This work summarizes my investigations into a pathological human condition in relation to the circulation of the brain. Additionally, I investigate the particle transport properties within the validated flow field of real aneurysm geometry, which is found to exhibit chaotic behavior.
Introduction
This clot usually forms either as a local narrowing of the vessel (stenosis), or it can arise from a ruptured stenosis, or clotted blood from further upstream (embolism). A broader overview of the current theories of the relevant pathogenesis is provided by Sforza et al. (2009).
In the treatment of stenosis, a stent is used from inside the vessel to support the wall and prevent any further local narrowing, while in intracranial aneurysms it is used to cover the neck of the aneurysm (see Figure 2.3c) to create artificial hydrodynamic resistance and prevent communication between the parent artery and aneurysm bulge.
Therefore, all information points in the direction that further improvements of the treatment methods and further development of the risk estimation will incorporate properties not only of the geometry, but also of the developed flow field inside the aneurysm. It highlights the techniques that can accurately model the emerging flow field inside a real aneurysm geometry.
A different method to obtain an adequate description is to understand and describe the microscopic nature of the fluid. Therefore, the macroscopic properties of the fluid are only the emergent features of the underlying small-scale dynamics.
Also, the structure of the vessel wall together with the parameters of the mentioned processes show strong spatial inhomogeneity. The results seem to indicate that although differences occur with respect to the rigid wall case, the main characteristics of the flow patterns (such as the location and size of the inflow jet and the complexity and stability of the intra-aneurysmal flow pattern) were not significantly altered.
More recently, in the work of Józsa and Paál (2014), two abdominal aortic aneurysm geometries were tested with a slightly modified version of the hyperelastic wall model commonly used in the literature (Khanafer et al.,2009;Leung et al. ,2006 ). They found only minor deviations in the qualitative image of the flow field between this hyperelastic model and the rigid wall model, if the rigid wall geometry was recorded at the time of the systolic peak, when the trapped volume of the vessel is greatest. .
Bernaschi et al. (2009) implemented a multiscale fluid flow simulation with the ability to couple LBM with molecular dynamics. Later Bisson et al. (2012) also demonstrated this multiscale implementation to exhibit linear scaling when run on a cluster of 32 GPUs.
The degrees of freedom of the distribution function are then reduced using a set of well-chosen approximations to finally obtain a discretized version of the Boltzmann equation that is applicable over a wide range of Knudsen numbers, including the range of continuity. In my opinion, the lattice Boltzmann method (in contrast to, for example, the Navier-Stokes equations) is constructed rather than derived using rigorous reduction of degrees of freedom at multiple scale levels.
Collisions are modeled as instantaneous events with no change in the total kinetic energy of the participating particles. The phase space of the distribution function can then be reduced to include only velocities.
LBM models and implementations for haemodynamic investigations
The meaning of the right-hand side is expressive: the collisions drive the system towards the equilibrium state with frequency ω. Applying the Chapman-Enskog analysis as in Chapman and Cowling (1992) in the long-wavelength (or low-frequency) limit, the system defined above can be related to the Navier-Stokes equation for.
The left-hand side can be written using the substantial derivative formulation for the single-particle distribution: The LBE formalism shown here is based on the BGK approximation for the collision operator whereω is a single scalar value, so all the moments relax with the same frequency.
Sometimes this can be achieved with the knowledge of the important limiting technological factors and the use of purely mathematical transformations, other times through the intelligent use of assumptions valid for the modeled physical problem. In the following I will outline some such key points which are not strictly part of the fundamental theory, but which are nevertheless necessary for the realization of a complete solution.
However, if the fluid has some velocity, even if it is a steady flow, the higher-order moments are not correctly recovered by using the equilibrium distribution alone (since even in steady state the resulting stress is not necessarily everywhere in the simulation domain). For a transient simulation, this increases the numerical stability since there will be no relevant initial state artifacts present at the beginning of the transient phase.
A commonly used method to define initial values is to calculate the feq equilibrium distribution values from the given macroscopic velocities and densities and use them as initial values for the f-values. If the macroscopic velocities are all 0 and the simulation is a steady state, then this initial value generation is correct.
