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Fluid mechanical aspects of blood flow in the thoracic aorta
Alexander Fuchs1
Department of Radiology, Karolinska University Hospital, Stockholm.
Department of Medical and Health Sciences, Linköping University, Linköping, Sweden.
FLOW & BioMEx, Department of Engineering Mechanics, Royal Institute of technology, KTH,
Osquars Backe 18, SE-100 44 Stockholm, Sweden.
e-mail: alex@mech.kth.se Niclas Berg
FLOW & BioMEx, Department of Engineering Mechanics, Royal Institute of technology, KTH,
Osquars Backe 18, SE-100 44 Stockholm, Sweden.
e-mail: niber@kth.se Lisa Prahl Wittberg
FLOW & BioMEx, Department of Engineering Mechanics, Royal Institute of technology, KTH,
Osquars Backe 18, SE-100 44 Stockholm, Sweden.
e-mail: prahl@mech.kth.se
1 Corresponding author
2 ABSTRACT
Arterial blood flow contains structures known to be associated with arterial wall pathologies (such as atherosclerosis and aneurysms) but also with helical motion reported to be atheroprotective.
Numerical simulation of the flow in a typical human thoracic aorta model was carried out for several heart- and flow-rates. The aim was to explore the presence and the underlying mechanism of the formation of helical flow, retrograde motion and the formation of smaller scale unsteady flow structures.
The main findings of the paper are as follows:
- Retrograde flow is induced during flow deceleration. Reversed flow may persist throughout the cardiac cycle in parts of the descending aorta. Retrograde flow may lead enhanced risk of upstream transport of thrombi from the descending aorta to the branches of the aortic arch.
- Helical flows are induced by bend and torsion of the aorta and through non-uniformity in the spatial distribution of the inlet flow (aortic valve plane).
- Amplification of axial vorticity was shown to occur in the thoracic aorta. This convective instability is enhanced in the descending aorta.
- Transitional and turbulent flow may occur in the thoracic aorta under elevated flow- and heart- rate conditions also in healthy individuals.
- Under normal conditions, healthy individuals do not develop turbulent flow in the thoracic aorta.
A hypothesis for a possible mechanism for the atheroprotective effect of helical flow is suggested.
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INTRODUCTION
Flow structures in arteries have been associated with certain arterial pathologies. For example, atherosclerosis and arterial aneurysms have been correlated with Wall Shear Stress (WSS) and helical flow in arteries is observed in practically all arteries and under different flow conditions. Numerous studies have focused on large scale flow structures, believed to have an impact on arterial wall pathologies. In addition to helical motion, retrograde flows have also been reported in the aorta and its main branches. Advances in techniques for velocimetry in combination with computational fluid dynamics (CFD) has allowed for an increased
understanding of the flow structures present in the aorta to relate fluid mechanics to clinical observations.
A natural starting point for aortic blood flow has historically been the straight, fully developed pipe flow. However, the limitations with respect to the flow in the aorta is well recognized and consequently the focus was directed towards bent pipes. Starting with the work of Dean (defining the Dean number relating the pipe radius to the pipe radius of the curvature), extensive work on both laminar and turbulent flows in bent pipes are found in literature. In the seminal work by Womersley [1], low speed pulsating flow in a straight pipe was presented, introducing a dimensionless parameter relating the oscillating frequency and the viscous time scales, expressed in terms of the Womersley number ( d/ 2 2f / , with f and being the frequency and kinematic viscosity, respectively). Considering highly oscillatory, fully
developed flow inside curved pipes characterized by both small and large curvature ratios, Lyne
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[2] observed the formation of the flow structures later termed Lyne vortices. Such vortices are associated with large Womersley numbers (in Lyne's measurements, = 28), where the pipe core behaves as being inviscid, and the Dean vortices are confined to the thin viscous boundary layer close to the wall, resulting in an extra pair of counter-rotating vortices. More recently, steady and oscillatory flow in a 180° curved pipe, using a blood analog as fluid has been considered by van Wyk et al, Plesniak and Bulusu as well as Najjri et al [3-5]. In these studies, the details of the flow were assessed by combinations of Particle Imaging Velocimetry (PIV), MRI or PIV and numerical techniques whereby the limitations of each technique could be reduced.
In-vivo studies of the flow in the aorta has been investigated, where Frazin et al [6]
carried out echocardiographic studies in dogs. Systolic clockwise helical flow was observed in the thoracic and abdominal aorta down to immediately beyond the renal artery
branches. The diastolic flow pattern revealed multiple vortices, but no clear pattern. The authors argued that since the helical velocity component was at least 25% of the axial velocity in the aortic arch and the proximal parts of the descending aorta, the helical component contributes to the WSS and may thereby contribute to certain pathologies (such as development of atherosclerosis and aortic dissection). In humans, helical flow has been
observed in healthy individuals [7-8] as well as associated to different arterial pathologies, such as Turner associated aortic shape anomalies and bicuspid aortic valves [9-11]. Jin et al [12]
utilized MRI data in a computational model to study the effects of wall curvature and compliance on the formation of helical motion in the ascending aorta. The results from the
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model suggested that the observed flow patterns were due to both the curved geometry of the aortic arch and the motion of aorta itself. In the study by Hope et al (2007) [11], 4D-flow-MRI was applied to 19 healthy volunteers and 13 patients with ascending aortic aneurysms. The flow field was analyzed with respect to helical and retrograde flow patterns along the aortic arch. Helical flow was observed in both groups as well as retrograde flow. However, located along the inner curvature between the two helices, it was found to persist over a longer period of time in aneurysm patients. To quantify helical flow, the scalar product of the velocity and vorticity vectors was formed, defining helicity. Morbiducci et al [13] introduced the Helicity Flow Index, based on the local normalized helicity averaged over a (Lagrangian) particle path.
