Estimation of the mechanical properties of a transversely isotropic material from shear wave fields via artificial neural networks

Abstract

Artificial neural networks (ANN), established tools in machine learning, are applied to the problem of estimating parameters of a transversely isotropic (TI) material model using data from magnetic resonance elastography (MRE) and diffusion tensor imaging (DTI). We use neural networks to estimate parameters from experimental measurements of ultrasound-induced shear waves after training on analogous data from simulations of a computer model with similar loading, geometry, and boundary conditions. Strain ratios and shear-wave speeds (from MRE) and fiber direction (the direction of maximum diffusivity from diffusion tensor imaging (DTI)) are used as inputs to neural networks trained to estimate the parameters of a TI material (baseline shear modulus μ, shear anisotropy ϕ, and tensile anisotropy ζ). Ensembles of neural networks are applied to obtain distributions of parameter estimates. The robustness of this approach is assessed by quantifying the sensitivity of property estimates to assumptions in modeling (such as assumed loss factor) and choices in fitting (such as the size of the neural network). This study demonstrates the successful application of simulation-trained neural networks to estimate anisotropic material parameters from complementary MRE and DTI imaging data.

1. Introduction

In the last decade, neural networks have received a great deal of attention as a platform for understanding large data sets like those obtained from 3D imaging (Hosny et al., 2018). Much of the excitement is due to the apparent ability of neural networks to make decisions and draw conclusions when presented with complex, noisy, irrelevant, and/or partial information (Eberhart, 2014).

Mechanical properties of tissue are of great interest as indicators of disease or injury, as well as for their role in normal function. MR elastography (MRE), which is based on MR imaging and analysis of shear waves, has been widely used to assess tissue stiffness noninvasively since the introduction of the method by Muthupillai et al. (1995).

While the vast majority of MRE studies have assumed isotropic material behavior, recent investigators (Romano et al., 2012; Qin et al., 2013; Tweten et al., 2015; Guertler et al.,2020) have used MRE to estimate parameters of anisotropic material models. Many biological materials, like muscle (Schmidt et al., 2016; Guertler et al., 2020) and white matter in the brain (Romano et al., 2012; Schmidt et al., 2018), may be mechanically anisotropic due to their fibrous microstructure. Anisotropic MRE is complicated by the fact that speeds of shear waves differ in waves with different propagation and polarization directions (Tweten et al., 2015). Anisotropic materials for use in MRE phantoms have recently been developed to facilitate and assess the accuracy of anisotropic MRE (Guidetti et al., 2019; Qin et al., 2013). In these prior studies, the inversion approaches used to estimate anisotropic properties are mathematically involved (due to the complex physics of wave propagation in anisotropic materials) and susceptible to numerical errors and artifacts (due to discretization, numerical approximations, and limited domain size). Machine learning offers an alternative approach; artificial neural network-based regression has begun to be applied by multiple researchers to estimate mechanical properties of biomaterials (Li et al., 2006; Qiao et al., 2007; Jiang et al., 2008; Singh et al., 2010; Shi et al., 2018; Murphy et al., 2018; Abueidda et al., 2019).

We propose a neural-network-based approach to estimate anisotropic material parameters, leveraging unique data acquired by MR imaging of harmonic, ultrasound-induced motion (MR-HUM) (Guertler et al., 2020). These data, consisting of full 3D displacement fields from shear waves excited by the acoustic radiation force of focused ultrasound, provide a stereotypical pattern of radially-propagating, ellipsoidal wave fronts, localized near the ultrasound focus. This behavior is readily simulated in a finite element model of shear-wave behavior in a transversely isotropic (TI) material with a local body force (Guidetti et al., 2019). Simulations produce displacement fields that are strikingly similar to experiment, with distinct features that capture the stiffness and anisotropy of the material. These features, common to simulation and experiment, are used as inputs to the neural network fitting process.

In this study, we explore the use of neural networks trained with simulation results to estimate TI material parameters from experimental data obtained by MRE and diffusion tensor imaging (DTI). The materials of interest are muscle tissue (chicken breast), which is a soft fibrous tissue believed to exhibit anisotropic properties, and a gelatin/glycerol gel, which is isotropic. In samples of the two materials, the shear modulus in the plane of isotropy (the “baseline shear modulus”), μ = μ₂, the shear anisotropy parameter, $\varphi =\frac{{\mu }_1}{{\mu }_2}-1$, and the tensile anisotropy parameter, $\zeta =\frac{E_1}{E_2}-1$, are estimated. We also assess the robustness of this approach by evaluating the sensitivity of the estimated properties to various modeling assumptions and choices in the neural network fitting procedure. The results support the utility of simulation-trained neural networks to estimate anisotropic material parameters from complementary sets of imaging data.

2. Methods

2.1. Experiment

2.1.1. Experimental setup and samples

The experimental setup is shown in Fig. 1. Ten muscle tissue (chicken breast) samples and two gel samples were used. Fresh chicken breast was obtained from a local grocery store and refrigerated at 4°C. Each tissue sample (Fig. 1a) was embedded in a gelatin/glycerol gel (4.2% w/w gelatin in equal parts water and glycerol) in a 50mL conical tube (27mm dia., Fig. 1b). The sample was refrigerated until testing, then inserted into a modified 50mL tube with a cutout on the top surface (Fig. 1c) to facilitate ultrasound penetration. The sample was oriented so that fiber direction, estimated visually, would be roughly perpendicular to the excitation. Gel samples were prepared identically, except without tissue embedded. The tube was placed in a 30mm diameter RF coil (IGT, Image Guided Therapy, Pessac, France). The ultrasound transducer (IGT) was placed above the tube with a water-filled bladder between the transducer and the sample (Fig. 1d–g), and the entire assembly was positioned at the isocenter of a 4.7T small-bore animal MRI scanner (Agilent/Varian, Santa Clara, CA) (Guertler et al., 2020).

