves appeared merged. In these cases, no speeds were recorded and these propagation directions were omitted from further analysis.

C. Iterative Parameter Estimation of θ_tilt, ϕ_fiber, μ_L, and μ_T

A flowchart explaining the full algorithm workflow is provided for reader convenience in Fig. 4. The values of the full acquisition’s overall rotation angle ϕ_fiber) and tilt (θ_tilt) and the values of μ_L and μ_T can be estimated using a three step procedure as follows.

Fig. 4.

Flowchart of data processing steps. The green boxes show acquiring and initial processing of the experimental data (Sections III-A and III-B). The blue steps show the steps for determining θ_tilt, ϕ_rot, μ_L, and μ_T from Section III-C. The orange boxes show the iterative parameter fitting steps that use the simulated Green’s function data to determine E_T/E_L (Section III-D.)

First, preliminary estimates, as indicated by superscript p of ${\varphi }_{fiber}^p$, ${\mu }_L^p$, and ${\mu }_T^p$ can be found by assuming an elliptical shape based on (10).

$$V(\text{Δ}\varphi )={\left(\frac{1}{\rho }\frac{{\mu }_L^p{\mu }_T^p}{{\mu }_L^p{\sin }^2(\text{Δ}\varphi )+u_T^p{\cos }^2(\text{Δ}\varphi )}\right)}^{1/2}$$ (11)

where $\text{Δ}\varphi ={\varphi }_{rot}-{\varphi }_{fiber}$.

Nonlinear least square fitting was used to estimate ${\mu }_L^p$, ${\mu }_T^p$, and ${\varphi }_{fiber}^p$ by minimizing the sum of the squares difference between the measured SH speeds, and the predicted speeds from (11).

Second, the estimated fiber rotation ${\varphi }_{fiber}^p$ was used to determine the fiber tilt angle θ_tilt by examining the B-mode images for the transducer rotation nearest to ${\varphi }_{fiber}^p$. These B-mode images were analyzed using a two dimensional Fourier transform (2DFT) approach as seen in Fig. 5. We assumed that the fibers were represented by a series of parallel lines at a given tilt. In order to efficiently calculate this tilt, we took the 2DFT of the B-mode image over a region of interest and integrated radially within the 2DFT plane. The angle of peak signal was related to the B-mode tilt using the rotation property of 2DFT as seen in Fig. 5(b).

Fig. 5.

Fig. a shows an example B-modes for determining θ_tilt. The red box show sample 2DFT regions and resultant θ_tilt estimate on the B-mode. This tilt, θ_tilt, is then used to re-fit the SH mode ellipse with a geometric scale factor to account for θ_tilt using (12). Fig. b shows the 2D FFT of the selected region from Fig. a. The dotted red line shows the peak intensity representing the tilt of the fibers, θ_tilt, in this case 11°

Finally estimates of μ_L, μ_T, and ϕ_fiber were determined by repeating the the fitting procedure (11), but with the angle ψ determined relative to the material symmetry axis,

$$\psi ={\cos }^{-1}(\widehat{A}\cdot \widehat{n})={\cos }^{-1}\left[\cos ({\theta }_{tilt})\cos ({\varphi }_{rot}-{\varphi }_{fiber})\right]$$ (12)

D. Estimate of in vivo E_T/E_L Using Simulated Shear Wave Signals

The third parameter, E_T/E_L, needed to fully characterize an incompressible, elastic TI material was determined by simulating shear wave signals using a range of values for E_T/E_L and comparing the shear wave speeds from these simulations to the measured in vivo SV mode speeds. The shear wave signals were calculated using Green’s function techniques as described by Rouze et al. [30], using the ARFI excitation force calculated using Field II [57] modeling our experimental setup.

