The impact of heart rate and cardiac output on the flow in the human thoracic aorta
Alexander Fuchs1,2* (ORCID: 0000-0003-0112-9271) Niclas Berg2 (ORCID: 0000-0002-6881-2094)
Lisa Prahl Wittberg2 (ORCID: 0000-0001-9976-8316)
1. Department of Radiology, Karolinska University Hospital, Stockholm, Sweden.
Department of Medical and Health Sciences, Linköping University, Linköping, Sweden.
2. FLOW & BioMEx, Department of Engineering Mechanics, Royal Institute of technology, KTH,
Osquars Backe 18, SE-100 44 Stockholm, Sweden.
*Corresponding Author: alex@mech.kth.se
Abstract:
Purpose: The purpose of the study is to determine the effects of heart rate (HR) and cardiac output (CO), in the temporal variation of CO on flow structures and related biomechanical markers.
Methods: The pulsatile flow in the thoracic aorta was simulated for 15 combinations of HR (60- 150 beats per minutes, BPM), CO and cardiac temporal profiles. In all cases, the Quemada viscosity model was used. The results were analyzed in terms of biomechanical markers such as extent of retrograde flow in the lumen and close to the wall, helicity parameters, commonly used wall shear stress (WSS) indicators along with proposed Endothelial Activation Indices (EAIs).
Results: The simulations demonstrated the presence of helical motion in all cases. The helical motion depends on the spatial distribution of the flow by the aortic valve. Time- and space-averaged helicity indices were found to have smallest values in the aortic arch and largest in the descending part of the aorta. For all cases, retrograde flow was observed. The extent of separated flow close to the aortic wall depended strongly on the rate of decelerating CO during late systole as well as possible axial flow deceleration periods during diastole. At high HR and CO, small scale flow structures developed, indicating transition to turbulence. Time averaged WSS-related indicators were less distinctive in assessing the spatial and temporal impact as compared to the EAI indicators (EAINobili and EAISoares) accounting for both accumulated stress and the temporal behavior of the stress.
Conclusions: The results underpin the importance of temporal variation of the cardiac flowrate and the impact of the deceleration phase of systole on retrograde flow and formation of helical flow structures. As retrograde and helical flow has been found to be related to atherosclerosis, the temporal contribution of the flowrate must be maintained, since time averaged biomechanical indicators filter out information of potential diagnostic importance. Temporal flow behavior, up to cell response frequency, needs to be reflected by the biomechanical indicators as in the proposed EAISoares indicator.
Keywords: Aortic hemodynamics, retrograde- and helical-flow, wall shear-stress, CFD.
Declarations
Funding The research received no external funding (faculty funding only).
Conflicts of interest/Competing interests: The authors declare no conflict of interest, nor any competing interests.
Ethics approval The research 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).
Consent to participate Informed consent was obtained before patient CTA data was anonymized.
Consent for publication: Not applicable.
Availability of data and material: All data relevant to the research presented could be obtained from any of the authors upon request.
Code availability: The modified version of OpenFOAM used for the simulations can be downloaded from https://github.com/niclasberg/BOpenFOAM. The scripts used for postprocessing (MATLAB and python) are available upon request from the corresponding author.
Authors' contributions
A.F. worked on segmenting the aortic model, generating grids, simulations, post-processing, analysis of results, preparing figures and tables as well as writing the original draft of the manuscript.
N.B. worked on post-processing, implementations and maintenance of the necessary software as well as reviewing the manuscript.
L.P.W. worked on conceptualization of the study, interpretation of results, managing the research project and reviewing the manuscript.
All authors read and approved of the final manuscript.
Acknowledgements
We thank Associate Professor Chunliang Wang and Professor Örjan Smedby in the Royal Institute of Technology (School of Technology and Health) for allowing us to use the
segmentation software Mialab. The computations were enabled by resources provided by the Swedish National Infrastructure for Computing (SNIC) at NSC at Linköping University and HPC2N at Umeå University partially funded by the Swedish Research Council through grant agreement no. 2016-07213. The corresponding author acknowledges the support from the department of radiology at Linköping University Hospital.
Introduction/background
The heart works in a wide range of operating conditions for which the “operating points” may change rather quickly. Thus, observations carried out a given instant of time, or even under short time intervals, can be problematic and are insufficient if seeking to characterize the heart as a pump or the impact of the heart work on the arterial flow. In clinical situations, longer observation periods may be used during which variations in the heart operating conditions may or may not be induced.
On the other hand, such longer time observations are limited in terms of availability of more
detailed information regarding the flow conditions. Iwata et al. (2012) carried out ambulatory blood pressure studies to determine possible relation to atherosclerosis. It was found that the 24h
ambulatory Blood Pressure (BP) monitoring variables were clearly associated with the presence of plaques in the aortic arch. This approach provides less detailed information as compared to 4- Dimensional Magnetic Resonance Imaging (4D flow MRI) data (commonly taken under rest
condition during a period of some minutes). Due to resolution superiority over diagnostic tools such as MRI and CTA, numerical simulation of blood flow may provide details about the flow conditions (cf Binter et al. (2015)) However, the number of simulations is commonly limited to a single or a small number of operating conditions due to the required computational time. During the past decades, numerous studies have applied computational fluid dynamics to arterial flows to explain certain arterial pathologies such as atherosclerosis (cf Johnston et al. (2004)). The recent years increase in computational power has facilitated the use of highly resolved unsteady flow
simulations, an important development since if seeking to understand mechanisms with potential to lead to pathologies such as atherosclerosis, temporal flow characteristics are a necessity.
