Resistances and capacitances used for CFD simulations of adolescent patients .
Cardiovascular diseases (CVDs) afflict many people across the world; thus, understanding the pathophysiology of CVD and the biomechanical forces which influence CVD progression is important in the development of optimal strategies to care for these patients. Over the last two decades, cardiac magnetic resonance (CMR) imaging has offered increasingly important insights into CVD. Computational fluid dynamics (CFD) modeling, a method of simulating the characteristics of flowing fluids, can be applied to the study of CVD through the collaboration of engineers and clinicians. This chapter aims to explore the current state of the CMR-derived CFD, as this technique pertains to both acquired CVD (i.e., atherosclerosis) and congenital heart disease (CHD).
Cardiovascular disease (CVD) is a common cause of morbidity and mortality around the world . Each year, CVD afflicts more than 1.9 million in the European Union and at least 800,000 in the United States of America (USA). The cost of health care has been increasing exponentially over the years with estimates noting about 196 billion Euros per year and 207.3 billion dollars spent annually on direct/ indirect costs of cardiovascular disease . In the United States, almost 801,000 Americans died from heart disease, stroke, or other CVD in 2013 and ~85.6 million Americans live with some form of CVD [1, 2]. While far fewer have congenital heart disease (CHD), these structural problems still afflict about 1% of live-born children and many require intricate operations for treatment/palliation. American Heart Association data suggest that approximately two million Americans live with some form of CHD, as do millions of Europeans and other individuals globally .
Mechanical stimuli (such as pressure and strain) have been shown to influence the onset and progression of CVD. For example, wall tension can be estimated as the product of vessel radius and blood pressure (BP). Chronic changes in wall tension initially driven by increases in pressure are believed to be the stimuli for vessel thickening, which then restores wall stress to a preferred operating range . Strain also reflects aortic deformation as present with hypertension and aneurysm formation [5, 6].
Of particular interest is wall shear stress (WSS) (Figure 1), which can be generally defined as the frictional force exerted on the walls of a vessel as a result of flowing blood. Areas of low time-averaged WSS are known to correlate with sites of atherogenesis and inflammation from prior studies [7–12]. These studies suggest that specific alterations in mechanical stimuli manifesting from CVD may be the stimuli ultimately leading to morbidity. Hence, there is value in knowing how and when they lead to structural, as well as functional and hemodynamic vascular changes.
The flow patterns of fluids are governed by partial differential equations that represent conservation laws for quantities such as mass and momentum . Predicting the impact of such flows in biomedical applications, as well as within other scientific disciplines, is time consuming and costly without computational tools. Computational fluid dynamics (CFD) is a method of simulating fluid passing through or around an object, in this case blood vessels, by replacing the partial differential equations with algebraic equations that can be solved numerically using digital computers. There are several open source and commercially available CFD software packages that facilitate the completion of these calculations with user-friendly interfaces that accept various types of medical imaging data. The workflow for each software package then allows a user to generate hemodynamic results (as presented throughout this chapter) with an appreciation for the governing mathematical equations. However, in performing CFD modeling, there are several important clinical and engineering considerations that should be kept in mind.
The general requirements for studying blood flow for patients with CVD and/or CHD using CFD include first creating a model of the vessel geometry from three-dimensional (3D) medical imaging data. Typically, cardiac magnetic resonance (CMR) or computed tomographic (CT) imaging data are readily available 3D data that provide clear definition of anatomy. CFD also requires prescription of the flow information for the entrance and exit of vessels which is gleaned from phase contrast (PC) velocity-encoded CMR data acquired at these sites. It is also necessary to prescribe the hemodynamic state beyond the borders of the 3D imaging dataset in order to obtain physiologic results (e.g., setting downstream resistance to obtain a realistic range of pressure). Direct or indirect assignment of this inlet and outlet information is referred to as “setting the boundary conditions.” Rheological properties, such as blood density and viscosity, are then assigned. The last step in the process entails the use of a powerful computer or cluster of computers to solve the governing equations for fluid flow throughout a version of the vessel’s geometry, which is represented as a computational mesh.
