**Simulation of Pulsatile Flow in Cerebral Aneurysms: From Medical Images to Flow and Forces**

Julia Mikhal, Cornelis H. Slump and Bernard J. Geurts

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/47858

### **1. Introduction**

There is a growing medical interest in the prediction of the flow and forces inside cerebral aneurysms [24, 52], with the ultimate goal of supporting medical procedures and decisions by presenting viable scenarios for intervention. The clinical background of intracranial aneurysms and subarachnoid hemorrhages is well introduced in the literature such as [46, 51]. These days, with the development of high-precision medical imaging techniques, the geometry and structure of blood vessels and possible aneurysms that have formed, can be accurately determined. To date, surgeons and radiologists had to make decisions about possible treatment of an aneurysm based on size, shape and location criteria alone. In this chapter we focus on the role of Computational Fluid Dynamics (CFD) for therapeutic options in the treatment of aneurysms. The tremendous potential of CFD in this respect was already anticipated in [29]. The value of numerical simulations for treating aneurysms will likely increase further with better quantitative understanding of hemodynamics in cerebral blood flow.

Ultimately, we aim to support the medical decision process via computational modeling. In particular, CFD allows to add qualitative and quantitative characteristics of the blood flow inside the aneurysm to this complex decision process. We propose to compute the precise patient-specific pulsatile flow in all spatial and temporal details, using a so-called 'Immersed Boundary' (IB) method. This requires a number of steps, from preparing the raw medical imagery to define the complex patient-specific flow domain, to the execution of high-fidelity simulations and their detailed interpretation in terms of flow visualization and the extraction of quantitative measures of relevance to medical practice. We compute the flow inside the aneurysm to predict high and low stress regions, of relevance to the possible growth of an aneurysm. We also visualize vortical structures in the flow indicating the quality of local blood circulation. We show that, as the size of the aneurysm increases, qualitative transitions in the flow behavior can arise, which express themselves as high-frequency variations in the flow and shear stresses. These variations could quantify the level of risk associated with the

©2012 Mikhal et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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growing aneurysm. Such computational modeling may lead to a better understanding of the progressive weakening of the vessel wall and its possible rupture after long time.

In this chapter we present a numerical model for the simulation of blood flow inside cerebral aneurysms. A significant amount of work has been done on simulation of flow in the brain and in the cardiovascular system [2, 5, 13, 18, 24, 38, 40]. As a numerical approach, the finite element method is most commonly used to represent geometries of vessels and the flow of blood through them. Often, the data are obtained from rather coarse biomedical imagery. As a result, the highly complex vessel geometry is defined with some uncertainty, and considerable smoothing and interface approximation need to be included to prepare simulations with a body-fitted approach [2, 6, 11]. As an alternative, the IB method was designed primarily for capturing viscous flow in domains of realistic complexity [39]. In particular, we consider a volume penalization method. In this method, fluid is penalized from entering a solid part of a domain of interest by adding a suitable forcing term to the equations governing the fluid flow [21]. This method is also known as 'fictitious domain' method [1] and physically resembles the Darcy penalty method [42] or the Navier-Stokes/Brinkman equations describing viscous flow on the scale of individual pores in porous domains [49]. Here, we consider in particular the limit in which the porous domain becomes impenetrable and flow in complex solid domains can be represented. This method is discussed as one of the IB methods in the recent review chapter by [32], and in the sequel will be referred to as 'volume penalization immersed boundary method', a label that was also adopted in [20].

A primary challenge for any CFD method, whether it is a body-fitted method [16] or an IB method [17, 32, 39], is to capture the flow near solid-fluid interfaces. In this region the highest velocity gradients may occur, leading to correspondingly highest levels of shear stress, but also potentially highest levels of numerical error. In methods employing body-fitted grids, the quality of predictions is directly linked to the degree to which grid-lines can be orthogonal to the solid-fluid interface and to each other. Also, variation in local mesh sizes and shapes of adjacent grid cells is a factor determining numerical error. The generation of a suitable grid is further complicated as the raw data that define the actual aneurysm geometry often require considerable preprocessing steps before any grid can be obtained. These steps include significant smoothing, segmentation and geometric operations eliminating small side vessels that are felt not to be too important for the flow. On the positive side, the main benefit of a body-fitted approach is that discrete variables are situated also at the solid-fluid interface, which makes implementation of no-slip boundary conditions quite straightforward. Hence, in body-fitted approaches the no-slip property can be accurately imposed, but only on a 'pre-processed' smoothed and often somewhat altered geometry [6, 11]. These finite element based approaches can be used to predict the patient's main flow structures of clinical value as suggested by [5, 6, 15, 19].