This definition was used to analyze the flow in a thoracic aorta using MRI data. von Spiczak et al [14] defined the averaged Local Normalized Helicity (LNH) calculated along a path-line
trajectory. The data of all path-lines was averaged providing a measure of global helicity.
Regarding the relation between helical flow structures and WSS, De Nisco et al [15]
reported that “unfavorable conditions of WSS were strongly and inversely associated with helicity intensity”. Quantifying the helical blood flow in 30 swine coronary arteries and its possible associations with vascular geometry and with atherogenic WSS phenotypes, the results showed that the coronary artery flow was characterized by counter-rotating bi-helical flow structures and the formation of the helical structure was attributed to vascular torsion.
Moreover, three helicity characterizing parameters were defined; namely the average helicity, average helicity intensity and ratio of (sign of) average helicity relative to average helicity intensity. These parameters were used alongside with the WSS related parameters (Oscillatory
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Shear Index (OSI), Time Averaged Wall Shear Stress (TAWSS) and Relative Residence Time (RRT), Transversal Wall Shear Stress (transWSS) and Cross Flow Index (CFI)) to characterize coronary artery flows in quantitative terms. Appearing in both the central parts of the aorta and more importantly near the walls, retrograde flow (relative to the axial directions of the vessel) leads to negative WSS component relative to the vessel axis direction [7,16]. In this context, it should be mentioned that unsteady WSS has been suggested to correlate with atherosclerosis [17].
Moreover, Gulan et al (2012) [18] performed detailed in-vitro measurements using 3D Particle Tracking Velocimetry (PTV) and MRI in a compliant silicon aortic model, demonstrating the presence of bi-directional flow in diastole with retrograde flow along the inner wall of the aorta. Besides flow structures, flow losses was also assessed from the measurement, computing the viscous and turbulent kinetic energy and dissipation rates using an eddy viscosity type model. The two measuring techniques indicated that the contribution of turbulence to the total kinetic energy and the dissipation rates was minor. Stalder et al [19] studied the flow in the aorta in-vivo, also by MRI, probing for flow instabilities in physiological pulsatile blood flow.
Flow instabilities were prominent in the ascending and descending aorta, but found to be considerably less in the aortic arch where the lowest mean Reynolds number was located.
Many aspects regarding the details in arterial flow dynamics and its influence on arterial wall pathologies still remain to be elucidated. The aim of this study was to identify and expose the underlying fluid mechanical mechanisms governing aortic blood flow structures with respect to retrograde motion and helical flow as well as assessing the generation of vorticity during the cardiac cycle. In the following, numerical simulations of the flow in the thoracic aorta
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under different flow- and heart-rate conditions are presented. The focus is directed towards better understanding of the formation of helical and retrograde motion in the aortic lumen and near the wall. Additionally, the mechanism for formation of smaller scale unsteady flow
structures is discussed.
METHODOLOGY
Numerical model & Boundary conditions
The geometry under consideration was derived from a computed tomography (CT) scan of a healthy patient. The computational domain included the ascending, the arch and the descending parts of the thoracic aorta. The outlet parts of the head/neck arteries were
extended as compared to the original CT-image. Similarly, the descending aorta followed the CT data down to the level of the diaphragm. The computational geometry was further extruded (Figure 1).
The governing model equations include the conservation of momentum and mass:
( i j)
i i
j i j j
u u u p u
t x x x x
(1a)
i 0
i
u
t x
(1b)
where are the density and viscosity of the mixture, respectively, p is the pressure and ui is the Cartesian velocity component in the i-direction The blood is assumed to be
incompressible and non-Newtonian. The numerical methodology follows the approach used in
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Fuchs et al [20] and van Wyk et al [3]. For the simulations presented here, the Red Blood Cell (RBC) concentration remained fixed leading to a constant mixture density. The blood viscosity was accounted for by adopting the model of Quemada [21] accounting for the effect of local shear on the local blood viscosity. The RBC concentration was modeled by a time-dependent convection diffusion equation where the RBC diffusivity was modeled according to Casa and Ku [22].
At the inlet of the ascending aorta, a time-dependent velocity profile was given,
applying a “top-hat” shape or a profile having a cross-plane variation. The time-dependence (in the range of 60-150 beats per minute, BPM) and flow-rate (in the range of 6-15 liter per
minute, LPM) was similar to that as presented by Frydrychowicz et al (2009) [23], depicted in Fig 1c. The flow distribution in the different branches was as suggested by Benim et al [24], setting the flow rates so that the Brachiocephalic Artery (BCA) delivered 15% of the inlet flow rate whereas the Left Common Carotid Artery (LCCA) and the Left Subclavian Artery (LSCA) delivered each 7.5% of the inflow. The remaining blood (70%) flowed through the descending aorta. The pressure at the descending aorta was kept at a constant value (0 Pa) whereas the pressure gradient at the other non-wall boundaries was set to zero. The aortic lumen was assumed to be rigid.