The final value of E_T/E_L was estimated by calculating a series of XY plane simulated shear wave signals for a range of values of E_T/E_L and selecting the value which minimized the sum of the squared differences between the simulated and experimentally measured SV speeds.

$$MSE=\frac{1}{n}\sum _{i=1}^n{({(V_{SV}^{Exp})}_i-{(V_{SV}^{Sim})}_i)}^2$$ (13)

where n is the number of detected experimental SV speeds present. This mean squared error was calculated for each Green’s function plane (13) and an error minimization was performed using quadratic fitting to find the Green’s function most similar to the experimental data.

IV. Results

As the transducer rotates, as shown in Fig. 2, shear waves trajectories, such as those seen in Fig. 3, are seen at each rotation angle in the lateral and axial dimensions. Fig. 6 shows particle velocity shear wave signals from a single acquisition in an in vivo vastus lateralis muscle as a function of space and time for 18 transducer rotation angles spaced at 10° rotation steps. As the transducer only rotates around 180°, at each rotation angle the left moving propagation direction and right moving propagation direction signals are separated, giving two signals per acquisition (36 signals shown in Fig. 6 obtained from 18 transducer rotation angles). This set represents one half of the signals acquired in a complete data set with 5° rotation steps.

Fig. 6.

Space time plots and speeds at many angles. The subplots going around the ring show space-time plots in 10° increments of ϕ_rot (transducer rotation). An example axis label for each subplot is shown in the upper left. The white lines on each subplot identify the trajectory of speeds identified. Each space time plot contains 0, 1, or 2 trajectories. The number of waves identified and calculated speeds are seen in the inner circle plot. The slower SH mode wave speeds are seen in orange triangles, while the faster SV speeds are in red circles. The ellipse fit to the SH speeds is shown in green (using (11) and (12)), as is the speeds along the major axis of the ellipse (c_parallel), and the minor axis of the ellipse (c_{perpendicular}). The blue line shows the rotational angle of the fibers ϕ_fiber corresponding to the major axis of the ellipse. These values of c_parallel and c_{perpendicular} are after θ_tilt has been accounted for (see Section III-C.)

A key observation in Fig. 6 is that, as the transducer rotational position is varied, multiple shear wave signals are observed for rotational directions near ϕ_rot = −10°, 30°, 170° and 210°, while only a single shear wave signal is observed for rotational angles near ϕ_rot = −90° and 90°.

To see the difference between multiple trajectories better, enlarged space-time images of selected shear wave signals are shown in Fig. 3, corresponding to ϕ_rot = 120°, 40°, and 180°. Because the fiber tilt shown in Fig. 2b is small, it is expected that the largest amplitude signal is the SH propagation mode. When a second shear wave signal is seen, for example in Fig. 3b, this mode is identified as the SV propagation mode.

In the center of Fig. 6 the shear wave speeds are plotted as a function of propagation angle with the SH propagation mode shown as orange triangles, and the SV propagation mode shown in red circles. For propagation directions near ϕ_rot = 15° and 195° the SH and SV trajectories on the space-time images appear to merge, and it was not possible to assign an estimated speed to either mode. These cases were omitted from further analysis.

From the B-mode image shown in Fig. 2b and the distributions of shear wave speeds for the SH propagation mode shown in the center of Fig. 6, it is clear that the material symmetry axis $\widehat{A}$ in Fig. 1b is not aligned with the X-axis (0° → 180°) for this example acquisition. Instead this axis is rotated by angle ϕ_fiber about the Z-axis and is also tilted by an angle of θ_tilt relative to the X-Y imaging plane.

Following along with the example acquisition shown in Fig. 6, the nonlinear least squares fitting found the preliminary estimates of $c_{parallel}^p=2.25\text{m}/\text{s}$, $c_{perpendicular}^p=1.37\text{m}/\text{s}$ resulting in ${\mu }_L^p=5.07\text{kPa}$, ${\mu }_T^p=1.89\text{kPa}$, and ${\varphi }_{fiber}^p={16.5}^{\circ }$. Note that these are not the values given in Fig. 6 as they are preliminary estimates. For this same acquisition, the 2DFT derived θ_tilt value from the ϕ_rot = 15° and 20° were weighted by the results according to the proximity to ϕ_fiber, with the result for the case shown in Fig. 6 of θ_tilt = 11° ± 2.5°. Once θ_tilt was accounted for geometrically, the final values of c_parallel = 2.33 m/s, C_{perpendicular} = 1.37 m/s, meaning μ_L = 5.41 kPa, μ_T = 1.89 kPa, and ϕ_fiber = 11° were found. These results are indicated on Fig. 6.