More specifically, the first step when performing CFD involves creating a computer aided design (CAD) model within the vascular regions of interest from medical imaging data. The CMR imaging focus of the current work most often uses data from either breath-held electrocardiographically (ECG)-gated, magnetic resonance angiography (MRA) or a respiratory-navigated, ECG-gated 3D nongadolinium-enhanced, entire heart CMR sequence. CFD can be used with other sequences and imaging modalities as well. The models created can provide the geometry on a patient-specific basis when it is desirable to focus on a clinical question for a specific patient, or for a group of patients with similar pathology . Alternatively, representative or idealized models are also sometimes used, where the geometry within the idealized model is informed by measurements taken from data within a collection of CMR scans across one or more patients. To date, our workflow has primarily used SimVascular (simvascular.github.io, latest version, La Jolla, CA, USA), but other commonly used software packages that facilitate the import and segmentation of CMR data include Cardiovascular Integrated Modeling and Simulation (CRIMSON, www.crimson.software), the Vascular Modeling Toolkit (VMTK, www.vtnk.org), and Mimics (biomedical.materialise.com/mimics, Plymouth, MI, USA), just to name a few. Each of these programs then facilitates discretization of the CAD model created from CMR data by interacting with some type of meshing software (e.g., MeshSim, www.simmetrix.com, Clifton Park, NY, USA). The parameters selected during the segmentation and meshing steps used in creating a computational version of the vasculature from CMR can have a large impact on the results obtained. For example, the accuracy of WSS indices (see below for details on specific WSS indices of interest) depends greatly on vessel radius, and therefore on the confidence with which segments or 3D boundaries for the CAD model are created from CMR data. In its simplest form (e.g., plane Couette flow), WSS is the product of viscosity and the near-wall velocity gradient (Figure 1) . This change in velocity from the wall of an artery to the next nearest location is largely determined by details of the computational mesh that are established. Note that the velocity on the wall is often zero due to a no-slip condition. Unfortunately, the computational costs of obtaining CFD results increase as a function of mesh density. The trade-off is often managed in today’s CFD studies by using adaptive-meshing approaches [15, 16] yielding smaller meshes that strategically place more elements where they are most needed, such as near the wall, for improved accuracy when determining WSS.
In practice, the density for blood is typically selected from the literature, and a Newtonian assumption (i.e., constant blood viscosity) is most often employed. Although blood is a shear-thinning fluid, meaning that its viscosity decreases as it is deformed, approximating its behavior as Newtonian is generally thought to be reasonable for the range of shear rates experienced by the portions of the vasculature that are highlighted in the CFD studies below. A unique aspect of CFD software packages designed specifically for biomedical applications is their ability to implement boundary conditions that replicate normal and CVD physiology . For example, the time-varying opposition to blood flow (i.e., impedance spectra) can be calculated from pressure and flow measurements made at the same location in the vascular system, but Windkessel models are often used as an approximation of the impedance given the impracticality of the necessary measurements within a clinical setting . It is an increasingly common CFD modeling standard to employ three-element Windkessel representations derived from CMR-acquired PC velocity-encoded (VENC) flow data for inlet and outlet boundary conditions. More recent boundary condition advancements include cardiac function through the use of closed-loop lumped-parameter networks (LPNs) with CFD models. Closed-loop LPNs were initially developed to model single ventricle physiology  and are now being used to characterize flow patterns in the coronary arteries and other vascular regions. These closed-loop LPN models are often tuned to match measured clinical data (i.e., cardiac output (CO), stroke volume, blood pressure, and ejection fraction), and then coupled to patient-specific simulations that use specialized computers to solve the conservation of mass, balance of momentum, and (in some cases) the vessel wall elastodynamics equations . The results are vascular hemodynamic indices that may aid the understanding and care of CVD, following detailed analysis of the resulting indices.
The application of CFD to clinical cardiovascular problems typically requires teamwork between clinicians and engineers, with a step-wise approach to gather and analyze specific data for the study of clinically important blood flow issues (Figure 2). As an example, understanding flow in the aortic arch by CFD requires CMR data for the anatomy of interest (MRA or other 3D data), as well as blood flow measurements (from PC velocity-encoded magnetic resonance imaging sequence, PC-MRI) for all major inflow and outflow vessels (in this case, the ascending and descending aortic flow, as well as the brachiocephalic vessels, as denoted by the black dotted lines in Figure 2). Blood pressure measurements from each limb are also used in assigning boundary conditions.
When creating CFD models for the aortic arch, the inlet is most often aortic flow. Typically, this can be imposed in one of several representations including plug flow (i.e., uniform flow), parabolic flow (as shown in Figure 1), or a patient-specific flow obtained from CMR phase contrast velocity-encoded (PC-MRI) data that intrinsically incorporates elements of both. In vivo, the velocity profile is determined by a ratio of inertial (i.e., those forces driving the flow) to viscous (those forces impeding the flow) forces . If the impact of viscous forces is large, such as in a smaller artery with lower velocity, then the velocity profile will be parabolic. However, in a larger artery such as the aorta, inertial forces are more pronounced, leading to a more uniform velocity profile except near the walls where the impact of viscous forces manifests. It is, therefore, desirable to use a retrospective ECG-gated phase contrast velocity-encoded sequence to sample the velocity profile, which should intrinsically capture these features, downstream of the valve for direct input into a CFD model, but this requires appropriate through-plane and in-plane velocity encoding to adequately resolve flow features. This approach may also be difficult to implement within the typical imaging time available for a clinical case, as it requires specialized sequences and obtains data that are more detailed than that commonly used in clinical imaging. An alternative approach is to construct CFD models with their inlet beginning at the aortic annulus, impose the measured blood flow waveform as an assumed velocity profile at the model inlet (with plug profile or patient-specific flow profile), and allow the curvature and related geometry of the arch to influence the resulting flow patterns . This approach does not require specialized sequences, minimizes the introduction of noise at the model inflow due to inadequate velocity encoding, and allows for improved temporal resolution compared to three-component PC-MRI, using multiple planes for flow assessment .
