Blood flow in vessels plays an important role in the physiological response of the vasculature in health and disease and in preserving the function of the end organs. While many of the descriptors that are important in evaluating the health of a vascular territory are well established, many others remain the domain of active investigation. The ability to establish the relationship between adverse hemodynamics and patient outcome has dramatically improved with the advent of robust high-resolution, noninvasive imaging measures of the local disease in the vessel wall, the prevailing flow conditions, and the status of the end organ. However, establishing causal connections between hemodynamic descriptors and physiological impact requires detailed knowledge of the spatial and temporal distribution of those descriptors. Computational Fluid Dynamics (CFD) methods are well suited to this task. The ever-increasing power of computational platform resources permits simulations of appropriately complex anatomic models in manageable compute times. In this chapter, the assumptions that underlie the CFD modeling approaches that are widely used in describing flow in the human vasculature are discussed. A description will be provided of the computational pipeline. Finally, examples of applications to patient-specific conditions will be presented.
Assessing the Velocity Field
In the same way that the motion of solid objectsin space is described by Newton’s equations of motion, fluid flow is described by the Navier-Stokes equations which include pressure terms and the fluid viscosity . These equations can be used to describe fluid flow in a broad range of conditions including external flows (e.g., air flowing around the exterior of a vehicle) or internal flows (such as blood flowing in a blood vessel.) However, it is not possible to provide analytic solutions for the Navier-Stokes equations for any but a few limited geometries and conditions. For this reason, most practical approaches to describing details of hemodynamics rely on numerical simulations, referred to as Computational Fluid Dynamics (CFD). Initial CFD studies were developed in an era of limited computational power and used rudimentary numerical schemes. In the early applications of CFD to the study of vascular physiology, idealized representations of relevant anatomical structures were used . Those began with 2D models which, while informative about important hemodynamic features, failed to capture important effects from secondary flow. The limitations of those models were recognized early on as the dominant role of geometry in governing critical features of hemodynamics became increasingly apparent. Extensions of the idealized models were developed to provide fully three-dimensional representations but were of limited use for patient-specific analyses [3, 4]. An example is presented in Fig. 17.1 of a schematic model of flow in the extracranial carotid arteries. The model is based on representative values of the bifurcation angle between the internal and external carotid arteries and uses typical values for the diameters of each vessel. General features of the flow in that territory include slow recirculating flow in the bulb of the internal carotid artery and a high velocity region in the medial aspect of the internal carotid artery. Planes transverse to the proximal internal and external carotid arteries are also useful to show those features. Such idealized models became increasingly sophisticated with inclusion of compliant walls, features of wall disease, and modeling of varying blood viscosity properties [5, 6].
More recently, advances in computational methods have been included into commercial solvers with much of that development being spurred by applications to the aerospace, auto, and other industries where flow conditions are significantly more extreme than in the human vasculature. These solvers are now sufficiently sophisticated to readily incorporate realistic geometries and physiological flow conditions and can therefore be applied to patient-specific anatomy and flow.
The interest in utilizing CFD methods in considerations of hemodynamics in vascular disease is manifold. While there are a variety of methods for assessing important features of hemodynamics in vivo, these have relatively coarse spatial and temporal resolution and suffer from technical and physiological challenges. Ultrasound is noninvasive and relatively inexpensive and has excellent temporal resolution for determining the full spectrum of velocities in a fixed insonation volume. It has strong abilities for quantifying peak velocities in flow jets which can be used to infer the degree of stenosis. It is unable to similarly map the velocity field through a three-dimensional volume and, in many cases, is obscured by bowel gas, calcifications, or overlying bone – as is the case for the brain. Furthermore, it is highly operator-dependent and is unsuited to measurement of volume flow. Catheter-injected angiography provides qualitative visualization of flow dynamics which is important for determining important physiological features such as vascular patency and the existence of collateral pathways. However, it is invasive and expensive and is not quantitative. MR imaging, particularly 4D Flow, has a number of desirable features: it is noninvasive and can depict the velocity field in space and time without limitations of overlying anatomy [7, 8]. This offers the possibility of determining derived descriptors such as volume flow and wall shear stress, the frictional force exerted by blood on the vessel wall. However, MR has moderate spatial and temporal resolution. Unlike ultrasound which determines the spectrum of velocities in the insonation volume, MR provides a voxel-averaged velocity measurement. Because of practical imaging constraints such as signal to noise ratios and acquisition times (which are often longer than 10 minutes), studies in a number of vascular territories are acquired with less than three or four voxels across the vascular lumen. The derivation of critical parameters such as the wall shear stress rely on an accurate measurement of the spatial gradient of velocities at the vessel wall, and MR-derived estimates of these measures must therefore be viewed with appropriate caution. A major attraction of CFD methods is the ability to specify very high resolution in space and time and calculate velocity fields with resolution far beyond anything that is currently achievable with in vivo imaging methods.
