Zener-type PDE model
To obtain the vibration properties of a biological tissue, a linear viscoelastic PDE was used to describe the displacement field of a wave in a tissue (Christensen and Richard 1982). The FEA model based on the Zener model was validated by comparing MRE data to FEA simulated data (Sect. 3.1). Although several viscoelastic PDEs exist, this study used the Zener-type model (three-element Maxwell type model) because its simulated wave images are similar to those obtained from MRE measured data (Jiang and Nakamura 2011). Figure 1 schematically depicts the 1D version of this model, where μ
0 and μ
1 are the spring constants, η
1 is the dashpot viscosity, and τ
1 = η
1/μ
1 is the relaxation time. The stress–strain relation of this model is expressed by:
$$\sigma = \int_{0}^{t} {2G} (t - \tau )\frac{{{\text{d}}e}}{{{\text{d}}\tau }}{\text{d}}\tau + I\int_{0}^{t} K (t - \tau )\frac{{{\text{d}}D}}{{{\text{d}}\tau }}{\text{d}}\tau ,$$
(1)
where σ is the Cauchy stress, t is the past time, e is the deviatoric strain, D is the volumetric strain, and I is the identity matrix; G(t) and K(t) are the shear and bulk-relaxation moduli, which are described by:
$$G(t) = \mu_{0} + \mu_{1} \exp ( - t/\tau_{1} ),$$
(2)
$$K(t) = \frac{2}{3}\frac{1 + \nu }{1 - 2\nu }\left[ {\mu_{0} + \mu_{1} \exp ( - t/\tau_{1} )} \right],$$
(3)
where ν is Poisson’s ratio. For the time harmonic vibration, Eq. (1) becomes:
$$\sigma = 2(G^{\prime} + iG^{\prime\prime})e\exp (i(\omega t + \delta )) + (K^{\prime} + iK^{\prime\prime})D\exp (i(\omega t + \delta )),$$
(4)
where G′ and G″ are the storage and loss moduli, respectively. K′ and K″ are the storage and loss bulk moduli, respectively, and ω and δ are the frequency and phase angle, respectively. The storage modulus and loss modulus are defined by:
$$G^{\prime} = \mu_{0} + \frac{{\mu_{1} (\omega \eta_{1} )^{2} }}{{\mu_{1}^{2} + (\omega \eta_{1} )^{2} }}$$
(5)
$$G^{\prime\prime} = \frac{{\mu_{1}^{2} (\omega \eta {}_{1})}}{{\mu_{1}^{2} + (\omega \eta_{1} )^{2} }}$$
(6)
Modified Stokes equation and modified integral method
Poisson’s ratio ν is close to ½ because the tissue is nearly incompressible, which means that K′ ≫ G′. Asymptotic analysis with respect to the large scaling parameter K′/G′ indicates that the displacement field u can be approximated as a solution for the following boundary value problem (7) or the modified Stokes equation (Jiang and Nakamura 2011; Jiang et al. 2011; Ammari et al. 2007), which is expressed as:
$$\left\{ {\begin{array}{*{20}l} {\nabla \cdot [2(G^{\prime} + iG^{\prime\prime})\varepsilon (u)] - \nabla p + \rho \omega^{2} u = 0,} \hfill \\ {\nabla \cdot u = 0,} \hfill \\ {u = f,} \hfill \\ {\partial_{\nu } u: = [2(G^{\prime} + iG^{\prime\prime})\varepsilon (u) - p]\nu = 0,} \hfill \\ \end{array} } \right.$$
(7)
where ρ is the tissue density, which can be taken as that of water (Fung 1993). p denotes the pressure from the longitudinal wave, and ε is the linear strain tensor defined by:
$$\varepsilon_{ij} (u) = \frac{1}{2}\left( {\frac{{\partial u_{i} }}{{\partial x_{j} }} + \frac{{\partial u_{j} }}{{\partial x_{i} }}} \right) .$$
(8)
Furthermore, the first two parts of Eq. (7) are considered in a domain that could be a human body or a phantom Ω. In contrast, the last two parts of Eq. (7) give the mixed type boundary condition with the traction zero boundary condition on part of the boundary of Ω with outer unit normal n and the displacement boundary condition with a given displacement f on the rest of the boundary where the vibration is given. It should be noted that the displacement field u is a complex vector.
