Figure 1: Angiography of cerebral arteries with an aneurysm, where the aneurysm is the spherical shaped part of the blood vessel close to the center for the picture.

Figure 2: A classical textbook illustration of the Circle of Willis, showing the major cerebral arteries. The blood flows in through the internal carotid arteries (ascending adjacent to the throat) and vertebral arteries (ascending through the neck), and out through the remaining arteries.

Figure 3: A simplistic view of the core components of an artery.

Aneurysm Formation and Rupture in the Circle of Willis

The goal of this project is to provide clinicians with new indicators, based on patient-specific simulations of the blood flow, for assisting the decision making on aneurysm treatment. Our simulations of basic physiological processes have already evolved into emerging clinical solutions documented in medical journals.

A new principal activity is to explore to what extent turbulence is present in the vicinity of aneurysms and what the importance of turbulence is. It is also an open question if the flow is really turbulent (chaotic) or if it is just a very complicated laminar flow with a wide range of scales. CBC@Simula has just started to investigate these tasks in strong collaboration with CBC@FFI/UCy. Preliminary findings point to significant increase in the wall shear stress, which is commonly believed to be a key factor influencing aneurysm growth and rupture. The frequency of oscillations is on the order of 100 Hz rather than the 1 Hz in laminar flow (heart rate 60 beats/min). Initial investigations must be based on high-resolution DNS, as well as carefully conducted LES studies, while a long-term goal is to tune an appropriate advanced turbulence model to account for the important effects that a standard laminar Navier-Stokes model is not able to reproduce.

So far we have only considered mechanical effects of the flow, but the tissue in the vessel wall is a dynamic material, adapting itself to the flow and stress at the wall. Incorporating dynamic physiological changes of the tissue is an exciting and important extension of scope.

We have recently established an effective collaboration with a world-leading researcher on aneurysms, Prof. Charles Strother at the Univ. of Wisconsin (Madison). He has a special focus on flow around aneurysms under rapid heartbeat and stressed conditions. Aneurysms have a tendency to rupture under such circumstances, but measurements and computations have hardly addressed this phenomenon so far. Consequently, this research focus will likely lead to novel results of considerable medical interest. Prof. Strother's high-resolution measurements on canine models provide boundary conditions for simulations as well as high-resolution data against which computer models can be validated. Through a collaboration with Dr. Søren Bakke's group at Oslo University Hospital we are conducting ultrasound measurements of blood flow in the brain during hard exercise. These measurements provide a lot of data for boundary conditions and for validation of human cerebral blood flow under rapid heartbeat.

We plan to significantly increase the emphasis on validation and uncertainty quantification of blood flow models, using tools developed in the Model Calibration subproject. A central question is to estimate the uncertainty due to inaccurate geometry and boundary conditions, and relate that uncertainty to the accuracy of the modeling and the influence of neglecting a series of physical and biological effects. This will be a major undertaking. Quantitative uncertainty assessment of patient-specific blood flow simulations has to our knowledge not yet been extensively studied in the science community. The results are fundamental to determine the usefulness of patient-specific blood flow simulations in clinical practice.

 

 

Example from PhD research in this field:

 

Computational Cerebral Hemodynamics

Cardiovascular diseases, like cerebral strokes, are among the main causes of death in the developed world. The number of incidents is expected to rise in the years to come. Some of the main causes of cerebral strokes are related to heavy alcohol consumption, smoking and high blood pressure. One particular type of stroke is the rupture of a damaged part of a blood vessel inside the head. The resulting bleeding which typically occur in the subarachnoid space that surrounds the brain is called subarachnoid hemorrhage (SAH). The subsequent lack of oxygen to the brain results in death within 30 days in 50% of the cases, or severe neurological deficits. The estimated incidence of SAH is roughly 1/10000 people annually. Our research addresses the issue of why some blood vessels become damaged and form a berry-like dilatation known as an aneurysm, and why some of these aneurysms rupture and cause SAH. An example of such an aneurysm is shown in Figure 1.