Heart contraction is characterized by the heart rate (HR) or pulse in terms of beats per minutes (BPM) and cardiac output (CO), commonly expressed in liters per minute (LPM). These two
parameters are dependent of each other as the amount of blood being pumped depends on the stroke volume times heart rate. Yet, the stroke volume depends on the amount of blood at end diastole in the left ventricle and the ability of the heart muscle to eject a certain portion of that blood, i.e.
ejection fraction (EF). In addition to these two parameters, the pumping characteristic of the heart requires characterization of the contraction and relaxation rates, essential for the flow in the systematic arteries. The relation between cardiac output and the formation of atherosclerosis was based on observations and in-vivo animal experiments as well as molecular studies at cellular level (REF). Over the years, different findings have indicated that high- and/or low- levels of shear stress and oscillatory shear stress could lead to formation of atherosclerosis. A recent review of Souilhol et al. (2020) discussed the endothelial response to shear-stress in atherosclerosis. The review summarizes the current understanding of the relation between shear stress and arterial pathology namely; that “low or low oscillatory shear stress” promotes atherosclerosis, supported by findings in animal studies carried out by introducing an artificial stenosis similar to those
associated with plaques (Souilhul et al. (2020), Cheng et al. (2006), Pedrigi et al. (2016)). Gijsen et al. (2013) conclude that coronary plaques grow in the distal regions exposed to low shear stress, leading to a stable plaque type. The regions upstream to or around the plaque itself experience higher shear stress and the plaque is softer. Even higher shear stress (higher than an estimated normal of 15 dyne/cm2 (1.5 Pa)), have been reported (Malek et al (1999) to be atheroprotective.
Similarly, the study by White et al. (2011) indicated that at five times higher shear stress than normal, the response of endothelial cells to elevated shear stress displayed atheroprotective effects, at least in terms of altered genetic expression.
Regarding the variability of the hear operating conditions, the study by Gemignani et al. (2008) testing 103 healthy individuals and patients performing semi-supine bicycle exercise, observed that at higher heart rates (i.e. 100 BPM), the systolic/diastolic time ratio was lower in the control
subjects as compared to different patient groups. The systolic/diastolic time ratio was found to be
about 0.5 at 60 BPM, increasing to almost one at 130-160 BPM range. No information about the changes in timing during the different phases of systole or diastole was provided. The data of Gemignani et al. (2008) yielded peak acceleration and deceleration ranging from 0.05 to 0.004 m/s2 and -0.02 to -0.002 m/s2, respectively. Munch et al. (2014) presented the effect of HR pacing on cardiac output during maximal cycling of 8 endurance trained cyclists, by measuring systemic and Weissler et al. (1961) studied the relationships between left ventricular ejection time, heart rate and stroke volume in human. Heart failure patients had short ejection time, related to the low stroke volume in these patients. The decrease in ejection time relative to heart rate was a consequence of the low stroke volume in congestive heart failure. Aortic stenosis patients had prolongation of ejection time i.e. slower velocity of myocardial contraction attributed to excessive intraventricular load induced by the high outflow resistance. In a recent article, Runte et al. (2019) carried out a systematic review and meta-analysis of the effects of hemodynamic changes curing physiological and pharmacological stress testing in healthy subjects and patients. The study provided some reference values and estimations of expected individual range of a circulatory response in healthy individuals and patients with aortic stenosis. The increase in HR/CO for healthy individuals with increasing intensity of the exercise; Light exercise: +32 BPM/2.7LPM; moderate;
+50BPM/4.7LPM; high dynamic exercise: +89BPM/10.45LPM.
A complicating factor in quantifying heart pumping is due to the variability of the heart activity. As pointed bout by Vignon-Clementel et al. (2010), flow and pressure in arteries are not necessarily periodic in time, due to heart-rate variability, respiration, complex, transitional flow, or acute
physiological changes. Additionally, aortic flow is significantly affected by the temporal behavior of the rate of injection under different loading for healthy individuals and for patients with heart failure.
Unfortunately, quantification of this factor is not found in the literature. On the cellular level, Janssen et al. (2010) studied the kinetics of cardiac muscle contraction and relaxation. Under normal
conditions muscle kinetics slows down with increasing contraction amplitude. Janssen et al. (2010) concluded that the contraction-relaxation kinetics depends on the sarcomere function itself and is not primarily dictated by Ca2+-handling processes. The findings also imply that flow structure (much) faster than cellular biochemical processes should be filtered out. On the other hand, slower flow structures affect the forces acting on endothelial cells.
Aortic flow depends on the cardiac pumping ability. As mentioned earlier, aortic flow has been increasingly and extensively studied using modern 4D flow MRI and Computational Fluid Dynamics (CFD) tools (cf Huang et al. (2016) to mention one of a huge number of publications). MRI is a widely used clinical tool applied for diagnostic purposes allowing for blood flow measurements in- vivo in four dimensions (space and time). However, the temporal and spatial resolution of MRI is limited and there are commonly finite number of scans per patient available. Thus, MRI is not used as a short-term monitoring tool. Taylor et al. (2002) studied blood flow and WSS in the human abdominal aorta using MRI. The effects of exercise on hemodynamic conditions in young, healthy subjects lead to an increase in heart rate, in average from 73 to 110 BPM, whereas the flowrate increased from about 3 LPM to about 7.2 LPM. During exercise, a substantial increase in WSS was observed; by a factor of about two in the supraceliac aorta and factor of about four in the infra-renal aorta.
