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: a.anssari-benam@ucl.ac.uk

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

3

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

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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

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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

19

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.

Reference

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1. Lloyd-Jones D, Adams RJ, Brown TM, Carnethon M, Dai S, De Simone G,

Ferguson TB, Ford E, Furie K, Gillespie C, Go A, Greenlund K, Haase N, Hailpern S,

Ho PM, Howard V, Kissela B, Kittner S, Lackland D, Lisabeth L, Marelli A,

McDermott MM, Meigs J, Mozaffarian D, Mussolino M, Nichol G, Roger V,

Rosamond W, Sacco R, Sorlie P, Stafford R, Thom T, Wasserthiel-Smoller S, Wong

ND, Wylie-Rosett J (2010) Heart disease and stroke statistics 2010 update: A report

from the American Heart Association, Circ, 121: e46-e215.

2. Nerem RM (1992) Vascular fluid mechanics, the arterial wall, and atherosclerosis.

J Biomed Eng 114: 274-282.

3. Frangos SG, Gahtan V, Sumpio B (1999) Localization of atherosclerosis, role of

hemodynamics. Arch Surg 134: 1142-1149.

4. Feldman CL, Stone PH (2000) Intravascular hemodynamic factors responsible for

progression of coronary atherosclerosis and development of vulnerable plaque. Cur

Opin Cardiol 15: 430-440.

5. Caro CG (2001) Vascular fluid dynamics and vascular biology and disease. Math

Methods Applied Sci 24: 1311–1324.

6. Cheng C, Helderman F, Tempel D, Segers D, Hierck B, Poelmann R, Van Tol R,

Duncker DJ, Robbers-Visser D, Ursem NTC, Van Haperen R, Wentzel JJ, Gijsen F,

23

van der Steen AFW, de Crom R, Krams R (2007) Large variations in absolute wall

shear stress levels within one species and between species. Atheroscl 195: 225-235.

7. Traub O, Berk BC (1998) Laminar shear stress: mechanisms by which endothelial

cells transduce an atheroprotective force. Arterioscl Thromb Vascular Bio 18: 677-

685.

8. Yamaguchi R (1999) Flow structure and variation of wall shear stress in

assymetrical arterial branch. J Fluids Struc 13: 429-442.

9. Gimbrone Jr. MA (1999) Endothelial Dysfunction, Hemodynamic Forces, and

Atherosclerosis. Thromb Haemostasis 82: 722-726.

10. Pohlman TH, Harlan JMM (2000) Adaptive responses of the endothelium to

stress. J Surg Res 89: 85–119.

11. Stone PH, Coskun AU, Kinlay S, Popma JJ, Sonka M, Wahle A, Yeghiazarians Y,

Maynard C, Kuntz RE, Feldman CL (2007) Regions of low endothelial shear stress

are the sites where coronary plaque progresses and vascular remodelling occurs in

humans: an in vivo serial study. Euro Heart J 28: 705-710.

12. Ku DN (1997) Blood flow in arteries. Ann Rev Fluid Mech 29: 399-434.

13. Liepsch D (2002) An introduction to biofluid mechanics-basic models and

applications. J Biomech 35: 415–435.

24

14. Giddens DP, Zarins CK, Glagov S (1993) The role of fluid mechanics in the

localization and detection of atherosclerosis. J Biomech Eng 115: 588–594.

15. Fujimoto S, Mizuno R, Saito Y, Nakamura S (2004) Clinical application of wave

intensity for the treatment of essential hypertension. Heart Vess 19: 19–22.

16. Papaioannou TG, Karatzis EN, Vavuranakis M, Lekakis JP, Stefanadis C (2006)

Assessment of vascular wall shear stress and implications for atherosclerotic disease:

Review. Int J Cardio 113: 12-18.

17. Varghese SS, Frankel SH, Fischer PF (2007) Direct numerical simulation of

stenotic flows. Part 1. Steady flow. J Fluid Mech 582: 253-280.

18. Taylor CA, Draney MT (2004) Experimental and computational methods in

cardiovascular fluid mechanics. Ann Rev Fluid Mech 36: 197–231.

19. Taylor CA, Cheng CP, Espinosa LA, Tang BT, Parker D, Herfkens RJ (2002) In

Vivo Quantification of blood flow and wall shear stress in the human abdominal aorta

during lower limb exercise. Ann of Biomed Eng 30: 402-408.

20. Vennemann P, Lindken R, Westerweel J (2007) In vivo whole-field blood velocity

measurement techniques: Review. Exp Fluids 42: 495-511.

21. Mekkaoui C, Friggi A, Rolland PH, Bodard H, Piquet P, Bartoli JM, Mesana T

(2001) Simultaneous measurements of arterial diameter and blood pressure to

25

determine the arterial compliance, wall mechanics and stresses In vivo. Euro J Vasc

Endovasc Surg 21: 208-213.

22. Borghi A, Wood NB, Mohiaddin RH, Xu XY (2008) Fluid–solid interaction

simulation of flow and stress pattern in thoracoabdominal aneurysms: A patient-

specific study. J Fluids Struc 24: 270-280.