Computational mechanics has a long and successful history in science and engineering. We know applications of this approach from, e.g., the structural design of bridges and cars, or simulation of air flow around aircrafts. Throughout the last decades computational mechanics has also been applied to biological tissue (biomechanics) and is today a promising field of research with a large potential in clinical medicine. For example, surgical planning can be performed to ensure the delivery of blood to all parts of the brain prior to surgery, and scientists can now use engineering tools to design artificial joints and heart valves. Computational biomechanics is also used to make complex models of the cranium and to simulate forces that occur during accidents and injuries, e.g., in forensic medicine.

Research projects have become increasingly interdisciplinary, and this thesis is no exception. To achieve the goal of describing and understanding cerebral blood flow, we have to include elements of computational fluid dynamics (CFD), turbulence, biology, medicine and computer science to mention some. The papers that make up this thesis have been published in or submitted to Gender Medicine, Stroke and Journal of Biomechanics and the book Automated Scientific Computing.

The purpose of this introduction is to describe the background for the performed studies, and how the studies relate to SAH. We also introduce the most important concepts to readers not familiar with the topic. This introduction is not a complete overview, and for more comprehensive reading on the topic, we can recommend. We will briefly describe some aspects of blood flow before proceeding to the blood vessel wall, including a damaged vessel wall where an aneurysm was initiated. Finally, we will present the major contributions from this thesis, and work proposed for the future.

Blood flow

The circulatory system is driven by the heart which contracts periodically, roughly one hundred thousand times a day. The heart is the size of a fist and ejects 70 ml of blood at 120 mmHg into the aorta where the average diastolic pressure is 80 mmHg. Blood undergoes a wide range of flow conditions in the body. In the largest artery, the aorta, the Reynolds number (describing the importance of intertia forces versus viscous forces) can get as high as 4500, but as small as 0.001 in the capillaries where the vessel diameter is just slightly larger than the diameter of the red blood cells. The flow from the heart is pulsatile and the large vessels close to the heart are compliant. In the intracranial arteries, the blood flow never falls below 40% of its maximum value, and the vessel deformation is less. Blood is a suspension of blood cells in a plasma, and shows a non-linear viscosity response to shear rate in the flowing blood, i.e., blood is a non-Newtonian fluid.

The vertebral arteries and the internal carotid arteries (ICAs) are the main suppliers of blood to the brain. They connect in the Circle of Willis (CoW), shown in Figure 2, which is a circle of blood vessels at the base of the brain. This circle of blood vessels ensures a reliable delivery of blood to the brain, even if something should happen to the other supplying blood vessels. The blood vessels contract or relax in order to distribute blood to the parts of the body which need it the most. For example, the flow of blood is altered to either the left or the right hand side of the brain during different mental activities, depending on what part of the brain is more active.

Blood vessels

Blood vessels can be divided into five major groups depending on their function and composition: arteries, arterioles, capillaries, venules and veins. The composition, diameter and wall thickness depend on the blood vessels distance from the heart. The vessels close to the heart experience a maximum pressure of about 120 mmHg, and naturally have a different composition than the veins where the pressure is about 5 mmHg. We will primarily consider the large intracranial arteries (illustrated in Figure 2) since this is where cerebral aneurysms typically develop.

Arteries consist mainly of three layers; the intima, media and adventitia, illustrated in Figure 3, and connect through an elastic lamina (or elastic membrane). In the lumen of an artery, blood is in contact with the endothelial cells of the intima. The endothelial cells are permeable and can sense external stimuli and chemically signal the hemodynamical stresses. The media mostly consists of smooth muscle cells and is the flexible layer of the vessel wall that takes most of the mechanical load. The outer layer, the adventitia, is composed only of connective tissue that contains nerves and nutrient capillaries. Compared to other arteries of similar size, the intracranial arteries have an `attenuated media and lack the external elastic lamina'. In some articles the description is slightly different, but the conclusion of weaker intracranial vessels compared to others, is the same.