If the storage and loss moduli are homogeneous, then applying the curl operator to Eq. (7) removes pressure p in Eq. (7). Thus, w = ∇ × u can be given as:
$$(G^{\prime} + iG^{\prime\prime})\Delta w + \rho \omega^{2} w = 0,$$
(9)
where Δ denotes the Laplacian. Because applying the curl operator to u may amplify the noise, we need to reduce the effect of noise included in u (Farahani and Kowsary 2010; Murio 1987). Herein the mollification method (Jiang and Nakamura 2011; Jiang et al. 2011; Ammari et al. 2007) was used for reducing the effect of noise.
Taking the complex conjugate of Eq. (9) gives:
$$(G^{\prime} - iG^{\prime\prime})\overline{\Delta w} + \rho \omega^{2} \bar{w} = 0$$
(10)
Additionally, integrating the inner product with w over a test domain R yields:
$$G^{\prime} - iG^{\prime\prime} = - \rho \omega^{2} \left( {\frac{{\int_{R} {\left| w \right|^{2} dx} }}{{\int_{R} {w\overline{ \cdot \Delta w} dx} }}} \right),$$
(11)
where ‘\(\cdot\)’ denotes the inner product. Due to the unique continuation property of the solution, the denominator cannot vanish (Jiang and Nakamura 2011). Because the usual algebraic method does not integrate over R, \(\overline{\Delta w}\) may vanish. This is an advantage of using the formula (11).
Equation (11), which can compute G′ − iG″ in R, is called the modified integral method (Jiang and Nakamura 2011). In particular,
$$G^{\prime} = - \rho \omega^{2} \text{Re} \left( {\frac{{\int_{R} {\left| w \right|^{2} dx} }}{{\int_{R} {w \cdot \overline{\Delta w} dx} }}} \right),$$
(12)
where Re() denotes the real part in the bracket.
Applying the Zener model (Jiang and Nakamura 2011) the following three important aspects were checked carefully. First, R must be at least half of the wavelength. Second, if the heterogeneity of the tissue is sufficiently smooth and the wavelength is sufficiently small, the tissue in a small test domain R can be assumed to be homogenous. Finally, even when there is discontinuity in G′ − iG″, the modified integral equation method still produces satisfactory results.
MRE experiment with micro MRI
To validate the numerical simulation, the simulated wave image was compared with the MRE measured wave image obtained by 0.3 T micro MRI (the Compact MRI series, MR Technology, Inc., Tsukuba, Japan). Figure 2 shows the micro MRI. The sample for the MRE measurement was a block agarose gel phantom (100 × 70 × 55 mm) placed in the micro MRI. A longitudinal wave was generated using an electro dynamic generator (C-5010 D-master, Asahi Factory Corp., Tokyo, Japan) as shown in Fig. 2. The wave propagated to the sample through a bar comprised of glass fiber reinforced plastics (GFRP). The diameter of the bar head was 8 mm. The motion of the nuclear spin induced by the local movement of a tissue phantom in a gradient magnetic field induces a phase shift θ at position x, which is given by:
$$\theta (x) = \gamma \int_{{t_{0} }}^{{t_{0} + N/F}} u (t,x) \cdot G(t){\text{d}}t,$$
(13)
where G is the magnetic field gradient, F is the excitation frequency, u is displacement fields, γ is the gyromagnetic ratio of characteristic of the nuclear isochromat, and N is the number of cycles. Equation (13) can compute the displacement fields u from the phase shift θ.