Fig. 1.

Experimental setup for MR-HUM of muscle tissue (chicken breast). (a) Cylindrical punched tissue sample. (b) Tissue sample embedded in the gelatin/glycerol mixture. (c) Tissue sample in a tube with cutout on the top surface to facilitate ultrasound penetration. (d) The ultrasound transducer (US) is placed above the tissue sample. (e-f) Schematic of MR-HUM experimental setup in YZ and ZX planes. The Y-axis is the (horizontal) axis of the scanner and sample tube; the Z-axis is the direction of ultrasound excitation. (g) Schematic showing fiber direction (thin lines) in the ZX plane, and vertical arrows indicating the excitation at the focus. The tissue sample is oriented so that the muscle fiber axis, as estimated visually before imaging, is roughly parallel to the X-axis. The actual fiber direction is measured later by DTI. Reproduced from Guertler et al. (2020).

2.1.2. Shear-wave excitation

Shear waves were produced by harmonically modulated acoustic radiation force from the focused ultrasound beam. Each tissue sample was exposed to a 1.5 MHz focused ultrasound beam directed vertically downward at an angle between 0° and 45° relative to the muscle tissue fiber direction (estimated visually, then measured by DTI as described below). The ultrasound focus was electronically set 2 mm below the natural focus of the transducer, and adjusted to maintain ~1.5 W peak power. The ultrasound amplitude was modulated by an acoustic frequency (300, 400, or 500 Hz) square wave to generate an amplitude-modulated acoustic radiation force at the focus at the frequency of the modulation signal (Fig. 2). Shear waves were excited at 300 Hz (gelatin/glycerol), 400 Hz (gelatin/glycerol and muscle), or 500 Hz (muscle); these frequencies were selected to produce waves with wavelength of 4–10 mm, less than the sample radius and with multiple (at least four) voxels per wavelength.

Fig. 2:

Amplitude modulation of focused ultrasound. The high frequency (1.5 MHz) ultrasound waveform is modulated by a low frequency (400 Hz) signal to produce an oscillatory force at the focus, resulting in shear waves at 400 Hz. Reproduced from Guertler et al. (2020).

2.1.3. MRE experiment

MRE data were acquired with a 2D multi-slice spin-echo sequence modified by the addition of sinusoidal motion-encoding gradients (Clayton and Bayly, 2011). Imaging parameter settings were: 1 mm isotropic voxels, repetition time (TR) = 1000 ms, and echo time (TE) = 33–36 ms, covering a field of view (FOV) of either 32 × 32 × 12 mm³ (matrix size 32×32×12) or 24 × 24 × 12 mm³ (matrix size 24×24×12) (Guertler et al., 2020). When data were acquired from the larger FOV, only data from the central 24 × 24 × 12 sub-array were analyzed. Sinusoidal motion-encoding gradients (1–3 cycles) of amplitude 20 Gauss/cm were synchronized with motion to induce phase contrast proportional to displacement. Motion encoding was applied in all three (X,Y,Z) directions. The resulting phase-contrast images were phase-unwrapped and rigid-body motioncomponents were removed to isolate shear waves. Displacement data were analyzed within a hemispherical region of interest (ROI) with a fixed radius (R_o =8, 10, or 12 mm; R_i =1.5mm) from the focus. Data outside the ROI were eliminated to reduce the effects of (i) relatively low displacement-to-noise ratio due to dissipation of shear waves and (ii) reflections or other interactions of waves with boundaries.

To obtain accurate estimates of fiber direction, DTI (Pierpaoli et al., 1996) was performed for all muscle tissue samples. Diffusion tensors were estimated from images obtained using 30 diffusion-weighting gradient directions. DTI scans had 2 mm³ isotropic voxel resolution over an imaging volume of 48 × 48 × 16 mm³ (matrix size 24 × 24 × 8) Fiber direction a was estimated from the first principal eigenvector of the diffusion tensor (Guertler et al., 2020).

2.2. Simulation

A finite element model (Fig. 3) was constructed to simulate the shear-wave behavior observed in the experiment. Modeling and simulation were performed using COMSOL Multiphysics (COMSOL, Inc., v.5.6, Burlington, MA). The model domain is a cylinder with 27 mm diameter and 30 mm length, and meshed with 26,454 tetrahedral elements (quadratic Lagrange interpolation). The average element quality is 0.77, the element volume ratio is 3.32×10⁻⁴, and the mesh volume is 1.67×10⁻⁵ m³. Resolution of the mesh was established by successive refinement until strain and wavelength estimates did not change. All three frequencies (300, 400 and 500Hz) separated in different simulations and each simulation took approximately 4 hours using 8 cores of Intel(R) Core(TM) i7–10875H.The material model is transversely isotropic, linear elastic with isotropic damping and the fiber direction is in the ZX-plane, and the deviation angle between the fiber direction and X-axis is θ. The range of simulation cases included several isotropic cases (ϕ = ζ = 0).

Fig. 3.

(a) Geometry used in simulation of the MR-HUM experiment. (b) The vertical (w) displacement field from simulation of focused ultrasound-induced waves in the muscle tissue sample. The ultrasound focus is above the geometric center of the cylindrical sample. The angle θ is the angle between the fiber direction aand the XY plane.

To simulate the acoustic radiation force, a harmonically-varying body force was applied to a small (R=1 mm) spherical volume at a location analogous to the experiment and simulation: 6 mm above the geometric center of the cylinder. The cylindrical domain was modified to include the shape of an indentation where the water-filled reservoir is used to couple the ultrasound transducer to the sample.