Vessel remodeling

Exercise is known to be beneficial to the health, and in humans, temporarily increased cerebral blood flow through physical activity has been shown to have a positive effect on the cerebral vasculature. Elderly people who reported high levels of aerobic activity showed less cerebral vessel tortuousness/curvature and more small-caliber vessels. It is common to lack or have one or more blood vessels in the CoW underdeveloped, and only 40% of the population fit the classical symmetric description shown in Figure 2. The reason might be that the vessels remodel over time, according to local flow conditions. One hypothesis is that the vessels remodel to maintain a constant flow induced shear stress on the vessel wall, and experiments have shown a good agreement with this hypothesis. This hypothesis is not true in bifurcations, where both too high, too low and oscillatory wall shear stress (WSS) might occur. It has also been shown that WSS values vary in the different bifurcations in CoW. Arterial remodeling occurs because of changed load on the endothelial cells. The process known as mechanotransduction is the chemical signaling between the cells due to stresses (or stretches) sensed by the endothelial cells at the vessel wall. Experiments have shown that endothelial cells themselves are greatly affected by the flow. The endothelial cells align with the main direction of flow, but show no distinct orientation in regions with disturbed or no flow.

Vessel remodeling has been demonstrated experimentally, see, where the flow of blood was surgically closed in the carotid arteries (located in the throat) of a rabbit. All the blood that entered the brain then had to travel through the basilar artery (in the back of the neck), see Figure 2. Immediately after the occlusion of the vessels, the flow in the basilar artery increased by 450%. The vessel became longer due to an elevated axial load, and the diameter increased because WSS increased, see Figure 4. When the carotid arteries were closed off, the flow increased immediately, but the vessel diameter expansion response was slightly slower. Later, the flow decreased but the vessel continued to grow. The peak WSS actually occurred after just 4 days and illustrates how fast blood vessels can remodel and adapt to new conditions. Note that the WSS in this study was not computed, but estimated proportional to the flow divided by the radius to the third power.

Figure 4: The figure shows the remodeling of a basilar artery in a rabbit after occluding the carotid arteries.

Aneurysms

The prevalence of cerebral aneurysms is 1-5% in the adult population, with an estimated annual risk of rupture of 1%. It is believed that an aneurysm is initiated because of an alteration of the hemodynamic stresses that can damage the endothelial cells. The damage causes an inflammatory and degenerative process in the vessel wall. In the arteries, places where the flow is disturbed are commonly associated with cardiovascular diseases such as aneurysms. Since disturbed flow typically occurs in bifurcations, this indicates that the forces acting on the vessel wall plays a significant role in the initiation of aneurysms. The basilar artery in the rabbit adapted to the new flow conditions, but the increased forces were too great in the bifurcation, and the endothelial cells were damaged. The damage to the wall after 84 days is shown in Figure 5.

Figure 5: The figure shows the basilar artery bifurcation from the rabbits in in a control (top row) and increased flow (bottom row). In the bottom row we can see the damage to the vessel wall in the three right most pictures. It is the same bifurcation in all three pictures (left most picture enlarged), but the different constituents of the wall are visualized using staining methods. (Van Gieson shows the collagen and connective tissue.Trichrome colors the erythrocytes orange, muscle red and collagen blue. H&E stain makes the nuclei of cells blue.

The upper row of pictures is from a healthy control specimen without additional flow in the basilar artery, while the bottom row of pictures shows the damaged vessel wall. In Figure 5 the three pictures furthest to the right (both top and bottom) show the same vessel wall but the different vessel constituents are visualized using different staining methods. Staining methods are techniques to distinguish parts of the tissue based on different color response of the wall constituents to chemicals. We can see that there is a loss of endothelial cells, internal elastic lamina disruption and smooth muscle cell depletion, as summarized on the right hand side in the figure. Due to the lack of smooth muscle cells, which is the primal source of the wall's structural integrity, the hemodynamic forces exceeded what the vessel wall could withstand, and the wall bulged out, initiating an aneurysm. Autopsy studies have shown that aneurysms in humans also have a different composition in the vessel wall, and are mechanically weaker when compared to healthy vessels. An ICA artery is roughly 0.2 mm thick, and for comparison, an aneurysm can be one quarter of this thickness , and sometimes so thin that the blood flow inside the aneurysm is visible.

