- Research
- Open Access
The effect of inlet and outlet boundary conditions in image-based CFD modeling of aortic flow
- Sudharsan Madhavan^{1}Email authorView ORCID ID profile and
- Erica M. Cherry Kemmerling^{1}
- Received: 22 June 2017
- Accepted: 10 May 2018
- Published: 30 May 2018
Abstract
Background
Computational modeling of cardiovascular flow is a growing and useful field, but such simulations usually require the researcher to guess the flow’s inlet and outlet conditions since they are difficult and expensive to measure. It is critical to determine the amount of uncertainty introduced by these assumptions in order to evaluate the degree to which cardiovascular flow simulations are accurate. Our work begins to address this question by examining the sensitivity of flow to several different assumed velocity inlet and outlet conditions in a patient-specific aorta model.
Methods
We examined the differences between plug flow, parabolic flow, linear shear flows, skewed cubic flow profiles, and Womersley flow at the inlet. Only the shape of the inlet velocity profile was varied—all other parameters were identical among these simulations. Secondary flow in the form of a counter-rotating pair of vortices was also added to parabolic axial flow to study its effect on the solution. In addition, we examined the differences between two-element Windkessel, three element Windkessel and the outflow boundary conditions. In these simulations, only the outlet boundary condition was varied.
Results
The results show axial and in-plane velocities are considerably different close to the inlet for the cases with different inlet velocity profile shapes. However, the solutions are qualitatively similar beyond 1.75D, where D is the inlet diameter. This trend is also observed in other quantities such as pressure and wall shear stress. Normalized root-mean-square deviation, a measure of axial velocity magnitude differences between the different cases, generally decreases along the streamwise coordinate. The linear shear inlet velocity boundary condition and plug velocity boundary condition solution exhibit the highest time-averaged wall shear stress, approximately \(8\%\) higher than the parabolic inlet velocity boundary condition. Upstream of 1D from the inlet, adding secondary flow has a significant impact on temporal wall shear stress distributions. This is especially observable during diastole, when integrated wall shear stress magnitude varies about \(26\%\) between simulations with and without secondary flow. The results from the outlet boundary condition study show the Windkessel models differ from the outflow boundary condition by as much as \(18\%\) in terms of time-averaged wall shear stress. Furthermore, normalized root-mean-square deviation of axial velocity magnitude, a measure of deviation between Windkessel and the outflow boundary condition, increases along the streamwise coordinate indicating larger variations near outlets.
Conclusion
It was found that the selection of inlet velocity conditions significantly affects only the flow region close to the inlet of the aorta. Beyond two diameters distal to the inlet, differences in flow solution are small. Although additional studies must be performed to verify this result, the data suggest that it is important to use patient-specific inlet conditions primarily if the researcher is concerned with the details of the flow very close to the inlet. Similarly, the selection of outlet conditions significantly affects the flow in the vicinity of the outlets. Upstream of five diameters proximal to the outlet, deviations between the outlet boundary conditions examined are insignificant. Although the inlet and outlet conditions only affect the flow significantly in their respective neighborhoods, our study indicates that outlet conditions influence a larger percentage of the solution domain.
Keywords
- Inlet boundary conditions
- Womersley
- Windkessel
- Outlet boundary conditions
Background
Cardiovascular computational fluid dynamics (CFD) models have the ability to aid physicians in non-invasive diagnostic decision making, and over the past decade, commercial, patient-specific modeling has become more common owing to numerous advancements in computing speed [1], medical image acquisition, and 3D data processing and visualization techniques [2–5].
Cardiovascular diseases (CVDs) are the leading cause of death globally [6], with the most common conditions including coronary artery disease (CAD), stroke, heart failure, rheumatic heart disease, heart arrhythmia, aortic aneurysms, and thromboembolic diseases [6, 7]. CAD and stroke account for about \(77\%\) of CVD deaths [6], but many other conditions contribute to impairment or decreased quality of life of the patient. As a means to diagnosing and understanding these conditions, commercial, patient-specific modeling of CVDs has become more common in recent years. For instance, HeartFlow, Inc., Redwood City, California has developed a non-invasive CFD-based tool to identify lesions causing ischemia [8, 9]. Another application of cardiovascular CFD is designing new surgical techniques and implantable medical devices [10, 11]. Procedures and devices have traditionally been validated via clinical trials, animal tests, and evaluation of patients post-surgery. Cardiovascular modeling is now increasingly aiding these developments [11–18]. For example, [10] designed a ‘virtual surgery’ for pediatric surgeons based on patient-specific images. Their framework also computed post-operative hemodynamics based on the virtual surgery, thereby aiding surgeons in surgical planning. Furthermore, hemodynamic alterations are known to be a significant cause of ischemic disease progression [19]. Owing to these uses and other promising applications, there is a substantial need for accurate modeling of cardiovascular flows.
