European Biophysics Journal
Wave propagation through viscous fluid-filled elastic tube under initial pressure:
theoretical and biophysical model
--Manuscript Draft-Manuscript Number:
EBJO-D-22-00016R1
Full Title:
Wave propagation through viscous fluid-filled elastic tube under initial pressure:
theoretical and biophysical model
Article Type:
Original Article
Keywords:
Navier-Stokes equations, pulse wave velocity, arterial wall elasticity, viscosity,
biophysical model
Corresponding Author:
Dejan Žikić, Ph.D.
Faculty of Medicine, University of Belgrade
Belgrade, SERBIA
Corresponding Author Secondary
Information:
Corresponding Author's Institution:
Faculty of Medicine, University of Belgrade
Corresponding Author's Secondary
Institution:
First Author:
Dejan Žikić, Ph.D.
First Author Secondary Information:
Order of Authors:
Dejan Žikić, Ph.D.
Katarina Žikić
Order of Authors Secondary Information:
Funding Information:
Science and Technology Development
Center, Ministry of Education
(32040, 41022)
Abstract:
The velocity of propagation of pulse waves through the arteries is one of the indicators
of the health condition of the cardiovascular system. By measuring the pulse wave
velocity, cardiologists estimate the elasticity of the wall of blood vessels and the
changes that occur with aging. Using the Moens-Korteweg equation, an erroneous
assessment is made in the analysis. The paper presents the solution of Navier-Stokes
equations for propagation of pulse waves through an elastic tube filled with viscous
fluid under initial pressure. The equation for pulse wave velocity depending on
viscosity, density and initial fluid pressure, density and elasticity of wall and geometry
of the tube is derived. The results of the equation were compared with the experimental
results measured on a biophysical model of the cardiovascular system.
Response to Reviewers:
C1: In my opinion, the readability of the paper could be improved by splitting and
moving parts of the lengthy derivations to an appendix of the paper.
A1:According to the reviewer's suggestions, we moved the derivation of the equation
for the pulse wave velocity to the Appendix of the paper.
Dr Dejan Žikić
C2:The value of the mathematical model would be increased if its sensitivity analysis is
performed, e.g., sensitivity and importance indexes, etc., could be calculated (see G.
Qian, A. Mahdi 2020 Mathematical Biosciences).
A2:The reviewer is right; the sensitivity analysis of the model is very important
especially in biological and biochemical models. With our model, it's a little different.
Since our model is in the field of fluid physics (or fluid biophysics), the sensitivity is
analyzed differently from the biological models described in the paper recommended
by the reviewer.
Our assumption was that before, during and after the experiment, the parameters of
the model do not change (unlike, for example: biochemical reactions, cell models or
cancer research where the parameters change over time).
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We analyzed the sensitivity by changing the parameters of the model (pressure,
density, viscosity) and monitoring the output signal at a constant input signal
(amplitude of pressure change and pulse wave frequency). Also, we changed the
sample rate of the ADC and found that the best measurement frequency is 1KHz
(higher frequency has no effect on the result). And of course, we repeated the
measurements under the same conditions dozens of times.
Our further research is to compare the mathematical model with the measured results
in patients and then we will apply some of the SA methods.
C3Page 3, l. 53-55, the text could be reformulated to make it more understandable.
A3:The reviewer is right; we accidentally misspelled. We corrected the text according
to the reviewer's suggestions
C4:Page 14, l. 7-14 the text could be reformulated to make it more understandable.
A4: We corrected the text according to the reviewer's suggestions.
C5: Page 16, l. 38-40, the text could be reformulated to make it more understandable.
A5:We corrected the text according to the reviewer's suggestions.
C6:Page 17, l. 16-20, the text could be reformulated to make it more understandable.
A6:We corrected the text according to the reviewer's suggestions.
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Authors' Response to Reviewers' Comments
Click here to access/download;Authors' Response to Reviewers'
Comments;Answers to reviewer.docx
ANSWERS TO REVIEWERS
We thank the reviewer for his very helpful comments and compliments for our work. We find
constructive suggestions very useful for improving manuscripts. As indicated below, we have
checked all comments provided by reviewer and made any necessary changes accordingly.
Reviewer #1:
C1: In my opinion, the readability of the paper could be improved by splitting and moving parts
of the lengthy derivations to an appendix of the paper.
A1: According to the reviewer's suggestions, we moved the derivation of the equation for the
pulse wave velocity to the Appendix of the paper.
C2: The value of the mathematical model would be increased if its sensitivity analysis is
performed, e.g., sensitivity and importance indexes, etc., could be calculated (see G. Qian,
A. Mahdi 2020 Mathematical Biosciences).
A2: The reviewer is right; the sensitivity analysis of the model is very important especially in
biological and biochemical models. With our model, it's a little different. Since our model is
in the field of fluid physics (or fluid biophysics), the sensitivity is analyzed differently from
the biological models described in the paper recommended by the reviewer.
Our assumption was that before, during and after the experiment, the parameters of the model
do not change (unlike, for example: biochemical reactions, cell models or cancer research
where the parameters change over time).
We analyzed the sensitivity by changing the parameters of the model (pressure, density,
viscosity) and monitoring the output signal at a constant input signal (amplitude of pressure
change and pulse wave frequency). Also, we changed the sample rate of the ADC and found
that the best measurement frequency is 1KHz (higher frequency has no effect on the result).
And of course, we repeated the measurements under the same conditions dozens of times.
Our further research is to compare the mathematical model with the measured results in
patients and then we will apply some of the SA methods.
C3
Page 3, l. 53-55, the text could be reformulated to make it more understandable.
A3: The reviewer is right; we accidentally misspelled. We corrected the text according to the
reviewer's suggestions
C4: Page 14, l. 7-14 the text could be reformulated to make it more understandable.
A4: We corrected the text according to the reviewer's suggestions.
C5: Page 16, l. 38-40, the text could be reformulated to make it more understandable.
A5: We corrected the text according to the reviewer's suggestions.
C6: Page 17, l. 16-20, the text could be reformulated to make it more understandable.
A6: We corrected the text according to the reviewer's suggestions.
Manuscript (with track changes)
Wave propagation through viscous fluid-filled elastic tube under initial
pressure: theoretical and biophysical model
Dejan Žikić1 & Katarina Žikić2
1
Institute of Biophysics, Faculty of Medicine, University of Belgrade, Serbia
2
Faculty of Physics, University of Belgrade, Serbia
Corresponding author:
Dejan Žikić, Associate Professor
Institute of Biophysics, Faculty of Medicine
University of Belgrade, 11000 Belgrade, Serbia
https://orcid.org/0000-0003-3247-9071
Tel:
+381 11 360 7159
Fax:
+381 11 360 7061
e-mail: dzikic@gmail.com
1
Abstract
The velocity of propagation of pulse waves through the arteries is one of the indicators of the health
condition of the cardiovascular system. By measuring the pulse wave velocity, cardiologists estimate the
elasticity of the wall of blood vessels and the changes that occur with aging. Using the Moens-Korteweg
equation, an erroneous assessment is made in the analysis. The paper presents the solution of Navier-Stokes
equations for propagation of pulse waves through an elastic tube filled with viscous fluid under initial
pressure. The equation for pulse wave velocity depending on viscosity, density and initial fluid pressure,
density and elasticity of wall and geometry of the tube is derived. The results of the equation were compared
with the experimental results measured on a biophysical model of the cardiovascular system.
Keywords: Navier-Stokes equations, pulse wave velocity, arterial wall elasticity, viscosity, biophysical
model
2
1. Introduction
Investigation of the waveforms of arterial blood pressure and flow through blood vessels plays an important
role in understanding the nature and disorders of the cardiovascular system. The beginning of the interest
of scientists in the wave nature of blood dates to the beginning of the 19th century when Young (1808)
published a paper on the wave motion of blood and derived the first formula for pulse wave velocity. In
1877, Moens and Korteweg simultaneously derived the equation for the velocity of a pulse wave
propagating through a fluid-filled elastic tube of thin walls (Moens 1877, Moens 1878, Korteweg 1878).
This equation is still used today in medicine to study the age of the cardiovascular system. The greatest
mathematical contribution was made by Womersley in the middle of the last century (Womersley 1955)
who solved the pulse wave velocity profile in a rigid tube filled with a viscous fluid. In addition, he offered
a solution for the pulse wave velocity profile through an elastic tube and developed a mathematical analysis
of blood flow in the arteries. In his work, he included viscosity in the pulse wave velocity calculation
(Womersley 1957).
To better understand the physics of cardiovascular processes, mathematical and experimental biophysical
models were mainly used. In mathematical models, the equations describe the principles of the physical
laws of a process, then the theoretical model is experimentally confirmed on the biophysical model.
Khir and Parker (2002) and later Li (Li et al. 2011), used the water-hammer equation to determine pulse
wave velocity. Westerhof et al. (1972) determined pulse wave velocity from waveforms of fluid pressure
and flow. Feng and Khir (2010) applied both equations to detect reflected waves in flow pressure signals.
In his work, Khir used a pressure-velocity loop to determine the arrival time of reflected waves (Khir et al.
2007). Lazovic et al. (2015), detected reflected waves that appear on the radial artery with aging.
Papageorgiou and Jones (1987a) investigated the various materials used in models for arteries and blood
vessels. They found that if elastic tubes with a nonlinear stress / strain ratio are used in the models, the
differences between real systems and physical models are minimal. They found (1987b) that the differences
between real systems and physical models are minimal when elastic tubes are used in the models where the
ratio of stress and relative elongation is not. The influence of viscosity on pulse wave velocity was
3
experimentally shown by Stojadinović et al. (2015). The influence of gravity on wave propagation was
experimentally shown by Žikić et al. (2019).
One of the parameters used in medicine to determine the age of the cardiovascular system is the pulse wave
velocity - PWV. The Moens-Korteweg equation is mainly used to estimate arterial elasticity from measured
PWV values. As this equation does not depend on blood viscosity, diastolic pressure, blood vessel wall
density as well as the influence of gravity, using this relation in medical research introduces an error in the
interpretation of results and can be misdiagnosed.
Morgan and Kiely (1954) offered a solution for calculating PWV through a viscous fluid in an elastic tube
but without initial pressure and without pipe wall density. Womersley (1955, 1957) also solved this problem
with viscosity but offered a solution in a complex equation that only physicists can solve. Also, this equation
cannot be applied if the fluid is under initial pressure and the wall density of the elastic tube is neglected.
Other scientists have also investigated how PWV changes with initial fluid pressure (Atabek et al. 1966;
Atabek 1968; Demiray and Akgün 1997) but the offered equations are very complicated especially for
medical doctors and the results of their equations do not agree with the experimental results.
One of the reasons for the interest in this phenomenon of wave propagation is related to research in
hemodynamics. The main area that cardiovascular physics deals with is the wave propagation of blood
through blood vessels. With aging, the biophysical parameters of the blood and blood vessels change, and
the correct interpretation of cardiovascular parameters can greatly contribute to accurate diagnosis and
successful therapy.
The aim of this paper is to show the mathematical derivation of the equation for pulse waves velocity
through a viscous fluid-filled elastic tube with initial fluid pressure and to prove experimentally the
accuracy of the equation.
2. Methods
2.1 Mathematical equations
Moens-Korteweg equation
4
Moens and Korteweg ingeniously derived the equation for PWV. They assumed that the pulse wave
increases the initial volume of the cylinder 𝑉𝑜 = 𝜋𝑅𝑜2 ∆𝑧 (Figure 1) of width ∆z to 𝑉 = 𝜋(𝑅0 +
∆𝑅0 )2 (∆𝑧 +
𝜕𝑢
∆𝑧) and
𝜕𝑧
derived three equations:
𝜕𝑢
2
2
∆𝑉 𝜋(𝑅0 + ∆𝑅0 ) (∆𝑧 + 𝜕𝑧 ∆𝑧) − 𝜋𝑅𝑜 ∆𝑧
=
𝑉𝑜
𝜋𝑅𝑜2 ∆𝑧
2
2
2∆𝑅0
∆𝑅0
𝜕𝑢 2∆𝑅0 𝜕𝑢
∆𝑅0 𝜕𝑢 2∆𝑅0 𝜕𝑢
∆𝑝
=
+( 2 ) +
+
+( 2 )
=
+
=−
𝑅0
𝜕𝑧
𝑅0 𝜕𝑧
𝜕𝑧
𝑅0
𝜕𝑧
𝐾
𝑅𝑜
𝑅𝑜
where ∆𝑉 is the change in volume, K- bulk modulus of fluid, (
∆𝑅0 2 2∆𝑅0 𝜕𝑢 ∆𝑅0 2 𝜕𝑢
) , 𝑅 𝜕𝑧 , ( 𝑅 2 ) 𝜕𝑧
𝑅𝑜2
0
𝑜
(1)
≈0
The difference in pressures on the front and back of the disk (Figure 2) can be expressed from the
pressure gradient, and the axial acceleration of the fluid is:
𝑚𝑎 = −
𝜕(∆𝑝)
𝜕2 𝑢
𝑙𝐴 = 𝜌𝑙𝐴 2
𝜕𝑧
𝜕𝑡
or
𝜕(∆𝑝)
𝜕2𝑢
= −𝜌 2
𝜕𝑧
𝜕𝑡
(2)
Due to the increase in pressure ∆p in the fluid, the force F1 tends to separate the two half-cylinders
(Figure 3a). On the other hand, this force is opposed by hoop stress 𝜎𝜃
𝐹1 = ∆𝑝𝑆1 = 2∆𝑝(𝑅0 + ∆𝑅0 )𝑙 = 2𝑆2 𝜎𝜃 = 2ℎ𝑙𝐸
2𝜋∆𝑅0
∆𝑅0
= 2𝐸
ℎ𝑙
2𝜋𝑅0
𝑅0
Arranging this equation gives the third Moens-Korteweg equation
∆𝑝 = 𝐸ℎ
∆𝑅0
∆𝑅0
~𝐸ℎ 2
(𝑅0 + ∆𝑅0 )𝑅0
𝑅𝑜
(3)
After differentiating the second and third equations and arranging using the first equation, a wave
equation is obtained
𝜕2𝑢
2𝜌𝑅0 𝜌 𝜕 2 𝑢
=
(
+ ) 2
𝜕𝑧 2
𝐸ℎ
𝐾 𝜕𝑡
For an elastic tube and an incompressible fluid ( 𝐾 → ∞), the equation becomes the Moens-Korteweg
equation
5
𝜕 2 𝑢 2𝜌𝑅0 𝜕 2 𝑢
1 𝜕2 𝑢
=
= 2 2
2
2
𝜕𝑧
𝐸ℎ 𝜕𝑡
𝑐0 𝜕𝑡
𝑐0 = √
𝐸ℎ
2𝜌𝑅0
(4)
Where c0 is the pulse wave velocity – PWV. As can be seen from the equation, viscosity, tube wall density
and initial fluid pressure do not affect the PWV.
Bramwell and Hill derived the equation for PWV without values for E, h, and R0 (since they are not constant
and depend on the artery itself). If we replace ∆R0 in equation (1) from equation (3) and assume that ∂u/∂z
is small it follows:
∆𝑉 2∆𝑅0
2 𝑅0 2 ∆𝑝 2𝑅0 ∆𝑝
=
=
=
𝑉𝑜
𝑅0
𝑅0 𝐸ℎ
𝐸ℎ
The rearrangement resulted in:
𝐸ℎ
𝑉𝑜 ∆𝑝
=
2𝑅0
∆𝑉
And by substituting in equation (4), the Bramwell-Hill equation is obtained:
𝑐0 = √
𝑉𝑜 ∆𝑝
𝜌 ∆𝑉
(5)
2.2 Pulse wave velocity when the fluid is under initial pressure
Let the initial pressure in the tube be p0. Due to the increase in pressure, the radius also increases:
R=R0+R and the length of the tube l=l0+l.