The model was solved using the COMSOL “Frequency Domain” solver, which finds the steady-state response to harmonic excitation.

The subdomain of the simulation corresponding to the imaged volume from the MRE experiment is a rectangular prismatic region 24 × 24 × 12 mm³ (X × Y × Z) divided into 1mm³ voxels. Excitation is applied at the center of the top surface of this region, and a hemispherical ROI is defined relative to the focus, as in the experiment.

2.3. Local frequency estimation (LFE) of shear wave speed

Shear-wave speeds are calculated for each voxel in the imaged region. Local frequency estimation (LFE) (Knutsson et al., 1994) is used to identify the wavelength (λ) of shear waves from the (dominant) w component of the displacement field; the wavelength is multiplied by frequency (f) to obtain the shear-wave speed (c = λf) in m/s. Example wave fields and corresponding maps of wave speed estimates are illustrated in Figs. 4–5 for isotropic and anisotropic samples; for visualization, the displacement field $\overline{w}$ is normalized by the mean displacement amplitude.

Fig. 4.

Example data from simulation and experiment illustrating estimation of shear wave speed in isotropic samples. (a) Normalized out-of-plane displacement $\overline{w}$ (dimensionless) in the ROI of one isotropic simulated sample (left, μ = 3000Pa, ϕ = 0, ζ = 0, f = 400Hz) and one gel experimental sample (right, sample 2, f = 400Hz). Displacement is normalized by the mean displacement amplitude 0.5 μm (simulation) and 0.2 μm (experiment). (b) Shear wave speeds C in the ROI of the isotropic simulated sample (left) and gel experimental sample (right). Images are linearly interpolated from 1 mm isotropic voxels to 0.01 mm pixel image resolution.

Fig. 5.

Example data from simulation and experiment illustrating estimation of shear wave speed in anisotropic samples. (a) Normalized out-of-plane displacement $\overline{w}$ (dimensionless) in the ROI of one anisotropic simulated sample (left, μ = 5000Pa, ϕ = 1.5, ζ = 2, f = 400Hz, fiber direction a = [1,0,0]) and one muscle experimental sample (right, sample A, f = 400Hz fiber direction a = [0.99,0,0.01]). Displacement is normalized by the mean displacement amplitude 0.3 μm (both simulation and experiment). (b) Shear wave speeds C in the ROI of one anisotropic simulated sample (left) and muscle experimental sample (right). Images are linearly interpolated from 1 mm isotropic voxels to 0.01 mm pixel image resolution.

$$\overline{w}=\frac{w}{mean\left|\mathit{u}\right|};\left|\mathit{u}\right|=\sqrt{u^2+v^2+w^2}$$ (1)

In the second panel (Fig. 4–5b), maps of shear-wave speeds are illustrated in one slice containing the center of acoustic force application. Shear-wave speeds calculated from LFE are different in each voxel (due to voxel-wise differences in polarization and propagation directions relative to fiber direction) and distributed between 2–4 m/s. The 10^th percentile wave speed in the ROI was used to estimate the minimal slow shear wave speed. The minimal slow shear wave speed is useful, as it is determined by the baseline shear modulus (the shear modulus governing shear in the plane of isotropy normal to the fiber directions) and not by either anisotropy parameter.

2.4. Strain and strain ratios

In a nearly-incompressible, transversely isotropic material model, the material behavior is governed by the bulk modulus, K, baseline shear modulus μ, tensile anisotropy ζ, and shear anisotropy, ϕ (Schmidt et al., 2016). The compliance and stiffness matrices can be shown in the matrix below by using these parameters to describe the strain-stress relationship:

$$\mathit{A}=\left[\begin{array}{c}{a}_{11}\\ a_{12}\\ a_{12}\\ 0\\ 0\\ 0\end{array}\begin{array}{c}{a}_{12}\\ a_{22}\\ a_{13}\\ 0\\ 0\\ 0\end{array}\begin{array}{c}{a}_{12}\\ a_{13}\\ a_{22}\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ \frac{a_{11}-a_{12}}{2}\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\\ a_{55}\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ a_{55}\end{array}\right]$$

The corresponding compliance matrix S can be calculated as:

$$S=\left[\begin{array}{cccccc}\frac{1}{E_1}& -\frac{{\nu }_{21}}{E_2}& -\frac{{\nu }_{21}}{E_2}& 0& 0& 0\\ -\frac{{\nu }_{12}}{E_1}& \frac{1}{E_2}& -\frac{{\nu }_2}{E_2}& 0& 0& 0\\ -\frac{{\nu }_{12}}{E_1}& -\frac{{\nu }_2}{E_2}& \frac{1}{E_2}& 0& 0& 0\\ 0& 0& 0& \frac{1}{2{\mu }_2}& 0& 0\\ 0& 0& 0& 0& \frac{1}{2{\mu }_1}& 0\\ 0& 0& 0& 0& 0& \frac{1}{2{\mu }_1}\end{array}\right]$$