Unfortunately, much of the information required to perform accurate cardiovascular CFD is usually unavailable due to the difficulty of making in vivo flow measurements on live patients. Consequently, in order to formulate a well-posed problem, most researchers must guess parameters such as flow boundary conditions, vessel wall properties, and sometimes even geometric vessel parameters if patient imaging is not of sufficient quality. It has been shown that these factors and others can significantly alter the flow solution [20–25]. For example, [25, 26] performed a numerical study to quantify the sensitivity of wall shear stress fields in the carotid bifurcation to geometric and secondary flow perturbations. They found that small geometric variations could significantly affect the flow solution. Sankaran et al. [27] quantified uncertainties due to geometry, boundary conditions, and blood viscosity in coronary blood flow simulations using a stochastic collocation method [28]. They concluded that solutions from modeling were most sensitive to variations in minimum lumen diameter. Sankaran et al. [29] developed a reduced-order model based on a machine learning approach to quantify uncertainties due to geometric variations. They found that larger arteries with significant stenosis were most sensitive to geometric variations. Liu et al. [19] modeled a patient-specific circle of Willis coupled with a zero-dimensional lumped parameter boundary condition. They determined that the accuracy and consistency of their method were improved relative to a resistance-based boundary condition. Steinman et al. [22] was a collective study by 25 research groups to predict the variability of pressure drop in a giant aneurysm model with a proximal stenosis. Various research groups performed CFD analysis with the same lumen geometry, flow rates, and fluid properties. However, the researchers were free to choose their own numerical methods, discretization, and solution strategies. They concluded that pressure could be predicted with reasonable accuracy by CFD in the giant aneurysm model but transitional patterns and derived quantities varied widely. Liu et al. [30] developed a new methodology for functional assessment of stenotic carotid arteries. Their methodology based on thresholding pressure gradient successfully delineated severe stenosis from mild-moderate ones. Xiong et al. [31] investigated the effect of blood pressure variability on carotid atherosclerotic plaques. They determined that beat-to-beat blood pressure variability could severely exacerbate long-term outcomes of atherosclerosis. Wong et al. [32] studied the effect of fluid structure interaction on carotid bifurcation models with varying degrees of atherosclerosis. They concluded that wall shear stress and geometric deformation are significantly influenced by the severity of the disease. Liu et al. [33] simulated fluid structure interaction of blood flow and elastic arteries with eccentric stenotic plaques. They showed that wall shear stress, pressure drop and von Mises stress were positively correlated with the degree of vessel occlusion via plaques. Pekkan et al. [23] examined variations between solutions from a first-order accurate commercial software and a second-order accurate in-house flow solver. Only the second-order methods could accurately match the three-dimensional flow features found in an experimental model. Recent studies [20, 21] showed the effect of mesh resolution on patient-specific models and concluded that a typical mesh resolution in comparison to a higher mesh resolution resulted in pronounced underestimation of quantities such as wall shear stress and oscillatory shear index. They also showed that higher resolution meshes were able to capture flow instabilities.
Since cardiovascular CFD simulations are used to make critical decisions in diagnosis [30], surgical planning [10], and medical device designs [12, 13, 15], it is essential to verify that the assumptions made by the researcher do not negatively impact the fidelity of the solution. In this paper, we focus on the impact on flow solution of assumed inlet velocity boundary conditions in the human aorta. Some have argued that researchers concerned about the choice of inlet conditions should merely extend the size of the simulation domain so the flow is fully-developed by the time it reaches the point of interest. However, this is rarely a realistic solution since real arteries are poorly approximated by long, straight tubes, thus the flow is never truly fully-developed within the body. Furthermore, it is often prohibitively complex to add realistic upstream sections of the vasculature, as in the case of the aorta, which is immediately distal to the heart.