The computational domain was discretized on a set of small (hexagonal and tetrahedral) cells. OpenFOAM was used to discretize and solve the governing equations, using formally second order accurate finite-volume discretization. Integration in time was carried out using a second order implicit scheme. Small time-steps were used (up to 0.1 ms) to allow for resolving
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fluctuations up to the kHz range. Monitoring points in the ascending aorta, the aortic arch and the descending aorta were used to store instantaneous data. Whole field data was acquired solely at certain given time intervals at a considerably lower rate as compared to the small time-steps used in the computations. The data was post-processed using Paraview. Spectral analysis, data filtering and further data evaluation, was carried out using MATLAB routines.
Vorticity generation and growth
In order to analyze the results in terms of formation of vortical structures, the vorticity transport equation was considered, obtained by taking the curl of equation (1a), and given by:
2 2
(1/ )
i i i j i
j j i ijk
j j j j k j
u u p
t u x x x x x x
(2.a)
where i is the vorticity vector. With the assumptions that the fluid mixture is incompressible with constant density, the 2nd and 3rd terms on the right hand side vanish identically. The main driving term for the generation of vorticity is the 1st term on the right- hand side of the equation (i.e. the so-called Vortex Stretching (VS) term), whereas the last term is viscosity related, commonly contributing with a smaller effect as compared to the VS term.
Assuming that the dominating term in the stress tensor is due to a gradient proportional to the shear length scale or the streamline curvature, a semi-quantitative assessment of the amplification rate of vorticity can be made. As the axial vorticity plays a role for the helical motion, the vorticity transport equation (2.a) was considered in terms of the axial components (a):
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i ai
a VS t
t D
D
(2.b)
where the time-derivative is taken in the Lagrangian frame following the fluid and ti and VSai are the tangent vector to the axis of the vessel and the (first) VS term on the right hand side of equation (2.a), respectively.
Helicity
The helicity (h= ui ωi) was computed from the local instantaneous velocity field.
Following Gallo et al [25] and the more recent paper of De Nisco et al [15], the helicity data was reduced by averaging in space. This was carried out to assess the temporal development of helicity relative to the cardiac cycle where hs is the space- time-average of the helicity and ha is the total mean helicity, defined as:
0
1 ( )
T
s i i
h u d dt
T
and0
1 ( )
T
a i i
h u d dt
T
(3)where T and are the integration time and aortic volume, respectively. An indicative index for directionality of rotation was suggested by defining h3 and h4 [15,25];
3 s 1 4 s
a a
h h
h and h
h h
(4)
The research was carried out in accordance with the Declaration of Helsinki and was approved by the Swedish regional ethical vetting board in Linköping (Project identification code: DNR 2017/258-31, Prof. Anders Persson). Informed consent was obtained before patient CT data was anonymized.
11 RESULTS
The results presented herein are all related to the same thoracic aorta geometry (Fig.
1a), applying an inlet flow rate scaled for different heart- and flow rates. The temporal variation of the inlet flow rate depicted in Fig. 1c represents the case of 75 BPM (beats per minute) and 6 LPM (liters per minute). For all cases investigated, the same set of outlet boundary conditions were applied as specified in the Methods section. The time-dependent flow rate at the inlet was scaled to yield the specified flow- and heart-rate, maintaining the same basic temporal shape.
Helical flow
A global view of the flow in the aorta was obtained by considering instantaneous stream tracers colored by the different parameters of interest. Figure 2 (90 BPM and 9 LPM) depicts stream-paths at the seven time instances indicated in Fig. 1c. These include peak flow (T2), mid- peak flow during acceleration/deceleration (T1 and T3, respectively), end systole (T4), and early-, mid- and end-diastole (T5, T6 and T7, respectively). The upper row of frames in Fig. 2 depicts the magnitude of the vorticity along the stream tracers. The middle row depicts the magnitude of the instantaneous flow velocity projected along the aorta centerline. The lower row is colored by the magnitude of the VS term in equation (2). The formation of a helical motion during most of the heart cycle is clearly shown. For other inlet profiles (differing in terms of temporal flow rate, not shown here), the helical motion begins with clockwise helical motion that changes direction in late systole/early diastole, returning to a weaker clockwise direction at mid- and late diastole. Animations of stream tracers colored by parameters as
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helicity (equation (3)), VS, axial velocity and vorticity, for some flow rates are provided in the supplementary material (S1a-S1e and S2a-S2e).
Quantification of helicity was carried out using the space-time-average of the helicity (hs), total mean helicity (ha) as well as two parameters indicating the directionality of rotation (h3 and h4), equations (3) and (4), presented for four heart and flow-rates in Table 1. The mean helicity increases with heart- and flow-rates. When HR increases, the duration of diastole, during which the helical motion is weaker, becomes shorter. Consequently, the contribution to the average helicity (equation (3)) gets smaller. For higher HR, the mean direction of the helical flow is maintained. A closer inspection of the flow field reveals that the local swirling direction may change during diastole (animations S1b and S2b in the Supplementary material).
Moreover, the results show the presence of helical motion in all bifurcation arteries throughout the cardiac cycle with the exception of a short period around peak systole.
The sign of helicity does not distinguish cases with the same signs or sign change in both the vorticity and the velocity. Hence, reversing a helical and flow motion leads to the same helicity sign. In order to assess changes in the vorticity and velocity independently, the helicity at each point in the domain was sorted depending on the signs of the velocity and vorticity.
Figure 3 depict the contribution of each component to the total helicity as function of the cardiac cycle. As noted, the main contribution in the three regions of the aorta is from the positive/positive and negative/positive (the signs of a and Ua, respectively) cases. The minimum and the corresponding peak in the curves (at about half of the cardiac cycle) correspond to end systole. The helicity in the ascending aorta and the arch decrease during
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systole and increase in diastole. The level of the main helicity components (positive/positive and negative/positive) dominate further downstream in the descending aorta.
