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An experimental model to simulate arterial pulsatile flow:
in vitro pressure and pressure gradient wave study
Afshin Anssari-Benam1,* and Theodosios Korakianitis2
1 Faculty of Engineering Sciences, University College London, Torrington Place,
London, WC1E 7JE, United Kingdom.
2 Parks College of Engineering, Aviation and Technology, Saint Louis University,
St. Louis, MO 63103, USA.
* Address for correspondence: Afshin Anssari-Benam,
Faculty of Engineering Sciences,
University College London,
Torrington Place,
London,
WC1E 7JE
United Kingdom.
Tel: +44 (0)20 7679 3836
Fax: +44 (0) 20 7383 2348
E-mail: [email protected]
Word count (Introduction to conclusion): 4791
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Abstract
A new experimental model developed to simulate arterial pulsatile flow is presented
in this paper. As a representative example, the flow characteristics and the properties
of brachial artery were adopted for the purpose of this study. With the physiological
flow of the human brachial artery as the input, the pressure and pressure gradient
waves under healthy and different scenarios mimicking diseased conditions were
simulated. The diseased conditions include the increase in blood viscosity (reflecting
the elevation of hematocrit), stiffening of the arterial wall, and stiffening of the aortic
root as the coupling between the heart and arterial tree, presented by the Windkessel
element in the setup. Each of these conditions resulted in certain effects on the
propagation of the pressure and pressure gradient waves, as well as their patterns and
values, investigated experimentally. The results suggest that the pressure wave
dampens at arterial sites with higher hematocrit, while the stiffening of the
Windkessel element elevated the diastolic pressure, and lowered the pressure drop,
similar to the results observed by stiffening the arterial wall. Based on these results, it
is hypothesised that the cardiovascular system may not function within the minimum
energy consumption criterion, contrary to some other physiological functions.
Keywords: Arterial flow, haemodynamics, pressure wave, pulsatile flow, simulator.
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An experimental model to simulate arterial pulsatile flow:
in vitro pressure and pressure gradient wave study
1. Introduction
Cardiovascular diseases (CVDs) remain the number one cause of human deaths in
industrialised countries, with a staggering annual sum of over $400 billion associated
with CVD treatment costs in the US alone [1]. There is considerable clinical evidence
that the initiation and development of many cardiovascular diseases are closely
associated with the arterial haemodynamic factors [2-6]. Clinical findings suggest that
haemodynamic parameters such as the blood pressure, the circumferential stress, and
in particular the shear stress applied to the arterial wall in each cardiac cycle, are the
key parameters regulating the function of the endothelial cells lining within the inner
wall of the arteries, both in healthy and diseased conditions [7-11]. This has prompted
a wide interest in the study of arterial fluid dynamics, receiving increasing attention
from both the fluid mechanics and the biological points of view [5,12,13]. A detailed
understanding of arterial blood flow parameters in healthy and diseased conditions
provides valuable information about the mechanisms involved in initiation and
development of CVDs, and may also assist in providing efficient clinical diagnosis
and treatment processes [14-17].
Various in vitro and in vivo experimental methods and modelling techniques have
been utilised to characterise different parameters of vascular fluid mechanics and
arterial haemodynamics [5,12,18]. In vivo experiments have been mainly associated
with non-invasive investigation of the blood flow patterns, velocity profiles and
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subsequently the shear stresses exerted by blood flow on the arterial wall [19,20].
Laser Doppler Velocimetry (LDV), Magnetic Resonance Imaging (MRI) and
ultrasound particle image velocimetry techniques have been extensively used in such
studies, to gain a better understanding of blood flow characteristics in the deeper
tissues of patients [20,21]. While these techniques have provided valuable information
and contributed significantly in understanding the arterial haemodynamics, they also
suffer from inherent drawbacks. Laser Doppler imaging implies low temporal
resolution due to scanning features, and the spatial resolution of magnetic resonance
imaging and ultrasound particle image velocimetry techniques is limited due to the
utilised wavelengths [20,22]. Applications of these techniques are therefore restricted
to macro and intermediate scale blood flows, and in regions not very close to the
arterial wall, as constructing the blood flow velocity profiles at regions closer to the
wall require more detailed spatially resolved measurements [20]. In addition, the
output data of such techniques suffer from the lack of both repeatability and
reproducibility, as the in vivo conditions assume obvious temporal subject-to-subject
and cycle-to-cycle variations.