The aorta is of particular interest not only due to its position proximal to all other arteries, but also because invasive and non-invasive experimental measurements on the aortic arches of animals and humans have reported wide variations in the shape of the velocity profile, including flat [34], skewed [35], and highly patient-specific [36]. Consequently, in cases where patient-specific profiles are unavailable, the optimal profile shape to assume is not clear, and researchers have made many different choices [37–44]. To our knowledge, it is thus far undetermined to what extent the researcher’s choice of aortic inlet boundary condition changes the solution, or how far distal to the inlet the flow is significantly affected by the choice of inlet condition. In addition, it is not always clear how the choice of outlet boundary condition affects the flow solution; most researchers choose between an outflow outlet condition, in which flowrate is specified at each outlet, and a Windkessel model, in which distal resistances and capacitances are modeled [45–49]. It is critical to answer these questions to determine the extent to which the hundreds of published studies with non-patient-specific inlet and outlet conditions are accurate. In the current study, we begin to address these issues by simulating aortic flow with a variety of idealized inlet and outlet conditions. At the inlet, we examine plug flow, parabolic flow with and without secondary flow, linear shear flows, skewed cubic profiles, and Womersley flow. At the outlet, we study the two-element and three-element Windkessel models and compare them with specified mass flow rate and zero diffusion flux (ANSYS® Academic Research [Fluent], release 16.2, outflow boundary conditions, ANSYS, Inc.). The overall goal is to quantify the differences in flow solution caused by choice of inlet and outlet conditions for the purposes of evaluating the impact of assumed boundary conditions on previously-published aortic flow studies.
Methods
Inlet sensitivity studies
In the first part of this study, the sensitivity of flow solutions to velocity inlet conditions was investigated. For these simulations, a zero diffusion flux for all flow variables at the outlets and an overall outlet flow rate were employed to impose specified \(\%\) mass flow splits (ANSYS® Academic Research [Fluent], release 16.2, 7.3.10, outflow boundary conditions, ANSYS, Inc.). The average outflow rates were obtained from [45, 67].
Outflow boundary conditions
Vessel | Outlet flow rate |
---|---|
B-right common carotid | 9.8 |
C-right subclavian | 9.5 |
D-left common carotid | 5.2 |
E-left subclavian | 6.4 |
F-descending aorta | 69.1 |
Parameters for the Windkessel outlet boundary conditions, adapted from [77]
Vessel | R_{ p } (dynes s/cm^{5}) | C (cm^{5}/dynes) | R_{ d } (dynes s/cm^{5}) |
---|---|---|---|
Right common carotid | 1180 | 7.70E−5 | 18,400 |
Right subclavian | 1040 | 8.74E−5 | 16,300 |
Left common carotid | 1180 | 7.70E−5 | 18,400 |
Left subclavian | 970 | 9.34E−5 | 15,200 |
Descending aorta | 188 | 4.82E−4 | 2950 |
Outlet sensitivity studies
For all simulations, flow was assumed to be laminar since the Reynolds number, \(\mathbf {Re}_{D}\) based on the inlet aortic diameter, D was about 1700 at peak systole. The simulations were performed until the fifth cardiac cycle. Wall shear stress (WSS), pressure, and vorticity contours were examined from the fifth cardiac cycle. The centerline of the model was computed and data slices perpendicular to the centerline were extracted at streamwise coordinates that were multiples of the aortic root diameter, D. Time-averaged wall shear stress (TAWSS) and other time-averaged flow quantities were also computed by averaging over the fifth cardiac cycle. Results comparing the various inlet and outlet boundary conditions are presented in the following sections.
Results and discussion
Effect of the shape of the inlet axial velocity profile
This subsection discusses the influence of the axial velocity profile shape on the solution. These flows had no secondary flow at the inlet.
Data slices perpendicular to the centerline of the model were extracted at various locations along the aorta. Figure 5 shows data at streamwise coordinates of 0.5D and 1D, where ‘D’ is the diameter of the aorta’s inlet. Axial velocity magnitudes are depicted by contours. In-plane velocities are represented by the vectors in Fig. 5. Surfaces closer to the inner and the outer arch are denoted by the letters ‘I’ and ‘O’ respectively. The effect of inlet boundary conditions is more pronounced closer to the inlet of the vessel. For instance, the peak in axial velocity is approximately at the center of the cross-section for the parabolic inlet boundary condition, as shown in Fig. 5a. Similarly, the contours in Fig. 5b, c show marked similarities to their respective inlet conditions, linear shear flows 1 and 2. Owing to inertia, flow inside the curved vessel gets pushed towards the outer side of the arch, labeled ‘O’. This effect is apparent in the in-plane velocity vectors of the parabolic velocity inlet cross section in Fig. 5a. The counter-rotating vortex (CRV) pair, formed because of the aforesaid effect [81, 82], is retained at a streamwise position of 1D for the parabolic inlet boundary condition. In addition to the CRV pair, there is a smaller vortex closer to the inner arch, ‘I,’ for the parabolic inlet boundary condition. A counterclockwise rotating vortex is present in the flow with the linear shear 1 inlet condition. However, the linear shear 2 inlet has a clockwise rotating vortex, observed in Fig. 5c. For linear shear flow inlet boundary conditions, there is a change in the direction of rotation of the tangential velocity vectors with increasing streamwise coordinate. This effect can be observed by comparing Fig. 5b, e for the linear shear 1 inlet condition. A similar trend is also noticeable in Fig. 5c, f for the linear shear 2 inlet boundary condition. It is also notable that both the primary and the secondary in-plane flows look considerably different for the three boundary conditions illustrated in Fig. 5, but in all three cases, secondary flows are only a small percentage of the total flow velocity.