The compliance matrix can also be expressed in terms of anisotropy parameters:

$$\mathit{S}=\left[\begin{array}{cccccc}\frac{1}{\mu \left(4\zeta +3\right)}+\frac{1}{9K}& \frac{-1}{2\mu \left(4\zeta +3\right)}+\frac{1}{9K}& \frac{-1}{2\mu \left(4\zeta +3\right)}+\frac{1}{9K}& 0& 0& 0\\ \frac{-1}{2\mu \left(4\zeta +3\right)}+\frac{1}{9K}& \frac{1+\zeta }{\mu \left(4\zeta +3\right)}+\frac{1}{9K}& \frac{-1}{2\mu \left(4\zeta +3\right)}+\frac{1}{9K}& 0& 0& 0\\ \frac{-1}{2\mu \left(4\zeta +3\right)}+\frac{1}{9K}& \frac{-1}{2\mu \left(4\zeta +3\right)}+\frac{1}{9K}& \frac{1+\zeta }{\mu \left(4\zeta +3\right)}+\frac{1}{9K}& 0& 0& 0\\ 0& 0& 0& \frac{1}{2\mu }& 0& 0\\ 0& 0& 0& 0& \frac{1}{2\mu \left(1+\varphi \right)}& 0\\ 0& 0& 0& 0& 0& \frac{1}{2\mu \left(1+\varphi \right)}\end{array}\right]$$

where $\varphi =\frac{{\mu }_1}{{\mu }_2}-1;\zeta =\frac{E_1}{E_2}-1.$The parameters μ₁ = μ (1 + ϕ) and μ₂ = μ (the “baseline” shear modulus) are the shear moduli governing deformation in planes parallel and perpendicular to the fiber direction, respectively, and E₁ and E₂ are the tensile moduli in directions parallel and perpendicular to the fiber direction, respectively.

The strain tensor can be described in terms of spatial derivatives of displacement:

$$E=\left[\begin{array}{ccc}{E}_{xx}& E_{xy}& E_{xz}\\ sym& E_{yy}& E_{yz}\\ sym& sym& E_{zz}\end{array}\right]=\left[\begin{array}{ccc}\frac{\partial u}{\partial x}& \frac{1}{2}\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)& \frac{1}{2}\left(\frac{\partial w}{\partial x}+\frac{\partial u}{\partial z}\right)\\ sym& \frac{\partial v}{\partial y}& \frac{1}{2}\left(\frac{\partial w}{\partial y}+\frac{\partial v}{\partial z}\right)\\ sym& sym& \frac{\partial w}{\partial z}\end{array}\right]$$

To characterize deformations with respect to the material symmetry axis, strains are estimated in the “fiber direction” and relevant “non-fiber” directions (Fig. 6). The “non-fiber” directions are chosen to have the same component in the Z-direction (loading direction) as the fiber direction, but rotated by ±90 degrees about the Z-axis. Loading needs to be consistent in fiber and non-fiber directions so that the loading has equivalent effects on fiber and non-fiber strain components, and thus any differences are due to material asymmetry.

Fig. 6.

Directions for defining strain components for MR-HUM in muscle tissue (chicken breast). (a-b) Fiber direction (a) and non-fiber direction (b, e) vectors, and perpendicular vectors c, d, f, are used to compute corresponding strain components (μ = 3000Pa, ϕ = 1, ζ = 1, θ = 30°, f = 400Hz).

Experiment:

The fiber direction at each voxel is obtained from the principal axis of the diffusion tensor corresponding to the direction of greatest diffusivity, measured from the diffusion tensor imaging (DTI) scan. The average fiber direction, a = [a_x, a_y, a_z], is obtained by spatially averaging this direction over all voxels in the ROI of imaged volume. The non fiber directions are b = [− a_y, a_x, a_z] and e = [a_y, −a_x, a_z] which have the same Z-component with a but rotated by 90 degrees about the z-axis. The vectors c, d, and f used to define the shear strains with respect to fiber and non-fiber directions are obtained by cross-products which are perpendicular to the fiber and non-fiber direction a, b, and e:

$$\mathit{c}=\mathit{a}\times \left(\mathit{a}\times \mathit{k}\right);\mathit{d}=\mathit{b}\times \left(\mathit{b}\times \mathit{k}\right);\mathit{f}=\mathit{e}\times \left(\mathit{e}\times \mathit{k}\right)$$ (2)

Simulation:

Fiber direction a and corresponding non-fiber directions b and e can be described as:

$$\mathit{a}=\left[cos\theta ,0,sin\theta \right];\mathit{b}=\left[0,cos\theta ,sin\theta \right];\mathit{e}=\left[0,-cos\theta ,sin\theta \right];$$ (3)

where θ is the angle between the fiber direction aand the XY-plane (Fig. 3). To obtain shear strains in planes containing the fiber direction or normal to the fibers, the following unit vectors are defined, c ⊥ a in the ZX-plane, and d ⊥ b, f ⊥ e in the YZ-plane (Fig. 6):

$$\mathit{c}=\left[sin\theta ,0,cos\theta \right];\mathit{d}=\left[0,sin\theta ,cos\theta \right];\mathit{f}=\left[0,-sin\theta ,cos\theta \right]$$

The strain components E_aa and E_bb are designated as “fiber” tensile strain and “non-fiber” tensile strain, respectively. Similarly, the strain components E_ac and E_bd are “fiber” and “non-fiber” shear strains. These components are the fiber directions and Cartesian strain:

$$E_{aa}=\mathit{a}E{\mathit{a}}^T;E_{bb}=\mathit{b}E{\mathit{b}}^T;E_{ee}=\mathit{e}E{\mathit{e}}^T$$ (4)

$$E_{ac}=\mathit{a}E{\mathit{c}}^T;E_{bd}=\mathit{b}E{\mathit{d}}^T;E_{ef}=\mathit{e}E{\mathit{f}}^T$$ (5)

Maps of strain components (normalized with the octahedral strain) in the ROI are shown in Fig. 7 and 8 for one slice near the actuation in an experimental and simulated sample, respectively. Strain fields exhibit key similarities between experiment and simulation; the out-of-plane (z) components are typically larger than those in-plane, and the “non-fiber” strains (E_bb, E_bd) are generally larger than the corresponding tensile and shear strains (E_aa, E_ac) in the planes containing the fiber direction. Strains are normalized by the mean octahedral strain in the ROI:

$$E_{ij}=\frac{E_{ij}}{mean\left(E_{oct}\right)};$$ (6)

$$E_{oct}=\frac{2}{3}\sqrt{{\left(E_{xx}-E_{yy}\right)}^2+{\left(E_{yy}-E_{zz}\right)}^2+{\left(E_{zz}-E_{xx}\right)}^2+6\left(E_{xy}^2+E_{yz}^2+E_{zx}^2\right)}$$ (7)

The relationships observed in simulations between the tensile strain ratio $\frac{E_{bb}}{E_{aa}}$, the shear strain ratio $\frac{E_{bd}}{E_{ac}}$, and input model parameters are shown in Fig. 9. Red points represent the special case of a fully isotropic material.

