Computational and Mathematical Methods in Medicine

Volume 2016 (2016), Article ID 9343017, 12 pages

http://dx.doi.org/10.1155/2016/9343017

## Rescaled Local Interaction Simulation Approach for Shear Wave Propagation Modelling in Magnetic Resonance Elastography

Department of Robotics and Mechatronics, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Krakow, Poland

Received 13 November 2015; Revised 16 December 2015; Accepted 17 December 2015

Academic Editor: Po-Hsiang Tsui

Copyright © 2016 Z. Hashemiyan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1, 2]. Elastography is used extensively in diagnostic applications (e.g., liver fibrosis or breast tumors detection [3–9]) due to flexibility and noninvasiveness. Since abnormal tissues are often stiffer than the normal ones, medical diagnosis can be achieved. Although the method was developed in the late 1980s [10–12] the major breakthrough came in the mid 1990s when a dynamic approach to elastography was proposed [13]. A Motion-Encoding Gradient (MEG) was introduced to a conventional MRI system leading to Magnetic Resonance Elastography (MRE) [13–17].

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 [18]. A similar approach used for irregular domains of elastically heterogeneous tissues is not possible in practice. Consequently, numerical simulations are used to ease this task. Modelling is used in MRE applications in order to create forward models that capture complex mechanisms of wave propagation in soft tissues. Previous studies in this field include various finite difference (FD) [17–19] and Finite Element methods (FE) [15, 20–24]. FE modelling has been used in previous studies for visualization of ultrasonic wave propagation [25–31], elasticity reconstruction [21, 32], and shear wave propagation analysis in gelatin phantoms [33–39].

The paper aims to develop a full three-dimensional (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) [40]. In contrast to the previous work, current investigation focuses on the guided wave propagation with rescaling procedure. The major novelty of the presented work relates to the application of the Local Interaction Simulation Approach (LISA) for guided wave propagation and a rescaling procedure for the LISA is proposed for shear wave propagation modelling. This major novelty is considered to tackle numerical problems.

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.

#### 2. Theoretical Background

Elastic wave propagation in an isotropic linear medium is governed by the momentum balance given aswhere is the divergence of stress tensor, is an external volume force, and represents particle acceleration vector. The constitutive equation that relates stresses to strains in a linear elastic solid is given aswhere is the Kronecker delta, represents material dilatation given by , and represent the Lamé constants for the material. The strain () is defined through the strain tensor using the following relationshipwhere represents particle displacement components. Combining (1)–(3) the equation of equilibrium, that is, [41],governs wave propagation in an infinite elastic space and for practical problems must be amended by appropriate boundary and initial conditions describing the problem. Boundary conditions increase the complexity of the problem since they give rise to the so-called guided wave propagation problem, where global wave propagation patterns, that is, modes, travel at different, and possibly frequency-dependent, speeds, as explained in [41]. It is well known that the solution to (4) can be found only for simple canonical problems. Numerical simulations are used for more complex scenarios.

#### 3. Numerical Models

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.

##### 3.1. Finite Element Model

The FE model used in the current investigations was developed using the* Marc Mentat 2013* software package. Following the work presented in [36], a 3D cylindrical container, with a diameter of 200 mm and thickness of 20 mm, was modelled using gel phantom material properties. The bottom of the cylinder was fixed in the direction (see Figure 1). Altogether 36 000 elements of 2 × 2 mm radial and axial element size and 200 element along the circle were used. The phantom was modelled as a homogenous isotropic elastic solid with Poisson’s ratio . Harmonic sinusoidal motion of 150 Hz was applied to the center of the top cylinder surface as an excitation. Three different elastic moduli () were investigated, that is, 30, 60, and 120 kPa, to study the relationship between shear wavelengths and shear moduli. Similarly, numerical simulations were performed using three different material density () values, , , and , for each Young’s modulus. Material damping was assumed to be zero.