To compliment patient and experimental studies, CFD is an excellent tool for providing detailed information to enable improved understanding of the dynamics of blood flow and its quantitative impact on the blood vessel. Over the past decades, large number of blood flow simulations in arteries have been carried out. Earlier papers considered steady-state flows without or with turbulence
models, rheological models for the blood and the interaction between the flow and the dispensability of the arterial wall. Nowadays, time dependent highly resolved arterial flow simulations are
customary resulting in large amount of data (velocity and pressure fields) that must be reduced to parameters relevant for the issue under consideration. As atherosclerosis is major health problem in
modern society, Wall Shear Stress (WSS) related indicators have been used to characterize the simulated flow field such as the Time Averaged WSS (TAWSS), Oscillatory Shear Index (OSI) and Relative Residence Time (RRT). Most CFD simulations have been based on so called patient- specific geometrical data along with empirically measured inlet boundary conditions, often focusing on different modelling related issues (wall properties of arteries, blood rheology) and/or clinical relevance (WSS, presence of turbulence) (e.g. Andersson et al (2019), Morbiduci et al. (2013), Pirola et al. (2018), Youssefi et al. (2018), Bocca Soulis et al. (2014), Mark et al. (2012), Gallo et al.
(2014), Pinto and Campos (2016), Fan et al. (2016) LaDisa et al. (2011), Condemi et al. (2017), De Nisco (2018), Fuchs et al. (2019), Vignon-Clementel et al. (2010), Zhu et al. (2018), Lantz et al.
(2013), Glimanov et al. (2019)).
Although arterial flow simulations have largely been investigated in literature, questions remain to be resolved with respect to the effect of variability in heart operating conditions on the flow structures developed and how it influences possible atheroprotective effects. In this study, the objective was to investigate the effect of the wide range of HR and CO that practically every individual experiences many times during life-time, characterizing the typical arterial flow
structures, i.e. helical motion and retrograde flow, and their impact on the wall of the thoracic aorta.
Numerical methodology
The geometry of the thoracic aorta was derived from a Computed Tomography (CT) scan of a healthy subject including the ascending aorta, the arch, the descending aorta, and the tree main branching arteries. The descending aorta followed the CT data only down to the level of the diaphragm. The computational geometry was further extended downstream along the descending aorta and the three head and neck branches (Brachiocephalic Artery (BCA), left Common Carotid Artery (LCCA) and left Subclavian Artery (LSCA)). The blood components were assumed to be incompressible although its bulk density may change due to spatial distribution of blood cells and plasma proteins. Assuming a constant density, the governing equations were simplified to include only conservation of mass (Eq. 1a) and momentum (Eq. 1b).
( i j)
i i
j i j j
u u u p u
t x x x x
(1a)
0
i 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 was assumed to be non-Newtonian implying that the bulk viscosity depends on the local blood composition and the local flow
conditions. Several empirical correlations have been proposed, e.g. Hund et al. (2019), for a recent review, to account for blood viscosity as function of shear-rate and in some cases also as function of hematocrit. The results presented here were computed assuming Quemada model with a hematocrit of 45%.
Considering boundary conditions, a time-dependent velocity profile was applied at the inlet of the ascending aorta, having a “top-hat” shape or characterized by cross-plane variation. The latter case was used to simulate an aortic valve stenosis (top-hat jet with reduced cross section or a smoothed jet with a given shear-layer thickness). The time-dependent flowrate was based on MRI data reported in the literature (e.g. data presented by Frydrychowicz et al. (2009)) and different heart- and flowrates were derived from the basic profile by:
i) Scaling the time by the cardiac cycle time (one over HR) and then scaling the inlet plane velocity to satisfy the specified CO,
ii) Specifying the heart rate and using the experimental data of e.g. Munch et al. (2014) or Rodeheffer et al. (2008) to set the CO,
iii) Specifying the ratio of systolic to diastolic time versus the heart rate. For a given heart rate, the duration of systole and diastole were computed, and the basic profile were scaled in time accordingly. The size of the flow rate was adjusted to satisfy the given CO,
iv) Explicitly setting the duration of the flow rate deceleration during late systole. This was carried out to simulate different types of heart-muscle contractility.
The temporal distribution of six flow rates is presented in Fig 1. Fig 1a displays the profile based on MRI data from Frydrychowicz et al. (2009). The CO in Figs 1b and 1c were derived from data by Liu et al. (2011) and Taylor et al. (1998), respectively. The profiles in Fig 1d-f were modified versions of Fig 1a, significantly changing the deceleration phase according to (iv).
a b c
d e f
Figure 1: Six inlet flow rate profiles derived from published data. Higher heart rates were derived by rescaling linearly the time, while flowrate was derived by scaling. The profiles in d-f were derived by altering the time-variation of the deceleration phase of systole.
The flow distribution in the different branches was as suggested by Benim et al. (2011), setting the flow rates as follows: The BCA branch delivered 15% of the inlet flow rate whereas the LCCA and LSCA delivered each 7.5% of the inflow. The remaining blood (70%) went through the descending aorta. The pressure at the distal descending aorta was kept at a constant value whereas the pressure gradient at the other non-wall boundaries was set to zero.
OpenFOAM 5.0 was used to discretized and solve the governing equation. The computational domain was discretized on a set of small (hexagonal and tetrahedral) cells. Grid sensitivity study was carried out with the number of computational cells varying between below 1 million and up to 20 million. For the results presented here mostly 5 million cells for the results presented here except for the highest HR and CO rates for which 15 million cells were used. The governing equations were approximated on the discrete nodes/cells of the computational domain using a formally second order (polynomial) approximation to the different terms of the equations. Small time-steps were used (in the range of 0.1 ms at HR=60 BPM and down to 0.01 ms for HR=150 BPM, to allow
resolving temporal fluctuation). The numerical methodology follows the one described in Fuchs et al. (2019) and van Wyk et al. (2015). The data was post-processed using Paraview and own python scripts along with corresponding MATLAB routines.