In the derivation of equation (3) it is shown that the hoop stress is equal
𝜎𝜃 =
𝑝0 𝑅0
2𝜋∆𝑅0
∆𝑅0
𝑅 − 𝑅0
= 2ℎ𝑙𝐸
= 2𝐸
ℎ𝑙 = 2𝐸
ℎ𝑙
ℎ
2𝜋𝑅0
𝑅0
𝑅0
By arranging, the final radius with the initial pressure in the tube is
𝑅=
𝑅0
𝑝 𝑅
1− 0 0
𝐸ℎ
The same pressure acts on the cross section of the cylinder (Figure 3b) and is opposed by longitudinal
stress
6
Formatted: Justified
𝐹2 = 𝑝0 𝜋𝑅02 = 𝜋[(𝑅0 + ℎ)2 − 𝑅02 ]𝜎𝑙 ~2𝜋𝑅0 ℎ𝜎𝑙
By arranging, the hoop stress is 2 times higher than the longitudinal stress
𝜎𝑙 =
𝑝0 𝑅0 𝜎𝜃
=
2ℎ
2
After substitution
𝐸
∆𝑅0
∆𝑙
= 2𝐸
𝑅0
𝑙0
𝑅 − 𝑅0
𝑙 − 𝑙0
=2
𝑅0
𝑙0
𝑙=
𝑙0
1
𝑝0 𝑅0
(
+ 1)~𝑙0 (1 +
)
2 1 − 𝑝0 𝑅0
2𝐸ℎ
𝐸ℎ
Due to the increase in fluid pressure in the tube by p0, the disk volume is:
𝑉 = 𝑅 2 𝜋𝑙
Or
2
𝑝0 𝑅0
𝑝0 𝑅0
∆𝑝
𝑅0
𝑝0 𝑅0 ∆𝑝
∆𝑝 (1 + 2𝐸ℎ )
∆𝑝 (1 + 2𝐸ℎ )
2
𝑉
= 𝑅 𝜋𝑙 = (
)
= 𝜋𝑙0 𝑅02
=
𝑉
) 𝜋𝑙0 (1 +
0
𝑝 𝑅
∆𝑉
2𝐸ℎ ∆𝑉
∆𝑉
∆𝑉
𝑝 𝑅 2
𝑝 𝑅 2
(1 − 0 0 )
(1 − 0 0 )
(1 − 0 0 )
𝐸ℎ
𝐸ℎ
𝐸ℎ
𝑝0 𝑅0
∆𝑝
𝐸ℎ 1 + 2𝐸ℎ
𝑉
=
∆𝑉 2𝑅0
𝑝 𝑅 2
(1 − 0 0 )
𝐸ℎ
(6)
Using equation (6) and Bramwell-Hill equation (5), the PWV can be calculated with the initial fluid
pressure in the tube:
√
𝑐=
𝐸ℎ
𝑝 𝑅
𝐸ℎ
𝑝
√
+ 0 0
+ 0
2𝑅0 𝜌 (1 2𝐸ℎ )
2𝑅0 𝜌 4𝜌
=
𝑝 𝑅
𝑝 𝑅
1− 0 0
1− 0 0
𝐸ℎ
𝐸ℎ
(7)
2.3 Wave propagation through an elastic tube filled with a viscous fluid
Fluid motion is governed by continuity equation (8) and Navier–Stokes equations (9,10):
𝑑𝜌
+ ∇(ρv) = 0
𝑑𝑡
𝜕𝑣𝑧
𝜕𝑣𝑧
𝜕𝑣𝑧
𝜕𝑝
1 𝜕
𝜕𝑣𝑧
𝜕 2 𝑣𝑧
𝜌(
+ 𝑣𝑟
+ 𝑣𝑧
)=−
+𝜇(
(𝑟
)+
) + 𝜌𝑔𝑧
𝜕𝑡
𝜕𝑟
𝜕𝑧
𝜕𝑧
𝑟 𝜕𝑟
𝜕𝑟
𝜕𝑧 2
(8)
(9)
7
𝜌(
𝜕𝑣𝑟
𝜕𝑣𝑟
𝜕𝑣𝑟
𝜕𝑝
1 𝜕
𝜕𝑣𝑟
𝜕 2 𝑣𝑟 𝑣𝑟
+ 𝑣𝑟
+ 𝑣𝑧
)=−
+𝜇(
(𝑟
)+
− )
𝜕𝑡
𝜕𝑟
𝜕𝑧
𝜕𝑟
𝑟 𝜕𝑟
𝜕𝑟
𝜕𝑧 2 𝑟 2
(10)
where  is density, 𝑣𝑧 is the axial velocity of the fluid, 𝑣𝑟 is the radial velocity of the fluid,  is the
dynamic viscosity. By solving these equations (Appendix) we obtain pulse wave velocities for the case of
small viscosity and high viscosity:
𝐸ℎ
𝑝
+ 0
1
𝑡
1
𝜇
𝜎 2 3𝑡
2𝑅0 𝜌 4𝜌
𝑐=
− ))
(1 + − √
(1 − 𝜎 +
2
𝑝 𝑅
4 𝑅0 2𝜔𝜌
4
8
1 − 0 0 (1 + 𝑡 )
𝐸ℎ
4
And for high viscosity
√
𝑐=
𝑐0
5 − 4𝜎 + 6𝑡
√ 𝑅 2 𝜔𝜌
0 𝜇
= ±√
𝐸ℎ
𝑝0
𝜔𝜌
1
+
𝑅 √
2𝑅0 𝜌 4𝜌 0 𝜇 5 − 4𝜎 + 6𝑡
(11)
(12)
The last equation (eq.12) has no application for medical research due to its high viscosity values, so it will
not be discussed further in the paper.
Equations of fluid motion
We will assume that the fluid is incompressible and viscous. It is also assumed that the fluid propagates
pulsatingly through a cylindrical tube of thin elastic walls. Fluid motion is governed by continuity
equation (8) and Navier–Stokes equations (9,10):
𝑑𝜌
+ ∇(ρv) = 0
𝑑𝑡
𝜕𝑣𝑧
𝜕𝑣𝑧
𝜕𝑣𝑧
𝜕𝑝
1 𝜕
𝜕𝑣𝑧
𝜕 2 𝑣𝑧
𝜌(
+ 𝑣𝑟
+ 𝑣𝑧
)=−
+𝜇(
(𝑟
)+
) + 𝜌𝑔𝑧
𝜕𝑡
𝜕𝑟
𝜕𝑧
𝜕𝑧
𝑟 𝜕𝑟
𝜕𝑟
𝜕𝑧 2
𝜕𝑣𝑟
𝜕𝑣𝑟
𝜕𝑣𝑟
𝜕𝑝
1 𝜕
𝜕𝑣𝑟
𝜕 2 𝑣𝑟 𝑣𝑟
𝜌(
+ 𝑣𝑟
+ 𝑣𝑧
)=−
+𝜇(
(𝑟
)+
− )
𝜕𝑡
𝜕𝑟
𝜕𝑧
𝜕𝑟
𝑟 𝜕𝑟
𝜕𝑟
𝜕𝑧 2 𝑟 2
(8)
(9)
(10)
where  is density, 𝑣𝑧 is the axial velocity of the fluid, 𝑣𝑟 is the radial velocity of the fluid,  is the
dynamic viscosity. The assumption is that the motion is axisymmetric without tangential velocity 𝑣𝜃 = 0.
We will also assume that the fluid is not affected by gravity, i.e. that the z axis is in a horizontal position,
so the last term in eq. 9 can be ignored. Next, we describe the movement of the tube wall with two
equations: longitudinal and radial
8
𝜕2𝜂
𝐸ℎ
𝜂
𝜕𝜁
=𝑝−
( +𝜎 )
(11)
(1 − 𝜎 2 )𝑅0 𝑅0
𝜕𝑡 2
𝜕𝑧
𝜕2𝜁
𝐸ℎ
𝜕 2 𝜁 𝜎 𝜕𝜂
𝜕𝑣𝑧 𝜕𝑣𝑟
𝜌0 ℎ 2 =
+
+
)
(
)−𝜇(
(12)
(1 − 𝜎 2 ) 𝜕𝑧 2 𝑅0 𝜕𝑧
𝜕𝑡
𝜕𝑟
𝜕𝑧
Where 𝜂 and 𝜁 are the components of the displacement of the point on the tube wall in the axial and radial
𝜌0 ℎ
directions, respectively, E – Young's modulus of elasticity, ρ0 –the density of the tube wall. The
assumption is that the pulse wave has a sinusoidal shape and that 𝜂, 𝜁, 𝑣𝑟 𝑎𝑛𝑑 𝑣𝑧 change in the same way.
Let them be
𝑝 = 𝑃𝑒 𝑖(𝐾𝑧−𝜔𝑡) , 𝜂 = Ω𝑒 𝑖(𝐾𝑧−𝜔𝑡) , 𝜁 = Λ𝑒 𝑖(𝐾𝑧−𝜔𝑡) , 𝑣𝑧 = 𝑢(𝑟)𝑒 𝑖(𝐾𝑧−𝜔𝑡) 𝑖 𝑣𝑟 = 𝑤(𝑟)𝑒 𝑖(𝐾𝑧−𝜔𝑡)
(𝑤, 𝑢 – functions of 𝑟, 𝑃, Ω, Λ – constants). After substitution in equations (9), (10), (11) and (12) and
𝑖𝜔𝜌
𝜇
differentiation (
= 𝛼2,
𝜕 2 𝑣𝑟
𝜕𝑧 2
≈ 0,
𝜕2𝑢
𝜕𝑧 2
≈ 0 ) follows:
𝜕 2 𝑢 1 𝜕𝑢
𝑖𝐾
+
+ 𝛼2𝑢 = 𝑃
𝜕𝑟 2 𝑟 𝜕𝑟
𝜇
𝜕 2 𝑤 1 𝜕𝑤
1
1 𝜕𝑝
+
+ (𝛼 2 − 2 ) 𝑤 =
𝜕𝑟 2 𝑟 𝜕𝑟
𝑟
𝜇 𝜕𝑟
𝐸ℎ
Ω 𝑖𝐾𝜎
−𝜌0 ℎ𝜔2 Ω = 𝑃 −
Λ)
( +
(1 − 𝜎 2 ) 𝑅02 𝑅0
𝐸ℎ
𝑖𝐾𝜎
𝜕𝑢
−𝜌0 ℎ𝜔2 Λ =
(−𝐾 2 Λ +
Ω) − 𝜇 ( |𝑟=𝑅0 + 𝑖𝐾𝑤)
(1 − 𝜎 2 )
𝑅0
𝜕𝑟
(13)
(14)
(15)
(16)
Equations (13) and (14) are partial Bessel equations.
