Rotational 3D Shear Wave Elasticity Imaging (3D-SWEI) offers many opportunities for the development of biomechanical biomarkers of muscle health. However, methods for material parameter reconstruction to date have involved manual processing steps, which are impractical for clinical implementations. In this work we describe and optimize algorithms for the automatic processing of 3D-SWEI skeletal muscle data and explore the most important factors affecting the success of automatic estimation of the shear wave speed (SWS) along and across the fibers. We compare the results of several automatic algorithms to the results obtained using manual processing across 130 in vivo 3D-SWEI datasets, evaluating the percent of valid acquisitions (non-gross-error), percent bias, and absolute percent error for each algorithm. We evaluate three major factors for algorithm optimization: wave detection method (comparing Radon Sum, time to peak, and cross correlation), lateral range selection (comparing a single range for all acquisition angles, changing the range with fiber orientation, and dynamic range selection, in addition to analyzing the start and end positions of the lateral range), and methods to improve robustness (exploring detecting the wave through 3D fits, and methods to isolate individual waves when multiple waves are detected). We conclude the optimal algorithm for passive muscle is a fixed lateral range (2-18 mm) Radon Sum method with multiwave detection and SWS fitting to the ellipse predicted in transversely isotropic materials, and we provide results that can guide algorithm choice for other applications. Additionally, we have made the implementation of all our algorithms publicly accessible at: DOI 10.5281/zenodo.18883176.

Ultrasound shear wave elasticity imaging (SWEI) quantifies the mechanical properties of soft tissue by relating measurements of shear wave speed (SWS) to a material model. Many SWEI algorithms estimate the time of flight (TOF) over some fixed distance along a given trajectory and calculate the wave velocity as the ratio of that distance to the TOF. To balance spatial resolution and SWS precision, most high-resolution 2D-SWEI algorithms average together multiple noisy velocity estimates to generate the final SWS. However, velocity calculated over a fixed distance with an estimated TOF can be prone to divide-by-near-zero (DBNZ) errors, especially as the true TOF approaches zero. The reciprocal of velocity, slowness, will never experience DBNZ errors as long as the fixed distance over which the TOF is estimated is greater than zero. In this work, we modify an existing 2D-SWEI algorithm to be based on point estimates of slowness. We propose a statistical model assuming normal TOF distributions and use it to evaluate the accuracy and the precision of the velocity- and slowness-based SWS algorithms in Monte Carlo simulations. We then extend our analysis of TOF data to empirically describe the bounds of the proposed model in the presence of speckle bias. Finally, we evaluate both algorithms in heterogeneous media and quantify the achievable lesion conspicuity across a range of shear wave SNR and SWS algorithm reconstruction parameters. We conclude that slowness-based SWS can achieve similar accuracy and precision as velocity-based SWS at less than one-tenth of the TOF SNR with no additional computational cost. For a fixed shear wave and TOF SNR in isotropic homogeneous media, we experimentally demonstrate a reduction in bias greater than 5% and a reduction in variability greater than 15%. We also demonstrate that the slowness-based approach achieves superior or equal lesion conspicuity as the velocity-based approach, regardless of imaging condition or algorithm parameters.

OBJECTIVE:Ultrasonic 3D rotational shear wave elasticity imaging (3D-RSWEI) has previously been used to quantify the mechanical properties of muscle with good precision in single timepoint studies. However, to date, the longitudinal variability of these measurements has not been evaluated. Herein, we report the longitudinal measurements of muscle stiffness and anisotropy assessed using 3D-RSWEI over a 3 month period. METHODS:3D-RSWEI measurements were made in vivo in the healthy vastus lateralis of 10 participants. Over 3 mo, we quantified the shear wave speed (SWS) along and across the fibers, along with fractional anisotropy (FA) at multiple knee flexion angles (≤45∘,45∘,60∘,90∘ and 110∘), and growth coefficients parametrizing the dependence of SWS and fractional anisotropy on applied knee flexion (βSWS,‖,βSWS,⊥ and βFA). The coefficient of variation (CoV) of each SWEI-based metric of interest was also quantified over 3 months. RESULTS:Throughout the duration of the study and across all participants, SWS measurements were consistent to within 10%CoV, with FA measurements consistent to within 13%. Fitting SWS growth versus knee flexion yielded a βSWS,‖ of 0.010±0.001(deg-1) and a βSWS,⊥ of 0.004±0.001(m/s/deg). Fitting FA growth with knee flexion yielded a βFA of 0.0029±0.0007(deg-1). Growth parameters had an average CoV of 8.3% for βSWS,‖,18.7% for βSWS,⊥ and 14.8% for βFA across all participants over 3 months. CONCLUSION:The low variability of these SWEI-based metrics over 3 months in healthy volunteers provides a reference for designing longitudinal studies to evaluate their sensitivity to histopathologic changes associated with muscle disease progression.