To overcome the restrictions associated with the quantification of flow velocity and
shear stress profiles mentioned above in in-vivo studies, computational and numerical
modelling techniques have been widely employed as alternative/complimentary tools
[12,18,23]. These techniques have provided a powerful means to investigate the blood
flow characteristics in different healthy or diseased conditions, e.g. hypertension,
different scales of stenosis, heart valves dysfunction etc., in a reproducible manner.
However, models are often subjected to simplifying assumptions which could limit
the scope of their application, and may potentially lead to unrealistic physiological
4
conditions and results [17,18,24,25]. Steady or simple oscillatory flows, rigid wall
tubes and neglecting the Fluid-Solid Interaction (FSI) effects are some of those
simplifying assumptions that may not correlate with the real physiological conditions
of blood flow in arteries [12,17,18].
In vitro experimental setups have therefore been employed as useful and reliable
means for studying arterial fluid dynamics. Because of the high complexity of the
constitutive equations characterising the mechanical behaviour of arterial wall and the
blood flow velocity fields [26,27], experimental models have mainly been developed
in order to investigate the correlation between the pressure and the arterial wall
displacement, and characterising the velocity profiles of the flow, in conjunction with
numerical models. Some have been used to generate oscillatory flows with low
Reynolds numbers ( ) where the velocity profiles were monitored by data
acquisition systems containing velocimeter sensors and flowmeters [26,28,29]. Others
have contained compliant tubes and have been used to generate steady flows in
various inlet and outlet pressures [27,30]. Although such set-ups have made important
contributions in understanding aspects of blood flow characteristics, the scope of their
function and application has not been extended to simulating more realistic
physiological pulsatile flow in arteries, and therefore the results may not be suitable
for clinical implications. For example, arterial haemodynamics is known to be
markedly influenced by coupling of the heart with the aorta, i.e. aortic root, and
coupling of the arterial tree with distal arteriolar and capillary network, i.e. the
peripheral resistance [31-33]. Such couplings necessitate design and assembly of
elements to simulate aortic compliance and peripheral resistance in the experimental
models. These effects have often not been considered in the presented experimental
5
models to date, being regarded as of secondary importance [12,17,27,30], and their
effects and influence on the overall simulated dynamics of the blood flow have
therefore remained rather elusive and less well characterised.
Furthermore, while investigation of shear stress values and patterns on the arterial
wall have been of primary interest in designing the experiments in the relevant
studies, wall shear stress is not a first principle diagnosis parameter in practice.
Instead, measuring and monitoring the pressure pulse is a common clinical protocol
which can be achieved through non-invasive or minimally invasive diagnostic
procedures. It may therefore be of more practical benefit to characterise the pressure
wave in different haemodynamic conditions, reflecting different stages of
cardiovascular pathologies, to gain a better understanding of the effects of each
condition on a clinically relevant parameter.
To address these, a new experimental system is presented in this paper, which is
designed and developed for emulating physiological pulsatile arterial flow,
considering both the elastic coupling of the aortic root to the arteries and the
peripheral resistance. The Windkessel theory has been adopted to simulate these
effects, as the coupling is made by the elastic element of the Windkessel theory, and
the peripheral resistance is included as a resisting element to blood flow. The set-up
enables monitoring and measurement of the pressure and pressure gradient waves in
real time, in different haemodynamic and geometric configurations. The effects of
changes in fluid viscosity, the elastic coupling and the wall elasticity, representing
different anatomical and diseased conditions, on the values and patterns of pressure
and pressure gradient waves are experimentally investigated. Correlations with the
6
relevant physiological and pathological arterial flow conditions are further described
and discussed.
2. Materials and Methods
2.1. Experimental model
The experimental model, illustrated in block diagrams in Figure 1a, is an open loop
hydraulic system comprising of five major components: programmable pulsatile flow
pump, an elastic element (the Windkessel element) placed before and coupled to the
elastic tube, the elastic tube, data acquisition and processing system, and the resistant
element. It was set to simulate a model of the brachial artery flow as a representative
example in this study. The brachial artery was chosen due to the availability of its
physiological flow wave and pressure pulse data, and its mechanical and geometrical
specifications matching the commercially available elastic tubes. A description of
each component of this model, including the characteristics and functions, is
presented in the following.