In our workflow for CFD simulations of the thoracic aorta to date, which is shown schematically in Figure 2, outlet boundary conditions have used measured BP and the PC-MRI flow data from each brachiocephalic vessel and the descending aorta together with an approach called the “pulse pressure method” to assign elements to the Windkessel parameters [21, 22]. The regional resistances, arterial capacitances, and distal resistances are estimated for each case and used in patient-specific modeling (Table 1) . Given the very small nature of the intercostal arteries, and the fact that their flow distributions are not typically characterized by imaging, these are not usually employed in the modeling. While flow to the intercostal arteries is important physiologically, characterizing the flow distributions requires multiple phase contrast image acquisition frames or 4D (four-dimensional) blood flow imaging using CMR that is often beyond the time available and needs of the clinically ordered session.
|Young’s modulus||E (dyn/cm2)||3.99E+06–3.57E+07||3.23E+06–1.17E+07|
|Innominate artery||Rc (dyn·s/cm5)||429–489||320–939|
|Left common carotid artery||Rc (dyn·s/cm5)||687–1720||702–1390|
|Left subclavian artery||Rc (dyn·s/cm5)||662–1830||500–1480|
|Descending aortic outlet||Rc (dyn·s/cm5)||99–231||101–232|
Fluid-structure interaction (FSI) simulations represent a specialized version of CFD modeling that considers the pulsatility and elastic nature of the arterial system. FSI, therefore, has the potential for introducing more clinically relevant features when determining indices such as instantaneous WSS, time-averaged WSS, and oscillatory shear stress, by including more realistic local deformations (Figure 3). For example, considering again the simple case of WSS calculated as the product of the near-wall velocity gradient and viscosity, the movement of the vessel wall as occurs in vivo will impact this calculation. Including local deformations in WSS calculations therefore has the potential to provide more realistic results.
Indices of WSS are calculated from the time-varying velocity field representing flow patterns within the aorta. Blood flow in the aorta, for instance, has long been recognized to be helical and can be replicated by CFD (Figure 4) . A helical flow pattern has many positive features including (1) facilitating blood flow transport and suppressing turbulent blood flow, (2) preventing the accumulation of atherogenic low-density lipoprotein (LDL) particles on arterial luminal surfaces, (3) enhancing oxygen transport from the blood to the multilayered arterial wall, (4) reducing the adhesion of platelets and monocytes on the arterial surface, and (5) optimizing flow patterns within origins of the brachiocephalic vessels . Helical blood flow patterns may therefore aid in protecting the arteries from the pathologic mechanisms of atherosclerosis, thrombosis, and intimal hyperplasia, as well as from dilation, aneurysm formation, and dissection .
CFD may offer predictive capabilities for difficult clinical problems as will be discussed in this chapter. For example, MRI during exercise has been pursued since the late 1990s, but is not trivial to implement and is therefore not conducted routinely. More specifically, CMR has been used to quantify blood flow during supine cycle ergometry in the ascending aorta, pulmonary artery [26, 27], abdominal aorta [28, 29], and left ventricle . Indices of WSS and cardiac output were quantified in combination with CFD modeling from a recumbent cycling protocol developed for imaging of the abdominal aorta [31–33]. CFD was also used for patient-specific models of blood flow in the thoracic aorta to quantify indices of WSS under simulated exercise conditions using changes in blood flow and resistance estimated from various literature sources [34, 35]. More recent exercise protocols for use with CMR go one step further. A pilot study developed a protocol to obtain PC-MRI blood flow measurements in the thoracic and brachiocephalic arteries during a three-tiered supine pedaling, and then related these measurements to noninvasive tissue oxygen saturation levels acquired with near-infrared spectroscopy (NIRS) during assessment using the same protocol . The goal of this work was to use NIRS data as a surrogate for exercise PC-MRI data when setting boundary conditions for future CFD studies of the thoracic aorta under simulated exercise conditions. Relationships and ensemble-averaged PC-MRI inflow waveforms are provided in an online repository for this purpose .