We present a volumetric characterization method for 3D-RSWEI data in in vivo skeletal muscle, in which shear wave speeds (SWS) are quantified at multiple rotational angles through multiple axial depths within the depth-of-field. We have shown in the vastus lateralis of 10 healthy volunteers that regions-of-interest 9.0 +/- 2.1 mm and 2.1 +/- 0.4 mm from the top and bottom boundary of the muscle yield reliable SWS estimates for the muscle body. We apply this volumetric characterization to longitudinally collected data and quantify coefficient of variation of SWEI-based metrics for the biomechanical properties of muscle from four visits over two months to be <18% for each parameter.

We present a 3D-RSWEI characterization of a polyvinyl alcohol phantom with applied uniaxial stretch at multiple transducer tilt angles (0 degrees, 15 degrees, 30 degrees, and 45 degrees), building on previous work [1]. We have observed shear horizontal (SH) and complex shear vertical (SV) wave mode propagation at large stretch ratios and at 15 degrees and 30 degrees of tilt. We have compared analytically calculated shear wave speeds assuming an Isihara material model to experimentally measured speeds in the PVA phantom to re-derive estimates of uniaxial stretch ratios. This analysis indicates the potential for 3D-RSWEI in higher order nonlinear elasticity characterization methods.

Elastic properties of materials can be measured by observing shear wave propagation following localized, impulsive excitations and comparing the spatiotemporal signals with signals calculated using a mechanical model of the material. These calculations are difficult in anisotropic materials because of the complex relations among the material symmetries, propagation directions, and wave polarizations. This study presents a new analytic model of wave propagation in an incompressible transversely isotropic (ITI) material, which is a commonly used model for anisotropic biological tissues such as skeletal muscle. The model solves the Cauchy equation of motion in the spatial and temporal Fourier domains by separating the system response function that relates the excitation force and shear wave signal into two terms corresponding to shear horizontal and shear vertical shear wave polarizations. Sample results are reported for an ITI material with model parameters similar to parameters measured in in vivo muscle and for model parameters determined by fitting the model to experimentally measured shear wave speeds in in vivo muscle. Calculation times for the analytic model are significantly shorter compared to similar calculations based on finite element simulations or Green's function calculations, and thus, the analytic model is well-suited for iterative fitting of model parameters.

We present ultrasonic 3D rotational shear wave elasticity imaging (3D-RSWEI) characterization of anisotropic 3D-printed hydrogel lattice phantoms that were originally developed for magnetic resonance elastography (MRE) applications. Shear wave speeds versus rotational angle were measured for two 3D-printed hydrogel lattices while submerged in water and after embedding in different stiffnesses of isotropic poly-vinyl alcohol (PVA). Shear wave propagation in the lattices was anisotropic, with faster wave speeds parallel to the scaling direction of the lattices. We observed that shear wave speeds and shear anisotropy decreased when the lattices were embedded in PVA versus submerged in water. Shear wave dispersion was observed in the lattice+PVA phantoms, with greater dispersion in the direction parallel to the scaling direction versus perpendicular. Dynamic shear testing results of the lattice+PVA phantoms were compared to 3D-RSWEI characterization with differences of 8.4% and 24.1% for propagation parallel and perpendicular to the scaling direction. These results demonstrate the feasibility of ultrasonic SWEI characterization of novel anisotropic MRE lattice phantoms.