Assuming that P is in the form: 𝑃 = 𝐴1 ∙ 𝐽0 (𝑦𝑟), where 𝐴1 is a constant and a non-homogeneous solution
in the form 𝑢(𝑟) = 𝐵1 ∙ 𝐽0 (𝑦𝑟), by substituting in equation (13) and differentiating, a particular solution is
obtained
𝑢(𝑟) =
𝑖𝐾𝐴1
1
∙ 𝐽 (𝑦𝑟)
𝜇 (𝛼 2 − 𝑦 2 ) 0
Assumption that a homogeneous solution in the form 𝑢(𝑟) = 𝐶1
𝐽0 (𝛼𝑟)
,
𝐽0 (𝛼)
where C1 is a constant to be
determined and in the final form:
𝑢(𝑟) = 𝐶1
𝐽0 (𝛼𝑟) 𝑖𝐾𝐴1
1
+
∙ 𝐽 (𝑦𝑟)
𝐽0 (𝛼)
𝜇 (𝛼 2 − 𝑦 2 ) 0
(17)
9
Similar mathematical operations are repeated to solve equation (14). Let non-homogeneous solution be in
the form 𝑤 = 𝐷1 ∙ 𝐽1 (𝑦𝑟), and by changing to equation (14) and differentiating, a particular solution is
obtained
𝑤=−
𝑦𝐴1
1
∙ 𝐽 (𝑦𝑟)
𝜇 (𝛼 2 − 𝑦 2 ) 1
Here, too, the assumption is that the homogeneous solution is in the form 𝑤 = 𝐶2
𝐽1 (𝛼𝑟)
,
𝐽0 (𝛼)
where C2 is a
constant, so it is in the final form:
𝐽1 (𝛼𝑟) 𝑦𝐴1
1
−
∙ 𝐽 (𝑦𝑟)
𝐽0 (𝛼)
𝜇 (𝛼 2 − 𝑦 2 ) 1
From the equation of continuity (8) =>
(18)
𝜕𝑤 𝑤
+ + 𝑖𝐾𝑢 = 0
𝜕𝑟 𝑟
By substituting equations (17) and (18) into equation (19) and arranging it, we get
(19)
𝑤(𝑟) = 𝐶2
− (𝑖𝐾𝐶1
𝐽0 (𝛼𝑟) 𝑖 2 𝐾 2 𝐴1
1
+
∙ 𝐽 (𝑦𝑟))
𝐽0 (𝛼)
𝜇 (𝛼 2 − 𝑦 2 ) 0
= 𝛼𝐶2
𝐽0 (𝛼𝑟) 𝑦 2 𝐴1
1
𝐶2 𝐽1 (𝛼𝑟) 1 𝑦𝐴1
1
−
∙ 𝐽 (𝑦𝑟) +
−
∙ 𝐽 (𝑦𝑟)
𝐽0 (𝛼)
𝜇 (𝛼 2 − 𝑦 2 ) 0
𝑟 𝐽0 (𝛼) 𝑟 𝜇 (𝛼 2 − 𝑦 2 ) 1
By solving we get
𝐶2 = −
𝑖𝐾
𝐶
𝛼 1
and after series expansion of Bessel coefficients:
𝑦𝑟 2
(2)
𝑦𝑟 4
(2)
𝐽0 (𝑦𝑟) = 1 −
+ 2 2 +⋯ ≈1
(12 )
(1 )(2 )
1
𝑦𝑟
𝑦𝑟 3
𝑦𝑟 5
(2)
(2)
(2)
𝑦𝑟
𝐽1 (𝑦𝑟) =
−
+
+⋯ ≈
1
2
12 ∙ 3
22 2
𝑖𝜔𝜌
𝑖𝜔𝜌
𝑖 𝜔
As well as (𝛼 2 − 𝑦 2 ) =
− 𝑖2𝐾2 =
− 2 ≈ 𝛼2
𝜇
𝜇
𝑐
Due to the viscosity the velocities of the fluid along the inner wall of the vessel (when r = R0) are
𝑢|𝑟=𝑅0 = 𝐶1
𝐽0 (𝛼𝑟) 𝑖𝐾𝐴1
1
𝐽0 (𝛼𝑟) 𝐴1
+
∙ 𝐽 (𝑦𝑟) = 𝐶1
+
𝐽0 (𝛼)
𝜇 (𝛼 2 − 𝑦 2 ) 0
𝐽0 (𝛼) 𝜌𝑐
𝑤|𝑟=𝑅0 = 𝐶2
𝐽1 (𝛼𝑟) 𝑦𝐴1
1
𝑖𝐾
2 𝐽1 (𝛼𝑟) 𝐴1
−
∙ 𝐽 (𝑦𝑟) = − (𝐶1
+
∙𝑅 )
𝐽0 (𝛼)
𝜇 (𝛼 2 − 𝑦 2 ) 1
2
𝛼 𝐽0 (𝛼) 𝜌𝑐 0
10
and we have
𝑑𝑢
𝐽1 (𝛼𝑅0 ) 𝑖𝐾𝐴1
1
𝐽1 (𝛼𝑅0 ) 𝐴1 𝐾 2 𝑅0
|𝑟=𝑅0 = −𝐶1 𝛼
+
∙ (−𝑦)𝐽1 (𝑦𝑅0 ) = −𝐶1 𝛼
+
2
2
𝑑𝑟
𝐽0 (𝛼)
𝜇 (𝛼 − 𝑦 )
𝐽0 (𝛼)
𝜌𝑐 2
𝐸ℎ
With substitutions: 𝐵 = (1−𝜎2 ) , 𝑃 = 𝐴1 ∙ 𝐽0 (𝑦𝑟) and 𝐽0 (𝑦𝑟) ≈ 1 in equation (15) we have
1
𝐵
𝑖𝐾𝜎𝐵
𝐴 + (𝜔2 −
Λ=0
)Ω −
𝜌0 ℎ 1
𝜌0 𝑅0
𝜌0 𝑅02
Since the velocity of the wall is equal to the velocity of the fluid along the wall of the vessel, equalizing
the equations
𝑢=
𝑑𝜁
𝐽0 (𝛼𝑟) 𝐴1 𝑖(𝐾𝑥−𝜔𝑡)
|
= −𝑖𝜔Λ𝑒 𝑖(𝐾𝑥−𝜔𝑡) = (𝐶1
+ )𝑒
𝑑𝑡 𝑟=𝑅0
𝐽0 (𝛼) 𝜌𝑐
𝐴1
𝐽0 (𝛼𝑅0 )
+ 𝐶1
+ 𝑖𝜔Λ = 0
𝜌𝑐
𝐽0 (𝛼)
𝑑𝜂
𝑖𝐾
2 𝐽1 (𝛼𝑅0 ) 𝐴1
𝑤=
|
= −𝑖𝜔Ω𝑒 𝑖(𝐾𝑥−𝜔𝑡) = − (𝐶1
+
∙ 𝑅 ) 𝑒 𝑖(𝐾𝑥−𝜔𝑡)
𝑑𝑡 𝑟=𝑅0
2
𝛼 𝐽0 (𝛼)
𝜌𝑐 0
𝑖𝐾𝑅0
𝑖𝐾 𝐽1 (𝛼𝑅0 )
𝐴 + 𝐶1
− 𝑖𝜔Ω = 0
2𝜌𝑐 1
𝛼 𝐽0 (𝛼)
And the last equation, with substitutions in (16) =>
(21)
(22)
−𝜔2 Λ = −𝐾 2
𝐵ℎ
𝑖𝐾𝜎 𝐵ℎ
𝜇
𝐽1 (𝛼𝑅0 ) 𝐴1 𝐾 2 𝑅0 𝑖 2 𝐾 2
2 𝐽1 (𝛼𝑅0 ) 𝐴1
Λ+
Ω−
+
−
+
∙ 𝑅 ))
(−𝐶1 𝛼
(𝐶1
(𝛼)
𝜌0 ℎ
𝑅0 𝜌0 ℎ
𝜌0 ℎ
𝐽0
𝜌𝑐 2
2
𝛼 𝐽0 (𝛼)
𝜌𝑐 0
𝜔2
𝑐2 ≪ 1 ,
𝛼
𝐾2
2 𝐽1 (𝛼𝑅0 ) 𝐴1
+
∙𝑅 )≪1
(𝐶1
2
𝛼 𝐽0 (𝛼)
𝜌𝑐 0
We obtain
−
𝜇
𝐾 2 𝑅0
𝜇𝛼 𝐽1 (𝛼𝑅0 )
𝐵ℎ
𝑖𝐾𝜎 𝐵ℎ
∙
𝐴1 +
𝐶 + (𝜔2 − 𝐾 2
)Λ +
Ω=0
𝜌0 ℎ𝜌𝑐
2
𝜌0 ℎ 𝐽0 (𝛼) 1
𝜌0 ℎ
𝑅0 𝜌0 ℎ
(23)
The system of equations (20), (21), (22) and (23) is solved by setting the coefficients A1, C1, Λ and Ω to
be equal to 0:
11
1
𝜌𝑐
|
𝑖𝜔𝑅0
2𝜌𝑐 2
|
1
𝜌0 ℎ
|
𝜇
𝜔2 𝑅0
−
∙
3
𝜌0 ℎ𝜌𝑐
2
𝐽0 (𝛼𝑅0 )
𝐽0 (𝛼)
𝑖𝜔 𝐽1 (𝛼𝑅0 )
𝛼𝑐 𝐽0 (𝛼)
0
𝑖𝜔
−𝑖𝜔
0
𝐵
𝜌0 𝑅02
𝑖𝜔𝜎 𝐵
𝑅0 𝑐 𝜌0
𝜔2 −
0
𝜇𝛼 𝐽1 (𝛼𝑅0 )
𝜌0 ℎ 𝐽0 (𝛼)
|
𝑖𝜔𝜎𝐵 | = 0
𝜌0 𝑅0 𝑐
𝜔2 𝐵 |
𝜔2 − 2
𝑐 𝜌0
−
By simplifying this determinant, we obtain
|
1−
2 𝐽1 (𝛼𝑅0 )
𝛼𝑅0 𝐽0 (𝛼𝑅0 )
1
|
−
1 𝐽1 (𝛼𝑅0 )
𝛼𝑅0 𝐽0 (𝛼𝑅0 )
2
𝐵ℎ
𝜌𝑐 2 𝑅0
𝜎𝐵
− 2
𝜌𝑐 𝑅0
1
|
𝜎𝐵ℎ
=0
𝜌𝑐 2 𝑅0
|
𝜌0 ℎ
𝐵ℎ
−
𝜌𝑅0 𝜌𝑐 2 𝑅0
1+
Using substitutions
𝐸ℎ
𝜌0 ℎ
= 𝑥,
=𝑡
(24)
2𝜌𝑐 2 𝑅0
𝜌𝑅0
and after solving the determinant, the so-called frequency equation in term of variable x was obtained
(4 − 2𝛼𝑅0
𝐽0 (𝛼𝑅0 ) 2
𝛼𝑅0 𝐽0 (𝛼𝑅0 )
𝐽0 (𝛼𝑅0 )
− 1) − 4𝜎 + 1 + 2𝛼𝑅0
]𝑥
) 𝑥 + [2𝑡 (
𝐽1 (𝛼𝑅0 )
2 𝐽1 (𝛼𝑅0 )
𝐽1 (𝛼𝑅0 )
𝛼𝑅0 𝐽0 (𝛼𝑅0 )
− 2𝑡(1 − 𝜎 2 )
− (1 − 𝜎 2 ) = 0
2 𝐽1 (𝛼𝑅0 )
Using the values for viscosity of blood, diameter of aorta or main arteries, blood density and density of
artery wall, we obtained that |𝛼𝑅0 | ≫ 1, ie that they are in the ranges of 𝛼𝑅0 = 3 − 12 and 𝑡 = 0.1 −
0.4.
After asymptotic expansion of the Bessel function
𝐽0 (𝛼𝑅0 )
1
~−𝑖+
𝐽1 (𝛼𝑅0 )
2𝛼𝑅0
and by substituting in the frequency equation and arranging, the quadratic equation is obtained
(−2𝛼𝑅0 𝑖 − 3)𝑥 2 + [4𝜎 − 2 + 2𝛼𝑅0 𝑖 + 𝑡𝛼𝑅0 𝑖 +
3𝑡
𝑡
] 𝑥 + (1 − 𝜎 2 ) (1 + − 𝑡𝛼𝑅0 𝑖) = 0
2
2
12
This quadratic equation can be solved for 𝜎 = 0, 𝜎 =
1
2
and 𝜎 = 1. For an artery 𝜎 = 0.49, so the roots of
1
2
quadratic equation for 𝜎 = , are:
𝑡
𝑖
𝜎 2 3𝑡
𝑖(1 − 𝜎 2 )
𝑥1 = 1 + +
− ) and 𝑥2 =
(2 − 2𝜎 +
2 𝛼𝑅0
2
4
2𝛼𝑅0
Substituting in (24)
𝐾 2 𝐸ℎ
𝐾2
= 2 𝑐02 = 𝑥
2
𝜔 2𝜌𝑅0 𝜔
We look for solutions in the following form and only for x1 (for 𝜇 → 0, 𝑥2 → ∞)
𝐾
𝐾1 + 𝑖𝐾2
𝑐 =
𝑐0 = √𝑥1
𝜔 0
𝜔
After substitution √𝑖 =
√2
(1
2
+ 𝑖) and solving the root of a complex number
𝜎 2 3𝑡
𝜎 2 3𝑡
𝑡 2 (1 − 𝜎 + 4 − 8 ) 2𝑖 (1 − 𝜎 + 4 − 8 ) 𝐾1 + 𝑖𝐾2
+
=
𝑐0
√𝑥1 = √1 + +
2
𝜔
2𝜔𝜌
2𝜔𝜌
√
√
𝑅0
𝑅0
𝜇
𝜇
𝜎 2 3𝑡
− )
4
8
2𝜔𝜌
√
𝐾2
𝜇 𝑅0
𝑐0 =
𝜎 2 3𝑡
𝜔
𝑡 (1 − 𝜎 + 4 − 8 )
1+4+
2𝜔𝜌
√
𝑅
𝜇 0
(1 − 𝜎 +
𝜎 2 3𝑡
𝐾1
𝑡 (1 − 𝜎 + 4 − 8 )
𝑐 =1+ +
,
𝜔 0
4
2𝜔𝜌
√
𝑅0
𝜇
By substituting c0 with equation (7) which is a function of the initial pressure and arranging, we obtain the
final equation (25) for the pulse wave velocity
𝐸ℎ
𝑝
√
+ 0
𝜔
1
𝑡
1
𝜇
𝜎 2 3𝑡
2𝑅0 𝜌 4𝜌
𝑐=
=±
− ))
(1 + − √
(1 − 𝜎 +
2
𝑝
𝑅
𝐾1
4 𝑅0 2𝜔𝜌
4
8
1 − 0 0 (1 + 𝑡 )
𝐸ℎ
4
And
(25)
2
𝑝0 𝑅0 (1 − 𝜎 + 𝜎 − 3𝑡)
4
8
𝐸ℎ
𝐾2 = ±𝜔
𝐸ℎ
𝑝0
𝑡
2𝜔𝜌
√
√
2𝑅0 𝜌 + 4𝜌 (1 + 4)
𝜇 𝑅0
1−
13
If we assume that t is very small it can be neglected (𝑡 = 0) 𝑎𝑛𝑑 the initial fluid is zero (𝑝0 = 0) we get
the same equation derived by Morgan and Kiely for small viscosity
𝑐=
𝜔
𝐸ℎ
1
𝜇
𝜎2
= ±√
(1 + − √
(1 − 𝜎 + ))
𝐾1
2𝑅0 𝜌
𝑅0 2𝜔𝜌
4
(26)
And if we neglect both σ and μ we get the Moens-Korteweg equation (4) for PWV.
In case of high viscosity, or |𝛼𝑅0 | ≪ 1, we use serial expansions
𝐽0 (𝛼𝑅0 ) ≈ 1 −
𝛼 2 𝑅02
,
4
𝐽1 (𝛼𝑅0 ) ≈
𝛼𝑅0
𝛼 2 𝑅02
(1 −
)
2
8
After substituting in the frequency equation and solving the quadratic equations we obtain
𝑐=
𝜔
=
𝐾1
𝐾2 = ±𝜔√
𝑐0
5 − 4𝜎 + 6𝑡
√ 𝑅2 𝜔𝜌
0 𝜇
= ±√
𝐸ℎ
𝑝0
𝜔𝜌
1
+
𝑅 √
2𝑅0 𝜌 4𝜌 0 𝜇 5 − 4𝜎 + 6𝑡
(27)
1
𝜇 √5 − 4𝜎 + 6𝑡
𝐸ℎ
𝑝0 √𝜔𝜌
𝑅0
+
2𝑅0 𝜌 4𝜌
Equation (27) has no application for medical research due to its high viscosity values, so it will not be
discussed further in the paper.
2.4 Biophysical model
A biophysical model of the cardiovascular system was used for the experimental validation of the equation
(Figure 4). A similar model was used in Stojadinović et al. (2015) and Žikić et al. (2019). In the first
experiment, pulse wave velocities were measured with increasing initial fluid pressure. The transmural
pressure (difference in pressure between two sides of a wall of a tube) was changed from 0 mmHg to 120
mmHg with a step of 20 mmHg. A silicone transparent isotropic tube with an inner diameter of 10 mm and
a wall thickness of 1 mm with a modulus of elasticity E = 5.19 MPa was used. The fluid was distilled water
with a density of 998 kg/m3 and a viscosity of 0.99 mPa·s at 23 °C. PWV was determined from the equation
d/t where d is the distance between the sensors (Figure 4) and t is the propagation time of the pulse wave
between the sensors. In all experiments, the distance d between the sensors was 1m.PWV was determined
14
by measuring pulse pressure waveforms of fluid at 1 m and the time difference between the waves was
determined. In Stojadinović et al. (2015) and Žikić et al. (2019) is a detailed explanation of the model and
the determination of PWV.
In another experiment, pulse wave velocities were measured with fluids of different densities and
viscosities. The fluid pressure was 0 mmHg. Aqueous alcohol solutions with a concentration of 10-60%
were used. Silicone tube of the same dimensions and properties as in the first experiment was used.
3.
Results
Table 1 shows the densities and viscosities of the fluids used in the experiments.
The results of PWV measurements with increasing fluid pressure are shown in Figure 5. The figure also
shows the calculated PWV values using the Moens-Korteweg equation, the Morgan-Kiely equation, and
the model using equation (2511) in both cases: when the parameter is neglected (t = 0) and when used for
the calculation (t ≠ 0).
The results of measuring PWV with a change in alcohol concentration and calculated PWV using MorganKiely equation, Moens-Korteweg equation and equation (2511) are shown in Figure 6. Figure 7 shows the
measured values of PWV and calculated PWV using Morgan-Kiely equation and equation (1125) for better
visibility and comparison.
4. Discussion
With age, the wall thickness of blood vessels increases. One of the most frequently observed is an increase
in the thickness of the intima-medium of the carotid artery (Homma et al. 2001; Tanaka et al. 2001; van
den Munckhof et al. 2012; Engelen et al. 2013; Dinenno et al. 2000; Green et al. 2010). The increase in
artery wall thickness was found to be 5m/year in both women and men (Homma et al. 2001; Tanaka et al.
2001; van den Munckhof et al. 2012; Engelen et al. 2013). Increasing the wall thickness leads to a decrease
in the lumen and inner radius of the blood vessel, so to compensate, the diameter of the artery gradually
15
Formatted: Justified
increases with age in both the central (van den Munckhof et al. 2012; Engelen et al. 2013; Dinenno et al.