2.1.1. Programmable pulsatile flow pump
A pulsatile pump was specifically designed and built to generate pulsatile flow for a
wide range of cardiovascular applications. The mechanical unit of the pump is
composed of four components: a servo-motor, a planetary gearbox, a ball screw, and a
cylindrical tank (Figure 1b).
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The servo-motor (MDFKS 056-23 190, Lenze, Germany) is connected to the ball
screw (SFI2005, Comtop, Taiwan) by a planetary gearbox (MPRN 01, Vogel,
Germany), enabling the screw to rotate in clockwise or anti-clockwise directions. A
piston is placed upon the ball, converting the rotation of the screw to
forward/backward sliding movement within the cylindrical tank. The gearbox
eliminates potential loosening between the rotating shaft of the servo-motor and the
screw. As the piston slides, it exerts force to the fluid inside the cylindrical tank and
pushes it to flow outward, into the elastic element and the rest of the setup. The flow
rate and the flow pattern of the fluid are adjusted by controlling the movement of
piston, i.e. its moving speed and rotational pattern of the screw, inside the cylindrical
tank. This is done by the electronic unit of the pump.
The electronic unit contains a microcontroller which controls the rotational pattern
and speed of the servo-motor, and subsequently the movement of the piston. The
programmable microcontroller (ATMega128, Atmel AVR®, USA) is programmed in
C to produce the desired arterial flow pattern. Because of the frequency response of
the servo-motor and the sampling rate of the microcontroller (1000 Hz), the pump is
capable of producing any physiological pulsatile arterial flow. The pulsatile flow
simulated by this setup was the brachial artery flow wave in a healthy individual, as
shown in Figure 1c [34].
2.1.2. Elastic element
Coupling of the heart with the arterial tree is of great importance in the
cardiovascular system. The highly distensible aortic root plays an essential role in this
8
coupling. Acting as an elastic buffering chamber between the heart and arterial tree, it
stores about 50% of the left ventricular stroke volume during systole. In diastole, the
elastic forces of the aortic wall force this 50% of the volume to the peripheral
circulation, thus creating a nearly continuous blood flow. This systolic-diastolic
interplay represents the Windkessel function theory, proposed by Otto Frank, in which
the aortic root has been considered as an elastic element placed after the heart pump
and peripheral arteries [31,35,36]. To simulate this effect in our experimental model,
an elastic element was placed before the elastic tube, as shown schematically in
Figure 1a.
2.1.3. Elastic tube
To study the effects of arterial wall elasticity on blood flow parameters, an elastic
tube with a defined stiffness modulus was utilised. Similar to other biological
applications, a medical grade silicon tube (D-34209, B.Braun®, Switzerland) was
chosen, with similar diameter and elasticity to a normal brachial artery [37], as
presented in Table 1.
The elastic properties of the tube were determined experimentally using a tensile
test unit (HCT 25/400, Zwick-Roell, Germany). After evaluation of non-linear load-
displacement curve, the stress-strain relationship was calculated based on the
dimensions of the tube under test, and large deformation theory. The stress-strain
behaviour of the tube was similar to that of typical arterial tissue, becoming stiffer by
increase in strain [38]. Arterial walls within human body typically experience
circumferential strains between 0-10% throughout the arterial system, within the
9
physiological arterial pressure pulse [30,37]. Within this strain range, the stress-strain
behaviour of the elastic tube could be approximated by a linear stress-strain relation.
Thus the elastic modulus of the tube was considered as the slope of the stress-strain
line within 0-10% of strain.
2.1.4. Data acquisition and processing system
The data acquisition and processing system consists of two pressure transducers and
a processing unit connected to a computer. The system is capable of measurement and
detection of pressure and pressure gradient waves in real time. The utilised pressure
transducers (MLT0670, ADInstrumentsTM, Australia) were blood pressure transducers
for in vivo applications with operational range of -50 mmHg to 300 mmHg. They
were placed at both ends of the elastic tube, detecting and measuring the inlet ( )
and outlet ( ) pressure waves of the tube (Figure 1a). The pressure gradient wave
was then calculated by the processing unit, using the difference between the inlet and
outlet pressures measured by pressure transducers at each time point during the flow
pulse.
Each transducer was connected to a separate amplifier (ML117 BP Amp,
ADInstrumentsTM, Australia), for amplification of output signals before processing.
The amplifiers were connected to the main data processing unit (Powerlab/4SP,
ADInstrumentsTM, Australia). This is a hardware unit connected to the computer,
processing the amplified output signals of the two transducers and converting them
into numerical data and graphs. For this study, three waves ( , and pressure
gradient) were monitored and calculated in real time.
