**5. Concluding remarks**

18 Will-be-set-by-IN-TECH

We compute the pulsatile flow in a range of Reynolds numbers 100 ≤ *Re* ≤ 400 while keeping the viscosity of the blood and the size of the vessel constant. In Section 3 we discussed the uncertainty in physical parameters, when consulting literature. This leads to a range of Reynolds number 175 *Re* 300 of physiologically realistic values. Including also unhealthy changes in the blood vessels, e.g., narrowing of the vessel diameter, or the development of an aneurysm and corresponding increase in the size of the vessel, but also changes in the viscosity of blood, or in the velocity of the flow, we propose to also compute the flow at *Re* = 100 and *Re* = 400. This total range of Reynolds numbers gives a complete set of flow conditions relevant to flow in the Circle of Willis. For all simulations we use a spatial resolution of 64 × 32 × 64, which we showed to be the lower limit at which quantitatively reliable results

The translation 'back' from non-dimensional units to physical units requires scaling of the time and of the shear stress values. For the indicated range of Reynolds numbers *Re* a single pulse of 0.82 *s* requires #*<sup>t</sup>* dimensionless time steps with (*Re*, #*t*)=(100, 65.6), (200, 131.2), (250, 164), (300, 196.8) and (400, 262.4). Moreover, the change in *Re* corresponds to a change

shear stress in *Pa* and time measure in *s*, which allows a more direct assessment than the fully

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 <sup>3</sup> <sup>0</sup>

t \* (s)

After these preparations, we show the dynamic response of the maximum shear stress in Figure 12 for 4 full pulse cycles. The initial condition is that of quiescent flow, i.e., *u* = *v* = *w* = 0 - we observe that this leads to a short transient, e.g., seen because the response in the first cycle differs slightly from that in later cycles. After this transient we observe for our reference case *Re* = 250 (solid bold line) that the maximum shear stress closely follows the

**Figure 12.** Maximal shear stress for realistic pulsatile flow in the aneurysm geometry. The reference *Re* = 250 is shown (bold, solid). Lower values are shown as *Re* = 200 (dot), *Re* = 100 (dash-dot), while higher value are displayed as *Re* = 300 (dash) and *Re* = 400 (thin, solid). The average stress levels

*<sup>r</sup>* (*u*<sup>∗</sup>

*<sup>r</sup>* )2. The final result is a wall

can be obtained for the case considered here.

dimensionless representation.

1

decrease with decreasing Reynolds number.

2

3

τ

(Pa)

\*

4

5

6

7

*<sup>r</sup>* , which affects the scale for the shear stress which is *ρ*<sup>∗</sup>

in *u*∗

We presented computational modeling of cerebral flow based on a volume penalizing IB method, aimed at understanding the flow and forces that emerge in aneurysms that may form on the Circle of Willis. We sketched how medical imagery can serve as point of departure for a sequence of numerical representations and modeling steps, ultimately leading to the full simulation of pulsatile flow in a realistic cerebral aneurysm, which was used as a case study in this chapter.

Taking data from literature we identified physiologically relevant flow conditions and their general uncertainty. Data concerning sizes of vessels, kinematic viscosity of blood and flow speeds in the region of the Circle of Willis during the cardiac cycle, can not be obtained with very high accuracy. This leaves considerable uncertainty as to the precise flow conditions. However, consensus seems to exist that the Reynolds number, which is the crucial parameter for incompressible flow, should be in the range 175 ≤ *Re* ≤ 300 for non-diseased situations. This is a rather wide range, but throughout this range the flow in the blood vessels is pulsatile and laminar, i.e., sharing quite comparable dynamics. The main challenge for computational modeling is not just to predict a certain flow under specified conditions, but to reckon also with the variability in flow conditions due to a variety of possible changes in key parameters. This approach was taken in this chapter.

Settling for *Re* = 250 as characteristic point of reference, we analyzed a particular, realistic cerebral aneurysm in detail. First, the main flow features and the reliability of predictions were considered for steady flow at fixed volumetric flow rate. This is not a realistic flow condition, as in reality interest is with pulsatile flow, but it does allow to investigate the sensitivity of the predicted solution on things such as spatial resolution. We visualized both qualitatively and quantitatively the steady flow in the aneurysm, as well as the shear stress field that emerges. It was shown that the main flow follows a path that is close to what used to be the original vessel before the formation of the aneurysm. Next to this 'main' flow, a complex circulation was shown to develop inside the aneurysm bulge. By considering contour plots and also profiles of velocity and shear stress at different spatial resolutions, the degree of reliability of the numerical simulation was discussed. The current IB method is first order accurate. Developments in which sub-grid forcing is included [42] can be used to increase the formal

#### 20 Will-be-set-by-IN-TECH 218 Aneurysm

order of accuracy to two - this appears a relevant extension of the IB approach and will be considered in more detail in the near future, allowing to cut down on the computational cost and/or increase the accuracy of flow predictions.