The initiation of aneurysms are multi factorial and complex. In the case of the rabbit, it was the greatly increased WSS in the bifurcation, which lead to a local inflammatory response and damage to the vessel wall. The WSS exceeded 30 Pa in the straight part, which indicates that the WSS in the bifurcation must have been much higher than the 40 Pa which is the limit for endothelial cell damage. Inflammatory processes can also be triggered by heavy alcohol consumption, smoking and hypertension, or even too low WSS. There are many unknown factors. For example, women are nearly twice as likely to experience SAH, and aneurysms arise in different locations in men and women. It has also been shown that Japanese and Finnish women are more likely to develop aneurysms, than other women. People with an asymmetric CoW are also more prone to harbor an aneurysm, and roughly 80% of the aneurysms are located in the anterior circulation of the CoW. However, it is maybe more intuitive that the neck to fundus ratio increases the risk of rupture, as well as the aneurysm diameter divided by the parent artery diameter, and the size of the aneurysm itself. Aneurysms with irregular shapes are also more prone to rupture and aneurysms grow in the direction of low WSS. These statistical facts give clues on where to start looking for answers to understand the pathogenesis of aneurysms. The cause and effects of many of these relations may be investigated non-invasively on a computer.

Saccular cerebral aneurysms are mainly treated in two ways depending upon the patient and the location of the aneurysm. Previously, the most common treatment was to open the skull and apply a clip over the aneurysm. This procedure, however, is associated with great risk to the patient and a long recovery time. It has become more common to treat aneurysms by inserting a detachable platinum coil into the aneurysm which causes the blood to coagulate in the aneurysm. The coil is inserted via a catheter from the femoral artery in the groin which is navigated to the cerebral arteries using real-time X-ray (endovascular coil embolization).

Computational science

Computational science is the use of computers to simulate phenomena and thereby increase physical insight. Many complex phenomena arising in nature can be described by partial differential equations (PDEs). To solve PDEs on a computer, the continuous PDEs have to be converted into a discrete system of equations using methods such as the Finite Element Method. The solution is then approximated in the computational domain and the error depends on the chosen resolution in both space and time, given a stable scheme. It is desirable to have a good resolution in order to minimize errors, but the cost is computing time. The computing time is dependent upon the number of unknowns and the method used to solve a problem. As shown in Paper I of this thesis, the cost may vary greatly between commonly used methods to solve flow problems.

Given an appropriate model with initial and boundary conditions, simulations can provide information that experiments can not provide. Computational fluid dynamics (CFD) can capture turbulence in intracranial aneurysms and provide spatial and temporal resolution superior to what is currently possible to measure with high resolution computational tomography or magnetic resonance imaging techniques. In a simulation both the pressure and velocity at any given point is known, without inserting measurement equipment into the flow which could affect the actual flow that one wants to measure. For instance, when performing CFD calculations, we can compute derivatives and visualize certain parts of the flow, to gain knowledge about a specific quantity. Examples of this include the study of turbulent kinetic energy and turbulent kinetic energy dissipation, which could illustrate how turbulence is developed and dissipated. Another advantage with computational methods is that it is possible to perform numerical experiments relatively cheaply, while laboratory equipment and staff might be expensive. Experiments can be conducted on a computer with different parameters to, e.g., find an optimal solution to a problem, at a fraction of the effort compared to a laboratory experiment. On the other hand, simulations might need long computational time, code can contain bugs and education takes a long time.