The top row of Fig. 7 shows sample simulated shear wave particle velocity signals at a time of 3 ms calculated using the angles θ_tilt and ϕ_fiber and the material properties μ_L = 5.41 kPa and μ_T = 1.89 kPa estimated by our algorithms for three cases E_T/E_L = 0.199, 0.148, and 0.127. The middle row of Fig. 7 shows corresponding shear wave speeds. An important observation for the simulated signals is that the signal amplitude varies with propagation direction, particularly for the SV propagation mode. Even with the same underlying μ_L, μ_T, and ϕ_fiber, small changes in E_T/E_L affect the intensity of the SV wave.

Fig. 7.

Comparison of Simulated and Experimental Signals. The top row shows the simulated Green’s function signals in the XY plane at 3 ms at three different values of E_T/E_L. In the middle row and bottom row, the simulated SH (black dots) and SV speeds (blue dots) at each angle are shown. In the bottom row the experimental SH mode is shown in the open orange triangles, and the experimental SV mode is shown in open red circles. In the bottom row the color of the circles show the SV mode strength relative to the other SV mode speeds present (similar to the top row), as calculated by normalizing each by the maximum Radon sum value found across all SV mode speeds in a given Green’s function plane. Yellow indicates a strong SV mode relative to blue. The fact that most of the high amplitude signals (yellow dots) are concordant with the experimental data adds confidence to the biofidelity of this E_T/E_L estimation approach.

The bottom row of Fig. 7 shows the shear wave speeds from experimental data (Fig. 6) overlaid on the calculated speeds from the middle row. The simulated SV speeds are color coded by the amplitude of the shear wave signal in the top row. We observe that the speeds determined from the signals with the largest shear wave amplitude occur at transducer angles of roughly ϕ_rot = −10°, 40°, 170° and 220°, which are roughly the same directions where two shear wave signals, both SH and SV, are observed experimentally. The iterative parameter estimation technique using Green’s Functions gave E_T/E_L = 0.151 for the acquisition shown in Fig. 6. The fact that most of the large amplitude signals in the simulated data are concordant with the experimental data adds confidence to the biofidelity of this E_T/E_L estimation approach.

In four acquisitions on two different days in the same healthy volunteer, four estimates of μ_L, μ_T, and E_T/E_L were calculated as seen in Fig. 8. Mean and standard deviation values for μ_L, μ_T, E_T/E_L and mean squared error are given in Table II.

Fig. 8.

μ_L, μ_T, and E_T/E_L from four acquisitions. On the left, in red circles, μ_L in kPa, and in blue squares μ_T in kPa, both vs acquisition number. On the right, in black triangles E_T/E_L vs acquisition number. Note that E_T/E_L is limited from 0 to 0.5 so as to highlight the variability in the measurements.

TABLE II.

Mean and Standard Deviation Across 4 Acquisitions

μL	5.77 ± 1.00 kPa
μT	1.93 ± 0.41 kPa
ET/EL	0.151 ± 0.051
MSE	0.244 ± 0.10 (m/s)2

Fig. 9 shows for each acquisition the value of the ratio of the two shear moduli μ_T/μ_L plotted vs the ratio of the Young’s moduli E_T/E_L. The blue line represents E_T/E_L = μ_T/μ_L which would be the case for an isotropic material.

Fig. 9.