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2.1.5. Resistant element
The resisting element is placed distal to the elastic tube assembly and before the
outlet reservoir (Figure 1a). It represents the existing hemodynamic peripheral
resistance of the circulatory system. Such resistance is designed to vary for simulation
of blood flow in different arterial sites with different peripheral resistances, to obtain
the desired pressure pulse relevant to the model artery. The element acts as a valve to
control the cross-sectional area of the outlet tube into the outlet reservoir (Figure 1a).
Since the inlet flow to the setup is set by the pulsatile flow and hence remains
constant once it is set, the valve alters the outlet flow velocity to the reservoir, and
enables control of the pressure in the setup, while maintaining a certain flow.
2.2. Experiments
2.2.1. Design of experiments
Experiments were designed to study the effects of change in viscosity of the fluid,
the stiffness of the tube’s wall, and stiffening of the Windkessel elastic element, with
specifications of each experiment as follows:
(1) Blood flow through a healthy brachial artery, simulating healthy physiological
flow conditions. This was designed to validate the model performance and as a
criterion to compare the results of other experiments with the healthy condition. The
flow wave was set to be the original physiological flow wave of the brachial artery
(Figure 1c), with a mean flow rate ( ) of 3.66 mLs-1 and the wave frequency ( ) of
11
1.16 Hz. The working fluid used in this experiment was chosen to be a Newtonian
fluid with a viscosity of mPa s and density of 1050 Kgm-3, to mimic the
density and viscous properties of blood.
(2) Blood flow through a healthy brachial artery with three different fluid viscosities.
This was used to study the impact of changes in hematocrit on flow characteristics.
Experiments were performed with two other working fluids, in addition to the one in
previous experiment, with viscosities of mPa s and mPa s, simulating the
effects of elevated hematocrit. The flow wave was set to be the same as original wave
used in previous experiment.
(3) Blood flow through a healthy brachial artery with stiffened Windkessel element.
This was used to investigate the effect of stiffening of the aortic root. In this
experiment, the elastic Windkessel element was replaced with a rigid segment of the
same geometry, with its elastic modulus two orders of magnitude higher than that of
the elastic element. The flow pattern and the working fluid were the same as those
used in the first experiment, and
(4) Blood flow through brachial artery with stiffened wall, to study the effect of
arterial wall stiffening. For this purpose, the tube representing the normal
physiological brachial artery was replaced with a rigid tube of the same geometry,
having a higher elastic modulus by two orders of magnitude. The flow pattern and
fluid characteristics were the same as those in the first experiment.
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2.2.2. Protocol
Before running the experiments, a defined procedure was employed to reduce and
eliminate probable errors in producing the original flow and in the measurement
system. The procedure included discharge of any remaining air bubbles from the
assembly of tubes and transducers, as well as calibration of measurement system,
including both the transducers and the processing unit. The discharging was done
through the discharge valves arrayed on both transducers. The calibration was also
essential because of the effect of electronic hysteresis on the measuring units. The
procedure was performed every time before starting a new experiment.
3. Results
3.1. Blood flow through a healthy brachial artery at physiological flow conditions
The (inlet pressure) wave measured and recorded in the experiment at the steady
state is presented in Figure 2a. The pressure varies in a range of 83 mmHg to 123
mmHg, within a period of 0.86 s, and the pressure gradient wave is between 0.3
mmHg to 3.8 mmHg with an average of 0.8 mmHg, in the same period. , and
the pressure gradient waves at the steady state within one cardiac pulse are shown in
Figures 2b & 2c. The Reynolds ( ) and Womersley ( ) numbers for flow in this
experiment are summarised in Table 2.
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3.2. Blood flow through a healthy brachial artery with three different fluid
viscosities
Working fluids with different viscosities were used in this experiment as described
in section 2.2.1. All other settings, i.e. the flow, the peripheral resistance and the
elasticity of the Windkessel element and the tube, were kept unchanged. The result for
wave at the steady state is shown in Figure 3. The pressure was raised with the
increase in viscosity, to maintain the same flow rate. The pressure rise was more
noticeable at higher viscosities. By contrast, the amplitude of pressure wave decreased
with increase in viscosity (Figure 3b). The pressure drop increased with increase in
the viscosity, as the pressure gradient wave possessed higher values (Figure 4).