In computational biomechanics, CFD can be of great importance in cases where both sick and healthy patients must be included in studies to distinguish relevant parameters. When an aneurysm is formed, there are many individual and complex factors involved, and it is necessary to conduct large population studies to single out causes. Harmless magnetic resonance imaging and CFD simulations of blood flow can be a good combination to determine the patient-specific intracranial flow conditions, that would not otherwise be possible due to strict ethical regulations of experiments on healthy subjects. One can never cure aneurysms or cardiovascular diseases with CFD, but one may learn a lot from the simulations, e.g., to provide the patient-specific optimal treatment or prevent diseases from occurring at all.

Limitations

Describing the flow of blood in arteries with deforming vessel walls is a fluid structure interaction (FSI) problem, and computationally very demanding since the equations governing the interaction between a fluid and a structure are often solved iteratively. It is known that intracranial arteries exhibit a 5-10% deformation, and in a computational study the middle cerebral artery (MCA) aneurysm wall was shown to deform by a maximum of 10-15%. This deformation has not been included in this thesis, primarily because FSI simulations are computationally complex and expensive.

The vessel walls have been treated as rigid in all of our studies. We assume that the effects of a deforming vessel wall can be neglected compared to other effects, such as the resolution of the medical images (the resolution can be as coarse as 0.5 mm between the pixels), segmentation of the images and uncertainties related to patient-specific boundary conditions.

The patient-specific MCA boundary conditions have not been available for the aneurysms used in Paper IV and Paper V. Instead, we have used a realistic waveform from another patient. The values have been fitted to match average values. On the outflow boundaries we have used resistance boundary conditions in Paper IV, which models the resistance occurring in the downstream vasculature, see. The effect of the resistance boundary condition is that the flow is nearly evenly distributed between the outlets, which is the reason for setting the flux division equal on the daughter vessels in Paper V. Blood is a non-Newtonian fluid, i.e., it shows a non-linear response to shear rates. Modeling blood as a non-Newtonian fluid is of importance in smaller arteries, such as the coronary arteries, but can be neglected in the larger intracranial arteries.

As pointed out in and Paper V, a direct numerical simulation (DNS) based on a single phase Newtonian fluid is not the correct approach for modeling turbulent flows in blood when the characteristic turbulent length scales approach the size of red blood cells. In our DNS simulation in Paper V, the smallest eddies are roughly eight times the size of red blood cells. Additional effects due to red blood cell interaction can therefore to first order be neglected, especially since the eddies that appear at these scales contain the least energy.

Summary of papers

Below we summarize some recent papers on Computational Hemodynamics. The first two papers of the thesis appears as book chapters in Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book, edited by Logg, Mardal, and Wells, Springer 2012. The third, fourth, and fifth papers are published in Gender Medicine, Stroke and Journal of Biomechanics, respectively.

Paper I : A comparison of some common finite element schemes for the incompressible Navier-Stokes equations

There is a jungle of numerical methods for solving the Navier-Stokes equations for incompressible fluid flow, and consequently a challenge to select the best compromise between accuracy and computational efficiency for a given problem class. Surprisingly, very few studies perform critical evaluations and comparisons of methods for incompressible flow. One reason is that a wide variety of methods are implemented in many different types of software, which makes a fair comparison of CPU time difficult. To conduct an objective study of accuracy and efficiency, the various methods should be implemented in the same software framework and differ as little as possible. The FEniCS software framework is ideal for this purpose, but restricts the attention to finite element methods and meshes with triangles or tetrahedrons (though with any degree of the approximating polynomials).

We have implemented six well-known and acclaimed schemes for the Navier-Stokes equations in the FEniCS framework, and performed detailed comparisons of these schemes in six flow cases with differing characteristics.

The conclusion is that there are large differences between the solvers with respect to accuracy and efficiency. Therefore, to obtain the best compromise of accuracy and efficiency when studying a class of problems by CFD techniques, it is highly recommended to first perform method comparison of the type we have done in this paper.