Fig. 7.

Example experimental strain components with respect to scanner coordinates (x, y, z) and fiber/non-fiber (a, b, c, d) directions. Strains on the top surface of the ROI of one experimental muscle tissue sample (sample A, chicken breast, fiber direction a = [0.99,0,0.01]). (a-c) Normal strains (E_xx, E_yy, E_zz) (d-e) Fiber tensile strain and non-fiber tensile strain (E_aa, E_bb). (f-h) Shear strains (E_xy, E_yz, E_zx). (i-j) Fiber and non-fiber shear strain (E_ac, E_bd). (f = 400Hz). Strains are normalized with respect to mean octahedral shear strain in the ROI: ${\overline{E}}_{oct}=1.62\times {10}^{-4}$.

Fig. 8.

Example simulated strain components with respect to scanner coordinates (x, y, z) and fiber/non-fiber (a, b, c, d) directions. Strains on the top surface of the ROI of one simulated anisotropic sample with fiber direction a = [1,0,0]). (a-c) Normal strains (E_xx, E_yy, E_zz). (d-e) Fiber tensile strain and non-fiber tensile strain (E_aa, E_bb) (f-h) Shear strains (E_xy, E_yz, E_zx). (i-j) Fiber and non-fiber shear strain (E_ac, E_bd) (μ = 5000Pa, ϕ = 1.5, ζ = 2, θ = 0, f = 400Hz). Strains are normalized with respect to mean octahedral shear strain in the ROI: ${\overline{E}}_{oct}=2.89\times {10}^{-4}$.

Fig. 9.

Tensile strain ratios E_bb/E_aa(non-fiber/fiber) and shear strain ratios E_bd/E_ac in simulations increase with increasing tensile anisotropy ζ and increasing shear anisotropy ϕ, respectively. The relationships diminish with the increasing fiber angle (deviation of fiber from the horizontal plane normal to the applied loading). Different surfaces are obtained for different value of μ from 1000Pa to 9000Pa. Red points represent the special case of a fully isotropic material.

With other parameters fixed, the tensile strain ratio E_bb/E_aa exhibits an approximately linear relationship with tensile anisotropy ζ; likewise, with other parameters fixed, the shear strain ratio E_bd/E_ac exhibits an approximately linear relationship with the shear anisotropy ϕ. The relationship between strain ratio E_bb/E_aa and E_bd/E_ac and model parameters ϕand ζ is clearest when the fiber deviation angle from the XY-plane is less than 30°. In the ideal case (i.e., simulations) the ratios E_ee/E_aa and E_ef/E_ac will be equal to E_bb/E_aa and E_bd/E_ac, respectively. However, in experiments, in which each strain ratio may be affected by noise or measurement error, inclusion of additional strain ratios should increase robustness to such non-ideal effects.

2.5. Artificial Neural Network (ANN) Development

In the current implementation, we exploit the axisymmetric features of the MR-HUM experiment and its similarity to simulation by using simulated data to train the network, which is then used to estimate parameters from experimental data. The neural network approach is implemented in MATLAB (R2019b, The Mathworks, Natick, MA; Neural Network Toolbox).

2.5.1. Basic setup

The input features of the shear-wave fields are X = [cos θ, sinθ, E_bb/E_aa, E_bd/E_ac, E_ee/E_aa, E_ef/E_ac, C_s] and the output features are Y = [μ, ϕ, ζ]. In training, the input features are taken from simulation results and the output features are the known model parameters. Bayesian regularization backpropagation is applied to avoid over-fitting, and there are 10–30 elements in one hidden layer (Fig. 10). To improve performance and accuracy, parameters are normalized to be on the same scale [−1 1], (Jayalakshmi and Santhakumaran., 2011). These steps were implemented using the MATLAB functions fitnet with the trainbr option to set up the fitting network with Bayesian regularization (“net = fitnet(20, ‘trainbr’)” ) and train (“net = train(net,X,Y)”) to train the network. The maximum number of epochs was set to 1000 in all cases.

Fig. 10.

Neural network structure. There are seven input features (cos θ, sinθ, E_bb/E_aa, E_bd/E_ac, E_ee/E_aa, E_ef/E_ac, C_s) and three output features (μ, ϕ, ζ). There is one hidden layer with 10, 20, or 30 nodes.

Simulations were performed to span the range of expected parameters, as described above. All 2520 simulated data sets with excitation frequency 400Hz and 500Hz are divided randomly into three groups: training sets, validation sets and testing sets, respectively 70%, 20% and 10% of the data sets. The range of input features and output features and min/max value in the data are shown in Table 1.

Table 1.

Original and normalized (in parentheses) input features for neural network training.