Computational Fluid Dynamics (CFD)
In most vascular territories and under a broad range of healthy and pathologic states, a numerical solution of the Navier-Stokes equations can be attained with limited and reasonable assumptions. While methods exist to accommodate each of the following, they are often neglected in conventional CFD calculations of hemodynamics: the vessel wall is assumed to be rigid; blood is assumed to be a Newtonian fluid; and blood flow is considered to be laminar without the presentation of turbulence. With those assumptions, CFD calculations can be conducted if the surface boundary of the vascular structure of interest is specified, if the inlet flow waveform is defined, and if the outlet flow conditions are appropriately conditioned.
Returning to the major assumptions, the importance of neglecting vascular complianceis not fully understood given the difficulty in conducting a simulation that includes wall motion – a so-called fluid-structure interaction (FSI) problem . Results of FSI compared to CFD with rigid walls indicate that neglecting compliance in a number of vascular territories (such as the intracranial vessels) has little effect . In other territories such as the aorta, larger differences are reported . It is, however, difficult to assess the extent to which the wall motion is correctly incorporated into the FSI models given their reliance on unreliable in vivo imaging to condition their boundary values. Furthermore, in conditions such as atherosclerosis, aneurysmal disease, or in the elderly population in general, resorting to an FSI simulation is likely unwarranted since vessels lose their compliance under those conditions, and conventional CFDmethods are likely to suffice.
Fluid viscositydescribes how the shear stress varies with changes in the shear rate, and if shear stress changes linearly with shear rate, the viscosity is constant. It is then referred to as a Newtonian fluid. Most CFD models assume that blood is a Newtonian fluid. There are in vivo situations where this condition is violated. On one limit, when blood recirculates slowly, red blood cells can aggregate, resulting in an increase in viscosity. Regions of slowly recirculating flow can occur in regions of aneurysmal dilatation. There are a number of analytical formulas that can be incorporated into CFD solvers that attempt to provide more physical models of blood viscosity at low shear rates . There are reports that these effects are relatively small. On the other limit, the viscosity of blood decreases substantially when passing through narrow (<300 micron) vessels when red blood cells move to the center of the vessel leaving only plasma near the wall of the vessel (the Fåhraeus–Lindqvist effect). These vessels are of the scale of arterioles and capillaries and are generally not of current interest for the determination of detailed featuresin their velocity fields.
