Computational and Mathematical Methods in Medicine

Volume 2016 (2016), Article ID 6143126, 17 pages

http://dx.doi.org/10.1155/2016/6143126

## Lattice Boltzmann Model of 3D Multiphase Flow in Artery Bifurcation Aneurysm Problem

^{1}School of Mechanical Engineering, Universiti Sains Malaysia, Engineering Campus, 14300 Nibong Tebal, Penang, Malaysia^{2}School of Aerospace Engineering, Universiti Sains Malaysia, Engineering Campus, 14300 Nibong Tebal, Penang, Malaysia

Received 6 January 2016; Revised 5 March 2016; Accepted 31 March 2016

Academic Editor: Giuseppe Pontrelli

Copyright © 2016 Aizat Abas 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

This paper simulates and predicts the laminar flow inside the 3D aneurysm geometry, since the hemodynamic situation in the blood vessels is difficult to determine and visualize using standard imaging techniques, for example, magnetic resonance imaging (MRI). Three different types of Lattice Boltzmann (LB) models are computed, namely, single relaxation time (SRT), multiple relaxation time (MRT), and regularized BGK models. The results obtained using these different versions of the LB-based code will then be validated with ANSYS FLUENT, a commercially available finite volume- (FV-) based CFD solver. The simulated flow profiles that include velocity, pressure, and wall shear stress (WSS) are then compared between the two solvers. The predicted outcomes show that all the LB models are comparable and in good agreement with the FVM solver for complex blood flow simulation. The findings also show minor differences in their WSS profiles. The performance of the parallel implementation for each solver is also included and discussed in this paper. In terms of parallelization, it was shown that LBM-based code performed better in terms of the computation time required.

#### 1. Introduction

Aneurysm is a condition in which a blood vessel wall is pathologically dilated. It normally happens at the vicinity of the circle of Willis, the main cerebral arteries, and has also been found to develop in the iliac and intracranial arteries [1]. Aneurysm could lead to artery rupture resulting in stroke or even death. The arterial wall may rupture when the wall itself is not strong enough to withstand the stresses exerted on it during the blood flow. As the treatment methods for aneurysm carry a high degree of risk [2], understanding the vessel’s internal fluid flow is important. It can be utilized to improve aneurysm diagnosis and treatment methods. Hashimoto et al. [3] pointed out that the aneurysm growth can be associated with the intra-aneurysmal hemodynamics. Hence, good understanding and knowledge of hemodynamic parameters are crucial. Various computational fluid dynamics (CFD) studies [4–9] have been carried out to investigate the relationship between hemodynamic factors and the risk of an artery wall’s rupture. These factors were the impingement force, the WSS, the pressure, and the blood velocity.

For the past two decades, the simplicity and the effectiveness of the Lattice Boltzmann method (LBM) in solving flow related problem has created awareness in the CFD community. LBM-based computations have been applied to simulate different complex and irregular geometries presented in flow problems. Some of these experiments demonstrated that LBM performed better than the classical CFD tools in certain applications [10–12]. The LB simulation is based on the Boltzmann theory instead of the expensive and time-consuming Navier-Stokes equation. LBM did have its limitations, despite the increasing popularity in solving flow problems. For example, though LBM is capable of solving high-Mach number flow in aerodynamics, it still lacks a consistent thermohydrodynamic scheme. However, some efforts were concerted to sort out these limitations [13].

The past few years have seen many researchers resorting to LBM in an attempt biomedical problem. In the work by Liu, biviscosity constitutive relations and control dynamics based on a D2Q9 lattice are used to solve a stenosis blood flow problem [14]. Sun and Munn predicted the blood flow by developing an LBM model that considers the blood as a suspension of particles in the plasma [15]. In addition, Leitner et al. introduced new LBM boundary conditions for the cardiovascular domain using a D2Q9 lattice model [16]. The application of LBM to three-dimensional (3D) problems is rather limited and noticeable applications can be found in the work by [17], featuring a 3D simulation of intracranial aneurysm geometry and in the work by [18] on wall orientation and shear stress in the artery.

In general, the application of LBM is not restricted to the fluid flow simulation. It has been found to be an efficient solver in sound absorption applications [19]. Another interesting use is in the simulation of floating rigid bodies on a free surface [20]. The LBM and the FVM solver can be combined together to solve the rigid body problem. Following this development, various other simulations were conducted utilizing this coupling method [4, 21–23].

Vast amount of performance analysis and optimization techniques has been found conducted to utilize the computation efficiency of LBM software. Kopta et al. [24] and Tian et al. [25] try to evaluate the parallel performance of LBM in terms of its scalability and efficiency. In addition, a number of optimization steps can be constructed on the LB code to improve the simulation efficiency [26]. Normally, OpenMPI and OpenMP are the two principle programming models often used in high-performance computing (HPC). The parallelization usually includes a pure message-passing-interface (MPI) algorithm that can be used along with LBM solver. A study has been found that combines both of these models in an effort to reduce the computation resources [26], especially involving complex geometries. Parallel performance can be further improved by replacing the CPU with Graphics Processing Units (GPU) algorithms during numerical calculation. GPU was originally developed for computer games and will provide better computation power for scientific applications compared to the conventional CPU unit [27].

In the current work, the prediction and the investigation of the complex, unsteady blood flow inside aneurysm geometry is explored by using LBM-based software, Palabos. To optimize the results obtained, three different collision models, namely, the incompressible BGK model, the regularized BGK model, and the multiple relaxation time (MRT) model, are compared to investigate the efficacy of each method in modelling blood flow problem [28]. The accuracy of these different LB-based scheme in solving aneurysm blow flow simulations will be examined by comparing the hemodynamic parameters with the conventional finite volume method based solver. To the best of the authors’ knowledge, there is no study conducted to compare the capability of both FVM and LBM models typically in the simulation blood flow problems in terms of the solution accuracy and parallel performance. The current study attempts to compare the distribution of mesh for both LBM and FVM models in the detection of high wall shear stress (WSS) region in the arterial wall. The parallel speedup with increasing number of cores is also compared between both LBM and FVM solvers in terms of its parallel efficacy.

#### 2. Lattice Boltzmann Models

The results shown in this section are formulated using D3Q19 lattice model. The LBM equation can be summarized in in which the right-hand side represents the streaming step. The left-hand term denotes the collision term which can be represented using the well-known Bhatnagar-Gross-Krook (BGK) model as given in and denote the relaxation frequency and time. represents the equilibrium function that relates to the lattice arrangement.

The equilibrium function, , can be described as in which represents weighting function across different lattice links. For the case of D3Q19 lattice model as depicted in Figure 1, the weighting functions can be described in Table 1.