The sign of helicity does not distinguish cases with the same signs or sign change in both the vorticity and the velocity. Hence, reversing a helical and flow motion leads to the same helicity sign. In order to assess changes in the vorticity and velocity independently, the helicity at each point in the domain was sorted depending on the signs of the velocity and vorticity.
Figure 3 depict the contribution of each component to the total helicity as function of the cardiac cycle. As noted, the main contribution in the three regions of the aorta is from the positive/positive and negative/positive (the signs of a and Ua, respectively) cases. The minimum and the corresponding peak in the curves (at about half of the cardiac cycle) correspond to end systole. The helicity in the ascending aorta and the arch decrease during systole and increase in diastole. The level of the main helicity components (positive/positive and negative/positive) dominate further downstream in the descending aorta.
Retrograde flow
As observed in Fig. 2 (and in the Supplementary material, animations S1a-S1e and S2a- S2e) the flow is well organized at early stages of systole (i.e. during T1 - T3). However, during the later stages of deceleration (T4) and diastole, smaller scale eddies appear. Figure 2b shows the stream tracers colored by the axial velocity component (i.e. the projection of the velocity vector on the tangent to the centerline of the vessel). The main observations are:
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- During the accelerating phase of systole (T1 and T2) the flow is largely unidirectional, with no clear separation bubbles present (excluding the aortic sinus and the bifurcations of the head-neck arteries).
- During flow deceleration (T3) flow reversal occurs in the ascending aorta.
- During diastole, retrograde flow occurs over larger parts of the aorta including the near aortic wall region.
These effects become relatively more pronounced with decreasing flow- and heart- rates. A quantification of the extent of retrograde flow inside the aorta and along its walls are presented in Fig. 4, comparing four cases with heart-rates of 60, 75, 90 and 150 BPM, having corresponding flow rates of 5, 7.85, 9 and 15 LPM, respectively. Fig. 4a depicts the ratio of the aortic volume having retrograde flow (relative to the aortic centerline axis) to the total volume, as function of normalized heart-cycle time. Fig. 4b depicts the corresponding ratio of the
arterial wall area having WSS characterized by negative direction relative to the direction of the aorta axis. The main peak of retrograde flow, both in terms of volume and wall area, is found at end-systole and early diastole. A second peak is observed at mid-diastole at the end of the second deceleration phase (T6). The total cycle averaged relative volume and surface area of retrograde flow for the four cases is provided in Table 2. Increasing cardiac rate leads to a decrease in the volume flow experiencing retrograde flow along with a corresponding decrease in the wall area with negative WSS.
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To better understand the formation and impact of retrograde flow during the cardiac cycle, the behavior of the axial velocity component (i.e. the local velocity vector projected onto the tangent to the aortic centerline) along the ten lines depicted in Fig. 1a for different time instances of the cardiac cycle as depicted in Fig. 1c was considered. The spatial velocity variations along the lines and temporal evolution of the axial velocity profiles were
investigated, as depicted in Figs. 5 and supplementary animations S1f and S2f), respectively.
Lines 2 and 6 in Figs. 5 correspond to the ascending aorta and the distal part of the aortic arch (just downstream of the bifurcating arteries of head, neck and arms), respectively. During systole, the axial velocity is higher in the near proximity of the inner aortic wall (i.e. inner curvature) and with slower towards the outer wall. Towards end-systole and early diastole the flow close to the outer wall of the aorta, decreases and ultimately changes direction.
Retrograde flow starts during the deceleration (late systole) of the cardiac output and
accentuates during early and mid-diastole. Initially, retrograde flow occurs closer to the inner wall of the aortic arch. During larger parts of diastole, multiple regions of retrograde flow can be observed together with increasing level of temporal- and spatial-fluctuations. The shape of the axial velocity profile becomes less smooth as the cardiac cycle evolves. Increasing heart-rate and flow-rate leads to stronger fluctuations and further increase in instantaneous spatial- and temporal-variations. The spatial oscillations of the axial velocity results in the formation of multiple inflection points.
To quantify the extent of the retrograde flow, the aorta was divided into 20 segments.
In each segment different quantities (axial velocity, helicity, the terms in the vorticity transport
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equation (equation (2)) were extracted. To save details, three groups of segments are
considered belonging to the ascending and descending aorta and the aortic arch, respectively.
In each of these regions, the spatial, temporal or combined averages were computed. Figure 6 depict the extent of the time averaged axial velocity in the three regions of the aorta for different instances in the cardia cycle. Retrograde flow is present throughout the cycle in the ascending aorta and the arch. In the descending aorta, no mean retrograde flow is present close to peak systole (close to 0.3 of the cardiac cycle). As the decrease of systolic flow rate starts (flow deceleration) retrograde flow is maintained throughout diastole. In terms of the whole aorta, the space averaged axial velocity is non-negative until the beginning of the deceleration phase (Fig. 4a). By considering the different segments, it is shown that retrograde flow is present in some parts of the aorta also during early systole. In the ascending aorta this is the effect of the aortic sinus with its reversed flow and the entrainment of the central parts of the flow. The velocity distribution in the ascending aorta in particular contains some peaks at certain speeds. These peaks are smoothed out in the descending aorta, where the velocity distribution is rather uniform.