2000; Green et al. 2010; Schmidt-Trucksass et al. 1999) and peripheral vessels (Green et al. 2010;
Sandgrenet al. 1998; van der Heijden 2000). An increase of 0.017 mm/year with atherosclerosis and 0.03
mm/year in those with pre-existing disease (Eigenbrodt et al. 2006) or 0.5% per year was observed, despite
maintaining a constant body surface area. Therefore, the elastic tube that most closely corresponds to the
ratio of the thickness of blood vessels of the radius of arteries in persons aged 40 and over was used in the
biophysical model, i.e. the ratio h/R0≈0.2. Also, medical PWV measurements are performed in middle-aged
and elderly people, when the structure of the blood vessel wall begins to change.
A comparison of the experimental results and the model shows that the results of the equation presented in
this paper, (eq. 2511), almost overlap. In addition to the measured values of PWV with pressure change in
Figure 5, the calculated values using other models and equations are also shown. If the Moens-Korteweg
equation (eq. 4) is used PWV has the same value for any initial fluid pressure. This can be seen from the
equation (eq. 4) because PWV does not depend on pressure. If the equation derived by Morgan and Kiely
(eq. A16 eq. 26 for t = 0 and p0 = 0) is used, the same PWV value is obtained again for all initial pressure
values, but the values will be less since the equation depends on the viscosity of the fluid than the values of
the Moens-Korteweg equation.
The slope of the measured PWV values is 0.00388 m/s•mmHg while the slope of the values from equation
(11) is 0.00342 m/s•mmHg. If t in equation (eq. 2511) is neglected (t = 0), then higher values of PWV are
obtained, and for the value of the initial transmural pressure of 0 mmHg it has the same value as the Morgan
and Kiely model (eq. A1626), so they start from the same point on the graph. The lower values obtained by
equation (11) compared to the experimental ones are most likely due to the increase in fluid temperature,
as a result of which the viscosity and density of the fluid changed.
Figure 6 shows the measured PWV values for fluids of different viscosities and densities with the same
initial pressure. The figure also shows the values obtained by the Morgan Kiely model (eq. A1626) and the
model shown in the paper (eq. 2511). If equation (eq. 4) is used (Figure 7), increasing values of PWV are
obtained, since only the density value changes in the equation. The measured values of PWV have the same
16
form of change as calculated by the Morgan Kiely model and with equation (11). The measured PWV
values at 60% alcohol are slightly lower compared to the models, which is most likely either due to alcohol
evaporation, so the viscosity value has changed or due to a change in fluid temperature. The values of
equation (11) and the measured values in this case also almost overlap, while the Morgan Kiely model has
higher PWV values and a larger range of values. In equation (11), the coefficient t was used, so the PWV
values are more similar to those measured compared to the Morgan Kiely model. And this proves, as with
measurements with a change in initial pressure, that the density of the tube wall plays an important role in
the propagation of waves. In this case, too, slightly lower PWV values were obtained by equation (11)
compared to the measured ones. One of the reasons is that the temperature affected the experimental setup.
These experimental results and the results obtained by the presented model in the paper show that when
analyzing the values of PWV measurements in medicine, blood viscosity, artery wall density and especially
diastolic arterial pressure should be taken into account. If the effect of fluid pressure on PWV were
neglected, then the age of the cardiovascular vessel wall in persons with hypertension would be incorrectly
estimated.If the same PWV value is measured in two patients, one of whom has hypertension, and uses the
Moens Korteweg equation for analysis, the same values of artery wall stiffness will be estimated, which is
incorrect according to the presented results. Also, in people whose blood viscosity is increased due to
diabetes, medications, cytostatic therapy and other conditions already mentioned, the medical diagnosis
will be errorness and may slow down therapy and recovery. Since the diameter of the artery narrows from
proximal to distal, the mean radius of the artery and the value of diastolic pressure when the person is lying
down should be used to calculate PWV.
5. Conclusion
The solution of Navier-Stokes equations for pulse propagation of waves through an elastic tube filled with
viscous fluid is presented. Derivation of equations for calculating the pulse wave velocities depending on
the viscosity, density and initial fluid pressure, density and elasticity of the pipe wall as well as the geometry
of the tube itself is presented. The results of the equation were compared with the measured experimental
17
values of pulse wave velocities on a biophysical model using fluids of different viscosities, densities and at
different values of initial pressure. The comparison shows that the model presented in this paper agrees
with the measured values in both cases: when the initial fluid pressure changes and when the viscosity and
density of the fluid change. Compared to other models used in medical research today, the presented model
uses all variables that affect the result. Using the equation for calculating the pulse wave velocities presented
in this paper, the age of the blood vessel wall will be more accurately estimated .Using the equation for
calculating pulse wave velocities presented in this paper, the age of the blood vessel wall will be more
accurately estimated, and therefore more precise therapy will be determined and the progress of the disease
or the effect of the therapy will be monitored.
Appendix
Equations of fluid motion
As already mentioned, fluid motion is governed by continuity equation (8) and Navier – Stokes equations
(9,10). We will assume that the fluid is incompressible and viscous and that the fluid propagates pulsatingly
through a cylindrical tube of thin elastic walls. We will also assume that the motion is axisymmetric without
tangential velocity 𝑣𝜃 = 0, as well as that the fluid is not affected by gravity, i.e. that the z axis is in a
horizontal position, so the last term in eq. 9 can be ignored. Next, we describe the movement of the tube
wall with two equations: longitudinal and radial
𝜌0 ℎ
𝜕2𝜂
𝐸ℎ
𝜂
𝜕𝜁
=𝑝−
( +𝜎 )
(1 − 𝜎 2 )𝑅0 𝑅0
𝜕𝑡 2
𝜕𝑧
(A1)
𝜌0 ℎ
𝜕2𝜁
𝐸ℎ
𝜕 2 𝜁 𝜎 𝜕𝜂
𝜕𝑣𝑧 𝜕𝑣𝑟
=
+
)
( 2+
)−𝜇(
2
2
(1 − 𝜎 ) 𝜕𝑧
𝜕𝑡
𝑅0 𝜕𝑧
𝜕𝑟
𝜕𝑧
(A2)
where 𝜂 and 𝜁 are the components of the displacement of the point on the tube wall in the axial and radial
directions, respectively, E – Young's modulus of elasticity, ρ0 –the density of the tube wall. The assumption
is that the pulse wave has a sinusoidal shape and that 𝜂, 𝜁, 𝑣𝑟 𝑎𝑛𝑑 𝑣𝑧 change in the same way. Let them be
18
𝑝 = 𝑃𝑒 𝑖(𝐾𝑧−𝜔𝑡) , 𝜂 = Ω𝑒 𝑖(𝐾𝑧−𝜔𝑡) , 𝜁 = Λ𝑒 𝑖(𝐾𝑧−𝜔𝑡) , 𝑣𝑧 = 𝑢(𝑟)𝑒 𝑖(𝐾𝑧−𝜔𝑡) 𝑖 𝑣𝑟 = 𝑤(𝑟)𝑒 𝑖(𝐾𝑧−𝜔𝑡)
(𝑤, 𝑢 – functions of 𝑟, 𝑃, Ω, Λ – constants). After substitution in equations (9), (10), (A1) and (A2) and
𝑖𝜔𝜌
𝜇
differentiation (
= 𝛼2,
𝜕 2 𝑣𝑟
𝜕𝑧 2
≈ 0,
𝜕2𝑢
𝜕𝑧 2
≈ 0 ) follows:
𝜕 2 𝑢 1 𝜕𝑢
𝑖𝐾
+
+ 𝛼2𝑢 = 𝑃
𝜕𝑟 2 𝑟 𝜕𝑟
𝜇
𝜕 2 𝑤 1 𝜕𝑤
1
1 𝜕𝑝
2
+
+ (𝛼 − 2 ) 𝑤 =
𝜕𝑟 2 𝑟 𝜕𝑟
𝑟
𝜇 𝜕𝑟
𝐸ℎ
Ω 𝑖𝐾𝜎
−𝜌0 ℎ𝜔2 Ω = 𝑃 −
Λ)
( +
(1 − 𝜎 2 ) 𝑅02 𝑅0
𝐸ℎ
𝑖𝐾𝜎
𝜕𝑢
−𝜌0 ℎ𝜔2 Λ =
(−𝐾 2 Λ +
Ω) − 𝜇 ( |𝑟=𝑅0 + 𝑖𝐾𝑤)
(1 − 𝜎 2 )
𝑅0
𝜕𝑟
(A3)
(A4)
(A5)
(A6)
Equations (A3) and (A4) are partial Bessel equations.
Assuming that P is in the form: 𝑃 = 𝐴1 ∙ 𝐽0 (𝑦𝑟), where 𝐴1 is a constant and a non-homogeneous solution
in the form 𝑢(𝑟) = 𝐵1 ∙ 𝐽0 (𝑦𝑟), by substituting in equation (A3) and differentiating, a particular solution is
obtained
𝑢(𝑟) =
𝑖𝐾𝐴1
1
∙ 𝐽 (𝑦𝑟)
𝜇 (𝛼 2 − 𝑦 2 ) 0
Assumption that a homogeneous solution in the form 𝑢(𝑟) = 𝐶1
𝐽0 (𝛼𝑟)
,
𝐽0 (𝛼)
where C1 is a constant to be
determined and in the final form:
𝑢(𝑟) = 𝐶1
𝐽0 (𝛼𝑟) 𝑖𝐾𝐴1
1
+
∙ 𝐽 (𝑦𝑟)
𝐽0 (𝛼)
𝜇 (𝛼 2 − 𝑦 2 ) 0
(A7)
Similar mathematical operations are repeated to solve equation (A4). Let non-homogeneous solution be in
the form 𝑤 = 𝐷1 ∙ 𝐽1 (𝑦𝑟), and by changing to equation (A4) and differentiating, a particular solution is
obtained
𝑤=−
𝑦𝐴1
1
∙ 𝐽 (𝑦𝑟)
𝜇 (𝛼 2 − 𝑦 2 ) 1
Here, too, the assumption is that the homogeneous solution is in the form 𝑤 = 𝐶2
𝐽1 (𝛼𝑟)
,
𝐽0 (𝛼)
where C2 is a
constant, so it is in the final form:
19
𝐽1 (𝛼𝑟) 𝑦𝐴1
1
−
∙ 𝐽 (𝑦𝑟)
𝐽0 (𝛼)
𝜇 (𝛼 2 − 𝑦 2 ) 1
From the equation of continuity (8) =>
(A8)
𝜕𝑤 𝑤
+ + 𝑖𝐾𝑢 = 0
𝜕𝑟 𝑟
By substituting equations (A7) and (A8) into equation (A9) and arranging it, we get
(A9)
𝑤(𝑟) = 𝐶2
− (𝑖𝐾𝐶1
𝐽0 (𝛼𝑟) 𝑖 2 𝐾 2 𝐴1
1
+
∙ 𝐽 (𝑦𝑟))
𝐽0 (𝛼)
𝜇 (𝛼 2 − 𝑦 2 ) 0
= 𝛼𝐶2
𝐽0 (𝛼𝑟) 𝑦 2 𝐴1
1
𝐶2 𝐽1 (𝛼𝑟) 1 𝑦𝐴1
1
−
∙ 𝐽 (𝑦𝑟) +
−
∙ 𝐽 (𝑦𝑟)
𝐽0 (𝛼)
𝜇 (𝛼 2 − 𝑦 2 ) 0
𝑟 𝐽0 (𝛼) 𝑟 𝜇 (𝛼 2 − 𝑦 2 ) 1
By solving we get
𝐶2 = −
𝑖𝐾
𝐶
𝛼 1
and after series expansion of Bessel coefficients:
𝑦𝑟 2
(2)
𝑦𝑟 4
(2)
𝐽0 (𝑦𝑟) = 1 −
+ 2 2 +⋯ ≈1
(12 )
(1 )(2 )
𝑦𝑟 1
𝑦𝑟 3
𝑦𝑟 5
(2)
(2)
(2)
𝑦𝑟
𝐽1 (𝑦𝑟) =
−
+
+⋯ ≈
1
2
12 ∙ 3
2
𝑖𝜔𝜌
𝑖𝜔𝜌
𝑖 2 𝜔2
As well as (𝛼 2 − 𝑦 2 ) = 𝜇 − 𝑖 2 𝐾 2 = 𝜇 − 𝑐 2 ≈ 𝛼 2
Due to the viscosity the velocities of the fluid along the inner wall of the vessel (when r = R0) are