Another piece of information required is the direction of the inlet velocity, although it is usually taken to be perpendicular to the inlet surface. The prescribed shape of the inlet velocity profile can have significant influence on the resulting flow field.
For example, if for a given problem B = 0.1, then even if we have an infinite number of processing units, the maximum achievable speedup is Smax = 10. Therefore, the LBM algorithm is called memory bandwidth limited, even if the collision step is extremely parallel, step streaming (whose execution time relies on memory bandwidth) contains inherently serial components due to current memory access implementations.
Since the problem fits the special bounds of the GPU hardware (and LBM does have such a formalism), very high execution performance can be achieved. Unlike with the previous implementation, one implication is that the full size of the simulated model must fit in the memory of the master node.
At the inlet, the LBM simulations had a parabolic velocity profile set to be parallel to the initial section of the inlet pipe. The runtime of this stationary phase is also included in all the LBM runtime measurements.
The greatest deviation in the results can be observed at the neck of the aneurysm sac during the high flow velocity phases near the systolic peak (see Fig. 6.5). In the top image: at the neck of the aneurysm (indicated by (b) in Figure 6.1); and in the lower picture: inside the bag (indicated by (a) in Figure 6.1).
I tested the scaling of the CPU and GPU LBM implementations on the low resolution geometry using the MRT approach. In the GPU portion of the image, there is a slight drop in performance scaling when using three Tesla cards.
Regarding the preparation of the calculations, one of the advantages of the lattice Boltzmann method is the lack of the need for the tedious creation of the numerical grid, which significantly simplifies the preprocessing phase of the simulations. I tested four different models of the lattice Boltzmann method: the regularized model, the entropic model, the incompressible BGK model, and the multiple relaxation time model.
Applications
The prediction of the occlusion process using numerical computational tools poses some difficult problems. Similar measurements were performed using different reagents to highlight the concentration of the main components of the formation (Falati et al., 2002; Furie and Furie, 2007).
Its magnitude is proportional to the magnitude of the emerging shear stress acting in the aforementioned plane. I used the forcing term proposed by Guo, Zheng, et al.(2002) to apply this force to the passive scalar field of the platelets.
The continuous red curve shows the results of the steady-state simulation plotted over the experimental results of Woldhuis et al. (1992) (Re=1). The results of the simulation are also compared with the experimental results from the same authors in Fig.
The top image shows the emergent geometry of the thrombus, with the site of vessel injury (yellow box at the base of the thrombus). The yellow dotted line shows the geometry of the thrombus recorded with video microscopy by Nesbitt et al. (2009).
I have developed a simple model for the simulation of the hemostasis process that uses only the properties of the blood flow together with two variables, namely the local concentration of the platelets and of an inhibitor, the adenosine diphosphate out of the more than two dozen factors which are currently known to affect the actual biological cascade process. I simulated a case of induced vessel injury in two dimensions and this model was able to produce qualitatively good results compared to experimental video-microscopy images of a real thrombus formation.
The appearance of filamentary fractal patterns is the result of the stretching and folding action of the mixing fluid generated by the strong sensitivity to the initial conditions, the main characteristics of chaos. In open flows, another important number is the escape velocity κ, which gives the rate of exponential decay of the number of particles still in the observation area at time t: n(t) = n(0)e−κt.
Based on the average flow velocity (approximately 0.3 m/s), tracers starting from the inlet and following the centerline should leave the geometry through one of the outlets in about 1.1 s. As a result of the sensitivity to the initial conditions, a very small time step must be used during the integration process.
Figure 8.2b shows the distribution of residence times for the same particles followed in Figure 8.6. I have chosen several line segments in the plane of the inlet cross-section shown in Figure 8-6.
1992 "Intracranial aneurysms: flow analysis of their origin and progression." American Journal of Neuroradiology, vol.13,1, pp.181-188. 1978 "Clinicopathological study of cerebral aneurysms: origin, rupture, repair and growth", Journal of neurosurgery, vol.48,4, pp.505-514.