The presence of inflection points is a necessary, but not sufficient, for formation of flow instability. However, it can be used as a qualitative indicator. A quantitative assessment of instability can be found by using for example the vorticity transport equation (2). Assuming constant density, the dominating term driving the growth of vorticity is the VS term balanced by the advection- and temporal-derivative terms (i.e. material derivative of vorticity). By coloring the tracer lines with the size of the VS term (Fig. 2c and the animations in the Supplementary
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material), it possible to relate the formation of the flow structures to the underlying mechanism. The strength of the VS term increases with increasing flow rate during the accelerating phase of systole. The VS term is large in the boundary layer near the wall of the aorta and in particular at the aortic sinus. During flow decelerating, the VS term maintains its strength in the proximal parts of the aorta while it is weakened in the descending aorta during T4 and T5. Further into diastole, the VS term becomes weaker in the ascending aorta
meanwhile the vorticity is advected downstream into the descending aorta (T6 and T7). In diastole, the size of all terms is reduced, although higher frequency oscillations can be observed at higher flow rates.
By considering the sign of the tangential VS term parallel to the axis of the aorta, the convective stability of the flow may be determined. The spatial distribution of that term is depicted for the three aortic regions in Fig. 7. The distribution of the VS term seems to be almost symmetric. Quantitatively, the mean VS-term is about 21, 35 and 73 s-2, for the
ascending aorta, the arch and the descending aorta, respectively. The instantaneous values do fluctuate strongly, but the positive mean value indicates the increase in the instability of the flow along the aorta. This effect is also noted in Fig. 2c, where the stream tracks are colored by VS. VS is strong near the boundaries, artery bifurcation and other geometrical irregularities such as the aortic sinus. As the VS terms is non-linear (in terms of the velocity), formation of vortices of different sizes is the underlying mechanism for the formation of small scale structures and ultimately turbulence.
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Formation of turbulence
As indicated by Figs. 5 and 7 (and animations in the Supplementary material),
fluctuations develop in the flow. In order to assess the character of the unsteadiness of the flow in different locations in the aorta, 572 monitoring points were used (some of which are
depicted in Fig. 1b) and placed throughout the aortic computational domain. The instantaneous data was phase averaged, yielding a phase mean component and a fluctuating component. In the calculations presented, between 20 and, at most, 50 cardiac cycles were used for the low (<9 LPM) and the highest flow rates (15 LPM), respectively. Phase average velocity was used to compute the mean mechanical kinetic energy (MKE). The difference between the instantaneous and phase average values was used to compute the fluctuating kinetic energy (FKE). Such a decomposition for two flow- and heart-rate case is depicted in Fig. 8. These parameters were computed separately for the ascending, arch and descending parts of the aorta. Table 3 details the level of FKE relative to the mean kinetic energy for the three segments and the entire aorta for four cases (different heart- and flow rates). The portion of kinetic energy is very low (<
0.05%) for flow rates below 9 LPM in the ascending aorta and the aortic arch. The level of FKE increases with heart- and flow-rates. Also, increasing heart-rate implies that the fluctuations are maintained over a larger portion of the cardiac cycle. These effects are depicted in Figures 5a and b, for the case of 60 and 90BPM, respectively.
Flow features may be characterized by considering the spectrum of FKE. Typical power spectra for higher flow conditions (heart- and flow-rates of 90 BPM/9 LPM and 150 BPM/15 LPM, respectively) are depicted in Fig. 9. The spectrum identifies the heart-rate frequencies and
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contains a clear set of harmonics of the base frequency (heart rate). At frequencies larger by more than an order of magnitude, the spectrum also contained, at lower amplitude, modes with less regular character. The classical slope of -5/3, characteristic to turbulent flows was added. Further analysis of the monitoring points in the aorta, the spectrum showed that the modes with strength of less than two orders of magnitude than the base mode, contain also non-harmonic fluctuations (“turbulence”). Thus, modes with larger frequency than 25-30 Hz seem to contain the irregular-turbulent, spectrum. In order to assess the fluctuating
components of the flow, a high-pass filtering threshold was set at 30 Hz. The high-pass filter enabled separating the MKE from its fluctuating counterpart, i.e. FKE. It should be emphasized that FKE is NOT reflecting turbulence, it rather displays the strength of fluctuations for
frequencies above 30Hz. Fig.10 depict the ratio of FKE to MKE for eight different heart-rates and flow rates as noted on monitoring points spread within the aorta. For low flow rates (6 LPM with 75 BPM) the FKE/MKE ratio was less than 0.1% but it increased to about 1% (10 LPM at 120 BPM) and to over 5% for the highest flow rate that was considered (15 LPM at 150 BPM).
The level of MKE increases along the aorta. Fig. 10 depict MKE and FKE in the three parts of the aorta for two of the extreme cases considered here (60 BPM/5 LPM and 150 BPM/15 LPM). The level of MKE increases along the aorta, reaching the largest values in the
downstream parts of the descending aorta. The level of MKE in the latter case (higher heart and flow rates) is almost an order of magnitude larger than for the former case. Similarly, the level of FKE is negligible whereas it is significant for the high flow rate.
20 DISCUSSION
This study exposes the fluid mechanical foundation of the mechanisms leading to observed flow structures in arterial flows. As noted in the literature [26-28], helical flow primarily depends on the geometry and the profile of the inlet conditions. These two parameters also determine the direction of the swirling motion. Hence, time- and space dependent inlet flow profile, the presence of stenosis and other geometrical disturbance (such as arterial bifurcation and aneurysm) may promote some natural swirling mode(s). Helical flow requires flow non-uniformity, such as streamline curvature, that implies non-vanishing vorticity.