Changes in viscosity affected the peak values in pressure waves, but did not alter the
shape or the pattern of the pressure gradient waves.
3.3. Blood flow through a healthy brachial artery with stiffened Windkessel element
The elastic element of the Windkessel theory was replaced by a rigid segment with
similar dimensions for this experiment. All other settings were the same as those in
the first experiment described in section 2.2.1. Stiffening of this element dramatically
reduced the amplitude of the pressure wave compared to the elastic state (Figure 5a),
resulting in a higher diastolic pressure. In addition, the pressure gradient became
slightly lower in value, leading to slightly less pressure drop in the rigid state to the
initial elastic state (Figure 5c).
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3.4. Blood flow through brachial artery with stiffened wall
Stiffening of the elastic tube wall representing the brachial artery model is
analogous to arterial wall stiffening by aging or other risk factors such as smoking and
atherosclerosis. To study this effect, the elastic tube was replaced by a rigid tube of
the same geometrical specifications in this experiment. As shown in Figure 6, the
amplitude of wave was reduced in the rigid tube, in addition to a time shift
detected between the two waves in the rigid and the elastic states, designated by
(Figure 6a). The pressure gradient wave possessed lower values in the rigid tube,
implying a lower pressure drop compared to the elastic tube (Figure 6d). Recalling
from the results of section 3.3, lower pressure drop was also observed in the case of
rigid Windkessel element compared to the elastic element.
4. Discussion
4.1. Analysis of the results and their relevance to the physiological and pathological
conditions
A new experimental model, designed to simulate and study the arterial blood flow
was presented in this paper. The current design allows study of the effects of coupling
of the heart with the aorta, and the arterial system with distal arteriolar and capillary
network, as the elastic element of the Windkessel theory and the resistant element
respectively, on the pressure and pressure gradient waves. Understanding the values
and patterns of the pressure wave and subsequently its propagation in the arterial tree,
and how they vary in different normal and diseased conditions, can provide insightful
15
information about the flow indications of each associated pathology, and clinical
diagnosis [15,39].
The physiological flow in a normal brachial artery was simulated, and the outcomes
were used to validate the experimental model, and as a control to compare with
altered conditions. The resulting pressure wave (Figure 2) was within the range of
reported physiological data on healthy brachial artery [34,40]. The values of fluid
dynamics parameters, i.e. the Reynolds and Womersley numbers (Table 2) are in a
good agreement with the published data for physiological flow conditions of arteries
[41].
Results of the experiments on the effects of changes in viscosity show that while the
mean value of the pressure gradient increases with increase in the viscosity, the
amplitude of the inlet pressure pulse is reduced. The difference between the systolic
and the diastolic pressures is equal to 39 mmHg at the viscosity of 2 mPa s, however
it reduces to 32 mmHg at the viscosity of 16 mPa s, implying that the viscosity may
act as a damping element on the wave characteristics of the flow. Hematocrit of blood
is defined as the percentage of total volume of the blood occupied by the red blood
cells, and higher hematocrit results in higher blood viscosity [42]. It is therefore
reasonable to conclude that the pressure wave tends to dampen at arterial sites with
higher hematocrit, or in pathological conditions which lead to higher overall blood
viscosity, such as the ‘sickle cell’ disease.
Comparing the results for elastic and rigid Windkessel elements indicated that
stiffening of the element resulted in reduced mean pressure gradient, and pressure
16
gradient amplitude. However, the diastolic pressure was raised noticeably. This
implies that in a pathological state of stiffened aortic root, higher diastolic pressure
may be observed in every heart beat. The same results were also obtained when the
elastic tube became rigid, implying that stiffening of the arterial wall due to aging or
other risk factors correlates to the same alterations in the pressure wave.
Characterisation of the variation in pressure and pressure gradient waves in different
haemodynamic conditions reflecting the healthy and diseased states can also have
important implications in understanding other aspects of the cardiovascular mechanics
such as the function of the heart valves. Indeed, it has been shown that the pressure
wave is an important parameter influencing the opening and the closure of the heart
valves, and their stretch levels [43]. Modelling studies have also highlighted the
effects of aortic root extensibility on the valvular function [44]. The current setup
allows for these effects to be investigated experimentally in real-time. Additionally,
the observed changes in the pressure wave with variation in the elasticity of the
Windkessel element can also be coupled with the models for further detailed analysis
of the dynamics of heart valves functions.