A complete model was obtained by incorporating realistic pulsatile flow, obtained from direct TCD measurements of the velocity of blood flow in the brain. The recorded velocity in a vessel near the Circle of Willis was used to impose a proper time-dependent volumetric flow rate, representing a cardiac cycle. Repeating a characteristic pulse periodically, leads to a model with which the time-dependent flow and shear stresses can be determined. As regards the flow structures, for pulsatile flow at *Re* = 250 one basically notices the same dominant flow features as in case of steady flow forcing, with the exception that the 'amplitude' of the motion becomes time-dependent. As an example, the recirculating flow in the aneurysm was observed throughout the entire cycle, but with a time-dependent intensity and a slight 'meandering' of the precise vortical structures. We sketched the extension of the pulsatile flow model to other flow conditions, i.e., other *Re* and other time-scales. If the flow is in the physiologically relevant regime *Re* ≤ 300, the response of, e.g., the shear stress, closely follows that of the input flow-rate forcing. At higher Reynolds numbers, indicative of possible diseased states, the flow develops considerable complexity and shows a transition toward much higher frequencies. This goes hand in hand with increased levels of shear stress and may be monitored as a potential indication of increased risk to the patient. By recording the spectrum of these frequencies an easy monitoring concept may become available.

The application of computational support in the monitoring and treatment of cerebral aneurysms is a field of ongoing research. Accessibility of time-dependent flow fields in all relevant detail is a crucial point from which to depart toward developing predictive capability for the associated slow growth of the aneurysm bulge. This requires a new kind of 'flow-structure interaction' in which degradation of endothelial cells due to reduced quality of blood circulation typically triggers a further expansion of the aneurysm bulge, and generally leads to an increase in the risk of rupture. The latter type of 'flow-structure interaction' is subject of ongoing research. In order to achieve a closer connection with medical practice several computational modeling steps still need to be taken, such as the development of higher order accurate methods, multi inflow/outflow configurations, flow-structure interaction in which a full coupling to slow degenerative processes of endothelial cells is made, as well as modeling of mechanical properties of brain tissue to also address aneurysm compliance during the pulsatile cycle. This brief listing shows the various developments that are still needed in order to make the pathway from medical imagery to quantitative decision support both reliable as well as fully automated.

### **Acknowledgements**

The authors gratefully acknowledge many fruitful discussions with Prof. Dr. Hans Kuerten (Eindhoven University of Technology and University of Twente). We are also grateful to Willem Jan van Rooij, MD, PhD and Menno Sluzewski, MD, PhD (St. Elizabeth Hospital, Tilburg, the Netherlands) for providing angiographic data and to Dr. Ir. Dirk-Jan Kroon (Focal, Oldenzaal, the Netherlands) for segmentation of the data. Computations at the Huygens computer at SARA were supported by NCF through the project SH061.

#### **Author details**

20 Will-be-set-by-IN-TECH

order of accuracy to two - this appears a relevant extension of the IB approach and will be considered in more detail in the near future, allowing to cut down on the computational cost

A complete model was obtained by incorporating realistic pulsatile flow, obtained from direct TCD measurements of the velocity of blood flow in the brain. The recorded velocity in a vessel near the Circle of Willis was used to impose a proper time-dependent volumetric flow rate, representing a cardiac cycle. Repeating a characteristic pulse periodically, leads to a model with which the time-dependent flow and shear stresses can be determined. As regards the flow structures, for pulsatile flow at *Re* = 250 one basically notices the same dominant flow features as in case of steady flow forcing, with the exception that the 'amplitude' of the motion becomes time-dependent. As an example, the recirculating flow in the aneurysm was observed throughout the entire cycle, but with a time-dependent intensity and a slight 'meandering' of the precise vortical structures. We sketched the extension of the pulsatile flow model to other flow conditions, i.e., other *Re* and other time-scales. If the flow is in the physiologically relevant regime *Re* ≤ 300, the response of, e.g., the shear stress, closely follows that of the input flow-rate forcing. At higher Reynolds numbers, indicative of possible diseased states, the flow develops considerable complexity and shows a transition toward much higher frequencies. This goes hand in hand with increased levels of shear stress and may be monitored as a potential indication of increased risk to the patient. By recording the

spectrum of these frequencies an easy monitoring concept may become available.

The application of computational support in the monitoring and treatment of cerebral aneurysms is a field of ongoing research. Accessibility of time-dependent flow fields in all relevant detail is a crucial point from which to depart toward developing predictive capability for the associated slow growth of the aneurysm bulge. This requires a new kind of 'flow-structure interaction' in which degradation of endothelial cells due to reduced quality of blood circulation typically triggers a further expansion of the aneurysm bulge, and generally leads to an increase in the risk of rupture. The latter type of 'flow-structure interaction' is subject of ongoing research. In order to achieve a closer connection with medical practice several computational modeling steps still need to be taken, such as the development of higher order accurate methods, multi inflow/outflow configurations, flow-structure interaction in which a full coupling to slow degenerative processes of endothelial cells is made, as well as modeling of mechanical properties of brain tissue to also address aneurysm compliance during the pulsatile cycle. This brief listing shows the various developments that are still needed in order to make the pathway from medical imagery to quantitative decision support

The authors gratefully acknowledge many fruitful discussions with Prof. Dr. Hans Kuerten (Eindhoven University of Technology and University of Twente). We are also grateful to Willem Jan van Rooij, MD, PhD and Menno Sluzewski, MD, PhD (St. Elizabeth Hospital, Tilburg, the Netherlands) for providing angiographic data and to Dr. Ir. Dirk-Jan Kroon (Focal, Oldenzaal, the Netherlands) for segmentation of the data. Computations at the Huygens

computer at SARA were supported by NCF through the project SH061.

and/or increase the accuracy of flow predictions.

both reliable as well as fully automated.

**Acknowledgements**

Julia Mikhal, Cornelis H. Slump and Bernard J. Geurts *Faculty EEMCS, University of Twente, P.O.Box 217, 7500 AE, Enschede, The Netherlands*

#### **6. References**


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#### 222 Aneurysm **Chapter 0 Chapter 11**