Such streamline curvature may be observed in the vicinity of bifurcating branches from the aortic arch and in general it may be observed at vessel bifurcations as observed in this work (Fig. 2 during diastole, T5-T7). Wall surface irregularities (with wall curvature) may also lead to formation of additional vorticity. Analyzing the size of vorticity and the VS term (equation (2)) enables assessing the vorticity formation process throughout the aorta. The presented results elucidate the presence of strong vorticity near the walls of the aorta during systole (T1-T3). In early diastole (T5), the vorticity is found to be strongest in the ascending aorta and proximal part of the arch. During late diastole, the vorticity is advected downstream. The aortic sinus is a site for elevated vorticity formation. In the aortic geometry under consideration, a slight
dilatation in the inner curved wall in the middle of the arch was observed, exemplifying the sensitivity of the results to shape irregularities and uncertainty in the segmentation process.
This location is also a site for stronger vorticity during early diastole (T5). Similar findings of vortex formation and elevated WSS were reported in the literature [8-11,29]. Frydrychowicz et
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al (2007) [8] noted that “minor geometric changes can result in major disturbances of local vascular hemodynamics”. The underlying mechanism for this phenomenon is explained in quantitative terms through vortex-stretching mechanism.
Helical flow in arteries have been reported to have positive, atheroprotective effect on the arterial wall. On the other hand, retrograde flow and WSS with direction opposite to the bulk flow direction has been reported to have negative effect of the arterial wall. With increasing flow-rate, the flow may exhibit instability leading to episodes of non-periodic and non-harmonic, flow structures. The protective nature of helical flow [15] may possibly be explained by the fact that the heavier HDL particles (density of >1.06 kg/m3) are subject to a (centrifugal) force towards the aortic wall whereas LDL and VLDL particles (less than 1 kg/m3) are subject to a force towards the center of the lumen. This potential mechanism is most important for low heart-rates since with increasing heart rate, the level of fluctuation increases that in turn would increase local mixing. If this hypothesis is viable, it may have a direct impact on drug delivery strategies. However, this hypothesis requires further scrutiny and in the detailed studies to be verified.
Retrograde flow and flow separation in the aorta was observed along with flow
separation due to formation of adverse pressure gradient. The results (Fig. 2, mid-row, Figs. 5 &
6 and supplementary animations S1c & S2c) show the extent and the strong time-dependence of flow separation both within the lumen and near the aortic wall. The main mechanism for retrograde was due to the flow-rate deceleration at the aortic ostium. During flow-rate deceleration, associated with an inverse pressure gradient, the flow direction is reserved as
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compared to the central part of the aorta (as in Womersley pipe flow). The high momentum, central stream entrains blood from the region closer to the arterial wall leading to enhanced retrograde flow in the artery and pronounced retrograde flow close to the wall, implying negative WSS. The relative volume of retrograde flow as well as the area of negative WSS are only weakly dependent on the heart- and flow-rate, when normalized in terms of cardiac cycle time (Fig. 4). It can be noted that the peak values of retrograde flow occur at normalized times of about 0.5 and 0.8 of the cycle for the given inlet flow-rate profile. At this points, the inflow reaches its end of a deceleration at (end) systole and at about mid-diastole, respectively. Once the flow starts to accelerate, the extent of retrograde flow reduces through the entrainment of fluid by the central stream. Thus, depending on the temporal behavior of inlet flow profile, it is possible to predict the time for peak retrograde flow. It is also plausible that the extent of retrograde flow is directly related to the strength of deceleration. This notion could be of clinical interest for assessing the impact of cardiac muscle function on the temporal
development and extent of negative WSS. Flow separation is commonly unsteady, leading to oscillatory WSS that is often associated with atherosclerosis. Such effect may be observed also in other arteries, such the carotid sinus [30] and renal artery stenosis [20]. Another clinical impact of reversed flow and in particular in regions where the flow is reversed throughout the cycle (descending aorta) is the risk of upstream transport of thrombi. Such possible mechanism for stroke was suggested by, for example Wehrum et al (2015) [31].
The vorticity formation and its amplification through VS was shown to generate transitional flow and a narrow turbulent spectrum at high heart- and flow-rates. Moreover,
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high-flow rate implies the presence of multiple inflection points in the axial velocity
component, promoting relatively strong fluctuation growth. At high heart-rate, the advection of vorticity generated in the arch is relatively slow as compared to duration time of diastole, implying that flow fluctuations remain in the aorta and feed the growth of fluctuation during the coming cardiac cycle. The onset of turbulence, however, also depends on presence of fluctuations with modes that can resonate with natural modes of strong amplification.