Experimental values of the ratio of shear moduli μ_T/μ_L (0.345 ± 0.111) vs ratio of Young’s moduli E_T/E_L (0.151 ± 0.051). Each experimental acquisition is plotted in black circles. The blue line represents where μ_T/μ_L = E_T/E_L.

Fig. 10 portrays the effect of variability in the 2DFT estimate of B-mode tilt (ϕ_tilt) on the resultant E_T/E_L found via the simulated shear wave signal method (III-D). The ϕ_tilt = 8.5° case represents one standard deviation in tilt below the mean, and ϕ_tilt = 13.5° represents one standard deviation greater in tilt as estimated on the 2DFT. The percent variability in μ_L = 6.5%, the percent variability in μ_T = 0.0%, and the percent variability in E_T/E_L = 11.3%

Fig. 10.

Effect of variation in θ_tilt on E_T/E_L. μ_L, μ_T, and E_T/E_L from the same acquisition at 3 different assumed θ_tilt values: 8.5°, 11°, 13.5°. On the left in red circles, μ_L in kPa and μ_T in blue squares, both vs tilt angle θ_tilt) in degrees. On the right, in black triangles E_T/E_L vs θ_tilt. Note that E_T/E_L is limited from 0.1 to 0.2 as compared to Fig. 8 so as to highlight the variability in the measurements. The percent variability in μ_L = 6.5%, the percent variability in μ_T = 0%, and the percent variability in E_T/E_L = 11.3%

V. Discussion

In this paper, we demonstrate that measuring both the SV and SH wave modes is possible using 3D SWEI in in vivo muscle by leveraging the tilted geometry of the muscle fibers relative to the transducer face. We also present a method of estimating the tensile anisotropy χ_E of muscle.

Fig. 7, bottom middle panel, demonstrates the agreement between the GF predicted SV and SH speeds and the in vivo experimental measurements. In the middle column, not only is the error between the GF and the experimental speeds minimized, but the location of the largest amplitude simulated SV signals (yellow and green dots) align with the angles where the experimental SV speeds were observed. The appearance of SV waves experimentally at the four regions where the GF predict the largest amplitude SV waves increases confidence that SV waves are not reflected waves.

As seen in Fig. 8, in each of four acquisitions on two days in the same healthy volunteer, the μ_L value measured was greater than μ_T. Additionally, the tensile anisotropy χ_E was greater than the shear anisotropy χ_μ in each acquisition.

As seen in Fig. 9, as the tilt angle θ_tilt increases, the calculated χ_E value also increases monotonically, indicating expected bias if the tilt angle is not correctly computed. However, across a 95% confidence interval of the θ_tilt = 11.4 ± 2.5° measured in the B-mode data, the variation of χ_E (0.344) represents only a 6% bias.

To quantify the expected accuracy of our Green’s function approach to estimating χ_E, for one acquisition, FEM simulations were computed to match the experimental setup with known inputs. As seen in Fig. 10, the matched GF and FEM simulated speeds at angles where there were experimentally observed SV waves varied from each other by only 0.11±0.13 m/s, while the SH wave speeds differences between the GF and FEM were 0.07 ± 0.07 m/s. Additionally, using the Green’s function reconstruction algorithm for this data set found an estimate of χ_E = 5.30, while the ground truth value was χ_E = 5.25, meaning a difference of 1%. Using a computational cluster, the GF plane calculations of Z direction displacement and resultant SWS in the XY plane necessary to estimate 120 possible χ_E values for a given θ_tilt, μ_L, and μ_T took approximately 150 CPU-hours on an Intel®Xeon Gold 3 GHz CPU. We are exploring methods to pre-populate look up tables to reduce this time, but this work is outside the scope of this paper.

The relationship between shear modulus and Young’s modulus in an isotropic, elastic, incompressible material is E = 3μ. However in ITI materials, there is no simple proportional relationship between E and μ in any given direction. As seen in Fig. 8, and as predicted by the equations modeling ITI materials (2)-(3), (5)-(7), χ_E is not equal to χ_μ and thus represents an additional independent parameter that may be explored for muscle health characterization.