What is New?

• A flexible high level scripting computational framework for solving the Navier-Stokes equations has been developed. The implementation consists of different solvers and problems. It is available for download at http://www.launchpad.net/nsbench.
• The IPCS method for solving the Navier-Stokes equations showed an overall good performance, with respect to minimizing computing time and errors, and was considered to be the best among the tested schemes. The GRPC solver was more accurate in some test cases, but was significantly slower.

Paper II : Computational hemodynamics

Based on the software developed in Paper I, this second paper demonstrates how FEniCS has been used to conduct studies of cerebral hemodynamics in canines and humans. This book chapter is devoted to the computation of blood flow in large cerebral arteries and how the blood flow affects the development of aneurysms. We describe the computation of stresses, and discuss boundary conditions and the process of generating geometries from medical imaging data to performing patient-specific simulations of hemodynamics. Specifically, we present three different applications: simulations related to a recently published study(Paper III) concerning gender differences in cerebral arteries, a study of the carotid arteries of a canine with an induced aneurysm, and a study of the blood flow in a healthy Circle of Willis, where patient-specific velocity measurements were compared with a model for the peripheral resistance.

What is New?

• Advanced code including multidisciplinary topics has been developed to conduct a series of studies.
• 4D high resolution magnetic resonance imaging velocity measurements of flow in an artificial canine aneurysm showed an overall good agreement compared with CFD simulations, but the computed WSS differed greatly.

Paper III : Sex Differences in Intracranial Arterial Bifurcations

Since SAH occurs more frequently in females than in males and since people who have large variations in their CoW are more likely to harbor aneurysms, we wanted to find out if there were any gender differences in the CoW, and if so, to quantify them. Vessel radii and bifurcation angles for MCA and ICA bifurcations were available from a previous study in our group. Statistical analysis revealed that the female vessel diameter of the MCA and ICA were smaller than that of the male (p \le 0.05), but the angles were inconclusive. Based on averaged data, we created four idealized geometries in total; male and female ICA and MCA bifurcations. The CFD simulations revealed that the peak WSS in the female ICA was 50% higher and WSS in the female MCA 19% higher when compared to the male bifurcations. This may partly explain why intracranial aneurysms and SAH are more likely to occur in females than in males. The study was simple, with idealized geometries and stationary flow, but it illustrated the gender differences clearly. The results were consistent with a previous study that demonstrated that there were more incidences of ICA aneurysms amongst women than men.

What is New?

• Women have smaller ICA and MCA diameters compared to men.
• Women have higher values of WSS in the ICA and MCA bifurcations (50% in ICA and 19% in MCA) compared to men.

Paper IV : A Quantitative Characterization of Differences in Flow Patterns in Ruptured Versus Unruptured MCA Aneurysms

Most aneurysms are asymptomatic (87%), but if detected, the physician has to evaluate the risk of rupture and determine the optimal treatment. All treatments of intracranial aneurysms are associated with a certain risk to the patient and choosing the right procedure can be of great importance. To estimate the risk of rupture is therefore highly clinically relevant, and has been the focus of many studies. The basis for this study is that cardiovascular diseases occur in regions with disturbed flow, and that hemodynamical forces affect the aneurysm wall. We are not only interested in the WSS strength, but also how the complexity of the flow influences the endothelial cells, mechanotransduction, wall composition, and therefore the rupture of aneurysms. Others have addressed the same topic, but the characterization has been performed subjectively and visually only. Here, we introduce new indicators for assessing risk of aneurysm rupture based on fluid mechanical properties such as the kinetic energy, vorticity and pressure drop over the aneurysm. The background is that a complex geometry might lead to a concentration of energy which locally results in a more complex flow. The complexity of the flow (increased vorticity) could affect the endothelial cell alignment and alter the structural integrity of the wall. The overall shear rates would increase and more energy would be required (pressure drop over the aneurysm) to drive the flow. This additional energy must be exerted on the wall and could thus possibly be related to rupture.