The outcome of the flow simulations under different heart-rate and flow-rates clearly shows that the level of fluctuations (measured in terms of high-pass filtered kinetic energy not containing modes below 30Hz), under rest or light cardiac load conditions, is of the order 0.1%
of the total kinetic energy of the flow in the aorta. The level of fluctuation is in agreement with the early data of Sabbah and Stein [32] and the recent results of Gulan et al (2018) [29]. For the highest cardiac load conditions (150 BPM/15 LPM), the level of the fluctuations could reach to order of more than 10% or higher at some isolated monitoring points, though with considerably lower mean value (Table A1). MKE increased by utilizing the potential energy of the pressure (leading to pressure loss) and FKE is generated by extracting energy from the main flow through vortex stretching mechanism as discussed in the main body of the paper. The levels of MKE and FKE increased along the aorta (Figs. 11). The level of mean MKE for the highest flow rate (15 LPM) was about 20 J/m3 (equivalent to about 10 mJ in the volume of the aorta in Fig 1a). Ha et al [33] on the other hand, using 4D-flow-MRI, reported levels of FKE of the same order as the total kinetic energy (i.e order of 10 mJ). Assessing the level of FKE by 4D-flow-MRI is very challenging due to the limitations of the data acquisition rate, i.e. the need for adequate
24
number of statistically independent samples and correcting/accounting for physiological variations in the cardiac output during the measurement period. The fluctuations may be generated within the aorta itself or being due to amplification of fluctuations upstream, in the left ventricle or at the aortic ostium. The estimated amplification due to vortex stretching indicate that strong perturbations would be required to generate turbulence levels of the order of 10mJ in healthy patients at rest. Since strong fluctuations are commonly detected by
auscultation only in pathological situation and at very high flow rates in normal individuals, some caution should be attached to reported 4D-flow-MRI based turbulence levels.
Limitations
The results presented herein were computed for a given aortic shape, given inlet flow rate profiles and given set of outlet boundary conditions. The aortic walls were assumed to be rigid which leads to stronger retrograde flow as compared to a compliant aorta. Further simplifying assumptions that were made include blood rheology (Quemada) model, perfect blood mixture (constant density and hematocrit). However, the rigid assumption is reasonable for patients suffering from cardiovascular diseases. When considering generation and
amplification of vorticity, constant viscosity was assumed. Also, the approach presented here is simplified and hence the tendency to convective instability should be seen as indicative only.
Flowing blood flow may result in spatial variations in the mixture density leading to density gradients and vorticity formation through the baroclinic term in equation (2). This mechanism is believed to be considerably smaller than the VS term and was therefore neglected here.
25
ACKNOWLEDGMENT
We thank Associate Professor Chunliang Wang and Professor Örjan Smedby in Royal Institute of Technology (Dept. of Biomedical Engineering and Health systems) for allowing us to use the segmentation software Mialab. The computations were carried out on computer resources at NSC at Linköping University and HPC2N at Umeå University through a SNIC grant. Fuchs, A., acknowledges the support from the department of radiology at Linköping University Hospital.
FUNDING
This research used faculty funding from the Royal Institute of Technology, KTH (Prahl Wittberg and Berg) and partial support by the regional county of Östergötland (Fuchs).
26
SUPPLEMENTARY MATERIAL
Animation series 1 (S1). Heart rate = 75BPM & flow rate = 7.85LPM.
a b
c d
e f
27
Animation series 2 (S2). Heart rate = 150BPM & flow rate = 15LPM.
a b
c d
e f
28
For both animation series, frames a-e shows stream tracers coloured by:
a: Norm of the vortex stretching (VS) term (equation (2)).
b: Helicity.
c: Streamwise tangential velocity magnitude (tangential to the aortic centreline).
d: Norm of the vorticity vector.
e: Norm of the velocity vector.
f: Streamwise tangential velocity component, along lines 2 & 6 (Lines from figure 1a and snapshot images in Fig 5a).
29
NOMENCLATURE
f Frequency
ui Velocity
p Pressure
ρ Density
Kinematic viscosity
ωi
h
Vorticity Helicity
α Womersley Number
αRBC
3D-PTV
Volume fraction of Red Blood Cells (hematocrit) Three Dimensional Particle Tracking Velocimetry BCA Brachiocephalic Artery
BPM CFD
Beats Per Minute
Computational Fluid Dynamics CFI Cross Flow Index
CTA Computed Tomography Angiography FKE
LCCA
Fluctuating kinetic energy Left Common Carotid Artery
30
LNH Local Normalized Helicity LPM Liters Per Minute
LSCA Left Subclavian Artery MKE Mechanical Kinetic Energy MRI Magnetic Resonance Imaging
OSI PIV
Oscillatory Shear Index Particle Imaging Velocimetry RBC Red Blood Cell
RRT Relative Residence Time
TAWSS Time Averaged Wall Shear Stress transWSS Transversal Wall Shear Stress VS Vortex Stretching
WSS Wall Shear Stress
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34 Figure 1
a b c
Figure 2
a T1 T2 T3 T4 T5 T6 T7
b T1 T2 T3 T4 T5 T6 T7
35
c T1 T2 T3 T4 T5 T6 T7 Figure 3
a b c
Figure 4
a b
36 Figure 5
a) Peak systole (T2)
b) Mid-diastole (T5 – T6)
c) Late diastole (T6 – T7)
37 Figure 6
38
39
a b c
Figure 7
a b c
40 Figure 8
Figure 9
a b
41 Figure 10
a b
c d
e f
42
g h
Figure 11
a b c
d e f
43
Figure Captions List
Fig. 1 The computational geometry and location of 10 monitoring lines (a, left frame). The line directions are colored with red indicating the outer wall of the arch. Location of monitoring points inside the thoracic aorta (b, mid-frame). The time variation of the inlet flow-rate (c, right frame) for a case with 75 BPM (beats per minute) and flow-rate of 6 LPM (liters per minute). The seven time instances in the heart cycle for which results are presented in the following are marked by T1-T7.
Fig. 2 Stream tracers colored by the vorticity magnitude (a, upper row), velocity component along the centerline (b, middle row) and by the magnitude of the vortex-stretching term in equation (2) (c, lower row).