In this work, we introduce novel shear wave excitation techniques to interrogate a broad range of shear wave trajectories with applications towards anisotropic material characterization. The two techniques discussed in this work, fixed geometry, divergent shear wave excitations and synthetic aperture shear wave excitations, interrogated a broad range of trajectories spanning more than 50 degrees, on par or exceeding the range of trajectories interrogated with steered linear SSI techniques across multiple steering angles. Further, each individual fixed geometry divergent shear wave excitation technique in this work interrogated trajectories spanning ranges greater than 30 degrees each.

In this work we propose and validate an algorithm to estimate phase velocity in anisotropic media using 3D rotational shear wave elasticity imaging. We propose a novel spatial and temporal weighting scheme to achieve millimeter-scale spatial and millisecond scale temporal resolution and apply it to the Eikonal wave equation. We demonstrate the accuracy of the algorithm in simulated and experimental datasets in hyper-elastic materials.

We present a 3D-RSWEI characterization of anisotropic lattice phantoms. Shear wave speeds versus rotational angle were measured while the lattices were submerged in water and after embedding in a softer, isotropic polyvinyl alcohol cryogel. In both cases, shear wave propagation was anisotropic and fit to an ellipse, with faster wave speeds observed along the material symmetry direction. Measured shear wave speeds and shear anisotropy decreased between lattices in water versus embedded in PVA. Linear wave dispersion slopes were higher along versus across the material symmetry direction. Measured speed of sound was higher and acoustic attenuation was lower than that of biological tissue. These results showcase applicability of 3D-RSWEI to characterize novel anisotropic lattice phantoms.

The stress-strain relation in a transversely isotropic (TI) material is described by five independent parameters. In the incompressible limit, only three parameters are required to describe shear wave propagation. Existing material parameterization models are not ideal for the analysis of wave propagation in the nearly incompressible TI (NITI) regime due to difficult-to-interpret parameters, complicated forms of the stiffness matrix elements, or the lack of five independent parameters. This study describes a new parameterization model for a general, TI material that uses the bulk modulus K, shear moduli μT and μL, a modulus-like term μE, and a new parameter η. In the proposed parameterization model, each parameter has a clear interpretation related to compressibility and shear wave propagation. The incompressible limit is represented by the limit K → ∞. Wave speeds and polarizations are derived and evaluated in both incompressible and NITI regimens. First-order NITI corrections are shown to be inversely proportional to the ratio of bulk modulus to shear moduli. In biological soft tissues, this ratio is approximately 106. NITI corrections depend on all five independent parameters; however, the small scale of these corrections validates previous studies that have assumed particular values for the parameter η.

Skeletal muscle is a complex tissue, exhibiting not only direction-dependent material properties (commonly modeled as a transversely isotropic material), but also changes in observed material properties due to factors such as contraction and passive stretch. In this work, we evaluated the effect of muscle passive stretch on shear wave propagation along and across the muscle fibers using a rotational 3D shear wave elasticity imaging system and automatic analysis methods. We imaged the vastus lateralis of 10 healthy volunteers, modulating passive stretch by imaging at 8 different knee flexion angles (controlled by a BioDex system). In addition to demonstrating the ability of this acquisition and automatic processing system to estimate muscle shear moduli over a range of values, we evaluated potential higher order biomarkers for muscle health that capture the change in muscle stiffness along and across the fibers with changing knee flexion. The median within-subject variability of these biomarkers is found to be <16%, suggesting promise as a repeatable clinical metric. Additionally, we report an unexpected observation: that shear wave signal amplitude along the fibers increases with increasing flexion and muscle stiffness, which is not predicted by transversely isotropic (TI) material simulations. This observation may point to an additional potential biomarker for muscle health or inform other material modeling choices for muscle.