Velocity boundary condition | Differences in TAWSS (\(\%\)) | Differences in WSS during peak systole (\(\%\)) | Differences in TAWSS up to 1D (\(\%\)) | Differences in WSS up to 1D during peak systole (\(\%\)) |
---|---|---|---|---|
Plug | 8.72 | 15.69 | 7.21 | 31.48 |
Womersley | 0.86 | 6.20 | 6.84 | 17.30 |
Linear shear flow 1 | 8.09 | 11.21 | 18.50 | 32.53 |
Linear shear flow 2 | 0.99 | 6.42 | 1.47 | 17.24 |
Cubic shear flow 1 | 5.08 | 0.63 | 12.73 | 2.67 |
Cubic shear flow 2 | 1.87 | 1.83 | 4.16 | 5.48 |
Differences in wall shear stress magnitudes of the parabolic inlet velocity boundary condition cases with and without secondary flow
Integration domain | Differences in TAWSS (\(\%\)) | Differences in WSS during peak systole (\(\%\)) | Differences in WSS during diastole |
---|---|---|---|
Entire wall surface | 3.58 | 0.19 | 5.35 |
Wall surface up to 1D | 7.74 | 1.50 | 26.56 |
Effect of adding secondary flow to the inlet
In this subsection, the effect of adding secondary flow to a parabolic axial inlet velocity profile is discussed. Only the parabolic axial flow is considered since it is the most commonly assumed inlet velocity profile shape in cardiovascular simulations.
Differences in wall shear stress magnitudes between the three-element Windkessel (RCR), the two-element Windkessel (RC), and the prescribed percentage flow rate outlet (outflow) boundary conditions
WSS comparision | Differences in TAWSS (\(\%\)) up to 1D (%) | Differences in WSS up to 1D (\(\%\)) during peak systole | Differences in TAWSS (\(\%\)) | Differences in WSS (\(\%\)) during peak systole |
---|---|---|---|---|
RCR and outflow | 0.3571 | 2.8515 | 18.2248 | 14.2861 |
RC and outflow | 0.3544 | 2.8528 | 18.2758 | 14.3076 |
RCR and RC | 0.0027 | 0.0013 | 0.0431 | 0.0250 |
The magnitude of these differences must be interpreted in the context of other uncertainties in cardiovascular flow simulation. For example, in an image-based coronary arterial model examined by [25, 26], different models of blood rheology accounted for about \(8\%\) variability in the solution, the effect of secondary inlet flow yielded \(13\%\) variability, and geometric uncertainties resulted in \(47\%\) variability in wall shear stress. It is notable that they generated secondary flow using an extension to their model with added curvature and helical pitch. Another study, [83], examined the effect of curvature and inlet velocity profile on a right coronary artery model. They concluded that inlet velocity profile had little effect on the flow compared with the effect of changing the curvature of the model. From our study, it is evident that the effect of changing the shape of the primary flow inlet velocity profile is not felt significantly beyond 1.75D, with D being the aortic root diameter. However, upstream of 1D, the shape of the axial flow can lead to as much as \(18\%\) variability in terms of time-averaged wall shear stress. Adding secondary flow on top of parabolic axial flow also results in significant variability in wall shear stress upstream of 1D, as high as \(26\%\) during diastole. Consequently, if accurate temporal modeling closer to the inlet and the aortic arch is desired, our results emphasize the need to model patient-specific inlet velocity conditions including secondary flow.
Effect of outlet boundary conditions
Conclusions and summary
This work investigated the variation introduced into a simulation of aortic blood flow by choice of inlet and outlet boundary conditions.