Data analysis & Post processing
The data was analyzed with respect to the presence of helical flow and its relation to retrograde flow and WSS. The helicity (h) is defined as the scalar product of the local velocity vector (ui) with the vorticity vector (ωi), i.e. h= uiωi The helicity data was reduced by averaging in space to assess the temporal development of helicity relative to the cardiac cycle (cf Gallo et al. (2012b),
Morbiducci et al. (2013) and De Nisco et al. (2018)) defined as:
0
1 ( )
T
s i i
h u d dt
T
0
1 ( )
T
a i i
h u d dt
T
(3a)where T and are the integration time and aortic volume, respectively. An indicative index for directionality of rotation was suggested by Gallo et al. (2018) by defining normalized parameters.
Here, the Helicity Oscillatory Index, denoted in the following by hosi, is introduced by us to assess the temporal oscillatory behavior of helicity. This parameter, equation 3b, is analogous to the Oscillatory Shear Index (OSI) commonly used to characterize the oscillatory character of the wall shear stress.
1 1 2
s s
r osi
a a
h h
h and h
h h
(3b)
For flows with unidirectional helical motion hs = ha implying that hr=1 and hosi=0. For highly oscillating helicity hs=0, and hence hr=0 and hosi=0.5. These descriptors are modified versions of Morbiducci et al. (2013).
Non-stationary retrograde flow leads to non-stationary negative WSS (according to definition below). The WSS in plane of the arterial wall can be defined as the projection of the WSS tensor into the plane tangent to the wall at a given point. If Ni is the normal vector to the wall at a given point, the projected WSSij tensor (i.e. Ni WSSij ) is a vector in plane tangent to the wall. For
convenience, in the following, this vector is also denoted as WSS. Natural local coordinate may be introduced by projecting the velocity vector onto the direction of the axis of the aorta and
components normal to the tangent. The local velocity component (ui) projected on the tangent to the centerline (Ti) is defined as utan = uiTi. Similar projection was done by Morbiducci et al. (2015) and Gallo et al. (2016).
Two parameters related to retrograde flow were defined:
- Relative Volumetric Retrograde Flow (RVRF)
The metric measured the volume of utan < 0 relative to the total aortic volume (V). Thus, RVRF is defined as:
tan tan
( ) 0 0
( ) ( )
1 0
e
V e
V
H u dV
RVRF t H u u
dV u
(9a)Martorell et al. (2014) normalized RVRF with a corresponding value for a straight pipe.
- Relative Negative WSS (RNWSS)
The wall points where WSS is negative (WSSneg) are defined as WSSneg= Tj Ni WSSij < 0.
Hence, RNWSS is defined on the aortic wall surface (S) according to:
( )
0 0
( ) ( )
1 0
e S neg
e
neg S
H WSS dS
RNWSS t H WSS WSS
dS WSS
(9b)WSS related indictors
As the WSS varies in space and time, it is customary to use WSS related parameters to capture different features of WSS. The most common parameters are Time Averaged Wall Shear Stress (TAWSS), Oscillatory Shear Index (OSI) and Relative Residence Time (RRT), defined as:
- Time Averaged Wall Shear Stress (TAWSS), (Suo et al. (2008) and Chen et al. (2016)) defined by;
0
1T
TAWSS WSS dti
T
(4)Note that TAWSS contains no information about WSS temporal variation.
- Oscillatory Shear Index (OSI) (cf. He et al. (1996), Chen et al. (2016)). OSI measures high- frequency variations of WSS and is defined as:
0
0
| |
1 1 2
T i T
i
WSS dt OSI
WSS dt
(5)
- Relative Residence Time (RRT) was defined by for example Rikhtegar et al. (2012) and Gallo et al. (2012):
0
1 (1 2 ) 1
T i
RRT WSS
OSI dt
T
(6)
In an earlier paper (Fuchs et al. (2019)), we proposed an approach to model activation of endothelial cells, denoted Endothelial Activation Index (EAI), analogous to activation of platelets as proposed by Nobili et al. (2008). In the Nobili platelet activation model, the target variable was named Platelet Activation State (PAS) and defined as the fraction of activated platelets relative to all platelets in the tested blood. Thus, PAS is a number varying between 0 and 1. The model has three empirically determined parameters; a, b and c (Nobili et al. (2008)). EAI was modelled using the same numerical values for the model parameters as suggested by the original model for platelets. The model assumes that PAS is a function of the scalar shear-stress () and time (t). The scalar stress () is defined as the Frobenius norm of the tensor.
/ 1 ; /
b a a b a
d PAS d D
ca D
dt dt (7)
The model parameters used were a=1.3198, b=0.6256, and c=10-5. The numerical values were determined from experimental data (cf Xenos et al. (2010)). Soares et al. (2013)
proposed a model explicitly including the effects of the temporal variations of the stress. The activation model resembles chemical reactions, leading to asymptotically to a final PAS value.
The model includes contribution from history effects (S), instantaneous stress (F) and time-derivative of the local stress (G):
( ) (1 ) d PAS
S F G PAS
dt (8a)
The three terms were defined by:
( , t) r ( ) t( )
S PAS H S PAS t H t (8b)
1 1
( , ) ( ) ( )
F PAS C PAS t t
(8c)
1 1
( , ) r ( ) ( )
G PAS C PAS t t
(8d)
Where Sr =1.5701 10−7; C=1.4854 × 10−7; =1.4854; =1.4401; Cr=1.3889 10−4; =0.5720,
= 0.5125. was defined by Soares et al. (2013) as the average of the absolute value of time-derivative of the stress. Here, we propose to instead use a low pass filtered value (i.e.
below 30Hz) of the scalar stress.