Parameter	min	max	interval
μ (Pa)	1000 (−1)	9000 (1)	2000 (0.5)
ϕ	−0.5 (−1)	2 (1)	0.5 (0.4)
ζ	−0.5 (−1)	2 (1)	0.5 (0.4)

2.5.2. Neural network ensemble

Neural network estimates vary because training sets, validation sets and testing sets are randomly selected each time a neural network is trained. Therefore, a single estimation is not sufficient to prove its veracity. However, it is possible to perform multiple cycles of neural network training and obtain distributions of estimates for each parameter. This provides a powerful method for estimating expected (mean) parameter values and confidence intervals. This concept, using an ensemble of neural networks with a plurality consensus, shows far better performance than a single copy of a neural network (Hansen and Salamon, 1990).

To achieve this, 100 cycles of models are applied to generate an ensemble of neural networks (NN₁, NN₂ … NN₁₀₀). The weight (w₁,w₂…w₁₀₀) of each predicted value (d₁,d₂…d₁₀₀₀) is defined and combined into the final result as the mathematical expectation:

$$\overline{d}={\sum }_{i=1}^{100}{w}_i{d}_i,{\sum }_{i=1}^{100}{w}_i=1$$ (8)

From the distribution of 100 cycles of estimation, assumed to be a normal distribution (Fig. 11), the mathematical expectation is the mean value:

$$\overline{d}=mean\left(d_i\right)i=\mathrm{1,2},3,\dots ,100$$ (9)

Fig. 11.

Example of estimated results distribution from one sample with 100 cycles. Columns (left, center, right) correspond to parameters (μ, ϕ, ζ). (a) Estimated results for muscle sample when fiber angle θ = 4.2° from the horizontal (XY) plane (f=400 Hz). (b) Estimated results for muscle sample when fiber angle θ = 6.6° ( =400 Hz). (c) Estimated results for tissue sample when fiber angle θ = 41.4° (f =500 Hz). (d-e) Estimated results for gels (two different samples, f=400 Hz).

2.5.3. Region of interest

To analyze data near the excitation site (the ultrasound focus) and minimize boundary effects, data within a hemispherical region of interest are included in the neural network fitting. The hemispherical ROI is defined by a “mask” with specific outer radius and inner radius. The effect of the choice of outer mask radius was investigated by using different values, 12mm, 10mm and 8mm; the inner radius was fixed at 1.5mm to reduce the influence of strains in the focal region.

2.5.4. Avoiding overfitting

Overfitting happens when the training set starts to memorize the noise or random fluctuations as concepts, providing a negative impact on the performance of the model on the test set; therefore, it is not able to generalize well to new data. Remedies include adding noise to the input features (Ying, 2019), simplifying the neural network, and early stopping (Jabber and Khan, 2015). In this study, we examine the effects of added noise and a non-zero regularization parameter using Bayesian regularization (MacKay, 1992). Bayesian regularization minimizes a linear combination of squared errors and weights. It also modifies the linear combination so that at the end of training the resulting network has good generalization qualities.

2.5.5. Robustness verification

Material parameter estimates were obtained with different model and fitting parameters (e.g., isotropic loss factor, mask radius, regularization parameter, noise level, and hidden layer size). To verify the robustness of the neural network approach, the influence of various modeling and fitting choices was assessed. These choices included

1. The isotropic loss factor in simulation (amount of damping). The loss factor in simulation describes shear-wave attenuation, the larger the loss factor, the faster shear waves attenuate. The loss factor was not known exactly, but was estimated from prior work. The choice of isotropic loss factor in the simulation affects the attenuation of the shear waves, and might influence estimates of elastic parameters. The true isotropic loss factor was not known, although it has been estimated previously to be near η_s = 0.4 (Riek et al., 2011; Schmidt et al., 2016).

2. ROI size, parameterized by mask radius. A larger ROI includes more waves, but would also increase effects of boundary interactions and noise.

3. Noise added to training data. Noise is added to training data for verification of robustness of the system. As discussed above, adding noise will help neural network avoid overfitting.

4. Regularization parameter. Regularization parameters modulate additional terms in the objective function of the neural network training that penalize parameter magnitudes to prevent overfitting.

5. Number of nodes in the single hidden layer. Increasing nodes in the hidden layer will increase the network’s ability to reproduce training data, but can lead to overfitting and requires more computational effort.

Parameter estimates were obtained using data from 400Hz shear waves in eight muscle tissue samples and at 500Hz from two muscle tissue samples. Parameter estimates in two isotropic gelatin/glycerol gel samples (same composition, one tested at 300 Hz and one at 400 Hz) were used to evaluate the neural network approach because shear anisotropy ϕ and tensile anisotropy ζ in these gels are theoretically zero.

Estimates of material parameters (μ, ϕ, ζ) were obtained by neural network fitting for the ranges of model and fitting choices in Table 2. In experimental samples of gelatin/glycerol, the estimation procedure was performed with various values of assumed (pseudo) “fiber direction.” Since there is actually no fiber, and the gel is isotropic, the influence of experimental error and training mismatch on strain and shear wave speed estimation are illustrated by the range of non-zero estimates of anisotropy parameters ϕ and ζ.

Table 2.

Default values of various conditions with bounds.

Parameter	min	default	max
ROI radius, Ro (mm)	8	10	12
Loss factor, ηs	0.3	0.4	0.5
Noise level, ψ	0	5%	10%
Hidden layer size, H	10	20	30
Regularization parameter, λR	0	0.2	0.5

3. Results

3.1. Isotropic gel sample

Fig. 12 shows the effects of assumed fiber deviation angles on estimated parameters in the two experimental gel samples, for the case in which the (pseudo) fiber direction lies in the ZX-plane and deviates from the XY-plane by various angles (Fig. 12a), and the case in which the (pseudo) fiber direction deviates from the ZX-plane by various angles (Fig. 12b). Shear modulus μ is not sensitive to deviations in assumed fiber angle. Values of the anisotropy parameters ϕ, ζ are relatively small in both gel samples, as long as the assumed fiber angle is near the training conditions (0–30° from the XY plane and 0° from XZ plane).