One of the advantages of a rotational 3D approach is that since all rotational angles are captured in 5° increments, the 3D SWEI methodology presented herein is completely agnostic to the starting position of the fiber rotation ϕ_fiber relative to the transducer. This means the 3D SWEI approach can be used without first identifying muscle fiber orientation with freehand B-mode imaging.

The values for the shear modulus along the fibers (μ_L = 5.77 ± 1.00 kPa) are similar but slightly higher than those found by other studies in the vastus lateralis using 2D SWEI (4.5 ± 1.0 kPa [28], 3.6 ± 0.45 kPa [7], 3.3 ± 0.4 kPa [58], and 5.1±1.3 kPa [59]). The use of 3D data herein ensures the μ_L and μ_T measurements account for both the rotation ϕ_fiber and tilt θ_tilt angles of the muscle fibers. Misalignment in either dimension would result in an underestimation of μ_L, which could be a source of bias in the results of 2D SWEI studies. In particular, accounting for the tilt angle θ_tilt in (12) increases our estimates. On average μ_L increased by 7.6% ± 3.5% when θ_tilt was incorporated, as compared to the assumption θ_tilt = 0°.

As seen in the center of Fig. 6, the speeds closest to the major axis of the SH mode ellipse were not included in the ellipse fitting. At those locations, visual assessment of the space-time plots made it clear that the SV and SH modes were spatially overlapping with each other. While the Radon sum algorithm was able to identify a trajectory, visual assessment could not determine if the trajectory was calculated using only SV mode data, only SH mode data, or a combination of both SV and SH mode, so the speed was excluded from further analysis. Supporting this, as seen in Fig. 7 bottom row, directly along the major axis of the ellipse, the SV speed predicted by simulation is very low amplitude compared to other SV speeds at other angles. When the inherent noise of experimental motion tracking is combined with this effect, it is unsurprising that it is difficult to separate the SH and SV modes for these propagation directions in experimental data.

The difficulty in separating the SH and SV waves along the main fiber axis could also be one of many factors contributing to the wide variability in speeds found using 2D SWEI techniques [24]. If the method used to identify the shear wave speed along the fibers does not account for the presence of both SV and SH mode speeds, it is unknown whether the speed measured along the fiber axis is from the SH mode or the SV mode, increasing variability in the measurements. The blue squares and highlighted area of Fig. 6 indicate what speed would be estimated by 2D SWEI if the delineation of SH and SV waves was not considered. Our work overcomes this by informing the estimate of μ_L with speeds around the ellipse rather than relying solely on speeds aligned with the fiber dimension, and inspecting space-time plots for multiple wave fronts.

As a proof of concept work, this paper presents measurements from only one healthy volunteer. However, the repeatability across multiple days and multiple acquisitions supports the feasibility of this method and motivates the development of 3D SWEI for muscle imaging and measuring mechanical properties of muscle.

The concordance between the Green’s function simulations and in vivo muscle data demonstrate that the linear, elastic, ITI model is a reasonable model in the vastus lateralis of a healthy volunteer. The ability to extend this model to other muscles is limited by the size of the muscle, as when the thickness of the muscle body approaches the shear wavelength it is likely that other wave guiding behavior and boundary effects would influence waves measured. Additionally, a limitation of this method is the requirement that the experimental and simulated parameters are exactly matched, as any mismatches would be expected to introduce bias in the χ_E estimates.

A notable limitation of this work is that it neglects viscosity. Viscosity introduces dispersion and attenuation in shear wave elastography data, which has been shown to bias estimated parameters based upon group speeds [60]. We have observed some dispersion in vastus lateralis data, primarily across the fibers with less dispersion along the fibers [61], which is consistent with reports from the literature [45]. Thus, it is possible that the SV mode analysis presented here has some bias, which might be addressed at the expense of computational complexity, but is outside the scope of this initial demonstration.