𝑢|𝑟=𝑅0 = 𝐶1
𝐽0 (𝛼𝑟) 𝑖𝐾𝐴1
1
𝐽0 (𝛼𝑟) 𝐴1
+
∙ 𝐽 (𝑦𝑟) = 𝐶1
+
𝐽0 (𝛼)
𝜇 (𝛼 2 − 𝑦 2 ) 0
𝐽0 (𝛼) 𝜌𝑐
𝑤|𝑟=𝑅0 = 𝐶2
𝐽1 (𝛼𝑟) 𝑦𝐴1
1
𝑖𝐾
2 𝐽1 (𝛼𝑟) 𝐴1
−
∙ 𝐽 (𝑦𝑟) = − (𝐶1
+
∙𝑅 )
𝐽0 (𝛼)
𝜇 (𝛼 2 − 𝑦 2 ) 1
2
𝛼 𝐽0 (𝛼) 𝜌𝑐 0
and we have
𝑑𝑢
𝐽1 (𝛼𝑅0 ) 𝑖𝐾𝐴1
1
𝐽1 (𝛼𝑅0 ) 𝐴1 𝐾 2 𝑅0
|𝑟=𝑅0 = −𝐶1 𝛼
+
∙ (−𝑦)𝐽1 (𝑦𝑅0 ) = −𝐶1 𝛼
+
2
2
(𝛼)
(𝛼
)
𝑑𝑟
𝐽0
𝜇
−𝑦
𝐽0 (𝛼)
𝜌𝑐 2
𝐸ℎ
With substitutions: 𝐵 = (1−𝜎2 ) , 𝑃 = 𝐴1 ∙ 𝐽0 (𝑦𝑟) and 𝐽0 (𝑦𝑟) ≈ 1 in equation (A5) we have
1
𝐵
𝑖𝐾𝜎𝐵
𝐴 + (𝜔2 −
Λ=0
)Ω −
𝜌0 ℎ 1
𝜌0 𝑅0
𝜌0 𝑅02
(A10)
20
Since the velocity of the wall is equal to the velocity of the fluid along the wall of the vessel, equalizing the
equations
𝑢=
𝑑𝜁
𝐽0 (𝛼𝑟) 𝐴1 𝑖(𝐾𝑥−𝜔𝑡)
|𝑟=𝑅0 = −𝑖𝜔Λ𝑒 𝑖(𝐾𝑥−𝜔𝑡) = (𝐶1
+ )𝑒
𝑑𝑡
𝐽0 (𝛼) 𝜌𝑐
𝐴1
𝐽0 (𝛼𝑅0 )
+ 𝐶1
+ 𝑖𝜔Λ = 0
𝜌𝑐
𝐽0 (𝛼)
𝑑𝜂
𝑖𝐾
2 𝐽1 (𝛼𝑅0 ) 𝐴1
𝑤=
|
= −𝑖𝜔Ω𝑒 𝑖(𝐾𝑥−𝜔𝑡) = − (𝐶1
+
∙ 𝑅 ) 𝑒 𝑖(𝐾𝑥−𝜔𝑡)
𝑑𝑡 𝑟=𝑅0
2
𝛼 𝐽0 (𝛼)
𝜌𝑐 0
(A11)
𝑖𝐾𝑅0
𝑖𝐾 𝐽1 (𝛼𝑅0 )
𝐴 + 𝐶1
− 𝑖𝜔Ω = 0
2𝜌𝑐 1
𝛼 𝐽0 (𝛼)
And the last equation, with substitutions in (A6) =>
(A12)
−𝜔2 Λ = −𝐾 2
𝐵ℎ
𝑖𝐾𝜎 𝐵ℎ
𝜇
𝐽1 (𝛼𝑅0 ) 𝐴1 𝐾 2 𝑅0 𝑖 2 𝐾 2
2 𝐽1 (𝛼𝑅0 ) 𝐴1
Λ+
Ω−
+
−
+
∙ 𝑅 ))
(−𝐶1 𝛼
(𝐶1
𝜌0 ℎ
𝑅0 𝜌0 ℎ
𝜌0 ℎ
𝐽0 (𝛼)
𝜌𝑐 2
2
𝛼 𝐽0 (𝛼)
𝜌𝑐 0
𝜔2
𝑐2 ≪ 1 ,
𝛼
𝐾2
2 𝐽1 (𝛼𝑅0 ) 𝐴1
+
∙𝑅 )≪1
(𝐶1
2
𝛼 𝐽0 (𝛼)
𝜌𝑐 0
We obtain
−
𝜇
𝐾 2 𝑅0
𝜇𝛼 𝐽1 (𝛼𝑅0 )
𝐵ℎ
𝑖𝐾𝜎 𝐵ℎ
∙
𝐴1 +
𝐶 + (𝜔2 − 𝐾 2
)Λ +
Ω=0
𝜌0 ℎ𝜌𝑐
2
𝜌0 ℎ 𝐽0 (𝛼) 1
𝜌0 ℎ
𝑅0 𝜌0 ℎ
(A13 - 23)
The system of equations (A10), (A11), (A12) and (A13) is solved by setting the coefficients A1, C1, Λ and
Ω to be equal to 0:
1
𝜌𝑐
|
𝑖𝜔𝑅0
2𝜌𝑐 2
|
1
𝜌0 ℎ
|
𝜇
𝜔2 𝑅0
−
∙
3
𝜌0 ℎ𝜌𝑐
2
𝐽0 (𝛼𝑅0 )
𝐽0 (𝛼)
𝑖𝜔 𝐽1 (𝛼𝑅0 )
𝛼𝑐 𝐽0 (𝛼)
0
𝜇𝛼 𝐽1 (𝛼𝑅0 )
𝜌0 ℎ 𝐽0 (𝛼)
0
𝑖𝜔
−𝑖𝜔
0
𝐵
𝜌0 𝑅02
𝑖𝜔𝜎 𝐵
𝑅0 𝑐 𝜌0
𝜔2 −
|
𝑖𝜔𝜎𝐵 | = 0
𝜌0 𝑅0 𝑐
𝜔2 𝐵 |
𝜔2 − 2
𝑐 𝜌0
−
By simplifying this determinant, we obtain
21
|
1−
2 𝐽1 (𝛼𝑅0 )
𝛼𝑅0 𝐽0 (𝛼𝑅0 )
1
|
−
1 𝐽1 (𝛼𝑅0 )
𝛼𝑅0 𝐽0 (𝛼𝑅0 )
2
1
𝐵ℎ
𝜌𝑐 2 𝑅0
𝜎𝐵
− 2
𝜌𝑐 𝑅0
|
𝜎𝐵ℎ
=0
𝜌𝑐 2 𝑅0
|
𝜌0 ℎ
𝐵ℎ
−
𝜌𝑅0 𝜌𝑐 2 𝑅0
1+
Using substitutions
𝐸ℎ
𝜌0 ℎ
= 𝑥,
=𝑡
(A14)
2𝜌𝑐 2 𝑅0
𝜌𝑅0
and after solving the determinant, the so-called frequency equation in term of variable x was obtained
(4 − 2𝛼𝑅0
𝐽0 (𝛼𝑅0 ) 2
𝛼𝑅0 𝐽0 (𝛼𝑅0 )
𝐽0 (𝛼𝑅0 )
− 1) − 4𝜎 + 1 + 2𝛼𝑅0
]𝑥
) 𝑥 + [2𝑡 (
𝐽1 (𝛼𝑅0 )
2 𝐽1 (𝛼𝑅0 )
𝐽1 (𝛼𝑅0 )
𝛼𝑅0 𝐽0 (𝛼𝑅0 )
− 2𝑡(1 − 𝜎 2 )
− (1 − 𝜎 2 ) = 0
2 𝐽1 (𝛼𝑅0 )
Using the values for viscosity of blood, diameter of aorta or main arteries, blood density and density of
artery wall, we obtained that |𝛼𝑅0 | ≫ 1, ie that they are in the ranges of 𝛼𝑅0 = 3 − 12 and 𝑡 = 0.1 − 0.4.
After asymptotic expansion of the Bessel function
𝐽0 (𝛼𝑅0 )
1
~−𝑖+
𝐽1 (𝛼𝑅0 )
2𝛼𝑅0
and by substituting in the frequency equation and arranging, the quadratic equation is obtained
(−2𝛼𝑅0 𝑖 − 3)𝑥 2 + [4𝜎 − 2 + 2𝛼𝑅0 𝑖 + 𝑡𝛼𝑅0 𝑖 +
3𝑡
𝑡
] 𝑥 + (1 − 𝜎 2 ) (1 + − 𝑡𝛼𝑅0 𝑖) = 0
2
2
1
This quadratic equation can be solved for 𝜎 = 0, 𝜎 = 2 and 𝜎 = 1. For an artery 𝜎 = 0.49, so the roots of
1
2
quadratic equation for 𝜎 = , are:
𝑡
𝑖
𝜎 2 3𝑡
𝑖(1 − 𝜎 2 )
𝑥1 = 1 + +
− ) and 𝑥2 =
(2 − 2𝜎 +
2 𝛼𝑅0
2
4
2𝛼𝑅0
Substituting in (24)
𝐾 2 𝐸ℎ
𝐾2
= 2 𝑐02 = 𝑥
2
𝜔 2𝜌𝑅0 𝜔
We look for solutions in the following form and only for x1 (for 𝜇 → 0, 𝑥2 → ∞)
𝐾
𝐾1 + 𝑖𝐾2
𝑐 =
𝑐0 = √𝑥1
𝜔 0
𝜔
22
After substitution √𝑖 =
√2
(1
2
+ 𝑖) and solving the root of a complex number
𝜎 2 3𝑡
𝜎 2 3𝑡
𝑡 2 (1 − 𝜎 + 4 − 8 ) 2𝑖 (1 − 𝜎 + 4 − 8 ) 𝐾1 + 𝑖𝐾2
+
=
𝑐0
√𝑥1 = √1 + +
2
𝜔
2𝜔𝜌
2𝜔𝜌
√
√
𝑅
𝑅
𝜇 0
𝜇 0
𝜎 2 3𝑡
𝐾1
𝑡 (1 − 𝜎 + 4 − 8 )
𝑐 =1+ +
,
𝜔 0
4
2𝜔𝜌
√
𝑅
0
𝜇
𝜎 2 3𝑡
(1 − 𝜎 + 4 − 8 )
2𝜔𝜌
√
𝐾2
𝜇 𝑅0
𝑐0 =
𝜎 2 3𝑡
𝜔
𝑡 (1 − 𝜎 + 4 − 8 )
1+4+
2𝜔𝜌
√
𝑅
𝜇 0
By substituting c0 with equation (7) which is a function of the initial pressure and arranging, we obtain the
final equation (A15) for the pulse wave velocity
𝐸ℎ
𝑝
√
+ 0
𝜔
1
𝑡
1
𝜇
𝜎 2 3𝑡
2𝑅0 𝜌 4𝜌
𝑐=
=±
− ))
(1 + − √
(1 − 𝜎 +
2
𝑝 𝑅
𝐾1
4 𝑅0 2𝜔𝜌
4
8
1 − 0 0 (1 + 𝑡 )
𝐸ℎ
4
And
(A15)
2
𝑝0 𝑅0 (1 − 𝜎 + 𝜎 − 3𝑡)
4
8
𝐸ℎ
𝐾2 = ±𝜔
𝐸ℎ
𝑝0
𝑡
2𝜔𝜌
√
+
+ √
𝑅
2𝑅0 𝜌 4𝜌 (1 4)
𝜇 0
1−
If we assume that t is very small it can be neglected (𝑡 = 0) 𝑎𝑛𝑑 the initial fluid is zero (𝑝0 = 0) we get
the same equation derived by Morgan and Kiely for small viscosity
𝑐=
𝜔
𝐸ℎ
1
𝜇
𝜎2
= ±√
(1 + − √
(1 − 𝜎 + ))
𝐾1
2𝑅0 𝜌
𝑅0 2𝜔𝜌
4
(A16)
And if we neglect both σ and μ we get the Moens-Korteweg equation (4) for PWV.
In case of high viscosity, or |𝛼𝑅0 | ≪ 1, we use serial expansions
𝐽0 (𝛼𝑅0 ) ≈ 1 −
𝛼 2 𝑅02
,
4
𝐽1 (𝛼𝑅0 ) ≈
𝛼𝑅0
𝛼 2 𝑅02
(1 −
)
2
8
After substituting in the frequency equation and solving the quadratic equations we obtain
23
𝑐=
𝜔
=
𝐾1
𝑐0
5 − 4𝜎 + 6𝑡
√ 𝑅2 𝜔𝜌
0 𝜇
= ±√
𝐸ℎ
𝑝0
𝜔𝜌
1
+
𝑅 √
2𝑅0 𝜌 4𝜌 0 𝜇 5 − 4𝜎 + 6𝑡
(A17)
and
𝐾2 = ±𝜔√
1
𝜇 √5 − 4𝜎 + 6𝑡
𝐸ℎ
𝑝0 √𝜔𝜌
𝑅0
2𝑅0 𝜌 + 4𝜌
Conflict of interest statement
No conflicts of interest, financial or otherwise, are declared by the authors.
Acknowledgments
This work was supported by Serbian Ministry of Education, Science and Technological Development
Grants 32040 and 41022.
Fig. 1. Increase in disk radius and width due to pulse wave: du/dz - the narrowing of the disk, u - disk
shift, ∆𝑅0 - radius change
Fig. 2. The propagation of the pulse wave increases the pressure on one side of the disk by ∆p
Fig. 3. Stresses in a circular cylindrical pressure vessel a - hoop stress, b - longitudinal stress
Fig. 4. A schematic diagram of the biophysical model of the cardiovascular system: H– automatic
hammer, P–pump, reservoir 1 (closed), reservoir 2 (adjustable height), M–manometer, S1, S2 –
pressure sensors mounted through the wall of the elastic tube, DAQ—data acquisition device, V –
one-way valve. The arrows indicate the flow direction of fluid through the valves and the tubes
Fig. 5. Comparison of models and measured values of PWV against initial fluid pressure: MoensKorteweg equation, Morgan Kiely equation and equation (2511) for t = 0 and t ≠ 0. 
Fig. 6. PWV against alcohol concentration
Fig. 7. Comparison of Morgan-Kiely equation, equation (2511) and measured values of PWV against
alcohol concentration
24
References
Atabek HB (1968) Wave Propagation through a Viscous Fluid Contained in a Tethered, Initially Stressed,
Orthotropic Elastic Tube. Biophys J. May; 8(5): 626–649.
Atabek HB, Lew HS (1966) Wave Propagation through a Viscous Incompressible Fluid Contained in an
Initially Stressed Elastic Tube. Biophys J. Jul; 6(4): 481–503.
Demiray H, Akgün G (1997) Wave propagation in a viscous fluid contained in a prestressed viscoelastic
thin tube. International Journal of Engineering Science, 35(10–11): 1065-1079
Dinenno FA, Jones PP, Seals DR & Tanaka H (2000) Age-associated arterial wall thickening is related to
elevations in sympathetic activity in healthy humans. Am J Physiol Heart Circ Physiol 278, H1205–H1210.
Eigenbrodt ML, Bursac Z, Rose KM, Couper DJ, Tracy RE, Evans GW, Brancati FL & Mehta JL (2006)
Common carotid arterial interadventitial distance (diameter) as an indicator of the damaging effects of age
and atherosclerosis, a cross-sectional study of the Atherosclerosis Risk in Community Cohort Limited
Access Data (ARICLAD), 1987–89. Cardiovasc Ultrasound 4, 1.
Engelen L, Ferreira I, Stehouwer CD, Boutouyrie P, Laurent S & Reference Values for Arterial
Measurements C (2013) Reference intervals for common carotid intima-media thickness measured with
echotracking: relation with risk factors. Eur Heart J 34, 2368–2380.
25
Feng J, Khir A (2010) Determination of wave speed and wave separation in the arteries using diameter and
velocity. Journal of Biomechanics. 43,455–462
Green DJ, Swart A, Exterkate A, Naylor LH, Black MA, Cable NT & Thijssen DH (2010) Impact of age,
sex and exercise on brachial and popliteal artery remodelling in humans. Atherosclerosis 210, 525–530.
Heijden van der Spek JJ, Staessen JA, Fagard RH, Hoeks AP, 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.
Hypertension 35, 637–642.
Homma S, Hirose N, Ishida H, Ishii T & Araki G (2001) Carotid plaque and intima-media thickness
assessed by b-mode ultrasonography in subjects ranging from young adults to centenarians. Stroke 32, 830–
835.
Khir A, Parker K (2002) Measurements of wave speed and reflected waves in elastic tubes and bifurcations.
Journal of Biomechanics. 35,775–783
Khir AW, Swalen MJP, Parker KH (2007) The simultaneous determination of wave speed and the arrival
time of reflected waves in arteries. Med Biol Eng Comput 45,1201–1210.
Korteweg D J (1878) Ueber die Fortpflanzungsgeschwindigkeit des Schalles in elastischen Röhren.
https://doi.org/10.1002/andp.18782411206
Lazović B, Mazić S, Zikich D, Žikić D (2015) The mathematical model of the radial artery blood pressure
waveform through monitoring of the age-related changes. Wave Motion 56:14–21
Li Y, Ashraf W, Khir A (2011) Experimental validation of non-invasive and fluid density independent
methods for the determination of local wave speed and arrival time of reflected wave. Journal of
Biomechanics. 44,1393–1399
Moens A I (1877) Over de voortplantingssnelheid van den pols (Ph.D. thesis). Leiden: S.C. Van Doesburgh.
Moens A I (1878) Die Pulskurve. Leiden: E.J. Brill
Morgan GW, Kiely JP (1954) Wave Propagation in a Viscous Liquid Contained in a Flexible Tube. The
Journal of the Acoustical Society of America 26, 323
Munckhof van den I, Scholten R, Cable NT, Hopman MT, Green DJ & Thijssen DH (2012) Impact of age
and sex on carotid and peripheral arterial wall thickness in humans. Acta Physiol (Oxf) 206, 220–228.
Papageorgiou Gl, Jones NB (1987a) Physical modeling of the arterial wall. Part 1: testing of tubes of various
materials. J Biomed Eng 9,153- 156.