Results
Retrograde flow
The presence of retrograde flow was assessed in terms of two quantities: RVRF (equation (9a)) and RNWSS (equation (9b)). The former expresses the relative volume of retrograde flow relative to the total volume of the aorta whereas the latter expresses the relative aortic surface having WSS pointing against the center line tangent relative to the total aortic surface area. Table 1 contains data of 15 simulations. The inlet flow-rate profiles refer to those presented in Fig 1. The “3D” notation indicates that that plug-flow profile was applied with no-slip at/near the wall of the inlet plane.
Case 9 is a 3D profile with a blockage of the inlet plane area by 30%. The other profiles correspond to a uniform inlet axial velocity profile throughout the inlet plane. In all cases, the total retrograde flow volume was less than 20% and decreasing with heart- and flowrates. The surface area with negative WSS varied around 30% for the different HR and CO considered. Exceptions were Case 5 with about 40% negative WSS using Profile b. Case 6 with Profile c, had only 15% negative WSS.
The dependency of RVRF on HR and CO showed to be somewhat stronger than that of RNWSS.
The tabulated values include the averaged values and standard deviation. A graphical presentation of results in Table 1 are provided in Figure 2 where the mean values and corresponding standard deviations are marked as function of HR (Fig 2a) or CO (Fig 2b). The general trend is that RNWSS is more sensitive to both HR and CO as compared to RVRF. The extent of retrograde flow volume and negative WSS were smaller at higher HR/CO. There is a variability in the results also for the same HR and CO, implying the dependence on the details of the inlet flow-rate profile.
RVRF, RNWSS [-] RVRF, RNWSS [-]
Figure 2: RVRF and RNWSS vs a) HR and b) CO with the mean vales marked by a symbol and the standard deviation given by error bars. The mean RVRF and RNWSS decrease with increasing HR and CO. The large error bars indicate the presence of strong fluctuations, which are smaller at lower HR and CO, increase at intermediate HR and CO and decrease again at high HR and CO.
Table 1: Temporal averages of the relative retrograde volume (denoted as RVRF) and relative negative WSS (denoted as RNWSS). The different inlet flow profiles refer to those depicted in Figs 1. The mean values for RNRF is less than 20% for the low HR/CO cases decreasing to just above 10% for the high HR/CO cases. The corresponding vales for RNWSS are above 30% and are somewhat lower for higher HR/CO. The table also gives the standard deviation of the data from the mean. HR and CO are the heart rate (BPM) and flowrate (LPM), respectively.
The behavior of RVRF and RNWSS is better understood if considering the temporal development of the parameters over the cardiac cycle and the relation to the time development of CO. Figure 3 depict RVRF and RNWSS for six of the computed cases for HR = 60 BPM and CO =5 LPM (Fig 3a – c and e - f) and HR = 150 BPM and CO =15 LPM (Fig 3d). Each frame includes RVRF, RNWSS and flowrate (normalized by its peak value during the cardiac cycle). Fig 3a-d are based on the flowrate of Frydrychowicz et al. (2009) modified as follows: Fig 3a is related to HR = 60 BPM and CO = 5 LPM. Fig 3b was simulating the same conditions but with a prolonged
deceleration during systole and thereby also shorter diastole. Fig 3c shows a similar case but with an abrupt termination of the deceleration phase (i.e. a very strong deceleration) leading to an instantly almost complete negative WSS in the whole aorta. The corresponding response in terms of the extent of retrograde volume was equally strong. However, still more than 60% of the volume flow was in the normal streamwise direction. Fig 3d depicts a similar situation as Fig 3a but at higher heart- and flowrates (HR =150 BPM and CO = 15 LPM). The relative volumetric retrograde flow in the low HR/CO (Fig 3a) and the corresponding high HR/CO (Fig 3d) are similar when scaled by HR, having a peak RVRF of about 40%. The peak negative WSS, on the other hand, decreases from about 90% to about 70% with increasing HR/CO. Fig 3e and f show RVRF and RNWSS for HR = 60 BPM and CO = 5 LPM, but characterized by different inlet flow rate profiles (corresponding to those in Fig 1b and c, respectively).
Case Inlet Profile (Fig 1) HR CO RVRF RVRF-std RNWSS RNWSS-std 1 Profile a 3D 60 5 1,80E-01 1,45E-01 3,21E-01 2,73E-01 2 Profile a 1D 60 5 1,44E-01 1,22E-01 2,83E-01 2,70E-01
3 Profile f 60 5 1,57E-01 1,38E-01 2,77E-01 2,67E-01
4 Profile e 60 5 1,42E-01 1,23E-01 2,86E-01 2,53E-01
5 Profile b 60 5 2,29E-01 1,59E-01 4,01E-01 2,64E-01
6 Profile c 60 5 2,85E-02 4,16E-02 1,47E-01 2,07E-01
7 Profile a 75 6 1,76E-01 1,14E-01 3,07E-01 2,39E-01
8 Profile a 90 9 1,71E-01 1,16E-01 2,87E-01 2,01E-01
9 Profile a Stenosis 90 9 3,16E-02 3,68E-05 1,10E-01 8,07E-05 10 Profile d 3D 90 10 3,12E-03 6,39E-06 2,53E-02 4,36E-05 11 Profile a 120 10 1,39E-01 1,18E-01 2,97E-01 2,13E-01 12 Profile a 120 14,8 1,18E-01 1,16E-01 2,66E-01 1,91E-01
13 Profile a 130 16 1,26E-01 1,25E-01 2,78E-01 2,00E-01
14 Profile a 140 17,1 9,35E-02 1,10E-01 2,43E-01 1,82E-01
15 Profile a 150 15 1,24E-01 1,15E-01 2,85E-01 1,89E-01
An important observation is the clear relation between the peaks of both the retrograde flow and negative WSS and the time instant when a deceleration phase ends, and a positive acceleration begins. This effect happens not only at the beginning of diastole, but also during irregular variation in the flow-rate, clearly shown in Fig 3e and 3f, where in the former, the inlet profile has a low amplitude oscillation during diastole and three distinct occasions in the latter. As expected, the near wall response, as expressed in terms of negative WSS, is stronger than the corresponding
volumetric response.