Capturing flow near complex shaped solid-fluid interfaces is equally challenging in an IB method, as it is in body-fitted approaches. In the IB approach adopted here, the actual geometry of the aneurysm can be extracted directly from the voxel information in the raw medical imagery, without the need for smoothing of the geometry. Grid generation is no issue for IB methods since the geometry of the flow domain is directly immersed in a Cartesian grid. The location of the solid-fluid interface is known only up to the size of a grid cell, and the shape of the interface is approximated using a 'staircase' representation, stemming from the fact that any grid cell is labeled either entirely 'solid' or entirely 'fluid'. Refinements in which a fraction between 0 and 1 of a cell can be fluid-filled [7] are not taken into consideration here. In fact, the medical imagery from which we start has a spatial resolution that is not too high when small-scale details are concerned. This calls for a systematic assessment of the sensitivity of predictions to uncertainties in the flow domain [31] incorporating also the effects due to adaptations of the domain by smoothing and interface reconstruction as is considered in higher-order methods [10]. Without relaxing the staircase approximation, the problem of capturing near-interface properties can only be addressed by increasing the spatial resolution. This gives an insight into the error-reduction by systematic grid-refinement for flow in complex geometries.

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growing aneurysm. Such computational modeling may lead to a better understanding of the

In this chapter we present a numerical model for the simulation of blood flow inside cerebral aneurysms. A significant amount of work has been done on simulation of flow in the brain and in the cardiovascular system [2, 5, 13, 18, 24, 38, 40]. As a numerical approach, the finite element method is most commonly used to represent geometries of vessels and the flow of blood through them. Often, the data are obtained from rather coarse biomedical imagery. As a result, the highly complex vessel geometry is defined with some uncertainty, and considerable smoothing and interface approximation need to be included to prepare simulations with a body-fitted approach [2, 6, 11]. As an alternative, the IB method was designed primarily for capturing viscous flow in domains of realistic complexity [39]. In particular, we consider a volume penalization method. In this method, fluid is penalized from entering a solid part of a domain of interest by adding a suitable forcing term to the equations governing the fluid flow [21]. This method is also known as 'fictitious domain' method [1] and physically resembles the Darcy penalty method [42] or the Navier-Stokes/Brinkman equations describing viscous flow on the scale of individual pores in porous domains [49]. Here, we consider in particular the limit in which the porous domain becomes impenetrable and flow in complex solid domains can be represented. This method is discussed as one of the IB methods in the recent review chapter by [32], and in the sequel will be referred to as 'volume penalization immersed

A primary challenge for any CFD method, whether it is a body-fitted method [16] or an IB method [17, 32, 39], is to capture the flow near solid-fluid interfaces. In this region the highest velocity gradients may occur, leading to correspondingly highest levels of shear stress, but also potentially highest levels of numerical error. In methods employing body-fitted grids, the quality of predictions is directly linked to the degree to which grid-lines can be orthogonal to the solid-fluid interface and to each other. Also, variation in local mesh sizes and shapes of adjacent grid cells is a factor determining numerical error. The generation of a suitable grid is further complicated as the raw data that define the actual aneurysm geometry often require considerable preprocessing steps before any grid can be obtained. These steps include significant smoothing, segmentation and geometric operations eliminating small side vessels that are felt not to be too important for the flow. On the positive side, the main benefit of a body-fitted approach is that discrete variables are situated also at the solid-fluid interface, which makes implementation of no-slip boundary conditions quite straightforward. Hence, in body-fitted approaches the no-slip property can be accurately imposed, but only on a 'pre-processed' smoothed and often somewhat altered geometry [6, 11]. These finite element based approaches can be used to predict the patient's main flow structures of clinical value as

Capturing flow near complex shaped solid-fluid interfaces is equally challenging in an IB method, as it is in body-fitted approaches. In the IB approach adopted here, the actual geometry of the aneurysm can be extracted directly from the voxel information in the raw medical imagery, without the need for smoothing of the geometry. Grid generation is no issue for IB methods since the geometry of the flow domain is directly immersed in a Cartesian grid. The location of the solid-fluid interface is known only up to the size of a grid cell, and the shape of the interface is approximated using a 'staircase' representation, stemming from the fact that any grid cell is labeled either entirely 'solid' or entirely 'fluid'. Refinements in which a fraction between 0 and 1 of a cell can be fluid-filled [7] are not taken into consideration

progressive weakening of the vessel wall and its possible rupture after long time.

boundary method', a label that was also adopted in [20].

suggested by [5, 6, 15, 19].