Inlet plug flow, parabolic flow, linear shear flows, skewed cubic flows, and Womersley flow were simulated and the resulting flow solutions were compared to study the effect of inlet conditions. Parabolic flow with and without secondary flow at the inlet was also studied. All other parameters were identical among these simulations. While the parabolic inlet condition without secondary flow has the lowest time-averaged wall shear stress, linear shear flow and plug flow have the highest time-averaged wall shear stress, about \(8\%\) higher than parabolic inlet condition without secondary flow. The axial and in-plane velocities for the different flow solutions are considerably different across data slices extracted at 0.5D and 1D from the inlet, where D is the inlet diameter. Data slices at 1.75D and 2.5D are qualitatively similar but there are minor differences between secondary flows at 1.75D. Normalized root-mean-square deviation (NRMSD) evaluated between the parabolic inlet condition without secondary flow and other axial velocity boundary conditions generally decreases along the streamwise coordinate and is less than 0.03 at 2.5D for all cases. These statistics show that the effect of inlet conditions becomes less pronounced as the streamwise coordinate increases. Adding secondary inlet flow to parabolic axial flow results in a slight variation of about \(4\%\) in terms of the time-averaged wall shear stress. However, between the inlet and a streamwise coordinate of 1D, there are larger differences. This is especially noticeable during diastole when shear stress magnitude differences integrated up to 1D are as high as \(26\%\).
Outlet conditions prescribing a zero-diffusion flux with specified mass flow rate (ANSYS® Academic Research [Fluent], release 16.2, outflow boundary conditions, ANSYS, Inc.), two-element Windkessel, and the three-element Windkessel conditions were investigated. Both the two-element and the three-element Windkessel models don’t vary much near the inlet as seen from the time-averaged wall shear stress variations. For instance, both the two-element and the three-element models differ from the outflow boundary condition by 0.3544 and \(0.3571\%\) respectively in terms of time-averaged wall shear stress integrated up to 1D. However, in terms of time-averaged wall shear stress integrated throughout the model, they differ from the outflow boundary condition by as much as about \(18\%\). Normalized root-mean-square deviation (NRMSD) evaluated between the outflow boundary condition and the Windkessel models generally increases along the streamwise coordinate. However, beyond 3.5D NRMSD varies by less than \(2.5\%\) along the streamwise coordinate. These statistics indicate that NRMSD remains constant for more than 5 diameters proximal to the outlet and that the effect of outlet conditions are more pronounced as the streamwise coordinate increases.
Based on the current results along with other studies on the subject [70, 89, 90], it is reasonable to conclude that inlet conditions, including both primary and secondary velocity profile shape, significantly affect the solution up to about two inlet diameters distal to the inlet. Similarly, the type of outlet condition chosen affects the solution significantly up to five inlet diameters proximal to the outlet. This suggests that the outlet boundary conditions influence a larger percent of the solution domain. The amount of variation observed between the various flow cases in this study can be interpreted as a lower bound on the error that can be expected in aortic flow simulations that do not use patient-specific boundary conditions. Although this study is limited to one healthy model, the underlying mechanisms of flow over the curvature of the vessel and the effect of branches would likely render qualitatively similar results in other subject-specific models. Nevertheless, studying more subject-specific models along with corresponding physiologically realistic inlet velocity boundary conditions to verify our conclusions is of interest for future work.
List of symbols
Symbols
D: diameter of aorta at the inlet; \(\overrightarrow{U}\): axial velocity vector.
Greek letters
\(\tau _{}\): shear stress.
Non-dimensional numbers
\(\mathbf {Re}\): Reynolds number.
Subscripts or superscripts
w: wall.
Acronyms and abbreviations
NRMSD: normalized root-mean-square deviation; WSS: wall shear stress; TAWSS: time-averaged wall shear stress.
Declarations
Authors’ contributions
EK designed the study and evaluation procedures. SM worked on implementing the study; preparing the model, simulating, and post processing the data. SM wrote this manuscript. EK contributed in reviewing and revising it. Both authors read and approved the final manuscript.
Authors’ information
Sudharsan Madhavan received his bachelor's degree in mechanical engineering from Indian Institute of Information Technology Design \& Manufacturing Kancheepuram, India and his masters in applied mathematics from University of Washington, Seattle, USA. He is currently a doctoral candidate at Tufts university, Medford, USA, working on cardiovascular fluid dynamics.
Erica M. Cherry Kemmerling holds a B.S. in physics and an M.S. and Ph.D. in mechanical engineering from Stanford University. She is currently an assistant professor of mechanical engineering at Tufts University. Her research focuses on fluid flow and mass transport in the human body with an emphasis on the circulatory system.
Acknowledgements
The authors would like to acknowledge Stanford Cardiovascular Biomechanics Computation Lab for letting us use their subject-specific model.
Competing interests
The authors declare that they have no competing interests.
Consent for publication
Not applicable.
Data availability
Please contact sudharsan.madhavan@tufts.edu for data requests.
Ethics approval and consent to participate
Not applicable.
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Authors’ Affiliations
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