Fig. 12.

Effect of assumed (pseudo) fiber angles on estimated material parameters (μ, ϕ, ζ) in two gelatin samples (blue squares: sample 1, 300 Hz; orange circles: sample 2, 400 Hz). (a) Effect of assumed fiber angle from XY-plane on the estimation of (μ, ϕ, ζ). (b) Effect of assumed fiber angle from the ZX-plane on the estimation of (μ, ϕ, ζ). Neural networks were trained using data from simulations of shear waves at 400 Hz and 500 Hz with fibers at angles from 0–30° from the XY-plane and 0°−30° from the XZ-plane.

3.2. Muscle tissue samples

3.2.1. Loss factor

Parameter estimates are shown in Fig. 13 for different loss factors (green: η_s = 0.5; red: η_s = 0.4, blue: η_s = 0.3). The mean value of the estimated shear modulus μ increases slightly with the loss factor; estimates of shear anisotropy ϕ and tensile anisotropy ζ decrease. The apparent shear modulus μ at 500Hz is higher than at 400Hz (as expected for a viscoelastic material) (Lopez et al., 2008), whereas estimates of shear and tensile anisotropy are lower.

Fig. 13.

Effect of assumed loss factor in simulations on estimated material parameters (μ, ϕ, ζ) in gelatin/glycerol and muscle tissue. Box-and-whisker plots show median, 25^th and 75^th percentile and min/max values. Columns in each figure represent different excitation frequencies and materials: gel 300 Hz (n=1), gel 400 Hz (n=1), muscle 400 Hz (n=8) and muscle 500 Hz (n=2). Box colors indicate different values of the assumed loss factors (green: η_s = 0.5; red: η_s = 0.4, blue: η_s = 0.3). In this and subsequent figures, the (pseudo) fiber angle in gelatin/glycerol material is assumed to be 0° from both the XY- and ZX-planes.

3.2.2. ROI outer radius

Fig. 14 shows estimated parameters for different ROI outer radius (green: R_o = 12mm; red: R_o = 10 mm, blue: R_o = 8 mm). The size of the ROI affects the estimates, by influencing what data are included. Larger radii include more data, but decrease the average signal-to-noise ratio, and include more boundary effects. Estimates of shear modulus μ decrease with the mask radius, as do estimates of shear anisotropy and tensile anisotropy ϕ, ζ. The effect of frequency is consistent in these estimates.

Fig. 14.

Effect of ROI outer radius size on estimated material parameters (μ, ϕ, ζ) in gelatin/glycerol and muscle tissue. Box-and-whisker plots show median, 25^th and 75^th percentile and min/max values. Columns in each figure represent different excitation frequencies and materials: gel 300 Hz (n=1), gel 400 Hz (n=1), muscle 400 Hz (n=8) and muscle 500 Hz (n=2). Box colors indicate different values of the ROI outer radius (green: R_o = 12 mm; red: R_o = 10 mm, blue: R_o = 8 mm).

3.2.3. Number of nodes in hidden layer

Estimation results are shown in Fig. 15 for different hidden layer size (green: H = 30; red: H = 20, blue: H = 10). The influence of hidden layer size is interesting because the shear modulus μreaches minimum value at H=20 when ϕ and ζ reach maximum. Hidden layer size decides the node number, which will increase the speed at which the neural network finds an answer.

Fig. 15.

Effect of hidden layer size on estimated material parameters (μ, ϕ, ζ) in gelatin/glycerol and muscle tissue. Box-and-whisker plots show median, 25^th and 75^th percentile and min/max values. Columns in each figure represent different excitation frequencies and materials: gel 300 Hz (n=1), gel 400 Hz (n=1), muscle 400 Hz (n=8) and muscle 500 Hz (n=2). Box colors indicate different values of the hidden layer size (green: H = 30; red: H = 20, blue: H = 10).

3.2.4. Level of added noise

Estimation results are shown in Fig. 16 for different added noise (green: ψ = 10%; red: ψ = 5%, blue: ψ = 0). The added noise does not appear to have a large effect on parameter estimates.

Fig. 16.

Effect of noise level added to training data on estimated material parameters (μ, ϕ, ζ) in gelatin/glycerol and muscle tissue. Box-and-whisker plots show median, 25^th and 75^th percentile and min/max values. Columns in each figure represent different excitation frequencies and materials: gel 300 Hz (n=1), gel 400 Hz (n=1), muscle 400 Hz (n=8) and muscle 500 Hz (n=2). Box colors indicate different values of the added noise (green: ψ = 10%; red: ψ = 5%, blue: ψ = 0).

3.2.5. Regularization parameter

Estimation results are shown in Fig. 17 for different regularization parameter (green: λ_R = 0.5; red: λ_R = 0.2, blue: λ_R = 0). Shear modulus μ decreases with increasing regularization parameter, while ϕ and ζ increase. With higher regularization parameter, the neural network will tend to stop earlier to avoid over-fitting; the higher the regularization parameter, the more iterations the neural network runs.

Fig. 17.

Effect of regularization parameter on estimated material parameters (μ, ϕ, ζ) in gel and muscle tissue. Box-and-whisker plots show median, 25^th and 75^th percentile and min/max values. Columns in each figure represent different excitation frequencies and materials: gel 300 Hz (n=1), gel 400 Hz (n=1), muscle 400 Hz (n=8) and muscle 500 Hz (n=2). Box colors indicate different values of the regularization parameter (green: λ_R = 0.5; red: λ_R = 0.2, blue: λ_R = 0).