The frames from left to right correspond to the different time instants during the cardiac cycle; T1 to T7 as depicted in Fig. 1c. The color bars for each sequence is provided in the right most frame for each of the
parameters.
Fig. 3 The instantaneous helicity decomposed by the signs of the axial vorticity (a) and the axial velocity (Ua). WU-pp and WU-nn stand for the case with positive or negative a and Ua, respectively. WU-np and WU- pn stand for opposite signs of a and Ua, respectively. The three frames
44
(a-c) correspond to the helicity components in a) the ascending, b) arch and c) descending regions of the aorta, respectively.
Fig. 4 The instantaneous volume of retrograde flow relative to the total aortic volume (a). The corresponding negative WSS area relative to the total aortic wall area (b). Retrograde flow is defined by projection of the local velocity vector on the tangent to the centerline in the plane of the point perpendicular to the centerline, and is negative. The projected velocity component is denoted in the text as axial velocity. Similarly, negative WSS is defined as the projection of the WSS onto the local tangent to the local axis of the aorta. Note that the results are given in terms of fraction of the heart-cycle time.
Fig. 5 Velocity component (projected to the aortic centerline direction) distribution along Lines 2 and 6 for two different heart- and flow rates at three different instants of the cardiac cycle (using the notations in Fig.
1a) and 1c). 4a) correspond to peak systolic flow (T2), b) correspond to mid-diastole (T5-T6) and c) correspond to late diastole (T6-T7). The case in left column represents low heart- and flow rates (60 BPM and 5 LPM) whereas in right column the highest heart- and flow-rates (150 BPM and 15 LPM) is displayed. The line length is measured from the inner (lesser) curvature towards the outer (greater) curvature of the aorta.
45
Fig. 6 The axial velocity probability distribution in the three artic segments a) ascending, b) arch and c) descending aorta, respectively, at different time instances during the cardiac cycle. Systole starts at T = 0 and ends by T = 0.5.
Fig. 7 The contribution of the vortex-stretching term to growth of axil vorticity (right hand side of equation 2b). The distributions, correspond to the three regions of the thoracic aorta; a) ascending, b) arch and c) descending, respectively. The symmetry of the plot is visual only as the integral of the distribution yields total contributions: 21 s-2, 35 s-2 and 73 s-2, in the three regions (a-c) respectively.
Fig. 8 The phase averaged velocity (blue line) and the corresponding fluctuations (red line) versus time-step (=0.0001 s), as monitored in two points in the ascending aorta. Low heart- and flow rate (60 BPM and 5 LPM, left frame) and higher heart- and flow-rate (90 BPM and 9 LPM, right frame). Note that the fluctuations are scaled by a factor of 10.
Fig. 9 Power spectrum at a monitoring point in the aortic arch at heart- rate/flow-rate 90BPM/9LPM (a) and 150/15 LPM BPM (b). The red line denotes the line of theoretical (Kolmogorov) decay of high Reynolds number turbulence (-5/3). The low frequency part of the spectrum, in
46
both cases shows clearly the harmonics of the heart frequency implying also the lack of turbulence at frequencies below about 30Hz.
Fig. 10 The MKE and FKE (multiplied by a factor of 100) at 220
monitoring points in the aorta for eight different cases. The flow rate decreases from frame a to frame h. a) 150BPM & 15LPM; b) 120BPM &
10LPM; c) 90BPM & 10.4LPM; d): 90BPM & 9LPM with different flow distribution in the outlet faces (Catarino (2015); e: 90BPM & 9LPM; f:
90BPM & 6LPM; g: 75BPM & 7.85LPM; h: 75BPM & 6LPM.
Fig. 11 MKE and FKE at monitoring points in the ascending aorta (a, d), the aortic arch (b, e) and the descending aorta (c, f). The upper and lower rows correspond to heart-rate of 60 BPM (5 LPM) and 15 BPM (15 LPM), respectively. The numbering of the monitoring points is related to the distance along the aorta.
47 Table 1
Helicity parameter/Case hs [m/s2] ha [m/s2] h3 [-] h4 [-]
60BPM & 5 LPM 0.19 2.46 -0.92 0.08
90BPM & 9 LPM 2.66 30.03 -0.91 0.09
120BPM & 10LPM 5.31 50.70 -0.90 0.10
150BPM & 15LPM 23.45 200.47 -0.88 0.12
Table 2
Relative retrograde flow in the aortic volume
Relative negative WSS on the aortic walls
60 BPM & 5 LPM 0.155 0.324
75 BPM & 7.85 LPM 0.134 0.314
90 BPM & 9LPM 0.125 0.284
150 BPM & 15 LPM 0.103 0.290
48 Table 3
Kinetic energy/Case Ascending aorta Aortic arch Descending aorta Entire aorta
60BPM & 5 LPM 1.0e-05 0.0001 0.0001 1.8e-05
90BPM & 9 LPM 4.0-04 5.1e-04 3.9e-04 0.002
120BPM & 10LPM 2.1e-03 2.8e-03 1.7e-03 1.6e-03
150BPM & 15LPM 3.1e-03 2.1e-03 2.2e-03 1.1e-03
Table Caption List
Table 1 Helicity parameters, hs, ha, h3, and h4 as defined in equations (3) and (4).
Table 2 Cardiac cycle average of the relative aortic volume with retrograde flow and aortic wall area with negative WSS, for four combinations of cardiac heat- and flow-rates.
Table 3 The portion of the mean fluctuating kinetic energy (FKE) relative to the mean mechanical kinetic energy (MKE) in three segments of the aorta.