The caliber and flow rates in healthyvessels are such that flow is laminar with the components of blood moving along well-ordered and predictable trajectories. Flow patterns can still be extremely complex with strong components of vorticity, but flow patterns remain highly predictable. As inertial effects, characterized by the product of velocity and vessel diameter, begin to dominate drag forces, characterized by viscosity, flow transitions from the well-ordered laminar condition to a more chaotic and unpredictable state and manifests as turbulent flow [13, 14]. Conditions that lend themselves to turbulent flow include flow distal to stenoses. Although velocities through the stenosis may increase dramatically, the reduced diameter can ensure that flow remains laminar in the stenosis throat. However, distal to the stenosis, high velocity persists in the flow jet which is now located in a region with a much larger diameter. Flow can then be turbulent with chaotic eddies being shed from the boundaries of the jet, generally accompanied by a dissipation of energy and an audible bruit . Correct numerical simulation of this situation requires a far greater degree of complexity than is required for laminar flow. Models for including turbulence into CFD simulations range from imposition of some simplifying assumptions such as in the most widely used model of turbulence, the k-epsilon model, where it is assumed that the turbulence viscosity is isotropic. Although inclusion of turbulence models into CFD analysis of blood flow in vessels is computationally expensive, commercial codes generally provide a k-epsilon model option [1, 15, 16]. More accurate simulations can be rendered using large eddysimulations  or an approach referred to as direct numerical simulation (DNS) . An example of the manifestation of turbulence in a DNS simulation is presented in Fig. 17.2 for flow through an idealized model of an arterial stenosis for flow through a regular cylindrical vessel with a slightly eccentric stenosis representing a 75% reduction in cross-sectional area. This presentation of velocity fluctuations shows a breakdown of the orderly flow in the flow jet into chaotic vortices several diameters distal to the throat of the stenosis. DNS simulations obtain their accuracy by directly computing all flow effects down to the smallest scales needed to accurately describe the relevant flow effects. They are thus extremely computationally intensive, and, while important in regimes such as hypersonic flow over airplane wings, DNS is rarely used in application to physiologic flows. In situations of high flow rates, such as distal to stenoses, flow conditions can be such that flow is no longer truly laminar, and the physics of flow dictates that flow becomes transitional and finally manifests true turbulence. In that case, it is challenging, and sometime impossible – even with an extremely high resolution – for code defined for laminar flow to converge to a stable solution, a situation that is manifested by increasingly long computational run times and inconsistent data. To avoid these types of computational failures, commercial CFD solvers are often constructed to include an artificial damping term which ensures that the solution remains stable. In that case, the user needs to realize that although a solution is generated, it is likely inaccurate, and appropriate caution must be used in drawing conclusions from those results.
In general, many physiological conditionsof interest can be closely approximated by laminar flow through rigid-walled vessels with Newtonian viscosity. For those cases, conventional CFD simulations can then be applied to generate highly accurate estimates of the velocity field. However, in cases where those conditions are not met, in vivo imaging modalities, in particular 4D Flow MRI methods, provide the intriguing prospect of more accurately determining the velocity field than is possible with CFD as the true physiological behavior is inherently present on a patient-specific basis, and does not require modeling. In the remainder of this chapter, we will restrict ourselves to a discussion of the application of conventional CFDto the analysis of hemodynamics in vivo.
CFD in the Laminar Flow Regime
A CFD analysisprovides a numerical solution of the Navier-Stokes equations, the governing equations of fluid motion. The key components required as input to the numerical model are a description of the lumenal surface of the vessels of interest and specification of the inlet and outlet flow boundary conditions.
Patient-specific modeling requires in vivo images of the vessels of interest [17–19]. The resolution of the images must be sufficient to permit an accurate representation of the associated velocity fields – preferably with greater than five to ten voxels across the vascular lumen. Achieving this is most challenging in regionsof pronounced curvature such as at the neck of a saccular aneurysm or in stenotic vessels. For example, an 80% diameter stenosis of the extracranial carotid arteries corresponds to a residual lumenal diameter of little more than 1 mm. For 3D volumetric modalities (MRA or CTA), this provides at most 2–3 voxels across the stenosis. This is even more limiting in territories with stenoses of smaller caliber vessels such as the coronary arteries or the intracranial vessels. However, current imaging modalities provide 3D angiographic images of the vascular lumen with high contrast to noise ratio to the adjacent tissue and with adequate spatial resolution to support high-quality CFD in relatively smooth vessels whose caliber is 3 mm or larger. Segmentation of the lumenal surface from the 3D data set can then be performed and be provided as input into the solver for the geometric boundary condition. Care is needed to avoid misrepresentations of important vascular features as in cases where aneurysmal bulges fold back into close proximity with the parent vessel. Unless the imaging modality has sufficient resolution, the two distinct regions can appear to be merged into one volume. This is illustrated in Fig. 17.3 where a rotational DSA study of an aneurysm of the internal carotid artery clearly identifies the aneurysm as a distinct saccular structure, whereas the lower resolution CE-MRA study merges aneurysm and parent vessel giving the appearance of a fusiform dilatation.