4.2. Analysis of the time shift in pressure waves
As illustrated in Figure 6, a time shift was observed between the pressure waves of
the rigid and the elastic wall states ( in Figure 6a). The results showed that the
pressure wave in the rigid wall travels faster than that in the elastic wall of the same
geometric characteristics. To mathematically explain this phenomenon in arteries and
determine the value of this time shift, we refer to the well-known wave propagation
17
equation in a tube, known as the Moens-Korteweg equation [45]. Assuming that the
flow is one dimensional [45]:
(1)
where c is the wave speed, E is the elastic modulus, h is the wall thickness, ai is the
inner radius and is the density.
As can be inferred by the equation, with increase in elastic modulus of the wall, i.e.
the wall becoming stiffer, the pressure wave would be travelling faster along the tube.
In case of the elastic state, substituting the specifications of the silicon rubber material
used as the elastic tube in our experimental set-up given in Table 1 into equation 1,
the wave speed is calculated to be ms-1, and for the case of rigid tube
ms-1. Considering the length of the tube, the time shift between the
pressure waves of the two sates equates to s. The pressure wave speed
obtained in the elastic tube is in a good agreement with the available data in the
literature, as the pressure wave speed in arteries with similar flow and geometrical
characteristics is reported to be in the range of 5 ms-1 to 8 ms-1 [46].
Changes in the pressure wave amplitude and the wall elasticity can significantly
alter the stress waves in arteries, i.e. the wall shear stress (WSS) and circumferential
stress waves; however numerical models which consider the fluid-wall interactions
are also required for a more detailed understanding. For the case of arterial stenosis,
these effects will be addressed in a future work by the authors, by coupling the
experimental data with a numerical FSI model representing different scales of
stenosis.
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While the relationship between the elasticity of tube wall and the pressure drop
along the tube in pulsatile flows has been less well characterised experimentally,
analytical models have been developed and used extensively. A widely popular
approach has been presented by Womersley, in solving the linearised Navier-Stokes
equations for oscillatory flow in a tube, where a relationship between the ‘fluid
resistance’, as a non-dimensional parameter, and is presented [47]. The graph in
Figure 7a, adapted from Womersley (1957), shows this relation. For values of ,
resistance to flow is increased by the increase in . The elastic and rigid tubes used
in our study both possessed similar initial radius of 2.35 mm at zero stress state.
However, subject to pulsatile pressure, the elastic tube elongates circumferentially
more than the rigid tube, leading to a higher average radius of approximately 2.52 mm
during each pulse. This results to values of 11.53 and 9.99 for the elastic and the
rigid tube, respectively, thus higher fluid resistance and subsequently more pressure
drop in the elastic tube (Figures 5d and 6d).
In addition to the above Womersley approach, Fung (1997) has used the classic
laminar flow in an elastic tube analysis to formulate a pressure-flow relationship,
subjected to assumptions which make it applicable for blood flow in arteries, given as
[41]:
(2)
where E is the tube’s young modulus, h is wall thickness, r0 is the initial radius of the
tube at zero stress state, L is the length of tube, µ is the viscosity of the fluid, p(0) and
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p(L) are the inlet and outlet pressures respectively, and is the flow rate. The graph
in Figure 7b shows the numerical solution of equation (2) for versus E,
using the values provided in Table 2 for our experimental model. As the graph shows,
the pressure drop decreases with increase in the Young’s modulus of the wall,
indicating that the Fung’s solution suggests that there is a lower pressure drop in more
elastic tubes. The point designated by ( ) symbol on the graph indicates the calculated
pressure drop by equation (2) for the used elastic tube (E = 463 KPa), equal to 1.64
mmHg, which is close to the measured experimental value of 1.4 mmHg.
Faster propagation of the pressure wave and lower pressure gradients in the stiffer
Windkessel element and the rigid wall tube, observed in our experiments, suggest that
less pressure gradient is required for the blood to flow through stiffer arteries
compared to the elastic ones, and therefore less energy consumption by the
cardiovascular system to produce the required pressure gradient. On this basis we
hypothesise that the cardiovascular system may not function within the minimum
energy consumption criteria, contrary to the most of other physiological functions. We
further suggest that the cardiovascular system aims to maintain a certain pattern of
blood pressure, and consequently circumferential and shear stress waves.