Papageorgiou Gl, Jones NB (1987b) Physical modeling of the arterial wall. Part 2: Simulation of the nonlinear elasticity of the arterial wall. J Biomed Eng.9, 216–221.
Sandgren T, Sonesson B, Ahlgren AR & Lanne T (1998) Factors predicting the diameter of the popliteal
artery in healthy humans. J Vasc Surg 28, 284–289.
Sandgren T, Sonesson B, Ahlgren R & Lanne T (1999) The diameter of the common femoral artery in
healthy human: influence of sex, age, and body size. J Vasc Surg 29, 503–510.
26
Schmidt-Trucksass A, Grathwohl D, Schmid A, Boragk R, Upmeier C, Keul J & Huonker M (1999)
Structural, functional, and hemodynamic changes of the common carotid artery with age in male subjects.
Arterioscler Thromb Vasc Biol 19, 1091–1097
Stojadinovic B, Tenne T, Zikich D, Rajković N, Milošević N, Lazović B, Žikić D (2015) Effect of viscosity
on the wave propagation. Journal of Biomechanics. 48(15): 3969-3974
Tanaka H, Dinenno FA,Monahan KD, DeSouza CA & Seals DR (2001) Carotid artery wall hypertrophy
with age is related to local systolic blood pressure in healthy men. Arterioscler Thromb Vasc Biol 21, 82–
87
Westerhof N, Elzinga G, Sipkema P (1972) Forward and backward waves in the arterial system. Cardiovasc
Res 6,648–656.
Womersley R (1955) Method for the calculation of velocity, rate of flow and viscous drag in arteries when
the pressure gradient is known. J Physiol. Mar 28; 127(3): 553–563.
Womersley R (1957) An elastic tube theory of pulse transmission and oscillatory flow in mammalian
arteries. Wright Air Development Center (Technical Report WADC-TR)
Young T (1808) “Hydraulic investigations, subservient to an intended Croonian lecture on the motion of
the blood.” Philosophical Transactions of the Royal Society of London 98, 164-186 [Errata in Ref (26), p.
31].
Žikić D, Stojadinović B and Nestorović Z (2019) Biophysical modeling of wave propagation phenomena:
experimental determination of pulse wave velocity in viscous fluid-filled elastic tubes in a gravitation field
Eur Biophys J 48:407-411
27
Manuscript (Clean)
Wave propagation through viscous fluid-filled elastic tube under initial
pressure: theoretical and biophysical model
Dejan Žikić1 & Katarina Žikić2
1
Institute of Biophysics, Faculty of Medicine, University of Belgrade, Serbia
2
Faculty of Physics, University of Belgrade, Serbia
Corresponding author:
Dejan Žikić, Associate Professor
Institute of Biophysics, Faculty of Medicine
University of Belgrade, 11000 Belgrade, Serbia
https://orcid.org/0000-0003-3247-9071
Tel:
+381 11 360 7159
Fax:
+381 11 360 7061
e-mail: dzikic@gmail.com
1
Abstract
The velocity of propagation of pulse waves through the arteries is one of the indicators of the health
condition of the cardiovascular system. By measuring the pulse wave velocity, cardiologists estimate the
elasticity of the wall of blood vessels and the changes that occur with aging. Using the Moens-Korteweg
equation, an erroneous assessment is made in the analysis. The paper presents the solution of Navier-Stokes
equations for propagation of pulse waves through an elastic tube filled with viscous fluid under initial
pressure. The equation for pulse wave velocity depending on viscosity, density and initial fluid pressure,
density and elasticity of wall and geometry of the tube is derived. The results of the equation were compared
with the experimental results measured on a biophysical model of the cardiovascular system.
Keywords: Navier-Stokes equations, pulse wave velocity, arterial wall elasticity, viscosity, biophysical
model
2
1. Introduction
Investigation of the waveforms of arterial blood pressure and flow through blood vessels plays an important
role in understanding the nature and disorders of the cardiovascular system. The beginning of the interest
of scientists in the wave nature of blood dates to the beginning of the 19th century when Young (1808)
published a paper on the wave motion of blood and derived the first formula for pulse wave velocity. In
1877, Moens and Korteweg simultaneously derived the equation for the velocity of a pulse wave
propagating through a fluid-filled elastic tube of thin walls (Moens 1877, Moens 1878, Korteweg 1878).
This equation is still used today in medicine to study the age of the cardiovascular system. The greatest
mathematical contribution was made by Womersley in the middle of the last century (Womersley 1955)
who solved the pulse wave velocity profile in a rigid tube filled with a viscous fluid. In addition, he offered
a solution for the pulse wave velocity profile through an elastic tube and developed a mathematical analysis
of blood flow in the arteries. In his work, he included viscosity in the pulse wave velocity calculation
(Womersley 1957).
To better understand the physics of cardiovascular processes, mathematical and experimental biophysical
models were mainly used. In mathematical models, the equations describe the principles of the physical
laws of a process, then the theoretical model is experimentally confirmed on the biophysical model.
Khir and Parker (2002) and later Li (Li et al. 2011), used the water-hammer equation to determine pulse
wave velocity. Westerhof et al. (1972) determined pulse wave velocity from waveforms of fluid pressure
and flow. Feng and Khir (2010) applied both equations to detect reflected waves in flow pressure signals.
In his work, Khir used a pressure-velocity loop to determine the arrival time of reflected waves (Khir et al.
2007). Lazovic et al. (2015), detected reflected waves that appear on the radial artery with aging.
Papageorgiou and Jones (1987a) investigated the various materials used in models for arteries and blood
vessels. They found that if elastic tubes with a nonlinear stress / strain ratio are used in the models, the
differences between real systems and physical models are minimal. The influence of viscosity on pulse
wave velocity was experimentally shown by Stojadinović et al. (2015). The influence of gravity on wave
propagation was experimentally shown by Žikić et al. (2019).
3
One of the parameters used in medicine to determine the age of the cardiovascular system is the pulse wave
velocity - PWV. The Moens-Korteweg equation is mainly used to estimate arterial elasticity from measured
PWV values. As this equation does not depend on blood viscosity, diastolic pressure, blood vessel wall
density as well as the influence of gravity, using this relation in medical research introduces an error in the
interpretation of results and can be misdiagnosed.
Morgan and Kiely (1954) offered a solution for calculating PWV through a viscous fluid in an elastic tube
but without initial pressure and without pipe wall density. Womersley (1955, 1957) also solved this problem
with viscosity but offered a solution in a complex equation that only physicists can solve. Also, this equation
cannot be applied if the fluid is under initial pressure and the wall density of the elastic tube is neglected.
Other scientists have also investigated how PWV changes with initial fluid pressure (Atabek et al. 1966;
Atabek 1968; Demiray and Akgün 1997) but the offered equations are very complicated especially for
medical doctors and the results of their equations do not agree with the experimental results.
One of the reasons for the interest in this phenomenon of wave propagation is related to research in
hemodynamics. The main area that cardiovascular physics deals with is the wave propagation of blood
through blood vessels. With aging, the biophysical parameters of the blood and blood vessels change, and
the correct interpretation of cardiovascular parameters can greatly contribute to accurate diagnosis and
successful therapy.
The aim of this paper is to show the mathematical derivation of the equation for pulse waves velocity
through a viscous fluid-filled elastic tube with initial fluid pressure and to prove experimentally the
accuracy of the equation.
2. Methods
2.1 Mathematical equations
Moens-Korteweg equation
Moens and Korteweg ingeniously derived the equation for PWV. They assumed that the pulse wave
increases the initial volume of the cylinder 𝑉𝑜 = 𝜋𝑅𝑜2 ∆𝑧 (Figure 1) of width ∆z to 𝑉 = 𝜋(𝑅0 +
𝜕𝑢
∆𝑅0 )2 (∆𝑧 + 𝜕𝑧 ∆𝑧) and derived three equations:
4
𝜕𝑢
2
2
∆𝑉 𝜋(𝑅0 + ∆𝑅0 ) (∆𝑧 + 𝜕𝑧 ∆𝑧) − 𝜋𝑅𝑜 ∆𝑧
=
𝑉𝑜
𝜋𝑅𝑜2 ∆𝑧
2
2
2∆𝑅0
∆𝑅0
𝜕𝑢 2∆𝑅0 𝜕𝑢
∆𝑅0 𝜕𝑢 2∆𝑅0 𝜕𝑢
∆𝑝
=
+( 2 ) +
+
+( 2 )
=
+
=−
𝑅0
𝜕𝑧
𝑅0 𝜕𝑧
𝜕𝑧
𝑅0
𝜕𝑧
𝐾
𝑅𝑜
𝑅𝑜
2 2∆𝑅 𝜕𝑢 ∆𝑅 2 𝜕𝑢
0
, ( 𝑅20 ) 𝜕𝑧
𝑅0 𝜕𝑧
𝑜
𝑜
∆𝑅
where ∆𝑉 is the change in volume, K- bulk modulus of fluid, ( 𝑅20 ) ,
(1)
≈0
The difference in pressures on the front and back of the disk (Figure 2) can be expressed from the
pressure gradient, and the axial acceleration of the fluid is:
𝑚𝑎 = −
𝜕(∆𝑝)
𝜕2𝑢
𝑙𝐴 = 𝜌𝑙𝐴 2
𝜕𝑧
𝜕𝑡
or
𝜕(∆𝑝)
𝜕2𝑢
= −𝜌 2
𝜕𝑧
𝜕𝑡
(2)
Due to the increase in pressure ∆p in the fluid, the force F1 tends to separate the two half-cylinders
(Figure 3a). On the other hand, this force is opposed by hoop stress 𝜎𝜃
𝐹1 = ∆𝑝𝑆1 = 2∆𝑝(𝑅0 + ∆𝑅0 )𝑙 = 2𝑆2 𝜎𝜃 = 2ℎ𝑙𝐸
2𝜋∆𝑅0
∆𝑅0
= 2𝐸
ℎ𝑙
2𝜋𝑅0
𝑅0
Arranging this equation gives the third Moens-Korteweg equation
∆𝑝 = 𝐸ℎ
∆𝑅0
∆𝑅0
~𝐸ℎ 2
(𝑅0 + ∆𝑅0 )𝑅0
𝑅𝑜
(3)
After differentiating the second and third equations and arranging using the first equation, a wave
equation is obtained
𝜕2𝑢
2𝜌𝑅0 𝜌 𝜕 2 𝑢
=
(
+ ) 2
𝜕𝑧 2
𝐸ℎ
𝐾 𝜕𝑡
For an elastic tube and an incompressible fluid ( 𝐾 → ∞), the equation becomes the Moens-Korteweg
equation
𝜕 2 𝑢 2𝜌𝑅0 𝜕 2 𝑢
1 𝜕2𝑢
=
=
𝜕𝑧 2
𝐸ℎ 𝜕𝑡 2 𝑐02 𝜕𝑡 2
𝐸ℎ
𝑐0 = √
2𝜌𝑅0
(4)
5
Where c0 is the pulse wave velocity – PWV. As can be seen from the equation, viscosity, tube wall density
and initial fluid pressure do not affect the PWV.
Bramwell and Hill derived the equation for PWV without values for E, h, and R0 (since they are not constant
and depend on the artery itself). If we replace ∆R0 in equation (1) from equation (3) and assume that ∂u/∂z
is small it follows:
∆𝑉 2∆𝑅0
2 𝑅0 2 ∆𝑝 2𝑅0 ∆𝑝
=
=
=
𝑉𝑜
𝑅0
𝑅0 𝐸ℎ
𝐸ℎ
The rearrangement resulted in:
𝐸ℎ
𝑉𝑜 ∆𝑝
=
2𝑅0
∆𝑉
And by substituting in equation (4), the Bramwell-Hill equation is obtained:
𝑉𝑜 ∆𝑝
𝑐0 = √
𝜌 ∆𝑉
(5)
2.2 Pulse wave velocity when the fluid is under initial pressure
Let the initial pressure in the tube be p0. Due to the increase in pressure, the radius also increases:
R=R0+R and the length of the tube l=l0+l.
In the derivation of equation (3) it is shown that the hoop stress is equal
𝑝0 𝑅0
2𝜋∆𝑅0
∆𝑅0
𝑅 − 𝑅0
= 2ℎ𝑙𝐸
= 2𝐸
ℎ𝑙 = 2𝐸
ℎ𝑙
ℎ
2𝜋𝑅0
𝑅0
𝑅0
𝜎𝜃 =
By arranging, the final radius with the initial pressure in the tube is
𝑅=
𝑅0
𝑝 𝑅
1− 0 0
𝐸ℎ
The same pressure acts on the cross section of the cylinder (Figure 3b) and is opposed by longitudinal
stress
𝐹2 = 𝑝0 𝜋𝑅02 = 𝜋[(𝑅0 + ℎ)2 − 𝑅02 ]𝜎𝑙 ~2𝜋𝑅0 ℎ𝜎𝑙
By arranging, the hoop stress is 2 times higher than the longitudinal stress
𝜎𝑙 =
𝑝0 𝑅0 𝜎𝜃
=
2ℎ
2
6
After substitution
𝐸
∆𝑅0
∆𝑙
= 2𝐸
𝑅0
𝑙0
𝑅 − 𝑅0
𝑙 − 𝑙0
=2
𝑅0
𝑙0
𝑙=
𝑙0
1
𝑝0 𝑅0
(
+ 1)~𝑙0 (1 +
)
2 1 − 𝑝0 𝑅0
2𝐸ℎ
𝐸ℎ
Due to the increase in fluid pressure in the tube by p0, the disk volume is:
𝑉 = 𝑅 2 𝜋𝑙
Or
2
𝑝0 𝑅0
𝑝0 𝑅0
∆𝑝
𝑅
𝑝0 𝑅0 ∆𝑝
∆𝑝 (1 + 2𝐸ℎ )
∆𝑝 (1 + 2𝐸ℎ )
0
2
2
𝑉
= 𝑅 𝜋𝑙 = (
) 𝜋𝑙0 (1 +
)
= 𝜋𝑙0 𝑅0
2 = 𝑉0 ∆𝑉
𝑝0 𝑅0
∆𝑉
2𝐸ℎ
∆𝑉
∆𝑉
𝑝
𝑅
𝑝0 𝑅0 2
0
0
(1 −
)
(1
−
)
(1
−
)
𝐸ℎ
𝐸ℎ
𝐸ℎ
𝑝0 𝑅0
∆𝑝
𝐸ℎ 1 + 2𝐸ℎ
𝑉
=
∆𝑉 2𝑅0
𝑝 𝑅 2
(1 − 0 0 )
𝐸ℎ
(6)
Using equation (6) and Bramwell-Hill equation (5), the PWV can be calculated with the initial fluid
pressure in the tube:
𝐸ℎ
𝑝0 𝑅0
𝐸ℎ
𝑝0
√
√
2𝑅0 𝜌 (1 + 2𝐸ℎ )
2𝑅0 𝜌 + 4𝜌
𝑐=
=
𝑝 𝑅
𝑝 𝑅
1− 0 0
1− 0 0
𝐸ℎ
𝐸ℎ
(7)
2.3 Wave propagation through an elastic tube filled with a viscous fluid
Fluid motion is governed by continuity equation (8) and Navier–Stokes equations (9,10):
𝑑𝜌
+ ∇(ρv) = 0
𝑑𝑡
𝜕𝑣𝑧
𝜕𝑣𝑧
𝜕𝑣𝑧
𝜕𝑝
1 𝜕
𝜕𝑣𝑧
𝜕 2 𝑣𝑧
𝜌(
+ 𝑣𝑟
+ 𝑣𝑧
)=−
+𝜇(
(𝑟
)+
) + 𝜌𝑔𝑧
𝜕𝑡
𝜕𝑟
𝜕𝑧
𝜕𝑧
𝑟 𝜕𝑟
𝜕𝑟
𝜕𝑧 2
𝜕𝑣𝑟
𝜕𝑣𝑟
𝜕𝑣𝑟
𝜕𝑝
1 𝜕
𝜕𝑣𝑟
𝜕 2 𝑣𝑟 𝑣𝑟
𝜌(
+ 𝑣𝑟
+ 𝑣𝑧
)=−
+𝜇(
(𝑟
)+
− )
𝜕𝑡
𝜕𝑟
𝜕𝑧
𝜕𝑟
𝑟 𝜕𝑟
𝜕𝑟
𝜕𝑧 2 𝑟 2
(8)
(9)
(10)
7
where  is density, 𝑣𝑧 is the axial velocity of the fluid, 𝑣𝑟 is the radial velocity of the fluid,  is the
dynamic viscosity. By solving these equations (Appendix) we obtain pulse wave velocities for the case of
small viscosity and high viscosity:
𝐸ℎ
𝑝0
√
1
𝑡
1
𝜇
𝜎 2 3𝑡
2𝑅0 𝜌 + 4𝜌
𝑐=
+
−
−
𝜎
+
− ))
(1
(1
√
2
𝑝 𝑅
4 𝑅0 2𝜔𝜌
4
8
1 − 0 0 (1 + 𝑡 )
𝐸ℎ
4
And for high viscosity
𝑐=
𝑐0
𝐸ℎ
𝑝0
𝜔𝜌
1
= ±√
+
𝑅0 √
2𝑅0 𝜌 4𝜌
𝜇 5 − 4𝜎 + 6𝑡
5 − 4𝜎 + 6𝑡
√ 𝑅 2 𝜔𝜌
0 𝜇
(11)
(12)
The last equation (eq.12) has no application for medical research due to its high viscosity values, so it will
not be discussed further in the paper.