For years, three parameter models of ITI materials have been proposed for muscle SWEI modeling [29], [30], [46], but to date quantification of all three parameters has not been realized in in vivo ultrasound SWEI. Recent demonstration of the ability to resolve two wave fronts in phantoms [51] showed the feasibility of using a tilted excitation to reconstruct tensile anisotropy.

Additionally, Li et al. have investigated these phenomena in biceps brachii and gastrocnemius muscles using 2D SWEI and measurements at 3 specified orientations of the transducer, assuming that the group shear wave speed measured with the transducer at θ_tilt = 45° represents the SV mode [62]. They found values that result in a χ_μ of 1.53 in the biceps brachii and 0.31 in the gastrocnemius, and χ_E of 0.75 in the biceps brachii and 0.81 in the gastrocnemius. All of these values are about 4 times lower than the range of values reported for the vastus lateralis herein. However they also measured a mean v_SV of 1.8 m/s, while we found a mean v_SV of 3.8 m/s, and given the relationship between $v_{\mathit{SV}}^2$ and χ_E seen in (9), it is unsurprising that approximately doubling the SV speed leads to a quadrupling of χ_E, showing again the importance of accurately measuring the SV speed. The 3D SWEI method herein leverages the tilt concepts but also facilitates characterization in muscles with a range of interrogation angles and fiber orientations.

Three ITI material parameters have been measured in vivo by magnetic resonance elastography (MRE) [48], and in ex vivo animal tissue [49], [50]. Looking at the three muscles in the calf using MRE in vivo, Guo et al. [48] found lower χ_μ values (soleus 0.50, gastrocnemius 0.80, tibialis anterior 0.27) compared to χ_μ = 2.11 herein for vastus lateralis, and also χ_μ values lower than herein (soleus 1.47, gastrocnemius 2.26, tibialis 2.00, compared to χ_E = 4.67 for vastus lateralis). These lower values for both tensile and shear anisotropy indicate the muscles appeared more isotropic using MRE than with 3D SWEI. It is encouraging that χ_μ > χ_μ with both MRE and 3D SWEI. There are several factors that could account for differences between these results. The interrogated muscles reported are different. The nature of the external continuous wave vibration and inversion of the wave equation employed by MRE means the parameters reconstructed are not as localized within the muscle, whereas 3D rotational SWEI measurements of muscle apply to a smaller region of interest (4 cm²). Additionally, the frequency content of MRE in the cited studies (40 Hz, 50 Hz, and 60 Hz in Guo et al.) is lower than the frequency content of SWEI used here (100-400 Hz). As frequency increases shear dissipative effects increase. Thus, there is an expectation that moduli estimated based on an elastic assumption will have an increasing bias as frequency increases. For example, see Yasar et al. [63], which shows that typically both the shear storage and loss moduli monotonically increase with excitation frequency.

The ability to measure tensile anisotropy χ_E and χ_μ may provide a unique biomarker for muscle characterization. However, the relationship between these material parameters, as well as μ_L and μ_T, and muscle health remains an open question. While still only a proof of concept, determining these relationships has potential application in characterizing a wide variety of muscle diseases including myopathies, dystrophies, muscle spasticities, muscle atrophy, age related sarcopenia, and athletic injury repair and healing.

VI. Conclusion

In this study, we describe measurements of multiple shear wave propagation modes in in vivo muscle using a 3D SWEI experimental setup with a tilted material symmetry axis, shown in Fig. 1(b). Results are reported from measurements made in a healthy volunteer on two different days in the relaxed vastus lateralis muscle, where the fibers are naturally tilted with respect to the skin surface. We demonstrate 3D SWEI can improve estimation of μ_L and μ_T by accounting for the effects of fiber rotation ϕ_fiber and tilt θ_tilt when evaluating SH mode speeds, which is not possible in 2D SWEI. We also demonstrate 3D SWEI measurement of the SV propagation mode which, when combined with matched Green’s function simulations, allows for estimation of tensile anisotropy, χ_E.