a b
c d
e f
Relative Volume/WSS [-]
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Time [s]
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Time-dependent Retrograde flow HR60-5-Relax-aoraki
Relative Volume Relative WSS<0 Norm inlet
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Time [s]
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Relative Volume/WSS [-]
Time-dependent Retrograde flow HR150-15F
Relative Volume/WSS [-] Relative Volume/WSS [-]
Figures 3: Temporal variation of the relative retrograde flow volume and the relative aortic volume with negative WSS related to the volumetric flow rate (normalized with its peak value).
Frames a – c with HR = 60 BPM having the same early systole inlet profiles, but differing in deceleration phases: normal, prolonged, and sudden jump character, respectively (frames d – f).
Frames e – f corresponds different inlet flow rate profiles according to that provided in Fig 1b and c, respectively, for HR =60 BPM, whereas Frame d corresponds to 150 BPM.
Helicity
Related to the flow structure in the aorta is the helicity parameters (equations (3)). These parameters are averaged in space and time and provides therefore only a general view regarding the flow conditions. Fig 4 depict the helicity parameters, hs, ha versus HR for the three segments of the aorta as well as the entire aorta. Table 2 summarizes the numerical values of hs, ha and hosi. The results show that the helicity has lowest absolute values in the ascending aorta and largest values in the descending aorta. It is also noted that hs is weakly negative for cases with the inlet flow corresponding to a minor aortic stenosis. With the inlet flow rate as depicted in Fig 1b, hs is negative in the ascending aorta, whereas it is non-
negative in the aortic arch. For the entire thoracic aorta, the helicity increases with increasing heart- and flowrates. Also note Cases 9 and 10 with 3D inlet profiles with character of jets instead of plug-flow shape in most other cases. As heart- and flow-rates increase from low HR/CO (HR = 60; CO = 5 LPM) to high HR/CO (HR = 140 BPM, CO = 17 LPM) hs and ha
increase from about 0.2 and 2.4 m/s2 to about 28 and 263 m/s2, respectively. Thus, the increase in helicity is super-linear in terms of HR or CO. The signed helicity, hs has a fixed signed indicating that the sign of the helicity in terms of volumetric- and time-average
remains the same. By definition hs less or equal to ha always. In contrast to the variations in hs
and ha, hosi varies only marginally around 0.45 for most cases, indicating that the helicity is highly oscillatory with hs being about 0.1 of the size of ha.
a b
Figure 4: The strength of helicity, hs (a) and ha (b) at the different segments of the aorta as well as the entire aorta.
Table 2: Helicity parameters (equations (3)) in the ascending aorta, aortic arch, descending aorta, and the entire aorta for 15 inlet flow-rate profiles. The Profile in this table correspond to those in Figs 1.
Being interested in the presence and strength of helical motion in the thoracic aorta, the paths of stream-tracers at different times of the cardiac cycle were considered. Figure 5 depicts two cases corresponding to a low HR and CO rates and the highest flow rate considered. At all combinations of HR and CO, the aortic flow has a helical motion in the ascending aorta, with a minor separation region in the inner wall of the bend. This separation region
reduces/disappears during systole and diastole. The flow in the aortic arch and descending aorta is nearly parallel to the centerline axis near the peak of systole. The deceleration phase of systole implies, as shown in Fig 3, the formation of a near wall separation starting closer to the outer wall and reaching a peak at the end of deceleration of the inflow. Fig 5b depicts a counterclockwise helical structure in the descending aorta, which changes direction at mid- diastole. At high HR and CO, the flow becomes irregular after reaching peak systole leading to generation of small-scale structures containing higher frequencies (not harmonics of HR).
The non-harmonic fluctuations dissipate for most HR and CO except for HR > 120BPM and CO> 10LPM. The reduction in helicity for the larger HR and CO, noted in Table 2, is the result of the small-scale structures having different direction and it does not indicate the strength of the coherent, large scale helical motion.
