Central moments multiple relaxation time LBM for hemodynamic simulations in intracranial aneurysms: An in-vitro validation study using PIV and PC-MRI
Introduction
Intracranial aneurysms (IA) are local malformations of the cerebral vasculature, which occur with an estimated prevalence of approximately 3% in the western population [1]. It is known that aneurysms can grow and develop into a stable state or rupture, if the hemodynamic forces exceed the vascular resistance. Since a reliable and clinically applicable rupture risk assessment procedure remains challenging until now, increasing research effort is being put into developing one. Owing to their non-invasive nature resulting in risk-free (for the patient) assessment of rupture probability and the possibility to access highly resolved velocity fields (in both space and time) computational fluid dynamics (CFD) studies are becoming increasingly popular [2,].
While valuable insights have been gained through such numerical studies [3,4], a number of challenges and gaps remain to be bridged, before predictive and reliable models based on CFD for aneurysm dynamics can be developed [5,6]. To put these challenges into perspective, Berg et al. [7] summarized the individual working steps involved in image-based blood flow simulations and provided corresponding recommendations to avoid simulation inaccuracies. As for any numerical simulation, the choice of the numerical solver, resolution and grid configuration are of the utmost importance to ensure convergence of the obtained solutions. The latter two also being consequences of the choice of the numerical method, the former is a determining factor concerning both convergence and efficiency of the solver.
The performances of the employed solvers can only be assessed through a systematic benchmarking/validation procedure against reliable experimental data. One of the most frequently used sources of experimental reference data is the particle image velocimetry (PIV) relying on laser-based high-speed camera flow measurements [[8], [9], [10], [11], [12]]. PIV measurements allow for more accurate measurements and higher resolutions as compared to other data acquisition tools such as magnetic resonance imaging (MRI) [13] or cerebral angiography [14]. A number of studies have presented qualitative comparisons of velocity fields for patient-specific aneurysms as obtained from CFD and PIV measurements, and reported good qualitative agreement between the two [15,16].
While most of the early publications relied on discrete solvers for the Navier-Stokes-Poisson (NSP) equations, e.g. using finite volume (FV) or finite element (FE) methods, emergent numerical methods such as smoothed particle hydrodynamics (SPH) [17,18] and the lattice Boltzmann method (LBM) [[19], [20], [21], [22], [23], [24], [25], [26]] are increasingly applied to such flows. Contrary to classical NSP solvers they are not strictly incompressible, as they approximate the low Mach flow regime through appropriate isothermal flow manifolds, and as such rely on a system of purely hyperbolic equations, making the algorithm inherently local. The absence of the elliptic component of the NSP equations allows for dramatic computation cost reduction while the parabolic nature of the partial differential equation (PDE) describing pressure evolution makes the formulation suitable for unsteady simulations – contrary to other weakly compressible formulations such as the artificial compressibility method (ACM). Furthermore, the compressible nature of the formulation (involving a thermodynamic pressure) along with kinetic heuristic boundary closures (such as the bounce-back rule) provide for an efficient and consistent implementation of wall boundaries [[27], [28], [29]]. The drastic decrease in computation time makes the method more viable compared to approaches based on the NSP equations. Based on these observations, a number of dedicated LBM-based solvers for blood flow simulations have been developed over the past decade, e.g. Refs. [20,30,31]. For example a research prototype was developed by the Siemens Healthineers AG (Erlangen, Germany), which is increasingly used to address clinical questions [[32], [33], [34]]. However, it must be noted that the LBM in its simplest form, i.e. with a single relaxation time (SRT) collision operator and a second-order discrete equilibrium distribution function (EDF) has a rather limited stability domain. The SRT formulation also leads to a number of other well-documented numerical artifacts including, but not limited to, the (non-dimensional) viscosity-dependence of the solid wall position when used with bounce-back-type boundary treatments [35].
The aim of the present work is to assess the performances of a specific class of multiple relaxation time (MRT) operator based on Hermite central moments and a fully expanded EDF. This collision model is shown to alleviate some of the traditional shortcomings of the classical SRT-based solvers, while – through its much wider stability domain – allowing for stable low-resolution simulations. The performances of the central Hermite multiple relaxation time (CHMRT) model are assessed, via our in-house solver ALBORZ [36,37], through a variety of test-cases spanning ideal and patient-specific geometries considering steady and pulsatile flows. The numerical results are compared with high-resolution in-vitro measurements for validation. They are also compared to results from a SRT solver with second-order EDF to further showcase the added value of the CHMRT collision operator. The issue of under-resolved simulations is also considered by systematically conducting simulations at different (lower) resolutions. It is shown that at low resolution (inaccessible to the SRT collision operator), the proposed scheme is still able to capture properly the flow dynamics.
Section snippets
Lattice Boltzmann solver for the Navier-Stokes equations
The LBM is a solver for the Boltzmann equation in the limit of the hydrodynamic regime [38]. The time-evolution of the discrete probability distribution functions is written as [29]:where fα are the discrete distribution functions, cα the corresponding particle velocities, δt the time-step size and Ωα the discretized particle collision operator. The collision operator Ωα is usually approximated via a Batnagar-Gross-Krook (BGK) linear relaxation model defined as [39]:
Results
In the following, qualitative and quantitative comparisons between the hemodynamic simulations based on the LB approach and the in-vitro phantom measurement are presented.
Discussion
With an improved linkage of medical and engineering disciplines as well as increasing computational resources new potentials in interdisciplinary research are revealed. Specifically, the challenging question of rupture risk assessment for intracranial aneurysm remains unanswered until now; however, more and more insights into patient-specific hemodynamics become accessible [63]. On the other hand, insufficient data processing, the absence of reliable boundary conditions and mostly long
Conclusions
This validation study comprising multimodal measurement techniques and various experimental scenarios demonstrates a number of points. Chief among them the necessity for multimodal approaches to validation of results in this area. It was readily shown that experimental measurements using both PIV and PCMRI were prone to non-negligible sources of uncertainty and needed to be considered with care. Furthermore it was proven that the presented LBM solver relying on a CHMRT collision operator is
Declaration of competing interest
None.
Acknowledgements
S.A.H. would like to acknowledge the financial support of the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) in TRR 287 (Project-ID 422037413).
F.H. was supported by the State Scholarship Fund of the China Scholarship Council (grant number 201908080236).
Furthermore, P.B. acknowledges the funding by the German Federal Ministry of Education and Research within the Research Campus STIMULATE (13GW0473A) and the German Research Foundation (BE 6230/2-1).
The authors further thank
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