Blood flow in vessels with artificial or pathological geometrical changes P. Tibaut 1, B. Wiesler 1, M. Mayer 2 & R. Wegenkittel 3 1 AVL LIST GmbH, Graz, Austria 2 VRVIs, Vienna, Austria 3 Tiani Medgraph AG, Vienna, Austria Abstract For more than a decade numerical investigations of blood flow inside vessels have been performed. Due to an increase in software user-friendliness and computer performance, these simulations are gaining more and more practical importance. The influence of geometrical changes of the vessels, e.g. by stenotic lesions, aneurysm, or by-pass operations, are subject of these investigations. A number of special features have to be considered to accurately calculate blood flow. Mainly the transient flow behavior due to the heartbeat and the non-newtonian flow effects, especially the dependency of the blood viscosity on the local shear rate, has to be mentioned in that respect. The blood flow in a healthy natural junction, arteria vertebralis dextra and sinistra with ateria basilaris, is compared with the flow in an artificial junction, resulting from a by-pass operation. The areas for preferred formation of stenotic lesions are identified. The shape of the vessels is significantly different in many areas from one human being to another. A detailed geometrical representation of the particular situation is required in such cases to achieve useful results, i.e. the vessel geometry of a particular person has to be known. This is valid for pathological geometrical changes as well. The entire process and the subsequent flow calculation and discussion of results is demonstrated on the basis of a bifurcation area, arteria carotis communis with arteria carotis interna and externa, with a pathological dilatation. Although numerous studies of blood flow have been performed, only a few have considered realistic anatomies under correct flow conditions.
462 Design and Nature II 1 Introduction Vascular diseases such as arteriosclerosis and aneurysms are becoming frequent disorders in the industrialized world due to excess sedentary and rich food. Causing more deaths then cancer, cardiovascular diseases are the leading cause of death in the western world. Aneurysms are caused by weakness of arterial wall resulting in a bulge in a shape of a small balloon. The greatest danger is that the aneurysm could break, particularly if the patient has elevated blood pressure, and produce bleeding within the brain. On histological level, aneurysms are caused by damage to intima cells in the arterial wall. Damage is believed to be caused by shear stress due to blood flow. Shear stress generates heat that breaks down these cells. Once aneurysm is formed, fluctuations in blood flow within aneurysm are of critical importance because they can induce vibration of the aneurysm wall that contributes to progression and eventual rupture. Arterosclerotic disease of carotid artery is a leading cause of stroke. Arterosclerotic plaque in the carotid artery obstructs blood flow to a brain and stimulates the formation of thromoembli that occlude downstream vessels. Unusual shear stress patterns and disturbed flows are related to plaque rupture, plaque erosion, and the formation of thromboembli. The risk of stroke from carotid artery stenosis progressively increases with increasing degree of stenosis, but the degree of stenosis does not completely explain stroke risk; moderate carotid artery stenosis is associated with stroke, trough not quite as strongly as high-grade stroke. Given the well-recognized predisposition to arterosclerotic plaque formation of specific arterial regions with curvature, such as the carotid bifurcation, hemodynamics and arterial geometry may both offer useful information for carotid artery stenosis evaluation. To date there are no reliable methods to determine wall shear stress in the recirculation zones downstream of the stenosis in clinical evaluation [1]. Thus, patient-specific information on hemodynamics generated from computational fluid dynamics (CFD) simulations may complement the evaluation of carotid artery stenosis. Knowledge of the carotid hemodynamics could clarify the relationship between carotid artery stenosis and symptoms, and, ultimately, the risk of stroke. 2 Mathematical model Blood flow can be mathematically modeled using the time-dependent Navier- Stokes equations for an incompressible fluid. Assuming the flow is laminar, the governing equations are written as: u = 0 (1) u (2) ρ + u u = p + 2 u t µ
Design and Nature II 463 where u is the fluid velocity, ρ the density, µ the viscosity and p the pressure. Solution of these partial differential equations requires the geometric boundaries and Dirichlet and/or Neumann conditions to be specified, and the choice of boundary conditions is very important since the predicted velocity field and shear stress can be quite sensitive to the flow conditions imposed at the boundaries. The majority of previous numerical studies of arterial flows assumed fully developed flows at the inlet and outlets of the model concerned. However, fully developed flows rarely occur in human arteries due to their relatively short lengths and their non-planar curvature and branching. Solution of the governing equations is impossible by the traditional analytical techniques for such a complicated system involving 3D irregular geometry, pulsatile flow and non-newtonian viscosity. Numerical techniques are required, hence the need for CFD. By now CFD has matured in its original engineering applications, and the application of CFD has become more and more important in cardiovascular fluid mechanics. Figure 1: Typical time course of the flow velocity during the cardiac cycle. Values on x-axis denote the time (T) normalised to a period of cardiac cycle (TP). A number of special features have to be considered to calculate blood flow accurately. Mainly the transient flow behavior due to the heartbeat and the non- Newtonian flow effects, especially the dependency of the blood viscosity on the local shear rate, has to be mentioned in that respect. For the pulsatile flow the diagram on Fig. 1 has been assumed. Assuming that µ in Eq.2 is a constant is equivalent to adopting a Newtonian model for the blood rheology. However, a better approximation to the fluid behavior of blood is given by the Cross model (ref. Lit.). The difference in axial velocity profiles between Newtonian and non-newtonian fluid is shown on Fig. 2.