Physics-informed neural networks (PINNs) solve differential equations and give compressed, analytic solutions free of discretization. Previous work used PINNs to model shear wave propagation from a spherical Gaussian impulse in an incompressible, transversely isotropic (ITI) material. Here, we extend this method to model the more complex geometry of an acoustic radiation force impulse (ARFI) and subsequent 3D shear wave propagation in both isotropic and ITI materials. We find that PINNs produce high-fidelity 3D solutions across different material models. PINN solutions qualitatively matched paired finite element method (FEM) simulations and without artifacts seen in FEM related to spatial discretization. The frequency content of PINN solutions differed slightly from the FEM, depending on the initial excitation geometry. PINNs have the potential to help rapidly explore material model parameter spaces. We also investigate the feasibility of solving the inverse problem to reconstruct material parameters from observed data by setting these parameters as variable outputs of the PINN model. PINNs were able to reconstruct the shear modulus using data measured in an isotropic elastic phantom to within 3 percent. Lastly, we show that PINNs can reconstruct all three independent parameters of an ITI material model from data.

We are investigating the potential for shear wave elasticity imaging (SWEI) derived material parameters to serve as biomarkers for skeletal muscle health. We consider muscle as a transversely isotropic (TI) material and use rotational 3D-SWEI acquisitions to characterize the shear wave propagation in directions along and across the muscle fibers at various passive stretch states. Data were collected in the vastus lateralis of the dominant leg of 10 healthy volunteers at various knee flexion angles (controlled by a BioDex system). The 3D-SWEI acquisitions were analyzed for both shear wave speed and amplitude in the directions along and across the muscle fibers. Relative to the values at 45° knee flexion, the shear wave speed along the fibers changed an average of +78% at 105° knee flexion and −11% at 0° knee flexion, while the shear wave speed across the fibers changed + 17% at 105° knee flexion, with no clear change at 0° knee flexion. These values support the need to control for subject positioning (joint angle) during skeletal muscle SWEI. Shear wave amplitude along the fibers increases by +110% from 45° to 105° knee flexion and changes by −43% from 45 to 0° knee flexion. Interestingly, in the direction along the fibers, the shear wave amplitude increases with increasing flexion even while shear wave speed increases over this same flexion range, opposite the trend expected from isotropic materials. Across the fibers, there is no clear trend in shear wave amplitude from 45 to 105° knee flexion, but amplitude changes by an average of +52% at 0° knee flexion relative to 45° knee flexion. This observation could provide insight for optimization of in vivo imaging protocols and understanding the higher order properties of skeletal muscle. Additionally, we explore the quantification of muscle's response to stretch through the rate of change in SWS with knee flexion at flexion angle above 45° and find linear fits with high R-squared values and slopes of 0.023 ± 0.0034 m/s/°.

There is increasing interest in using ultrasound shear wave elasticity imaging to study tissues described as incompressible, transversely isotropic (ITI) materials, such as skeletal muscle. In silico modeling helps us predict and understand shear wave behavior in complex materials like the ITI model, which supports two shear polarizations with different, direction-dependent propagation speeds. Existing techniques, the finite element method (FEM) and Greens functions, are computationally expensive and generate large file sizes. Physics-informed neural networks (PINNs) is a relatively novel technique to solve partial differential equations and produces solutions that are compressed, analytic, and free of space-time discretization. Here, we solve the 3D wave equation for an ITI material using PINNs and show that solutions match FEM simulations to first order for material parameters based on skeletal muscle. Estimated shear wave speeds for the PINN and FEM solutions differed by an average of 4.7%. Unlike the FEM simulation, the PINN solution had no reflection artifacts at the boundaries. Second-order differences in frequency content and amplitude distribution suggest the need for further validation. PINNs can enable rapid exploration of the complex shear wave behavior in ITI materials and can be extended to different material models by adjusting the wave equation and initial conditions.

Ultrasonic rotational 3-D shear wave elasticity 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 (ITI) material with three parameters: the longitudinal shear modulus μL , the transverse shear modulus μT , and the tensile anisotropy χE . Measurement of the SV wave is necessary to characterize χE , but the factors that influence SV mode generation and characterization with ultrasonic SWEI are complicated. This work uses Green's function (GF) 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: μL , μT , χE , fiber tilt angle [Formula: see text], and F-number of the push geometry on SV mode speed, amplitude, and rotational distribution. These analyses demonstrate that the following hold: 1) as μL increases, SV waves decrease in amplitude so are more difficult to measure in SWEI imaging; 2) as μT increases, the SV wave speeds increase; 3) as χE increases, the SV waves increase in speed and separate from the SH waves; 4) as fiber tilt angle [Formula: see text] 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 [Formula: see text].