After peak systole after end systole mid-diastole a)
Case Profile HR CO hs [m/s2] ha [m/s2] hs [m/s2] ha [m/s2] ha [m/s2] ha [m/s2] ha [m/s2] ha [m/s2] hosi [-]
1 Profile a 3D 60 5 0,121 1,552 -0,011 2,370 0,346 2,993 0,185 2,413 0,462
2 Profile a 60 5 0,118 1,495 0,000 2,302 0,336 3,589 0,183 2,644 0,465
3 Profile f 60 5 0,140 1,703 -0,034 2,644 0,309 3,953 0,168 2,957 0,472
4 Profile e 60 5 0,100 1,280 0,026 1,980 0,321 2,631 0,178 2,070 0,457
5 Profile b 60 5 0,120 1,904 0,039 2,970 0,438 3,795 0,240 3,032 0,460
6 Profile c 60 5 0,026 0,722 0,069 1,065 0,158 1,275 0,096 1,061 0,455
7 Profile a 90 6 0,313 4,175 0,105 6,237 1,271 7,987 0,679 6,395 0,447
8 Profile a 90 9 0,652 9,433 1,274 16,893 1,775 15,919 1,313 14,258 0,454
9 Profile a Stenosis 90 9 0,003 0,267 0,026 0,229 0,052 0,298 0,031 0,270 0,442
10 Profile d 3D 90 10 0,004 0,167 0,026 0,213 0,044 0,267 0,028 0,224 0,438
11 Profile a 120 10 1,519 28,006 3,945 48,144 7,531 72,848 4,855 53,407 0,455
12 Profile a 120 14,8 2,941 66,268 14,659 116,969 28,617 149,579 17,518 116,985 0,425
13 Profile a 130 16 4,986 87,766 21,723 161,561 38,483 228,224 24,336 169,825 0,428
14 Profile a 140 17,1 6,253 125,633 27,841 229,237 38,643 348,941 27,777 263,294 0,447
15 Profile a 150 15 5,090 102,250 26,951 189,960 33,152 269,962 23,454 200,468 0,442
Ascending aorta Aortic arch Descending aorta Whole aorta
b)
Figure 5: Stream-tracers colored with the local absolute value of the helicity for a) HR = 60 BPM and CO = 5LPM and b) HR = 140 BPM and CO = 17.1 LPM. Left, mid and right-hand side figures correspond to just after peak systole, end systole and mid-diastole, respectively.
For clarity, different scales were used in the different frames.
Wall shear stress indicators
As mentioned earlier, the effects of WSS have traditionally been expressed in terms of TAWSS, OSI and RRT, equations 4-6, respectively. In addition, we introduced a parameter called EAI, computed using the two models: EAINobili and EAISoares according to equations (7) and (8), respectively. Figure 6 depict the surface values of TAWSS, OSI, RRT, and the results of the two EAI models, for the same two cases as shown in Figs 5. The five different indicators both differ and show similarities when compared. For example, the separation bubble produced by the aortic sinus leads to flow oscillations and longer residence time, in turn leading to larger OSI and RRT in that region. However, the WSS is relatively weak resulting in low TAWSS values.
The EAI parameters depend on the residence time and shear-stress (with EAISoares also
depending on the time-derivative of WSS), implying low EAI values at the aortic sinus. OSI and RRT detect the separation bubble (Figs 6a-c and 6f-h) in the ascending aorta/arch. Similarly, the EAI models along with TAWSS, OSI and RRT do detect the stronger WSS at the aortic arch, and near the bifurcation of the head-neck arteries. The stronger oscillation occurring at the high HR and CO case are not reflected in the OSI (Fig 6b and g). TAWSS increases with HR and CO due to increase in WSS while RRT increases also due to formation of multiple separation
bubbles (with total decrease in surface area). Longer residence time along with stronger WSS yield larger EAI values.
To enable detailed studying of the effects following different HR and CO on the WSS related parameters, a tabulated mean value of these parameters for different cases is provided in Table 3. The table shows that space averaged OSI is below 0.15. The OSI values may indicate a low- level oscillation since the mean value is much less than 0.5. However, examining the
instantaneous values, the different parameters are found to vary strongly in space and time.
TAWSS increases with HR and CO as WSS increases and hence its time average increases.
Increasing flow- and heart rate leads to a reduction RRT from about 10ms to less than 0.5ms. By comparing RRT for some CO it appears as if the heart rate has a stronger impact in reducing the residence time. The EAI parameters increase with residence time and shear strength by
construction (equation (7)). As EAISoares includes the stress (filtered) time derivative, yielding larger values as compared to EAINobili.
a b c d e
f g h i j
Figures 6: Summary of the WSS-related parameters for low HR/CO (a – e) and high HR/CO (d – j), corresponding to Fig 5 a and b, respectively. The high HR/CO cases implies considerably stronger and highly oscillatory WSS, reflected in larger average values of TAWSS and both EAI parameters. The longer residence time at higher HR/CO due to stronger fluctuations is observed in RRT and EAI. Note the different scales in the different frames.
Table 3: Summary of the behavior of endothelial (pathology) indicators.