4.3. Limitations of the experimental model
The working fluid used in our study was a Newtonian fluid, despite the fact that
blood shows characteristics of a non-Newtonian fluid, possessing higher viscosity in
smaller vessels with lower flow rates, i.e. arterioles and capillaries. However, studies
have suggested that assuming the blood as a Newtonian fluid may result in acceptable
20
analysis and reasonable accuracy for larger arteries [12,18]. Such assumption has been
widely employed in both the experimental and the computational simulations.
The hydraulic fittings used to assemble the pressure transducers downstream and
upstream of the elastic tube could be sources of flow turbulences and therefore
additional pressure drops. This may have unavoidable effects on the values and shape
of the waves. However, since this study was carried out in a Reynolds number below
that of a turbulent flow regime, which is the case for most of the arteries in healthy
conditions [17], resistance against the flow at the fittings are likely to be very small
and may thus be neglected. Furthermore, the total resistance of the hydraulic circuit
can be controlled by the resistant element for higher Reynolds numbers which may
result in a transient or a turbulent flow regime, minimising the sudden increase of
flow resistance and pressure drop within the setup.
While the measurement system in our experimental setup was capable of detecting
the time shift between the pressure waves, the limitations in sampling time resolution
(0.01s) prevented us of reporting the precise value of the time shifts observed between
the waves in rigid and elastic conditions.
Direct use of Moens-Korteweg equation for calculating the wave propagation times
in this study may be theoretically problematic, since the tube wall in our study was 0.9
mm and thus not theoretically a thin-wall tube. Additionally, silicone rubber is
theoretically a compressible material with a Poisson ratio of . Both of these
two characteristics violate the assumptions used in deriving the Moens-Korteweg
21
equation [45]. However the calculated values are in a good agreement with those
reported in the literature.
5. Conclusion
Flow conditions for brachial artery in healthy and diseased cases were simulated
experimentally. Pressure and pressure gradient waves were studied for each case.
Alteration of the haemodynamics parameter from the healthy state, such as viscosity,
wall stiffness and the stiffness of the Windkessel element, had certain effects on both
the pattern and the values of the waves, and also on their propagation, which were
experimentally characterised. Analytical approaches described in this paper supported
the experimental results, validating the capabilities of the experimental model in
simulating the physiological arterial flow. The results suggest that the pressure and
pressure gradient waves may also be regarded as indicators of the clinical condition of
the cardiovascular system, i.e. healthy or diseased, further addressing the implications
of haemodynamics in understanding the function of the arterial vasculature. In a
planned work for a future contribution, the experimental model described in this study
will be used to simulate blood flow in different scales of stenotic conditions, in
conjunction with a numerical model, to characterise other haemodynamic parameters
such as the circumferential and wall shear stress waves.
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Table Legends
Table 1- Characteristics of the brachial artery and the elastic tube used in the
experimental model. The Dimensions are at the zero stress state and the elastic
modulus is determined for a strain range of 0-10%.
Table 2- Reynolds and Womersley numbers of the simulated flow at the steady state,
and their values for the physiological arterial flow.
30
Table 1
Inside Diameter(mm)
Wall Thickness(mm)
Elastic Modulus (KPa)
Brachial Artery [37] 4.5 0.83 460
Elastic Tube 4.7 0.9 463
Table 2
(mPa s) (mLs-1) (Hz)
Values in the experiment 2 3.66 1.16 1254 3.16
Physiological values for arteries [41,44] 1.2 - 5 1-200 1 - 2 800 - 4500 3 - 13
31
Figure Legends
Figure 1- (a) Block diagram of the experimental setup. (b) Schematic view of the
pump and its mechanical components, (c) The physiological flow pattern of a healthy
brachial artery [34], utilised in experiments.
Figure 2- (a) Produced pressure wave versus elapsed time for the healthy brachial
artery. (b) and waves in one cardiac cycle. (c) Pressure gradient wave in the
same period. Graphs present the waves at the steady state.
Figure 3- Effects of viscosity on pressure wave: (a) Pressure is raised with increase in
viscosity. (b) Amplitude of the pressure wave decreases as the viscosity increases.
Figure 4- Effects of viscosity on pressure gradient wave: (a) The pressure gradient
wave in different viscosities. (b) Higher pressure drop was observed in higher
viscosities.
Figure 5- Effects of Windkessel elastic element on pressure and pressure gradient
waves: (a) The pressure wave in elastic and rigid states of the Windkessel element. (b)
The value of pressure wave amplitude: the amplitude is reduced in rigid state. (c) The
pressure gradient wave in elastic and rigid Windkessel element. (d) Values of the
mean pressure drop: the pressure drop is slightly lower in the case of the rigid
Windkessel element. Data represents the steady state.
