Properties of soft biological tissues are increasingly used in medical diagnosis to detect various abnormalities, for example, in liver fibrosis or breast tumors. It is well known that mechanical stiffness of human organs can be obtained from organ responses to shear stress waves through Magnetic Resonance Elastography. The Local Interaction Simulation Approach is proposed for effective modelling of shear wave propagation in soft tissues. The results are validated using experimental data from Magnetic Resonance Elastography. These results show the potential of the method for shear wave propagation modelling in soft tissues. The major advantage of the proposed approach is a significant reduction of computational effort.
Mechanical properties of tissues are one of the most significant indicators used for detection of various abnormalities in medical diagnosis. Tumors and other pathologies often exhibit values of elastic moduli that are significantly different from healthy tissues. It is well known that none of the classical medical approaches, such as Computed Tomography (CT), Magnetic Resonance Imaging (MRI), and Ultrasonography (US), are able to detect mechanical properties of tissues that are diagnosed by palpation [
Modelling in elastography relies on direct and inverse problems. The former relates to measurements of tissue responses to applied stresses. The latter is related to estimation of unknown mechanical properties from measured mechanical responses. Both problems are formulated using physical laws, which provide equations that relate biomechanical properties, such as shear modulus, Poisson’s ratio, viscosity, nonlinearity, and poroelasticity, to measured mechanical responses. Accurate models are required to predict displacement responses to different mechanical excitations to solve the inverse problem. For simple setups the equations that describe the direct problem have been solved analytically [
The paper aims to develop a full threedimensional (3D) model of shear wave propagation in a gelatin phantom for MRE applications. Some primary investigation has been performed for the bulk wave propagation model based on the Local Interaction Simulation Approach (LISA) [
Then the LISA model is developed to examine density, shear modulus, and shear wavelength in a gelatin phantom. This study proposes the rescaling solution method in order to avoid numerical problems, especially related to wave amplitude. Numerical simulation results are compared with FE simulation results and MRE experimental measurements from a soft tissue mimicking an agarose gelatin phantom.
Elastic wave propagation in an isotropic linear medium is governed by the momentum balance given as
This section describes numerical models used for shear wave propagation in soft tissues. Firstly FE model was developed as a reference. Then a LISA model is described. The major focus is on a rescaling procedure that is used to avoid numerical discrepancies.
The FE model used in the current investigations was developed using the
Elementary discretization scheme used for wave propagation modelling in the LISA 3D [
The shear wavelength (
The LISA, previously used for wave propagation in complex media [
The following iteration equations are acquired for each displacement component for a general orthotropic case [
Shear wave propagation in the 3D cylinder, already described in the previous section, was modelled using the LISA approach. Numerical simulations involved the same material properties, boundary conditions, and excitation frequencies as in the FE model described in Section
When a numerical technique is used, such as LISA, for wave propagation simulation, various numerical errors must be accounted for. It is well known that for certain material parameter values elastic waves are quickly damped out making results interpretation cumbersome. This is illustrated in Figure
(a) The original shear waveform exhibiting attenuated amplitudes for the density
Soft tissues are highly demanding from computational point of view. From physical perspective it is well known that mainly transversally polarized waves propagate in these structures [
This drawback can be resolved twofold: by reformulating constitutive relationships in order to eliminate the longitudinal wave component, or by manipulating model parameters to push the shear wave closer to the stability limit. In the following work the second approach was employed as this requires no intervention in the solver structure, maintaining the flexibility of the method to model wider class of materials (i.e., solid media and soft tissues).
The longitudinal and shear wave speeds can be expressed
When (
The amplification factor that governs the numerical damping is analysed in Figure
Numerical simulations of the gel phantom using the LISA approach were conducted and analysis was performed to investigate the effect of density scaling parameter on numerical stability for the 3D case. Various scaling values were selected and respective wave velocities calculated. The initial density was assumed as
The effect of scaled density on wave propagation velocities and numerical stability.
Density  Courant number  Longitudinal wave velocity ( 
Scaling factor in Figures 
Comments 