The intracranial vasculature from 12 patients with intracranial aneurysms were used for creating computational meshes. CFD simulations were carried out with a realistic MCA waveform inlet profile, and resistance outflow boundary conditions to model the downstream vasculature. The visualizations of the flow showed different flow patterns in ruptured versus unruptured subjects. In the ruptured aneurysms, the flow impinged with a more perpendicular angle at the opposing wall compared to the unruptured aneurysms. The ruptured aneurysms had both a higher maximum kinetic energy and a higher mean kinetic energy density, and resulted in a more unstable flow with higher values of maximum vorticity, mean vorticity density and higher pressure drop. Higher kinetic energy and vorticity values showed a stronger correlation with rupture when compared to values for WSS, and can possibly be better indicators for risk assessment of aneurysm rupture.

What is New?

• The ruptured MCA aneurysms exhibit elevated values of kinetic energy.
• The ruptured MCA aneurysms exhibit a more complex flow.
• The ruptured MCA aneurysms exhibit a greater pressure drop over the aneurysm due to a more complex flow pattern in the ruptured aneurysms.
• Higher kinetic energy levels and vorticity values show a stronger correlation with rupture when compared to values for WSS, and can possibly be better indicators for risk assessment of aneurysm rupture.

Paper V : Direct Numerical Simulation of Transitional Flow in a Patient-Specific MCA Aneurysm

In Paper IV, we computed the vorticity within 12 MCA aneurysms as a measure of flow complexity. One of the aneurysms with relative high values of vorticity was selected for a DNS to investigate the presence of turbulence. We used a realistic patient-specific waveform on the inlet and an equal flow division on the outlets. The numerical solver CDP was used to solve the Navier-Stokes equations. On 32 CPUs each cardiac cycle required 12 hours of computational time, and we simulated 13 cycles. The flow showed transition to turbulence just after peak systole, and the turbulent fluctuations increased in intensity until mid deceleration, before relaminarization occurred during diastole. The WSS magnitude had a maximum value of 41.5 Pa. The recorded velocity frequencies were predominantly in the range of 1-500 Hz. The peak pressure fluctuations had a frequency of roughly 100 Hz and 1.5 mmHg. The current study confirms, through properly resolved CFD simulations, that turbulence can occur in intracranial aneurysms. Because of the local arterial wall response to hemodynamical forces, the effects of oscillatory WSS caused by turbulence could be of great importance and correlated with aneurysm rupture.

Transient effects often seem to be neglected even though the spatial resolution is good. It is not unusual to have a temporal resolution of only 100 time steps per heart cycle. If one considers ordinary flow in the vicinity of a spherical medium MCA bifurcation aneurysm, the artery diameter is 2.5 mm and the aneurysm diameter is about 10 mm. With a maximum velocity approaching 100 cm/s, a mass-less tracer would travel the distance of nearly one aneurysm diameter per time step, or 4 artery diameters. With such a coarse resolution, one can not capture the fine structures in the flow, nor the recorded frequencies of about 500 Hz reported by former investigators. In contrast to a resolution in time of 100 time steps per heart cycle, we used 40,000 time steps in the best resolved simulation in this study. It should also be noted that this resolution in time was barely good enough to describe the most rapid velocity changes.

Even though we have only studied one aneurysm, the findings are similar to those obtained in a previous study where turbulence was found experimentally in 10 of 17 aneurysms. Similar frequencies were also found in another study, where ophthalmic artery (which goes from the ICA towards the eyes) aneurysm bruits were recorded with a microscope on the eyes. (The bruits disappeared after endovascular coil embolization and parent artery occlusion, but not after stenting.) Turbulence can not be excluded in the other aneurysms in Paper IV, but this has not been studied due to computational resources. However, if all scales would have been modeled with a proper resolution, one could expect to find turbulence in, percentage wise, the same number of aneurysms as in the previous studies.