There are several indices of WSS that have been associated with locations of atherosclerosis in various vascular beds. The most common of these is time-averaged WSS. WSS is actually represented by vectors that change with each fraction of time within the cardiac cycle. Most reports simply present time-average representations on the wall within the region of interest; this is done for simplicity and because the mechanisms and details by which a particular WSS index leads to neo-intimal thickening are not yet precisely known. Areas of low time-averaged WSS have also been found in a rotating pattern down the descending aorta , correlate with areas of plaque deposition , and are accentuated after correction of CoA . However, there is evidence suggesting that temporal and spatial changes in WSS may also serve as stimuli for neo-intimal thickening. Oscillatory shear index (OSI) is also commonly reported in CFD studies . OSI is a measure of WSS directionality in which lower OSI values indicate that WSS is oriented predominantly in the primary direction of blood flow, while a value of 0.5 is indicative of bidirectional WSS with a time-average value of zero. Theoretically, regions of low WSS magnitude and high OSI are less likely to experience fluid forces that promote washout of noxious and potentially atherogenic materials in contact with the arterial surface (e.g., LDL). In general, adverse values for these WSS indices (e.g., ~15 dyn/cm2 for the thoracic aorta) are expressed as thresholds for low magnitude instantaneous and time-averaged WSS. OSI greater than 0.1 are considered adverse, as are spatial and temporal WSS gradients greater than 100 dyn/cm2 and ±200 dyne/cm2/s, respectively) [12, 21, 39, 40]. Understanding these hemodynamic principles and differences between indices that use them, the practitioner may apply CFD modeling to describe blood flow patterns in the setting of typical clinical cardiovascular pathology—atherosclerosis and congenital heart disease.
Any discussion of CFD for use in generating flow patterns and indices of WSS would be remiss without a discussion of potential limitations, particularly with respect to advancements in CMR that have facilitated quantification of the same indices through direct processing of data from a more advanced clinical imaging session. For example, 4D blood flow imaging using CMR has been used to investigate flow disruptions in the thoracic aorta [37, 41, 42]. These methods do require specialized pulse sequences that may be outside of the clinical workflow for some centers, are complex to acquire and post-process, and prolong scan time (~15–20 min for a 4D-navigated flow-imaging dataset). Additionally, these methods suffer from low spatial/temporal resolution relative to CFD simulations, which could limit the precision of the WSS results compared to those WSS data calculated from a properly conducted CFD simulation where uncertainties in the processes employed were appropriately considered [43–45]. For example, the limitations and variability in the model creation process must be carefully controlled by acquiring a high-resolution 3D representation of the anatomy (by MRA or equivalent sequence) and PC-MRI data. Furthermore, the individuals building CAD models must be rigorous when using software, especially with regard to selection of a mesh of sufficiently high density. In this manner, there can generally be a high level of confidence in the results assuming that physiologic boundary conditions have been implemented. WSS results from 4D blood flow imaging can suffer from the same potential limitations of traditional CFD modeling techniques, if the spatial resolution and post-processing operations do not carefully capture the near-wall velocity gradient.
Additional limitations may exist when CFD modeling is attempted for biologic systems. First, clinical acquisition of suboptimal 3D anatomic data can adversely affect models. Acquisition of non-robust 3D data may lead to oversimplification of anatomy during modeling. Second, suboptimally acquired PC-MRI flow data, with inadequate temporal resolution or low dynamic range, may underestimate flow. Third, ignoring cardiac motion, which can affect the accuracy of flow determinations, may lead to errors in modeling . Finally, assigning accurate boundary conditions (especially resistances of the peripheral vasculature or the microcirculation when coronary modeling is attempted) can be challenging, as detailed information may have to be gleaned from the literature because patient-specific data may be difficult to attain. Such resistances can also vary over time and certainly may differ in healthy and diseased states .
2. Computational fluid dynamics modeling applied to the study of atherosclerosis
Atherosclerosis is a complex pathobiologic process which begins with endothelial dysfunction, involves a cascade of particles (including white blood cells, chemotactic factors, and smooth muscle cells), and leads to progressive changes in blood vessel walls (Figure 5) .