Original density  Green line in Figure 






Wavelength was rescaled in Figure 






Wavelength was rescaled in Figure 






Wavelength was rescaled in Figure 






Wavelength was rescaled in Figure 






Wavelength was rescaled in Figure 
The results in Table
The shear waveform patterns: (a) densityscaled models and (b) after inverse rescaling.
If the only influence on wave amplitude was due to amplification factor, amplitude drop should have been observed in the simulation. Previous work on the effect of the courant number on pulse distortion in 1D finite difference schemes [
It is clear that once wave propagation is simulated with scaled densities, an inverse spatial scaling procedure should be applied to the results to retrieve proper responses. This is accomplished by an inverse scaling procedure employed for the space sensor waveforms. Again (
To illustrate the approach, dispersion curves for respective rescaled models were calculated and used to recover the original waveforms. In the following analysis, the
The dispersion plot for the original density and three
Numerical simulations of shear wave propagation: (a) original waveform (density
Simulated shear waveforms: (a) after rescaling from
In summary, two interesting observations can be made after the analysis performed in this section. Firstly, the wave amplitude increases when density is rescaled towards larger values. Secondly, the inverse rescaling of waveforms allows one to reproduce accurately the original wavelengths.
The MRE data from the experiments reported in [
The experimental setup used for MRE tests: I. phantom; II. applicator of the electromechanical driver; III. electromechanical driver; IV. head coil [
The propagation of elastic waves in the phantom was imaged with an MRE pulse sequence sensitive to motion in the horizontal direction. The shear wavelength was estimated manually by calculating the distances between the adjacent wave peaks. Also, the mean of shear wavelength was measured by averaging the wavelength over the four phase offsets. Subsequently, for isotropic elastic infinite solid, an estimate of the local shear modulus
Numerical simulations of shear wave propagation in the phantom described in Section
Simulated FE, LISA, and experimental MRE shear wave propagation patterns are presented in Figures
Shear wave propagation patterns for the agarose gel phantom: (a) FE mode [
Subsequently, the outofplane displacement component responses were acquired from the simulated (FE and LISA models before and after scaling) and experimental (MRE measurements) data. The results, presented in Figure
Shear wave propagation, comparison of displacement waveforms for the FE model and LISA model before and after scaling and MRE measurements.
Following these investigations, simulated shear wavelengths, calculated for different values of elastic moduli and density, were compared with the relevant analytical values calculated from (
Comparison of shear wavelengths estimated from the FE and LISA models of bulk wave propagation with the relevant analytically estimated wavelengths for different elastic moduli and material densities (0.5, 1 and
Although the results are quite consistent for lower values of elastic moduli, significant discrepancies between numerically (FE and LISA) and analytically (bulk wave propagation solution) estimated results can be observed for higher values of elastic moduli (corresponding to larger wavelengths), particularly for lower densities. These discrepancies are further discussed in the next section.
Equations (
Figure
Example of shear wave propagation displacement profile in the phantom.
These numerically estimated values were compared in Figure
Comparison of shear wavelengths estimated from the semianalytical LISA guided wave model with the relevant analytically estimated wavelengths from bulk wave propagation model for different elastic moduli and material densities (
Since guided wave propagation is inevitably associated with wave interactions with boundaries, the effect of boundary conditions was investigated. Two different boundary conditions, namely, fixed and free ends, were examined. Altogether five different model scenarios, for both the FE and semianalytical LISA models, were analysed: (
Five different model boundary condition, for both the FE and semianalytical LISA models; (
When the 40 mm thick phantom is analysed, the wavelength is larger for free conditions, if compared with the relevant 20 mm thick phantom in the semianalytical model. The value of wavelength is then further increased by the constraint in the
A 3D rescaled LISA model has been proposed for shear wave propagation analysis. Numerical simulations have been performed to analyze the shear wavelength, that is, the primary parameter characterizing shear modulus, in order to examine several factors that influence shear modulus estimation in homogenous phantoms.
The results show that rescaled LISA can be used very efficiently for shear wave propagation modelling in MRE investigations. Good results agreements have been achieved between the LISAbased, FE model, and experimental MRE measurements. The major advantage of the proposed rescaled LISA method is computational efficiency. Significant reductions of computational effort have been achieved when compared with the classical FE modelling approach. The computational time was reduced more than 260 times for the case investigated.
The results also demonstrate that shear wavelength estimated from the presented LISA and FE models are reasonably close to the theoretical calculations, for homogenous elastic cylindrical phantoms investigated, for shorter wavelengths (i.e, for lower Young’s moduli and high densities). In contrast, the solutions based on guided wave propagation are more accurate for longer wavelengths. Also by the rescaling procedure which is presented in this paper, the wave amplitude problems related to numerical errors in soft tissues modelling can be avoided. This analysis can serve as an indicator of interfacial conditions for complex wave propagation in biological tissues.
There is no conflict of interests involved.
The work presented in this paper was supported by the Foundation for Polish Science under the research WELCOME Project no. 20103/2 (Innovative Economy, National Cohesion Programme funded by EU). The fourth author would like to acknowledge research financial support from the Faculty of Mechanical Engineering and Robotics, AGH University of Science and Technology. The authors would also like to thank Professor KaiNan An from Mayo Clinic College of Medicine in Rochester, USA, and Dr. Frank Chen from