What is New?

• Turbulence is not believed to occur in aneurysms, but we have shown computationally that turbulence can occur in an MCA aneurysm.
• Turbulence induced pressure fluctuations of 1.5 mmHg occurred at frequencies that were two orders of magnitude higher than the natural frequency of a heart beat.
• The effects of the elevated and oscillatory WSS on the endothelial cells should be studied conducting physical experiments in vivo.
• Future CFD simulations of intracranial blood flow should be performed with a sufficient resolution in space and time to capture possible turbulence. The smallest time and length scales should be computed to gain insight into the resolution of the simulation.

Future work

The rupture of an intracranial aneurysm is due to mechanical failure of the vessel wall, since the aneurysm wall contains less smooth muscle cells and is structurally weaker than the surrounding healthy vessels. Understanding the fundamental processes that leads to smooth muscle cell loss should be the primary goal of future studies. It is unclear whether increased understanding of turbulence or FSI will provide the best clinically relevant information. FSI is known to occur, but turbulence is mostly believed not to occur. We believe that it is important to challenge this assumption, since characteristic frequencies for velocity and pressure fluctuations caused by turbulence are two orders of magnitude greater than the frequencies at which FSI occurs.

Most CFD simulations are performed at heart rates in the range of 60-75 beats per minute. With an average age of only 51 years when SAH occurs, it is reasonable to assume that many people with aneurysms are physically active and on a daily basis exceed heart rates that are related to rest. This is why exercise, and the effects of blood flow on a vessel wall during physical activity would have been natural to include in a future study. To our knowledge, there are only two CFD studies of flow in an aneurysm at varying heart rates. The rule of thumb for a healthy person is that the individual maximum pulse is approximated as 225 minus the persons age. Given the average age of when SAH occurs, heart rates of 174 beats per minute, and peak velocities up to 180 cm/s in the MCA seem to be reasonable assumptions for a CFD simulation, which is in agreement with measurements from previous studies. We have transcranial Doppler velocity measurements from a healthy control in our group where the velocity in the MCA was measured to 200 cm/s during maximum physical activity. We have found turbulence in one patient-specific MCA aneurysm with the assumption of a heart rate of 60 beats per minute. The likelihood of intermittent turbulence increases greatly with physical exercise and turbulence can therefore be expected to be found in other aneurysms.

Due to increased stresses on the arterial wall caused by exercise, or by turbulence that has shown to occur in patients at rest, these elevated stresses must be considered. Turbulent flow must be considered an equally normal state for blood flow in an aneurysm, as is laminar flow. This calls for a whole new series of experiments to understand how endothelial cells and the vessel wall composition is affected by local turbulent flow and pressure fluctuations.

A DNS is computationally very expensive, which explains why we have only performed DNS for one flow case. However, one can obtain comparable results by performing a large eddy simulation. Turbulence may be absent in most cases in the circulatory system, but has been shown to occur in a stenosed carotid bifurcation, and the aorta. Caution must be taken such that numerical simulations are not under-resolved. This means that if turbulence is physically present in a specific case, the simulations must capture this.

Interdisciplinary research can be challenging with respect to communication. A computational scientist's work needs to be communicated properly for the medical community to understand the implications, and vice versa. The ability to communicate scientific results is of great importance to exchange information properly between the scientific branches of biomechanics and medicine. By using the results from CFD simulations, one can simulate the injection of a contrast fluid, by solving an additional PDE for a passive scalar. The results can be visualized like an angiography and are thus easily communicated to physicians, since the additional passive scalar appears as a contrast fluid in a blood vessel. Further examples can be to abandon vector arrows and to go in the direction of volume visualizations of structures in the flow. Examples of such structures could be kinetic energy or enstrophy, visualized using illustration-inspired techniques.

- Edited excerpt from "Computational Cerebral Hemodynamics A PhD Thesis at Center for Biomedical Computing" by Kristian Valen-Sendstad