2.4 Biophysical model
A biophysical model of the cardiovascular system was used for the experimental validation of the equation
(Figure 4). A similar model was used in Stojadinović et al. (2015) and Žikić et al. (2019). In the first
experiment, pulse wave velocities were measured with increasing initial fluid pressure. The transmural
pressure (difference in pressure between two sides of a wall of a tube) was changed from 0 mmHg to 120
mmHg with a step of 20 mmHg. A silicone transparent isotropic tube with an inner diameter of 10 mm and
a wall thickness of 1 mm with a modulus of elasticity E = 5.19 MPa was used. The fluid was distilled water
with a density of 998 kg/m3 and a viscosity of 0.99 mPa·s at 23 °C. PWV was determined from the equation
d/t where d is the distance between the sensors (Figure 4) and t is the propagation time of the pulse wave
between the sensors. In all experiments, the distance d between the sensors was 1m. In Stojadinović et al.
(2015) and Žikić et al. (2019) is a detailed explanation of the model and the determination of PWV.
In another experiment, pulse wave velocities were measured with fluids of different densities and
viscosities. The fluid pressure was 0 mmHg. Aqueous alcohol solutions with a concentration of 10-60%
were used. Silicone tube of the same dimensions and properties as in the first experiment was used.
8
3.
Results
Table 1 shows the densities and viscosities of the fluids used in the experiments.
The results of PWV measurements with increasing fluid pressure are shown in Figure 5. The figure also
shows the calculated PWV values using the Moens-Korteweg equation, the Morgan-Kiely equation, and
the model using equation (11) in both cases: when the parameter is neglected (t = 0) and when used for the
calculation (t ≠ 0).
The results of measuring PWV with a change in alcohol concentration and calculated PWV using MorganKiely equation, Moens-Korteweg equation and equation (11) are shown in Figure 6. Figure 7 shows the
measured values of PWV and calculated PWV using Morgan-Kiely equation and equation (11) for better
visibility and comparison.
4. Discussion
With age, the wall thickness of blood vessels increases. One of the most frequently observed is an increase
in the thickness of the intima-medium of the carotid artery (Homma et al. 2001; Tanaka et al. 2001; van
den Munckhof et al. 2012; Engelen et al. 2013; Dinenno et al. 2000; Green et al. 2010). The increase in
artery wall thickness was found to be 5m/year in both women and men (Homma et al. 2001; Tanaka et al.
2001; van den Munckhof et al. 2012; Engelen et al. 2013). Increasing the wall thickness leads to a decrease
in the lumen and inner radius of the blood vessel, so to compensate, the diameter of the artery gradually
increases with age in both the central (van den Munckhof et al. 2012; Engelen et al. 2013; Dinenno et al.
2000; Green et al. 2010; Schmidt-Trucksass et al. 1999) and peripheral vessels (Green et al. 2010;
Sandgrenet al. 1998; van der Heijden 2000). An increase of 0.017 mm/year with atherosclerosis and 0.03
mm/year in those with pre-existing disease (Eigenbrodt et al. 2006) or 0.5% per year was observed, despite
maintaining a constant body surface area. Therefore, the elastic tube that most closely corresponds to the
ratio of the thickness of blood vessels of the radius of arteries in persons aged 40 and over was used in the
9
biophysical model, i.e. the ratio h/R0≈0.2. Also, medical PWV measurements are performed in middle-aged
and elderly people, when the structure of the blood vessel wall begins to change.
A comparison of the experimental results and the model shows that the results of the equation presented in
this paper, (eq. 11), almost overlap. In addition to the measured values of PWV with pressure change in
Figure 5, the calculated values using other models and equations are also shown. If the Moens-Korteweg
equation (eq. 4) is used PWV has the same value for any initial fluid pressure. This can be seen from the
equation (eq. 4) because PWV does not depend on pressure. If the equation derived by Morgan and Kiely
(eq. A16 for t = 0 and p0 = 0) is used, the same PWV value is obtained again for all initial pressure values,
but the values will be less since the equation depends on the viscosity of the fluid than the values of the
Moens-Korteweg equation.
The slope of the measured PWV values is 0.00388 m/s•mmHg while the slope of the values from equation
(11) is 0.00342 m/s•mmHg. If t in equation (11) is neglected (t = 0), then higher values of PWV are obtained,
and for the value of the initial transmural pressure of 0 mmHg it has the same value as the Morgan and
Kiely model (eq. A16), so they start from the same point on the graph. The lower values obtained by
equation (11) compared to the experimental ones are most likely due to the increase in fluid temperature,
as a result of which the viscosity and density of the fluid changed.
Figure 6 shows the measured PWV values for fluids of different viscosities and densities with the same
initial pressure. The figure also shows the values obtained by the Morgan Kiely model (eq. A16) and the
model shown in the paper (eq. 11). If equation (eq. 4) is used (Figure 7), increasing values of PWV are
obtained, since only the density value changes in the equation. The measured values of PWV have the same
form of change as calculated by the Morgan Kiely model and with equation (11). The measured PWV
values at 60% alcohol are slightly lower compared to the models, which is most likely either due to alcohol
evaporation, so the viscosity value has changed or due to a change in fluid temperature. The values of
equation (11) and the measured values in this case also almost overlap, while the Morgan Kiely model has
higher PWV values and a larger range of values. In equation (11), the coefficient t was used, so the PWV
values are more similar to those measured compared to the Morgan Kiely model. And this proves, as with
10
measurements with a change in initial pressure, that the density of the tube wall plays an important role in
the propagation of waves. In this case, too, slightly lower PWV values were obtained by equation (11)
compared to the measured ones. One of the reasons is that the temperature affected the experimental setup.
These experimental results and the results obtained by the presented model in the paper show that when
analyzing the values of PWV measurements in medicine, blood viscosity, artery wall density and especially
diastolic arterial pressure should be taken into account. If the effect of fluid pressure on PWV were
neglected, then the age of the cardiovascular vessel wall in persons with hypertension would be incorrectly
estimated. Also, in people whose blood viscosity is increased due to diabetes, medications, cytostatic
therapy and other conditions already mentioned, the medical diagnosis will be errorness and may slow
down therapy and recovery. Since the diameter of the artery narrows from proximal to distal, the mean
radius of the artery and the value of diastolic pressure when the person is lying down should be used to
calculate PWV.
5. Conclusion
The solution of Navier-Stokes equations for pulse propagation of waves through an elastic tube filled with
viscous fluid is presented. Derivation of equations for calculating the pulse wave velocities depending on
the viscosity, density and initial fluid pressure, density and elasticity of the pipe wall as well as the geometry
of the tube itself is presented. The results of the equation were compared with the measured experimental
values of pulse wave velocities on a biophysical model using fluids of different viscosities, densities and at
different values of initial pressure. The comparison shows that the model presented in this paper agrees
with the measured values in both cases: when the initial fluid pressure changes and when the viscosity and
density of the fluid change. Compared to other models used in medical research today, the presented model
uses all variables that affect the result. Using the equation for calculating the pulse wave velocities presented
in this paper, the age of the blood vessel wall will be more accurately estimated.
11
Appendix
Equations of fluid motion
As already mentioned, fluid motion is governed by continuity equation (8) and Navier – Stokes equations
(9,10). We will assume that the fluid is incompressible and viscous and that the fluid propagates pulsatingly
through a cylindrical tube of thin elastic walls. We will also assume that the motion is axisymmetric without
tangential velocity 𝑣𝜃 = 0, as well as that the fluid is not affected by gravity, i.e. that the z axis is in a
horizontal position, so the last term in eq. 9 can be ignored. Next, we describe the movement of the tube
wall with two equations: longitudinal and radial
𝜌0 ℎ
𝜕2𝜂
𝐸ℎ
𝜂
𝜕𝜁
=𝑝−
( +𝜎 )
2
2
(1 − 𝜎 )𝑅0 𝑅0
𝜕𝑡
𝜕𝑧
(A1)
𝜌0 ℎ
𝜕2𝜁
𝐸ℎ
𝜕 2 𝜁 𝜎 𝜕𝜂
𝜕𝑣𝑧 𝜕𝑣𝑟
=
+
+
)
(
)−𝜇(
2
2
2
(1 − 𝜎 ) 𝜕𝑧
𝜕𝑡
𝑅0 𝜕𝑧
𝜕𝑟
𝜕𝑧
(A2)
where 𝜂 and 𝜁 are the components of the displacement of the point on the tube wall in the axial and radial
directions, respectively, E – Young's modulus of elasticity, ρ0 –the density of the tube wall. The assumption
is that the pulse wave has a sinusoidal shape and that 𝜂, 𝜁, 𝑣𝑟 𝑎𝑛𝑑 𝑣𝑧 change in the same way. Let them be
𝑝 = 𝑃𝑒 𝑖(𝐾𝑧−𝜔𝑡) , 𝜂 = Ω𝑒 𝑖(𝐾𝑧−𝜔𝑡) , 𝜁 = Λ𝑒 𝑖(𝐾𝑧−𝜔𝑡) , 𝑣𝑧 = 𝑢(𝑟)𝑒 𝑖(𝐾𝑧−𝜔𝑡) 𝑖 𝑣𝑟 = 𝑤(𝑟)𝑒 𝑖(𝐾𝑧−𝜔𝑡)
(𝑤, 𝑢 – functions of 𝑟, 𝑃, Ω, Λ – constants). After substitution in equations (9), (10), (A1) and (A2) and
𝑖𝜔𝜌
𝜇
differentiation (
= 𝛼2,
𝜕2 𝑣𝑟
𝜕𝑧 2
𝜕2 𝑢
≈ 0, 𝜕𝑧2 ≈ 0 ) follows:
𝜕 2 𝑢 1 𝜕𝑢
𝑖𝐾
+
+ 𝛼2𝑢 = 𝑃
2
𝜕𝑟
𝑟 𝜕𝑟
𝜇
𝜕 2 𝑤 1 𝜕𝑤
1
1 𝜕𝑝
+
+ (𝛼 2 − 2 ) 𝑤 =
2
𝜕𝑟
𝑟 𝜕𝑟
𝑟
𝜇 𝜕𝑟
𝐸ℎ
Ω
𝑖𝐾𝜎
−𝜌0 ℎ𝜔2 Ω = 𝑃 −
Λ)
( 2+
2
(1 − 𝜎 ) 𝑅0
𝑅0
𝐸ℎ
𝑖𝐾𝜎
𝜕𝑢
−𝜌0 ℎ𝜔2 Λ =
(−𝐾 2 Λ +
Ω) − 𝜇 ( |𝑟=𝑅0 + 𝑖𝐾𝑤)
2
(1 − 𝜎 )
𝑅0
𝜕𝑟
(A3)
(A4)
(A5)
(A6)
Equations (A3) and (A4) are partial Bessel equations.