Objective.Determining elastic properties of materials from observations of shear wave propagation is difficult in anisotropic materials because of the complex relations among the propagation direction, shear wave polarizations, and material symmetries. In this study, we derive expressions for the phase velocities of the SH and SV propagation modes as a function of propagation direction in an incompressible, hyperelastic material with uniaxial stretch.Approach.Wave motion is included in the material model by adding incremental, small amplitude motion to the initial, finite deformation. Equations of motion for the SH and SV propagation modes are constructed using the Cauchy stress tensor derived from the strain energy function of the material. Group velocities for the SH and SV propagation modes are derived from the angle-dependent phase velocities.Main results.Sample results are presented for the Arruda-Boyce, Mooney-Rivlin, and Isihara material models using model parameters previously determined in a phantom.Significance.Results for the Mooney-Rivlin and Isihara models demonstrate shear splitting in which the SH and SV propagation modes have unequal group velocities for propagation across the material symmetry axis. In addition, for sufficiently large stretch, the Arruda-Boyce and Isihara material models show cusp structures with triple-valued group velocities for the SV mode at angles of roughly 15° to the material symmetry axis.

Shear wave elasticity imaging (SWEI) usually assumes an isotropic material; however, skeletal muscle is typically modeled as a transversely isotropic material with independent shear wave speeds in the directions along and across the muscle fibers. To capture these direction-dependent properties, we implemented a rotational 3-D SWEI system that measures the shear wave speed both along and across the fibers in a single 3-D acquisition, with automatic detection of the muscle fiber orientation. We tested and examined the repeatability of this system's measurements in the vastus lateralis of 10 healthy volunteers. The average coefficient of variation of the measurements from this 3-D SWEI system was 5.3% along the fibers and 8.1% across the fibers. When compared with estimated respective 2-D SWEI values of 16.0% and 83.4%, these results suggest using 3-D SWEI has the potential to improve the precision of SWEI measurements in muscle. Additionally, we observed no significant difference in shear wave speed between the dominant and non-dominant legs along (p = 0.26) or across (p = 0.65) the muscle fibers.

Five material parameters are required to describe a transversely isotropic (TI) material including two Poisson's ratios that characterize the compressibility of the material. Both Poisson's ratios must be specified to model an incompressible, TI (ITI) material. However, a previous analysis of the procedure used to evaluate the incompressible limit in a two-dimensional (2D) space of Poisson's ratios has shown that elements of the stiffness tensor are not unique in this limit, and that an additional, fourth parameter is required to model these elements for an ITI material. In this study, we extend this analysis to the case of shear wave propagation in an ITI material. Shear wave signals are modeled using analytic Green's tensor methods to express the signals in terms of the phase velocity and polarization vectors of the shear horizontal (SH) and shear vertical (SV) propagation modes. In contrast to the previous result, the current analysis demonstrates that the phase velocity and polarization vectors are independent of the procedure used to evaluate the 2D limit of Poisson's ratios without the need to include an additional parameter. Thus, calculated shear wave signals are unique and can be used for comparison with experimental measurements to determine all three model parameters that characterize an ITI material.

We previously have demonstrated the use of ultrasonic rotational 3D SWEI to measure all three mechanical properties necessary to fully characterize the vastus lateralis muscle in vivo as an incompressible transversely isotropic material: transverse shear modulus $\mu_{T}$ , longitudinal shear modulus $\mu_{L}$ , and tensile anisotropy $\chi_{E}$ . In this work we investigate how varying knee flexion angle affects the mechanical properties of the vastus lateralis. As knee flexion angle increased, $\mu_{L},\ \chi_{E}$ , and $\chi_{\mu}((\mu_{L}-\mu_{T})/\mu_{T})$ all increased, while $\mu_{T}$ remained consistent across knee angles. At all nine knee flexion angles investigated in a healthy volunteer $\chi_{E} > \chi_{\mu}$ , indicating that these are independent parameters.