4. Discussion

In nearly-incompressible, transversely isotropic materials, three key parameters, baseline shear modulus (μ), shear anisotropy (ϕ), and tensile anisotropy (ζ), were estimated from images of ultrasound-induced shear waves using artificial neural network fitting. Neural networks were trained using data from simulations as input and known model parameters as outputs. Input features included fiber direction, strain ratios, and wave speed (cos θ, sinθ, E_bb/E_aa, E_bd/E_ac, E_ee/E_aa, E_ef/E_ac, C_s). In anisotropic materials, the strains in fiber and non-fiber directions caused by axis-symmetric loading are expected to be unequal, with greater strain in the non-fiber direction if the fibers provide stiffness. Estimates of parameters in fibrous muscle tissue (chicken breast) differed from those in gelatin/glycerol gel. This expectation was confirmed in simulations. Estimates of μ were generally the most consistent; this may be because μ can be estimated accurately from just the minimum shear-wave speed.

Modeling and neural network fitting choices can also affect estimates of material parameters. For example, when the assumed loss factor is increased, estimates of shear modulus μ increase slightly, while estimates of shear and tensile anisotropy ϕ, ζ decrease. If the size of the ROI is increased, estimates of shear modulus μ and anisotropy parameters (ϕ, ζ) all decrease. However, the effects of modeling and fitting choices are relatively small compared to the parameters of muscle tissue, providing evidence of robustness of the ANN approach. In addition, parameters estimated for gelatin/glycerol at 300 Hz, using networks trained with data at 400 Hz and 500 Hz, were reasonably accurate (as evidenced by small anisotropy parameters in this isotropic material), even though the experimental frequency was out of the range of the training data.

Excitation frequencies also affect the material mechanical response. Higher shear modulus μ, is observed at 500 Hz compared to 400 Hz; this increase in baseline shear modulus, which is typical of viscoelastic biological materials (Atay et al., 2008; Lopez et al., 2008), is accompanied by lower estimates of shear and tensile anisotropy ϕ, ζ.

The current, neural network-derived, estimates of tissue parameters in chicken breast ex vivo, using the default choices of model and fitting parameters, are μ = 6.56±0.17 kPa, ϕ = 1.09±0.05, and ζ = 1.19±0.08 at 400Hz. These are reasonably consistent with estimates from Guertler et al. (2020), (μ = 8.95±1.23 kPa, ϕ = 0.67±0.41, ζ = 1.37±0.68) for the same tissue samples obtained by using the phase gradient to calculate wave speed in different directions. Notably the neural-network-derived estimates have a lower standard deviation. Dynamic shear testing (DST) of chicken breast samples also provided estimates of μ = 6.19±1.71 kPa, ϕ = 0.84±0.30 at 25–45 Hz (Guertler et al., 2020) (DST is unable to provide information on tensile modulus). Neither previous MR-HUM nor DST is ideal; the phase gradient estimates of wave speed are susceptible to errors caused by numerical differentiation and phase wrapping, and DST is susceptible to variations in contact conditions and test order.

Previously Gennisson et al. (2003) found a ratio of 2.8 between maximum and minimum shear wave speed in beef muscle (biceps femoris tendinosus), corresponding to a value of ϕ = 1.8. In human subjects, Green et al. (2013) estimated shear moduli of medial gastrocnemius, soleus, and tibialis anteriori parallel and perpendicular to fibers using MRE at 60Hz; they found average shear anisotropy ϕ = 0.30 (gastrocnemius), ϕ = 0.28 (soleus), and ϕ = 0.18 (tibialis). Liu et al. (2016) performed an optical coherence tomography elastography measurement in ex vivo chicken breast and provided estimates of baseline shear modulus of 4.7 ± 1.1 kPa (by elastography) and 5.1 ± 1.1 kPa (by quasi-static mechanical testing) and tensile anisotropy ζ = 2.37±0.84. While there are inevitable differences between tissue from human, bovine, and chicken muscle, and between individual muscles and samples, the general agreement between results from the current approach and prior methods is encouraging.

Limitations of this study include the fact that only small-amplitude wave motion is obtained in MR-HUM, as in all MRE experiments, and thus the parameters only describe linear elastic or viscoelastic behavior. The nearly-incompressible, transversely isotropic material model is the only model considered. More complex, nonlinear, hyperelastic or hyper-viscoelastic models, such as the Holzapfel-Gasser-Ogden (HGO) model (Holzapfel, 2000) would be appropriate to characterize behavior of fibrous materials in the large-deformation regime. Wave motion would need to be superimposed on large deformations to probe nonlinear behavior (Hou et al., 2020).

The application of deep learning to this problem is a topic of future interest. Instead of using features of the displacement and strain fields as input, parameters can be estimated directly from the raw displacement fields. Deep learning typically requires more data and is more computationally intensive, but may ultimately be more powerful for non-invasive characterization of in vivo tissue.

5. Conclusion

This study demonstrates that estimates of anisotropic material parameters, as well as confidence intervals for these estimates, can be obtained from experimental MRE and DTI data using ensembles of neural networks trained on simulations. Estimates are robust to choices and assumptions involved in simulating the system or fitting the neural network. Future studies should explore the use of neural network-based methods to estimate anisotropic mechanical properties noninvasively from MR elastrography of tissue in vivo.

Highlights.

• Shear wave speeds and strains depend on fiber directions in anisotropic tissue.

• Artificial neural networks were used to estimate anisotropic material parameters.

• Neural nets were trained on simulations to estimate parameters from experiments.

• Anisotropic parameter estimates are robust to choices in modeling and fitting.