23. Korakianitis T, Shi Y (2006) Numerical simulation of cardiovascular dynamics

with healthy and diseased heart valves. J Biomech 39: 1964-1982.

24. Zendehboodi GR, Moayeri MS (1999) Comparison of physiological and simple

pulsatile flows through stenosed arteries. J Biomech 32: 959–965.

25. Matthys KS, Alastruey J, Peiro J, Khir AW, Segers P, Verdonck PR, Parker KH,

Sherwin SJ (2007) Pulse wave propagation in a model human arterial network:

Assessment of 1-D numerical simulations against in vitro measurements. J Biomech

40: 3476–3486.

26. Deplano V, Siouffi M (1999) Experimental and numerical study of pulsatile flows

through stenosis: Wall shear stress analysis. J Biomech 32: 1081-1090.

27. Tang D, Yang C, Kobayashi S, Ku DN (2001) Steady flow and wall compression

in stenotic arteries: a three dimensional thick-wall model with fluid-wall interactions.

J Biomed Eng 123: 448-557.

26

28. Sioufi M, Pélissier R, Farahifar D, Rieu R (1984) The effect of unsteadiness on

the flow through stenoses and bifurcations. J Biomech 17: 299-315.

29. Sioufi M, Deplano V, Pélissier R (1998) Experimental analysis of unsteady flows

through a stenosis. J Biomech. 31: 11-19.

30. Tang D, Yang C, Ku DN (1999) A 3-D thin-wall model with fluid-structure

interactions for blood flow in carotid arteries with symmetric and asymmetric

stenoses. Comput Struc 72: 357-377.

31. Belz GG (1995) Elastic properties and the windkessel function of the human aorta.

Cardio Drugs Therapy 9: 73-83.

32. Maurits NM, Loots GE, Veldman AEP (2007) The influence of vessel wall

elasticity and peripheral resistance on the carotid artery flow wave form: A CFD

model compared to in vivo ultrasound measurements. J Biomech 40: 427-436.

33. Beulen BWAMM, Rutten MCM, van de Vosse FN (2009) A time-periodic

approach for fluid-structure interaction in distensible vessels. J Fluids Struc 25: 954-

966.

34. Nichols WW, O’Rourke MF (2005) McDonald’s blood flow in arteries. 5th edn,

Oxford University Press Inc., New York.

27

35. Sagawa K, Lie RK, Schaefer J (1990) Translation of Otto Frank’s paper “Die

Grundform des Arteriellen Pulses”. Zeitschrift fur Biologie 37, 483–526 (1899). J

Molecul Cell Cardiol 22: 253–277.

36. Wang JJ, O’Brien AB, Shrive NG, Parker KH, Tyberg JV (2003) Time-domain

representation of ventricular-arterial coupling as a windkessel and wave system. Am J

Physio - Heart Circul Physio 284: H1358–H1368.

37. Heijden Spek JJ, Staessen JA, Fagard RH, Hoeks AP, Struiker Boudier HA, van

Bortel LM (2000) Effect of age on brachial artery wall properties differs from the

aorta and is gender dependent: a population study. Hyperten 35: 637-642.

38. Teng Z, Tang D, Zheng J, Woodard PK, Hoffman AH (2009) An experimental

study on the ultimate strength of the adventitia and media of human

atherosclerotic carotid arteries in circumferential and axial directions. J Biomech 42:

2535-2539.

39. Cruickshank K, Riste L, Anderson SG, Wright JS, Dunn G, Gosling RG (2002)

Aortic pulse-wave velocity and its relationship to mortality in diabetes and glucose

intolerance. Circ 106: 2085–2090.

40. Lowe A, Harrison W, El-Aklouk E, Ruygrok P, Al-Jumaily AM (2009) Non-

invasive model-based estimation of aortic pulse pressure using Suprasystolic brachial

pressure wave forms. J Biomech 42: 2111-2115.

28

41. Fung YC (1997) Biomechanics: Circulation. 2nd edn, Springer-Verlag New York

Inc., New York.

42. Fung YC (1993) Biomechanics: Mechanical properties of living tissue. 2nd edn,

Springer-Verlag New York Inc., New York.

43. Yap CH, Kim H-S, Balachandran K, Weiler M, Haj-Ali R, Yoganathan AP (2010)

Dynamic deformation characteristics of porcine aortic valve leaflet under normal and

hypertensive conditions. Am J Physiol Heart Circ Physiol 298: H395-H405.

44. Sachs MS, Yoganathan AP (2007) Heart valve function: a biomechanical

perspective. Phil Trans R Soc B 362: 1369-1391.

45. Li MX, Beech-Brandta JJ, John LR, Hoskins PR, Easson WJ (2007) Numerical

analysis of pulsatile blood flow and vessel wall mechanics in different degrees of

stenoses. J Biomech 40: 3715-3724.

46. Bronzino JD (1999) The biomedical engineering handbook. 2nd edn, Volume I,

CRC Press, Florida.

47. Womersley JR (1957) An elastic tube theory of pulse transmission and oscillatory

flow in mammalian arteries. Carpenter Litho & Prtg. Co., Springfield.

29

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