After the initial endothelial dysfunction, atherosclerotic plaque develops over many years [49, 50]. The initial feature in the evolution of plaque, seen on autopsy in children as young as 10–18 years old, is the fatty streak . Many studies have shown that intimal thickening, as assessed by ultrasound, can be detected as plaque burden increases. At first, plaque causes outward remodeling of a blood vessel, maintaining the lumen’s dimensions, followed by plaque encroachment on the vessel lumen . By the time atherosclerotic plaque causes stenosis (and decreases luminal dimensions by 60% or more), symptoms can be seen under conditions of great oxygen demand, such as exercise. Thus, if the carotid vasculature is affected, cerebral transient ischemic attacks might be manifested, whereas if the coronaries are affected, then myocardial ischemia can occur and be manifested as angina. Peripheral plaque can lead to symptoms of left pain—either claudication or rest pain.
Studies have emerged to characterize the vascular hemodynamic effects of plaque by employing computational modeling based on CMR data. This area of CFD research attempts to systematically study vascular WSS and OSI as a way to better understand where plaque forms and how plaque influences blood flow. In a study of adults with preexisting aortic plaque, for example, time-averaged WSS patterns existed in a rotating pattern down the thoracic aorta that correlated with areas of atherosclerotic plaque . WSS distribution, therefore, depended on the axial level and circumferential location in a given axial level of the aorta (Figure 6). Similarly, many other CFD studies have shown low WSS and high OSI are associated with atherosclerosis. In an adult coronary CT study, coronary segments with established plaque exhibited lower WSS compared to adjacent normal areas . Within plaques, WSS was lower, and plaque volume was higher in mid-plaque compared to upstream and downstream areas (Figure 7) . In a study of carotid atherosclerosis, low time-averaged WSS and high OSI were seen in areas of increased mature plaque volume (Figure 8) .
This prior work served as motivation for a novel pilot CMR study of early vascular changes in pediatric patients with type 1 diabetes. Twenty preadolescent and adolescent patients with type 1 diabetes (median age of 15.8 years, range of 11.6–18.4 years) were enrolled in this prospective CMR study and compared with eight control subjects (15.8 years with a range of 10.3–18.2 years). Using same-day brachial artery reactivity testing, lower flow-mediated dilation was seen for the subjects with diabetes (p = 0.036), as expected—indicating the presence of endothelial dysfunction in this group, as seen by others . When patient-specific CFD models were created from CMR data, those with diabetes had more aortic regions with high time-averaged WSS when compared with controls, although the groups had similar OSI (Figure 9) . Many cardiovascular risk factors, including type 1 diabetes, induce physiological outward arterial remodeling (dilation) that begins in response to overall higher initial laminar shear stress (Glagov phenomenon). With vascular inflammation, remodeling progresses, resulting in adverse shear stress in larger arteries . This pilot pediatric diabetes study may provide a glimpse into early vascular remodeling (i.e., where wall shear stress is still high and the aorta, which has begun to stiffen, has yet to dilate). Longitudinal studies are needed to understand how areas of WSS and OSI change with aging, and as atherosclerosis progresses. Understanding this may enhance therapies for early treatment of atherosclerosis by aiding the development of medications that favorably alter WSS and OSI .
3. Computational fluid dynamics modeling and congenital heart disease
CFD modeling is useful in understanding not only blood flow as it relates to atherosclerosis but also blood flow in a number of structural heart diseases, including aortic coarctation (CoA), aortic dilation, and aneurysms (as with bicuspid aortic valve and connective tissues diseases), and for more complex diseases, such as repaired Tetralogy of Fallot (TOF) (to aid percutaneous interventions) and single ventricle physiology (to optimize the Fontan operation’s total cavo-pulmonary palliation). Collaboration between clinicians and engineers is important, so that optimal workflow can be achieved to allow timely, patient-specific model creation for consideration in clinical decision making. Table 2 shows some situations where CFD simulations have been applied to clinical cardiovascular medicine; these will be discussed here.
|Area of interest||Applications||Clinical impact||Limitations|
|Coronary artery disease (CAD)||Models based on|
angiography or CT
scan may predict
develop optimal therapy
and predict outcome.
|Models may be|
used to predict
and risk of
may aid in
up by reducing
Models may aid
care by allowing
to stent design.
and low image
(FSI) may aid
|Arterial wall||Models determine|
shear stress (WSS)
(OSI) related to
|Models may predict|
sites likely of
of therapy with
devices (i.e., stents).
|Detailed anatomy of|
may be challenging
to model; using
|Congenial Heart Diseases|
Tetralogy of Fallot
of blood flow
may aid therapy.