32
Figure 6- Effects of change of wall elasticity on pressure and pressure gradient waves:
(a) The wave in the elastic and rigid tube wall states: a time shift ( ) is observed
between the two waves. (b) Pressure gradient wave in the elastic and rigid tube wall
states: no time shift is observed between the two pressure gradient waves. (c) wave
amplitude: the amplitude of wave decreases in rigid tube wall state. (d) Values of
the mean pressure drop: there is less pressure drop in rigid tube compared to the
elastic tube. Data represents the steady state.
Figure 7- Pressure drop of pulsatile flow in elastic tubes: (a) Womersley solution:
fluid resistance increases with the increase in , for (adapted from [46]), (b)
Fung’s solution: the symbol in the graph indicates the value of Young modulus for
the elastic tube (0.463 MPa), in E axis and the corresponding pressure drop of 1.64
mmHg, given by equation (2).
Figure 1
33
Figure 2
GearboxServo-Motor Screw Piston
(b)
(a)
0
12
34
5
67
8
0.00 0.17 0.34 0.52 0.69 0.86
Time (s)
Flow
(ml/s
)
(c)
34
(b)
(c)
80
100
120
0.00 0.17 0.34 0.52 0.69 0.86
Time (s)
Pres
sure
(mm
Hg)
P1 WaveP2 Wave
Figure 3
80
100
120
10 11 12 13 14 15
Elapsed Time (s)
Pres
sure
(mm
Hg)
0.86 s
(a)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.00 0.17 0.34 0.52 0.69 0.86
Time (s)
Pres
sure
Gra
dien
t (m
mHg
/cm
)
35
Figure 4
80
100
120
140
160
180
200
220
0.00 0.17 0.34 0.52 0.69 0.86
Time (s)
Pres
sure
(mm
Hg)
(a)
(b)
(a)
mPa s
mPa s
mPa s
3935
32
0
5
10
15
20
25
30
35
40
45
Viscosity (mPa s)
Pres
sure
Wav
e A
mpl
itude
(mm
Hg)
µ = 2 µ = 5 µ = 16
36
Figure 5
(b)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
0.00 0.17 0.34 0.52 0.69 0.86
Time (s)
Pres
sure
Gra
dien
t (m
mH
g/cm
)
mPa s
mPa s
mPa s
(a) (b)
1.4
5.4
15.6
0
2
4
6
8
10
12
14
16
18
Viscosity (mPa s)
Mea
n Pr
essu
re D
rop
(mm
Hg)
µ = 2 µ = 5 µ = 16
37
Figure 6
(a) (b)
(c)
80
90
100
110
120
130
0.00 0.17 0.34 0.52 0.69 0.86Time (s)
Pres
sure
(mm
Hg)
Elastic Element
Rigid Element
0.0
0.1
0.2
0.3
0.4
0.00 0.17 0.34 0.52 0.69 0.86Time (s)
Pres
sure
Gra
dien
t (m
mHg
/cm
)
Elastic ElementRigid Element
39
12.6
05
1015202530354045
Elastic Element Rigid Element
Am
plitu
de (m
mH
g)
1.4
1.2
00.20.40.60.8
11.21.41.61.8
Elastic Element Rigid Element
Mea
n Pr
essu
re D
rop
(mm
Hg)
(d)
38
Figure 7
80
90
100
110
120
130
0.00 0.17 0.34 0.52 0.69 0.86Time (s)
Pres
sure
(mm
Hg)
(c) (d)
(a)
Elastic Tube
Rigid Tube
0.00
0.10
0.20
0.30
0.40
0.00 0.17 0.34 0.52 0.69 0.86Time (s)
Pres
sure
Gra
dien
t (m
mHg
/cm
)
Elastic TubeRigid Tube
39
17.1
05
1015202530354045
Elastic Tube Rigid Tube
Pres
sure
Wav
e A
mpl
itude
(m
mH
g)
1.4
0.8
00.20.40.60.8
11.21.41.61.8
Elastic Tube Rigid Tube
Mea
n Pr
essu
re D
rop
(mm
Hg)
39
0
20
40
60
80
100
120
140
0.01 0.1 1 10 100 1000
E (MPa)
Pres
sure
Dro
p (m
mH
g)
(b)
40