We illustrate the process of predicting flow and forces based on incompressible Newtonian fluid to characterize blood properties. Non-Newtonian corrections can be readily included, however, these typically lead only to modest changes in the predictions and will hence be omitted here. Pulsatile flow forcing is obtained from the direct measurement of the time-dependent mean flow velocity in a vessel during a cardiac cycle. Transcranial Doppler (TCD) sonography is a non-invasive technique, which can be used for this purpose, allowing to measure cerebral blood flow velocity near the actual aneurysm [51]. We consider a full range of physiologically relevant conditions. Understanding flow patterns inside an aneurysm may help to describe long-term effects such as the likelihood of the growth [4] or even rupture [44] of the aneurysm, or the accelerated deterioration of the vessel wall due to low shear stress [8]. Regions of high and low shear stress are often visualized as potential markers for aneurysm growth. High shear stress levels were reported near the 'neck' of a saccular aneurysm, and may be relevant during the initiating phase [44]. Low wall shear stress has been reported to have a negative effect on endothelial cells and may be important to local remodeling of an arterial wall and to aneurysm growth [4]. A low wall shear stress may facilitate the growing phase and may trigger the rupture of a cerebral aneurysm by causing degenerative changes in the aneurysm wall. The situation is, however, more complex, as illustrated by the phenomenon of spontaneous stabilization of aneurysms after an initial phase of growth [25]. It is still very much an open issue what the precise relation is between shear stress patterns and general circulation on the one hand, and developing medical risks such as aneurysm rupture, on the other hand. In this complex problem, hemodynamic stimuli are but one of many factors.

Cerebral aneurysms are most often located in or near the Circle of Willis [51] – the central vessel network for the supply of blood to the human brain. Common risk-areas are at 'T' and 'Y'-shaped junctions in the vessels [15]. Treatment of cerebral aneurysms often involves insertion of coils. This coiling procedure represents considerable risk and uncertainty about the long-term stability of coiled aneurysms [45, 47]. Blood vessels and aneurysms are rather complex by their structure and geometrical shapes. The walls of blood vessels contain several layers of different types of biological cells, which provide elasticity to the vessels and play a role in the compliance [40]. The shape of cerebral aneurysms developing in patients can be inferred by using three-dimensional rotational angiography [33]. In this procedure a part of the brain can be scanned, and aneurysms even of a size less than 3 mm can be depicted [3, 48]. This technique allows a reconstruction of three-dimensional arteries and aneurysms and hence an approximate identification of the blood vessels and parts of the soft tissue in the scanned volume. In the IB approach the domain is characterized by a so-called masking function, which takes the value '0' in the fluid (blood) part and '1' in solid (tissue) parts of the domain. The raw angiography data allows for a simple 'staircase approximation' of the solid-fluid interface that defines the vessel and aneurysm shape. Individual voxels in the

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digital data form the smallest unit of localization of the solid-fluid interface. A computational cell is assigned to be 'solid' or 'fluid' on the basis of the digital imagery. We will adopt the 'staircase' geometry representation in this chapter and do not incorporate any additional smoothing of the geometry or sophisticated reconstruction methods.

For a more complete modeling of the dynamics in the vessel system, flow-structure interaction often plays a role [40]. In that case also parameters and models that characterize, e.g., mechanical properties of arterial tissue, influence of brain tissue and the influence of the cerebrospinal fluid are required. The amplitude of the wall motion in intracranial aneurysms was found to be less than 10% of an artery diameter. Despite the rather modest motion of the vessel, long time effects may accumulate. Even modest movement can affect the vessel walls, which might play a role in possible aneurysm rupture as was hypothesized in [35]. For realistic pulsatile flows some movement of the aneurysm walls was observed during a cardiac cycle [36]. In this chapter we take a first step and restrict to developing the IB approach for rigid geometries. This allows to obtain the main flow characteristics inside relatively large cerebral aneurysms for which the wall movement can be neglected [24].

The organization of this chapter is as follows. In Section 2 we present the computational model, discuss numerical discretization and introduce the IB method for defining complex vessel and aneurysm geometries. We also describe the process of the reconstruction of the geometry from medical imagery. We illustrate steady flow inside a realistic aneurysm geometry in Section 3 and discuss the reliability of numerical predictions. A pulsatile flow and qualitative impression of the flow and forces distribution inside a realistic aneurysm is presented in Section 4. Concluding remarks are in Section 5.