|Some have cited|
this as the “ultimate
nature of each
category of CHD,
the complex anatomy,
3.1. Aortic CFD modeling for congenital heart diseases
Building upon the CFD modeling efforts summarized above that focused on the thoracic aorta as a vascular surrogate for the coronaries in the study of atherosclerosis, CFD modeling research naturally extended to the diseases of the thoracic aorta—especially to the study of CoA. CoA is a relatively common congenital narrowing of the proximal thoracic aorta, occurring in about 8–11% of patients with CHD, and usually involves the thoracic aorta just after the origin of the left subclavian artery (juxta-ductal CoA). Patients with unrepaired CoA may suffer from ill-effects of hypertension (or in infancy, from cardiovascular collapse with ductal closure). Even after successful CoA repair, many are followed due to the presence of bicuspid aortic valve (up to 85%) with or without stenosis, residual hypertension (7–33% of patients) , re-coarctation (5–50%) , aortic aneurysm, aortic dissection, and possibly early atherosclerosis . Recent CFD modeling studies have shown altered time-average WSS after end-to-end anastomoses . CFD modeling has also been employed to assess WSS after Dacron patch repair of CoA —an operative procedure now known to be complicated by aneurysm formation. Aneurysms, in turn, introduce local geometric abnormalities leading to heterogeneity in WSS that have historically been linked to adverse consequences such as cellular proliferation and plaque progression (Figure 10). Bicuspid aortic valve is a frequently found coexisting CHD for patients with CoA—occurring in up to 85% of these individuals. Recently, sophisticated CFD simulations, using a custom MATLAB® program (MathWorks, Natick, MA, USA) to facilitate segmentation of a common variant (right-left cusp fusion) of bicuspid aortic valve, attempted to account for the influence of the valve on flow patterns and turbulence in the ascending aorta . This represents an added step toward realism, and constitutes an alternative to imposing measured PC-MRI data directly when creating patient-specific CFD models for patients with CoA and bicuspid aortic valve. Additionally, attempts to account for cardiac motion have been applied to the study of aortic flow patterns via CFD. Cardiac motion seems to be most prevalent in altering ascending aortic (AAo) flow rather than flow in the arch or descending aortic flow, likely due in part to less tissue tethering for AAo than the descending aorta . Differences in time-averaged WSS quantified in this study from simulations with the measured PC-MRI inflow waveforms, as compared to motion-compensated cardiac waveforms, were more pronounced than differences from the model creation or mesh dependent aspects of CFD discussed above. These results suggest that accounting for cardiac motion when quantifying blood flow through the aortic valve can lead to different conclusions for hemodynamic indices, which may be important, if these results are ultimately used to predict patient outcomes . CFD modeling may, thus, aid in optimal management of aortic and aortic arch diseases—whether by influencing operative technique or optimizing devices through material development.
Modeling can aid analysis of aortic dilation, which occurs in patients with bicuspid aortic valves, or in those with connective tissue diseases, such as Marfan, Loeys-Dietz, and Ehlers Danlos syndromes. It is unclear if this is causal, or, more likely, contributing to the underlying vascular pathology. In these populations, CFD has demonstrated adversely high OSI in the ascending aorta, an area prone to dilation . Many of these patients undergo an operation to treat an excessively dilated aorta, in order to prevent aortic rupture. Operative techniques used might include total aortic root and valve replacement (TAR), valve-sparing root replacement (VSRR), or the novel and less invasive procedure of placing a personalized external aortic root support (PEARS) introduced by Golesworthy et al. . Using MRI-derived data for CFD modeling, Marfan patients have recently been studied after operation for placement of PEARS, and although qualitative hemodynamic indices appeared similar, some small differences in quantitative measures of helical flow were seen pre- and post PEARS in a small cohort. Larger, longitudinal studies will be needed to understand the hemodynamic effects of these operations. CFD may be used in the future to optimize therapy  by aiding the creation of aortic “sleeves” from materials which impart better WSS properties to the patients.
3.2. Single ventricle CFD modeling
Over the last decade, CFD modeling has increasingly been applied to patients with single ventricle physiology (Table 3), who often require multiple palliative operations in infancy and childhood. The goal of operative therapy is to have the functioning ventricle as the systemic pump, and to secure a source of pulmonary blood flow. Unobstructed flow to the systemic and pulmonary circulations is the goal—to achieve widely patent branch pulmonary arteries and no residual aortic arch obstruction .
When the aorta is small, as in hypoplastic left heart syndrome (HLHS), shortly after birth, the infant is palliated with a Norwood operation to create a neo-aorta from side-side anastomosis of the main pulmonary artery (MPA) and native aorta. Usually, arch repair is undertaken in the same setting. Atrial septectomy occurs too ‐ to allow mixing of systemic and pulmonary venous flow, and pulmonary flow is guaranteed by either an aortopulmonary arterial shunt (i.e., either a Blalock-Taussig (BT) or a central shunt) or right ventricular (RV) to pulmonary shunt (i.e., the Sano shunt). Some time between 3 and 6 months’ of age, when the pulmonary vascular resistance is acceptable, the patient undergoes cavo-pulmonary shunt (Figure 11) to direct a portion of the systemic venous blood to the lungs for oxygenation. Finally, when the child is older (18 months to 4 years of age), the remaining systemic venous blood and hepatic blood are directed to the lungs via completion of the Fontan (Figure 12). CMR is often used in clinical follow-up of these patients.