FEA modeling
Figure 3 shows the FEA model and its boundary conditions. A 3D FEA model of the tissue phantom (100 × 70 × 55 mm) was created and analyzed to obtain its complex displacement fields using ‘harmonic analysis’ in ANSYS (Version 14.0). The steady-state response of the model to sinusoidal excitation was calculated by ‘harmonic analysis’. The model had eight node elements uniformly distributed, and each element measured 1.25 × 1.25 × 1.25 mm. To obtain appropriate wave fields, ten elements per a wavelength are typically necessary. The created FEA model satisfied this condition. All degrees of freedom of the nodes on the bottom surface (an x–y slice) were fixed (Fig. 3). The nodes included in the center circular 8-mm-diameter region on the y–z plane were vertically excited in the x direction at frequency f (=62.5, 125, and 250) [Hz] and 0.5 mm amplitude. The degrees of freedom of the other nodes were not restricted. The storage and loss moduli were computed as their true values of the inverse algorithm using Eqs. (5) and (6) when the mass density ρ, Poisson’s ratio ν, spring constants μ
0 = μ
1, and relaxation time τ
1 were 1000 kg/m3, 0.499 (nearly incompressible), 7.5 kPa, and 0.025 s, respectively. The complex displacement fields u in response to the excitation were analyzed with this model and the above conditions.
The original Zener model with the strain–stress relation given by Eq. (1) was used to simulate the displacement field data instead of the modified Stokes model because the Stokes model is an approximation of this Zener model when Poisson’s ratio is very close to 0.5. In this study, this assumption was only used for the inversion. Mollification of the modified integral method was employed for reducing the effect of noise when necessary.
FEA of the heterogeneous model
Next we evaluated the accuracy of the inversion algorithm with respect to the heterogeneous viscoelastic model (Fig. 4). A columnar phantom with a diameter of 10, 15, or 20 mm was embedded in the block model. All columnar phantoms had heights of 55 mm. Eight node elements were used and the element numbers of each columnar phantom were 1056, 2992, and 5456. The viscoelastic parameters of the background gels were the same as those of the homogeneous model. The columnar materials had a spring constant of μ
0 = μ
1 = 15 kPa, while the other parameters were the same as the background parameters. The storage moduli G′ of the background and columnar gels were computed from Eq. (5) as 15 and 30 kPa, respectively.
Validation of the inversion scheme
The inversion method was validated using both homogeneous and heterogeneous models. The viscoelastic moduli were recovered by applying the inversion scheme based on the modified integral method to the displacement field data computed by FEA of the homogeneous model at 62.5, 125, and 250 Hz. Data processing to recover the viscoelastic modulus fields from the FEA results was performed with programs written with MATLAB R2013a (Mathworks). For the inversion, the number of points for the numerical integration in each direction, N
x
, N
y
, and N
z
, was set to 3, and the computations were conducted with the data obtained by FEA (coordinate resolution = 1.25 mm). The numerical integration must have an appropriate number of points for reliable simulations of the MRE inversion scheme. If too few points are used in the numerical integration, a highly accurate storage modulus cannot be recovered. Thus, the parameters must be set appropriately.
To compare the 3D inversion to the commonly used 2D inversion, the 2D inversion method was applied to the computed displacement data on the MRI scan section (z = 32.5 mm) in Fig. 3. For the 2D inversion, the number of points for the integral computation was set to N
x
= N
y
= 3 and N
z
= 1. Additionally, the 3D inversion scheme was applied to the heterogeneous FEA results.
FEA modeling of a human liver
The 3D FEA of a human liver was conducted to simulate the in vivo MRE experiment of a healthy volunteer (male, age 22 years). The subject provided written informed consent, and the study was approved by the institutional ethics committee in the National Institute of Radiological Science. A spin-echo echo-planar imaging (SE-EPI) pulse sequence with a motion-encoding gradient (MEG) was used to visualize the shear wave pattern in the subject. The experiment was performed with a 3.0 T MRI scanner (Signa HDx; GE Healthcare) using the following parameters: field of view (FOV) = 288 × 288 mm, imaging matrix = 64 × 64, the number of slices = 7, slice thickness = 4.5 mm, TR = 448 ms, and TE = 41.7 ms. To evaluate this experiment by FEA, the 3D shape data of a real human liver extracted from an MRI scanner was used to create the FEA model. The FEA model was discretized via a four-node element with ICEM CFD 14.0. ANSYS 14.0 (ANSYS Incorporated 2011) was employed assuming that the liver model was homogeneous. The interface of the passive drive (diameter 40 mm) section was excited in the normal direction at 62.5 Hz (Fig. 5). The storage modulus was set as the mean of the measured elastogram in the region of interest (ROI) (Fig. 10).