464 Design and Nature II Figure 2: Comparison of axial velocity profiles of both: Newtonian and non- Newtonian case. Figure 3: Comparison between measured (a) and calculated (b) functional dependency based on Cross model. Dependency of blood viscosity on the local shear rate is described with equation of Cross, (Eq. (3)) where Rouleaux-effect is taken into account. For given formulation the comparison between calculated and measured functional dependency is shown on Fig. 3. The formulation of Cross describes well measured data.
Design and Nature II 465 η = η η η o ηo η + 1+ Aγ Ns = 0.13 2 m Ns = 0.005 2 m, A = 8 (3) Above expression was implemented in commercial CFD code AVL SWIFT. 3 Workflow A successful analysis incorporates a description of the process from (nonintrusive) computer tomography to the generation of surface data of the vessels in STL format, the smoothing of the surface and the generation of the computational meshes. The workflow consists of the following steps: Computer tomography Automatic generation of CAD surface data Smoothing of the surface using Surface Laplace smoothing algorithm, which does not conserve the volume of original model, but very small number of iterations (e.g. 2) does not change geometry significantly. Automatic generation of the computational meshes. Non-structured grids automate the process of fitting elements to the complex geometries of the human anatomy. One of the most important steps in any CFD simulations is to generate a good quality mesh, which represents accurately the fluid domain of interest. Here the local mesh refinement, number of boundary layer cells, proper smoothed and corrected cells play an important role. For the problem concerned here, the mesh has been generated with automatic meshing tool AVL FAME. Transient flow simulation considering pulsatile flow and physical properties of blood Result assessment, e.g. the influence of stenotic lesions, aneurysm, by-pass operations. 4 Results The procedures described above were applied to two cases in order to illustrate complete process of pre-processing, solving and post-processing the medical phenomena. A more detailed discussion of the flow results is beyond the scope of this paper.
466 Design and Nature II Arteria Vertebralis Dextra Arteria Vertebralis Sinistra Arteria Basilaris Natural carotid bifurcation Artificial bifurcation after by-pass i Figure 4: Natural carotid bifurcation and artificial bifurcation after by-pass operation. (a) (b) Figure 5: a) Carotid bifurcation in ascending and descending pulsatile cycle - distribution of velocity vectors, b) natural and artificial bifurcation distribution of wall shear rates during systolic acceleration.
Design and Nature II 467 The first case shows a healthy natural junction, arteria vertebralis dextra and sinistra with ateria basilaris compared with the flow in an artificial junction, resulted from a by-pass operation in temple area. Visualisation of the results of CFD simulation of the pulsating flow using the grids shown on Fig. 4 are presented in Fig. 5, where the distribution of velocity vectors and wall shear stresses are shown. From the results, it is evident, that the pulsatile flow doesn t change the nature of the flow behavior significantly, but the level of the flow quantities (e.g. velocity vectors, Fig. 5a). Bigger is the influence of the junction angle, where local wall shear rates (Fig. 5b) are significantly higher by increasing the bifurcation angle (artificial bifurcation after by-pass operation). The second case shows arteria carotis bifurcation geometry extracted from circulus arteriosus (willis). Pathological geometry deformation of the vessels is evident. Figure 6: MR image, generated CAD surface (circulus arteriosus and arteria carotis) and computational grid geometry of the aneurysm and parent vessel. Surface velocity values are mainly within 0.25m/s. Streamlines show uniform flow in the arteria communis as well as in arteria interna and externa. The flow in aneurysm has mixing and tumbling tendency. Kinetic energy dissipates there and induces higher pressure drop.
468 Design and Nature II Figure 7: Surface maps of the velocity magnitude, streamlines and total pressure. General characteristic of blood flow in the aneurysm are illustrated in Fig. 8. Streamlines spread and began to mix as they entered central part of the neck. Streamlines mixed as they impacted the posterior wall of the aneurysm and swirled in the left, right and inferior direction. Figure 8: Overview of the complex aneurysnm hemodynamics - velocity vectors and streamlines distribution in the center and on the surface of aneurysm. Flow in the upper part of the bifurcation is direct, whereas the lower part represents area of recirculation and stagnation. 5 Conclusion Realistic modeling can yield new insights into the arterial blood flow. Furthermore, for the purposes of diagnosis and surgical planning, realistic patient specific modeling is needed because of the large anatomic and physiologic variability among individuals. These examples demonstrate how CFD tools can
Design and Nature II 469 be used to assess the distributions of blood velocity and biochemical forces imposed on the arterial wall by the blood. This may provide better understanding of the relationship between hemodynamics and vascular disease, and may one day lead to a non-invasive tool for early perdition of carotid disease or surgical planning. References [1] Cebral J.R., Yim P.J., Loehner R., Soto O., Choyke P.L., Blood Flow Modeling in Carotid Arteries with Computational Fluid Dynamics and MR Imaging, Academic Radiology, Vol 9, No 11, pp 1286-1299. [2] Cebral J.R., Loehner R., Burgess J.E., Computer Simulation of Cerebral Artery Clipping: Relevance to Aneurysm Neuro-Surgery Planning, European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2000, Barcelona. [3] Xu X.Y, Zhao S.Z., Ariff B., Long Q., Hughes A.D., Thom S.A., CFD and FE Stress Analysis of the Carotid Bifurcation Based on Individual Anatomical and Flow Data Acquired Using MRI and Ultrasound, BED- Vol. 50, 2001 Bioengineering Conference ASME 2001, pp 621-622. [4] SWIFT Manuals, AVL LIST GmbH, Graz.