|RV morphology||LV morphology||RV or LV morphology|
Using CMR data, CFD modeling in single ventricle patients has focused on understanding energy losses occurring in these unique low-velocity flow Glenn and Fontan circuits. Modeling has aimed to understand blood flow distribution to the lungs by using branch pulmonary arterial PC-MRI data in CFD models. In this manner, CFD has also led to a better understanding of the pulmonary distribution of hepatic flow and the elusive “hepatic factor” which has been implicated in the development of pulmonary arteriovenous malformations [68–72].
Catheterization-based CFD modeling, such as that included in the work by Migliavacca et al. , has advanced the clinicians’ understanding of the unique Norwood circulation by showing that (1) larger shunts diverted an increased proportion of cardiac output to the lungs and away from systemic perfusion, resulting in poorer oxygen delivery and pulmonary overcirculation, and that (2) the systemic vascular resistance exerted more effects on hemodynamics than pulmonary vascular resistance. CMR-based CFD modeling efforts have been less common in this Norwood population, because of a number of challenges. First, highly turbulent flow from shunts (BT, central or Sano) can lead to inaccuracies in modeling, because of suboptimal PC-MRI data. Second, as this Norwood population is very young ages (i.e., neonates to a few months of age), often 3D anatomic imaging data (CMR or CT) are not readily available, as it is not typically acquired during routine clinical care, which relies predominantly on echocardiographic data. Finally, without catheterization data simultaneous with 3D anatomic imaging, accurate pulmonary and systemic resistance data may not be available, thereby leading to inaccuracies in CFD modeling. Furthermore, the influences of anesthesia (used during catheterization or 3D image acquisition) on such resistance data may lead to models that may not adequately represent real-life clinical conditions. Nonetheless, computational modeling of this unique Norwood physiology has been attempted and refined over the last two decades  with a nice review of the challenges provided by Pennati et al. .
CFD modeling of cavo-pulmonary shunts has been more limited relative to the CFD study of the total cavo-pulmonary shunt (Fontan). In pilot research, the Stanford Institute for Computational and Mathematical Engineering, though, systematically studied five Glenn patients, incorporating CMR data and catheterization data into their CFD models. Inflow data came from the superior vena cava (SVC) PC-MRI flow data, while outflow data were more complex and depended on the patient’s specific pulmonary tree. They first determined right and left lung flow split from the right pulmonary artery (RPA) and the left pulmonary artery (LPA) PC-MRI flow data and then determined the total resistance—i.e., Rp (proximal resistance) plus Rd (distal resistance) for the pulmonary tree. They calculated the mean flow in each branch as being proportional to the outlet surface area and calculated the downstream resistance by the mean pressure gradient (obtained at catheterization) between SVC and left atrium. The authors show low WSS, complex Glenn flow patterns at the caval-pulmonary anastomoses with a transition to laminar flow more distally in the lung, and a complex pressure waveform which dampens after the anastomoses. Power loss in this small cohort was low, and the efficiency of flow was high. The complexity of the pulmonary tree seems to add computational time, but more work is needed to understand how many pulmonary branches should be modeled for accuracy. This study highlights the complexity of assigning appropriate inlet and outlet boundary conditions—a problem that is compounded when patients are studied by either CMR or cardiac catheterization while they are under anesthesia which alters systemic and pulmonary resistances .
CFD may be useful for bilateral bidirectional Glenn connections , where patients have the persistence of the left and right superior vena cava. Case reports of CFD modeling in this unique Glenn circuit show differential lung flow due to differences in pulmonary resistance (which may result, e.g., from unilateral lung disease (e.g., pneumonia)) or differences in branch pulmonary arterial dimensions. Furthermore, de Zelicourt et al.  have noted that unbalanced lung perfusion may affect pulmonary arterial growth. Thus, constructing the best Glenn circuit with attention to downstream branch pulmonary arterial flow may be essential to long-term patient health.