Case HR [BPM]CO [LPM]TAWSS [Pa] OSI [-] RRT [-] WSS [Pa] dW/dt [s-2] EAI-N [-] EAI-S [-]
1 Profile a 3D 60 5 8,44E-01 1,32E-01 1,08E-02 9,21E-01 9,61E+00 3,17E-04 3,05E-03
2 Profile a 60 5 8,19E-01 1,27E-01 1,11E-02 8,96E-01 1,02E+01 3,25E-04 2,98E-03
3 Profile f 60 5 9,01E-01 1,54E-01 1,07E-02 9,83E-01 2,06E+01 3,11E-04 2,02E-03
4 Profile e 60 5 7,62E-01 1,13E-01 1,17E-02 8,47E-01 8,98E+00 2,72E-04 1,72E-03
5 Profile b 60 5 9,26E-01 1,43E-01 9,90E-03 1,02E+00 1,40E+01 2,53E-04 1,93E-03
6 Profile c 60 5 3,90E-01 1,45E-02 1,45E-03 6,38E-01 0,00E+00 3,38E-04 1,19E-03
7 Profile a 90 6 1,34E+00 1,36E-01 6,47E-03 1,49E+00 2,52E+01 3,16E-04 3,06E-03 8 Profile a 90 9 6,04E+00 2,56E-02 6,70E-05 2,54E+00 0,00E+00 4,24E-04 4,96E-03 9 Profile a Stenosis 90 9 2,64E-01 5,40E-05 4,45E-02 2,93E-01 1,62E-02 1,93E-04 2,01E-04 10 Profile d 3D 90 10 2,45E-01 2,80E-05 4,38E-02 2,71E-01 1,02E-02 1,67E-04 1,56E-04 11 Profile a 120 10 5,23E+00 1,10E-01 1,56E-03 5,99E+00 1,66E+02 1,11E-04 4,17E-04 12 Profile a 120 14,8 9,12E+00 9,78E-02 8,40E-04 1,05E+01 3,29E+02 1,34E-03 3,37E-02 13 Profile a 130 16 1,10E+01 1,09E-01 7,47E-04 1,29E+01 5,08E+02 1,93E-03 1,31E-01 14 Profile a 140 17,1 1,63E+01 8,84E-02 4,83E-04 1,92E+01 7,57E+02 1,93E-04 1,36E-03 15 Profile a 150 15 1,30E+01 1,02E-01 6,28E-04 1,50E+01 1,64E+03 0,00E+00 0,00E+00
Discussion
The effects of different combinations of HR, CO, and variations of the inlet flow rate profile on the resulting flow field (biomechanical) indicators were presented. The purpose was to
characterize the impact of helical motion and retrograde flow on aortic wall parameters derived from the WSS (including two endothelial activation models and their variability with various heart operating conditions. Both retrograde and helical flow structures are inertia driven
whereas WSS depends also on local viscosity. Therefore, the effects of varying heart- and flow rates and not least the temporal variation in CO on WSS indicators were studied.
The results show that details of the temporal development of the flowrate and its spatial
distribution have a major impact on the retrograde flow in the aortic volume and near the aortic wall. It is commonly accepted that low and oscillatory WSS contributes to atherogenesis. The results clearly demonstrate that oscillatory WSS is directly related to temporal irregularity (deceleration followed by acceleration) in the flow rate temporal profile, leading to the negative WSS. A similar observation was made by Dai et al. (2004) on an endothelial cell culture study, directly relating the sign changes in WSS as being “athero-prone” whereas non- negative, non-oscillatory WSS was found to be “athero-protective”.
Averaged quantities often used to characterize near wall flow behavior, i.e. TAWSS, OSI and RTT, are unable to reflect the impact of mechanical stress on the endothelial cells. These, like all cells, adapt to slow changes (hours and days). It is equally clear that cells as well as the flow cannot respond to too quick changes, due to inertial effects. Data on Ca2+ signaling in different cells (Boulware and Marchant 2008) indicates skeletal muscles, cardiac muscles and smooth muscles have response ranges of about 50 Hz, 10 Hz and 1-0.1 Hz, respectively. Ca2+ signaling is not the only time-limiting process in activating cellular processes. Intra-cellular
transport/diffusion of larger molecules (e.g. mRNA) and synthesis of larger molecules (proteins) is a much slower process unable to respond to high frequency mechanical stimuli.
These observations that cells have different windows of frequency responses may have several implications. As atherosclerosis is closely related to inflammatory processes, the temporal response becomes more intricate, and possibly implying the inability of cells to physiologically respond to high frequency (order of < 0.1 s) flow fluctuations.
Two parameters representing activation/excitation of endothelial cells (i.e. EAI defined by equations (7) and (8)) were introduced. The latter model employs the averaged time derivative of the scalar stress. The filter uses a low-pass filter that maintains fluctuations below 30 Hz. The rationale of this approach is that a cell can respond to temporal stress only up to certain frequency. It can be assumed that cell response depends on the diffusion time and cell deformability response time. The ratio of these two timescales is the so-called Weissenberg number (Thurston and Henderson 2006)). The Weissenberg number (Wi) is the product of the shear rate (or its RMS value in unsteady flows) and the cellular/fluid relaxation time. In oscillatory flow, the RMS value of the shear rate can be measured directly. If Wi >>
1, the elasticity of the cells dominates over the flow time, whereas when Wi << 1, the viscosity dominates. Cellular calcium response to stress is relatively slow (Hong et al. 2006) noted time-response in terms of seconds). Boulware and Marchant (2008) presented a review of timing of Ca2+ signaling. Smooth muscle cell (e.g. in the aorta) signaling was found to be in the range of about 1 to 10 Hz. The microfluidic experiments of DeStefano et al. (2017) also focused on slow response to shear stress indicating response times of the same range as mentioned above. The slow response implied above may have an impact on the rheological (viscoelastic) properties of blood, a property that was neglected here.
Two of the WSS related parameters, TAWSS and RRT are in dimensional form, an obvious shortcoming. Both these parameters should be scaled by relevant quantities to reflect the importance of the time averaged WSS or transport process for TAWSS and RRT, respectively.
The time-scale in RRT is related to viscous diffusion, whereby it would be reasonable to compare this time scale to the local convective transport, for example by the wall normal transport (vorticity) and/or molecular transport rate of relevant blood components (e.g. LDL).
The main limitations of the results presented here are due to the assumption that the wall of the thoracic aorta is rigid. The assumption implies that all pressure variations propagate instantaneously leading to an immediate response to pressure changes. Therefore, a backpressure could result in retrograde flow in regions with low momentum. This also may explain the extent of the negative WSS to backpressure. In this paper, only one rheological model was used. On the other hand, the larger flow structures such as the helical motion and retrograde flow are inertia driven and
determined primarily by the geometry and the temporal behavior of the driving pressure/inlet flow and not by viscous effects.
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