Surgeons have always tried to avoid “right angles” in the construct of the unique bidirectional Glenn and Fontan circuits, as they perceived unfavorable flow disturbances in these regions. de Leval et al. [79, 80] described various methods for palliation of HLHS. In 2007, Bove et al.  studied Fontan circuits of different varieties using computational simulations and showed that when either the total cavo-pulmonary connection (TCPC) or the extra cardiac connection (ECC) is performed after a bidirectional Glenn anastomosis, caval offset of the superior and inferior vena cavae (IVC) can be achieved by beveling the IVC portion of the connection to either the right or the left lung. They demonstrated that beveling the TCPC to the right conferred a significant advantage to the TCPC. Similarly, when the ECC was beveled toward the left lung, important differences were found in flow distribution, but not power losses. This research has been continued by many with continued scrutiny regarding possible power losses in this low-velocity circuit, which is prone to swirling of blood flow due to competitive flow from the superior and inferior vena cavae [81, 82]. Some centers, such as the Children’s Hospital of Philadelphia and Stanford University, among others, have conducted “virtual operative procedures” to study the hemodynamic effects of a given operation by employing CMR-based CFD modeling techniques. These computations have led some to alter the Fontan circuit by employing a “Y graft” for the Fontan , as first described by two groups—Marsden et al.  and Soerensen et al. . Not only is the energy loss reduced, especially during simulated exercise, but also IVC flow (and the seemingly critical “hepatic flow”) can be more equally distributed to the right and left lung fields, which is potentially protective against the development of pulmonary arteriovenous malformations (Figure 13) [83, 85, 86].
3.3. Tetralogy of Fallot CFD modeling
Patients with Tetralogy of Fallot (TOF) (i.e., subpulmonary and/or pulmonary valve stenosis with ventricular septal defect, overriding aorta, and right ventricular (RV) hypertrophy) typically undergo operative repair in infancy. Long term, their outcome is related to the degree of chronic pulmonary regurgitation (PR) (i.e., pulmonary valve leakage), and resulting RV dilation and RV dysfunction. Restrictive RV physiology correlates with larger RV and more PR after repair . Pulmonary arterial compliance impacts the amount of regurgitation . Thus, establishing and maintaining appropriate sized branch pulmonary arteries is essential, as elevated resistance distal to compliant arteries exacerbates PR. Furthermore, the initial type of TOF repair may impact how much PR a patient has—with those having RV outflow tract (RVOT) transannular patches having considerably more PR and more dilated RV than those with RV—pulmonary arterial conduits . Patients with repaired TOF, in the current era, are frequently studied by CMR to better understand when the PR has progressed sufficiently to warrant operative intervention for pulmonary valve replacement .
CMR-based CFD modeling has been applied to the study of TOF patients to understand how branch pulmonary arterial geometry (i.e., diameters and the bifurcation angles for the right and left pulmonary arteries) influences pulmonary regurgitation (Figure 14) [91, 92]. Chern et al. found that regurgitation occurs first from the LPA and suggested that it may be due to the small angle between LPA and MPA. The authors acknowledge the limitations of their CFD models which do not account for the influences of either distal pulmonary vascular resistance or ventricular hypertrophy (diastolic pressure) on pulmonary regurgitation.
Some researchers have proposed that a “reference geometry atlas” be available for the branch pulmonary arteries onto which specific patient data (derived from CMR 3D MRA imaging) may be mapped, so as to reduce computational time . This is a novel idea that may allow CFD to move more readily from the research realm to clinical setting. Others have described in silico models to permit “virtual surgery” for patients with TOF and left pulmonary arterial stenosis , an intriguing application that warrants further CFD study with larger cohorts. Another novel application of CFD for patients with repaired TOF is modeling of the RVOT flow to target those for whom transcatheter pulmonary valve replacement is not currently the ideal intervention due to RVOT dilation. In this manner, CFD may aid the design of novel percutaneously placed “pulmonary valve reducer” (for those with enlarged RVOT) , thus allowing nonoperative pulmonary valve replacement. Additional CFD research for TOF patients is warranted as better solutions are sought for this population that has chronic PR.
In summary, creating clinically relevant CFD models requires care and rigor by individuals at all steps in the process—from initial acquisition of clinical data through all the steps of mathematical modeling. CFD is moving beyond being simply an intriguing mathematical study of blood flow. Based on many pilot studies, CFD is poised to have an increasingly powerful role in the care of patients with CVD in the next decade, by allowing a more refined understanding of the hemodynamics of both acquired CVD due to atherosclerosis and CHD. Continued partnerships between clinician-scientists and engineers are essential to the successful achievement of this goal. Only with such collaborations will the complex process of patient-specific modeling be streamlined and successfully integrated into clinical decision making to optimize medical and interventional therapies for cardiovascular disease.
The authors would like to acknowledge Mara Koffarnus for her assistance in manuscript preparation. The authors also acknowledge Mary Draney PhD, Frandics Chan MD, PhD, Ronak Dholakia PhD, Laura Ellwein PhD, David Wendell PhD, and Joseph Cava MD PhD for technical assistance with some of the parts of the listed CFD results.