12
Assuming that P is in the form: 𝑃 = 𝐴1 ∙ 𝐽0 (𝑦𝑟), where 𝐴1 is a constant and a non-homogeneous solution
in the form 𝑢(𝑟) = 𝐵1 ∙ 𝐽0 (𝑦𝑟), by substituting in equation (A3) and differentiating, a particular solution is
obtained
𝑖𝐾𝐴1
1
∙ 𝐽 (𝑦𝑟)
2
𝜇 (𝛼 − 𝑦 2 ) 0
𝑢(𝑟) =
Assumption that a homogeneous solution in the form 𝑢(𝑟) = 𝐶1
𝐽0 (𝛼𝑟)
,
𝐽0 (𝛼)
where C1 is a constant to be
determined and in the final form:
𝑢(𝑟) = 𝐶1
𝐽0 (𝛼𝑟) 𝑖𝐾𝐴1
1
+
∙ 𝐽 (𝑦𝑟)
2
𝐽0 (𝛼)
𝜇 (𝛼 − 𝑦 2 ) 0
(A7)
Similar mathematical operations are repeated to solve equation (A4). Let non-homogeneous solution be in
the form 𝑤 = 𝐷1 ∙ 𝐽1 (𝑦𝑟), and by changing to equation (A4) and differentiating, a particular solution is
obtained
𝑤=−
𝑦𝐴1
1
∙ 𝐽 (𝑦𝑟)
2
𝜇 (𝛼 − 𝑦 2 ) 1
Here, too, the assumption is that the homogeneous solution is in the form 𝑤 = 𝐶2
𝐽1 (𝛼𝑟)
,
𝐽0 (𝛼)
where C2 is a
constant, so it is in the final form:
𝐽1 (𝛼𝑟) 𝑦𝐴1
1
−
∙ 𝐽 (𝑦𝑟)
𝐽0 (𝛼)
𝜇 (𝛼 2 − 𝑦 2 ) 1
From the equation of continuity (8) =>
(A8)
𝜕𝑤 𝑤
+ + 𝑖𝐾𝑢 = 0
𝜕𝑟 𝑟
By substituting equations (A7) and (A8) into equation (A9) and arranging it, we get
(A9)
𝑤(𝑟) = 𝐶2
− (𝑖𝐾𝐶1
𝐽0 (𝛼𝑟) 𝑖 2 𝐾 2 𝐴1
1
+
∙ 𝐽 (𝑦𝑟))
2
𝐽0 (𝛼)
𝜇 (𝛼 − 𝑦 2 ) 0
= 𝛼𝐶2
𝐽0 (𝛼𝑟) 𝑦 2 𝐴1
1
𝐶2 𝐽1 (𝛼𝑟) 1 𝑦𝐴1
1
−
∙ 𝐽0 (𝑦𝑟) +
−
∙ 𝐽 (𝑦𝑟)
2
2
2
𝐽0 (𝛼)
𝜇 (𝛼 − 𝑦 )
𝑟 𝐽0 (𝛼) 𝑟 𝜇 (𝛼 − 𝑦 2 ) 1
By solving we get
𝐶2 = −
𝑖𝐾
𝐶
𝛼 1
and after series expansion of Bessel coefficients:
13
𝑦𝑟 2
𝑦𝑟 4
(2)
(2)
𝐽0 (𝑦𝑟) = 1 −
+ 2 2 +⋯ ≈ 1
(12 )
(1 )(2 )
𝑦𝑟 1
𝑦𝑟 3
𝑦𝑟 5
(2)
(2)
(2)
𝑦𝑟
𝐽1 (𝑦𝑟) =
−
+
+⋯ ≈
1
2
12 ∙ 3
22 2
𝑖𝜔𝜌
𝑖𝜔𝜌
𝑖 𝜔
2
2
2
2
As well as (𝛼 − 𝑦 ) = 𝜇 − 𝑖 𝐾 = 𝜇 − 𝑐 2 ≈ 𝛼 2
Due to the viscosity the velocities of the fluid along the inner wall of the vessel (when r = R0) are
𝑢|𝑟=𝑅0 = 𝐶1
𝐽0 (𝛼𝑟) 𝑖𝐾𝐴1
1
𝐽0 (𝛼𝑟) 𝐴1
+
∙ 𝐽0 (𝑦𝑟) = 𝐶1
+
2
2
𝐽0 (𝛼)
𝜇 (𝛼 − 𝑦 )
𝐽0 (𝛼) 𝜌𝑐
𝑤|𝑟=𝑅0 = 𝐶2
𝐽1 (𝛼𝑟) 𝑦𝐴1
1
𝑖𝐾
2 𝐽1 (𝛼𝑟) 𝐴1
−
∙ 𝐽1 (𝑦𝑟) = − (𝐶1
+
∙𝑅 )
2
2
𝐽0 (𝛼)
𝜇 (𝛼 − 𝑦 )
2
𝛼 𝐽0 (𝛼) 𝜌𝑐 0
and we have
𝑑𝑢
𝐽1 (𝛼𝑅0 ) 𝑖𝐾𝐴1
1
𝐽1 (𝛼𝑅0 ) 𝐴1 𝐾 2 𝑅0
(−𝑦)𝐽
(𝑦𝑅
)
|𝑟=𝑅0 = −𝐶1 𝛼
+
∙
=
−𝐶
𝛼
+
1
0
1
𝑑𝑟
𝐽0 (𝛼)
𝜇 (𝛼 2 − 𝑦 2 )
𝐽0 (𝛼)
𝜌𝑐 2
𝐸ℎ
With substitutions: 𝐵 = (1−𝜎2 ) , 𝑃 = 𝐴1 ∙ 𝐽0 (𝑦𝑟) and 𝐽0 (𝑦𝑟) ≈ 1 in equation (A5) we have
1
𝐵
𝑖𝐾𝜎𝐵
𝐴1 + (𝜔2 −
Λ=0
)Ω −
2
𝜌0 ℎ
𝜌0 𝑅0
𝜌0 𝑅0
(A10)
Since the velocity of the wall is equal to the velocity of the fluid along the wall of the vessel, equalizing the
equations
𝑢=
𝑑𝜁
𝐽0 (𝛼𝑟) 𝐴1 𝑖(𝐾𝑥−𝜔𝑡)
|𝑟=𝑅0 = −𝑖𝜔Λ𝑒 𝑖(𝐾𝑥−𝜔𝑡) = (𝐶1
+ )𝑒
𝑑𝑡
𝐽0 (𝛼) 𝜌𝑐
𝐴1
𝐽0 (𝛼𝑅0 )
+ 𝐶1
+ 𝑖𝜔Λ = 0
𝜌𝑐
𝐽0 (𝛼)
𝑑𝜂
𝑖𝐾
2 𝐽1 (𝛼𝑅0 ) 𝐴1
𝑤=
|𝑟=𝑅0 = −𝑖𝜔Ω𝑒 𝑖(𝐾𝑥−𝜔𝑡) = − (𝐶1
+
∙ 𝑅 ) 𝑒 𝑖(𝐾𝑥−𝜔𝑡)
𝑑𝑡
2
𝛼 𝐽0 (𝛼)
𝜌𝑐 0
𝑖𝐾𝑅0
𝑖𝐾 𝐽1 (𝛼𝑅0 )
𝐴1 + 𝐶1
− 𝑖𝜔Ω = 0
2𝜌𝑐
𝛼 𝐽0 (𝛼)
And the last equation, with substitutions in (A6) =>
(A11)
(A12)
−𝜔2 Λ = −𝐾 2
𝐵ℎ
𝑖𝐾𝜎 𝐵ℎ
𝜇
𝐽1 (𝛼𝑅0 ) 𝐴1 𝐾 2 𝑅0 𝑖 2 𝐾 2
2 𝐽1 (𝛼𝑅0 ) 𝐴1
Λ+
Ω−
(−𝐶1 𝛼
+
−
+
∙ 𝑅 ))
(𝐶1
𝜌0 ℎ
𝑅0 𝜌0 ℎ
𝜌0 ℎ
𝐽0 (𝛼)
𝜌𝑐 2
2
𝛼 𝐽0 (𝛼)
𝜌𝑐 0
𝜔2
𝑐2 ≪ 1 ,
𝛼
𝐾2
2 𝐽1 (𝛼𝑅0 ) 𝐴1
+
∙𝑅 )≪1
(𝐶1
2
𝛼 𝐽0 (𝛼)
𝜌𝑐 0
14
We obtain
𝜇
𝐾 2 𝑅0
𝜇𝛼 𝐽1 (𝛼𝑅0 )
𝐵ℎ
𝑖𝐾𝜎 𝐵ℎ
−
∙
𝐴1 +
𝐶1 + (𝜔2 − 𝐾 2
)Λ +
Ω=0
𝜌0 ℎ𝜌𝑐
2
𝜌0 ℎ 𝐽0 (𝛼)
𝜌0 ℎ
𝑅0 𝜌0 ℎ
(A13 - 23)
The system of equations (A10), (A11), (A12) and (A13) is solved by setting the coefficients A1, C1, Λ and
Ω to be equal to 0:
1
𝜌𝑐
|
𝑖𝜔𝑅0
2𝜌𝑐 2
|
1
𝜌0 ℎ
|
𝜇
𝜔2 𝑅0
−
∙
𝜌0 ℎ𝜌𝑐 3
2
𝐽0 (𝛼𝑅0 )
𝐽0 (𝛼)
𝑖𝜔 𝐽1 (𝛼𝑅0 )
𝛼𝑐 𝐽0 (𝛼)
0
𝑖𝜔
−𝑖𝜔
0
𝐵
𝜌0 𝑅02
𝑖𝜔𝜎 𝐵
𝑅0 𝑐 𝜌0
𝜔2 −
0
𝜇𝛼 𝐽1 (𝛼𝑅0 )
𝜌0 ℎ 𝐽0 (𝛼)
|
𝑖𝜔𝜎𝐵 | = 0
𝜌0 𝑅0 𝑐
𝜔2 𝐵 |
𝜔2 − 2
𝑐 𝜌0
−
By simplifying this determinant, we obtain
|
1−
2 𝐽1 (𝛼𝑅0 )
𝛼𝑅0 𝐽0 (𝛼𝑅0 )
1
|
−
1 𝐽1 (𝛼𝑅0 )
𝛼𝑅0 𝐽0 (𝛼𝑅0 )
2
𝐵ℎ
𝜌𝑐 2 𝑅0
𝜎𝐵
− 2
𝜌𝑐 𝑅0
1
|
𝜎𝐵ℎ
1+ 2
=0
𝜌𝑐 𝑅0
𝜌0 ℎ
𝐵ℎ |
−
𝜌𝑅0 𝜌𝑐 2 𝑅0
Using substitutions
𝐸ℎ
𝜌0 ℎ
= 𝑥,
=𝑡
2
(A14)
2𝜌𝑐 𝑅0
𝜌𝑅0
and after solving the determinant, the so-called frequency equation in term of variable x was obtained
(4 − 2𝛼𝑅0
𝐽0 (𝛼𝑅0 ) 2
𝛼𝑅0 𝐽0 (𝛼𝑅0 )
𝐽0 (𝛼𝑅0 )
− 1) − 4𝜎 + 1 + 2𝛼𝑅0
]𝑥
) 𝑥 + [2𝑡 (
𝐽1 (𝛼𝑅0 )
2 𝐽1 (𝛼𝑅0 )
𝐽1 (𝛼𝑅0 )
𝛼𝑅0 𝐽0 (𝛼𝑅0 )
− 2𝑡(1 − 𝜎 2 )
− (1 − 𝜎 2 ) = 0
2 𝐽1 (𝛼𝑅0 )
Using the values for viscosity of blood, diameter of aorta or main arteries, blood density and density of
artery wall, we obtained that |𝛼𝑅0 | ≫ 1, ie that they are in the ranges of 𝛼𝑅0 = 3 − 12 and 𝑡 = 0.1 − 0.4.
After asymptotic expansion of the Bessel function
𝐽0 (𝛼𝑅0 )
1
~−𝑖+
𝐽1 (𝛼𝑅0 )
2𝛼𝑅0
15
and by substituting in the frequency equation and arranging, the quadratic equation is obtained
(−2𝛼𝑅0 𝑖 − 3)𝑥 2 + [4𝜎 − 2 + 2𝛼𝑅0 𝑖 + 𝑡𝛼𝑅0 𝑖 +
3𝑡
𝑡
] 𝑥 + (1 − 𝜎 2 ) (1 + − 𝑡𝛼𝑅0 𝑖) = 0
2
2
1
This quadratic equation can be solved for 𝜎 = 0, 𝜎 = 2 and 𝜎 = 1. For an artery 𝜎 = 0.49, so the roots of
1
quadratic equation for 𝜎 = 2, are:
𝑡
𝑖
𝜎 2 3𝑡
𝑖(1 − 𝜎 2 )
𝑥1 = 1 + +
− ) and 𝑥2 =
(2 − 2𝜎 +
2 𝛼𝑅0
2
4
2𝛼𝑅0
Substituting in (24)
𝐾 2 𝐸ℎ
𝐾2 2
=
𝑐 =𝑥
𝜔 2 2𝜌𝑅0 𝜔 2 0
We look for solutions in the following form and only for x1 (for 𝜇 → 0, 𝑥2 → ∞)
𝐾
𝐾1 + 𝑖𝐾2
𝑐0 =
𝑐0 = √𝑥1
𝜔
𝜔
After substitution √𝑖 =
√2
(1 +
2
𝑖) and solving the root of a complex number
𝜎 2 3𝑡
𝜎 2 3𝑡
2
(1
−
𝜎
+
−
)
2𝑖
(1
−
𝜎
+
𝑡
4
8
4 − 8 ) 𝐾1 + 𝑖𝐾2
+
=
𝑐0
√𝑥1 = √1 + +
2
𝜔
2𝜔𝜌
2𝜔𝜌
√
√
𝜇 𝑅0
𝜇 𝑅0
𝜎 2 3𝑡
𝐾1
𝑡 (1 − 𝜎 + 4 − 8 )
𝑐 =1+ +
,
𝜔 0
4
2𝜔𝜌
√
𝜇 𝑅0
𝜎 2 3𝑡
(1 − 𝜎 + 4 − 8 )
2𝜔𝜌
√
𝐾2
𝜇 𝑅0
𝑐0 =
𝜎 2 3𝑡
𝜔
(1
−
𝜎
+
𝑡
4 − 8)
1+4+
2𝜔𝜌
√
𝜇 𝑅0
By substituting c0 with equation (7) which is a function of the initial pressure and arranging, we obtain the
final equation (A15) for the pulse wave velocity
𝐸ℎ
𝑝
√
+ 0
𝜔
1
𝑡
1
𝜇
𝜎 2 3𝑡
2𝑅0 𝜌 4𝜌
𝑐=
=±
+
−
−
𝜎
+
− ))
(1
(1
√
2
𝑝0 𝑅0
𝐾1
4
𝑅
2𝜔𝜌
4
8
𝑡
0
1−
𝐸ℎ (1 + 4)
And
(A15)
16
2
𝑝0 𝑅0 (1 − 𝜎 + 𝜎 − 3𝑡)
4
8
𝐸ℎ
𝐾2 = ±𝜔
𝐸ℎ
𝑝0
𝑡 2𝜔𝜌
√
√
2𝑅0 𝜌 + 4𝜌 (1 + 4)
𝜇 𝑅0
1−
If we assume that t is very small it can be neglected (𝑡 = 0) 𝑎𝑛𝑑 the initial fluid is zero (𝑝0 = 0) we get
the same equation derived by Morgan and Kiely for small viscosity
𝑐=
𝜔
𝐸ℎ
1
𝜇
𝜎2
= ±√
(1 + − √
(1 − 𝜎 + ))
𝐾1
2𝑅0 𝜌
𝑅0 2𝜔𝜌
4
(A16)
And if we neglect both σ and μ we get the Moens-Korteweg equation (4) for PWV.
In case of high viscosity, or |𝛼𝑅0 | ≪ 1, we use serial expansions
𝐽0 (𝛼𝑅0 ) ≈ 1 −
𝛼 2 𝑅02
,
4
𝐽1 (𝛼𝑅0 ) ≈
𝛼𝑅0
𝛼 2 𝑅02
(1 −
)
2
8
After substituting in the frequency equation and solving the quadratic equations we obtain
𝑐=
𝜔
=
𝐾1
𝑐0
𝐸ℎ
𝑝0
𝜔𝜌
1
= ±√
+
𝑅0 √
2𝑅0 𝜌 4𝜌
𝜇 5 − 4𝜎 + 6𝑡
5 − 4𝜎 + 6𝑡
√ 𝑅 2 𝜔𝜌
0 𝜇
(A17)
and
𝐾2 = ±𝜔√
1
𝜇 √5 − 4𝜎 + 6𝑡
√
𝐸ℎ
𝑝
𝑅0
+ 0 𝜔𝜌
2𝑅0 𝜌 4𝜌
Conflict of interest statement
No conflicts of interest, financial or otherwise, are declared by the authors.
Acknowledgments
This work was supported by Serbian Ministry of Education, Science and Technological Development
Grants 32040 and 41022.
17
Fig. 1. Increase in disk radius and width due to pulse wave: du/dz - the narrowing of the disk, u - disk
shift, ∆𝑅0 - radius change
Fig. 2. The propagation of the pulse wave increases the pressure on one side of the disk by ∆p
Fig. 3. Stresses in a circular cylindrical pressure vessel a - hoop stress, b - longitudinal stress
Fig. 4. A schematic diagram of the biophysical model of the cardiovascular system: H– automatic
hammer, P–pump, reservoir 1 (closed), reservoir 2 (adjustable height), M–manometer, S1, S2 –
pressure sensors mounted through the wall of the elastic tube, DAQ—data acquisition device, V –
one-way valve. The arrows indicate the flow direction of fluid through the valves and the tubes
Fig. 5. Comparison of models and measured values of PWV against initial fluid pressure: MoensKorteweg equation, Morgan Kiely equation and equation (11) for t = 0 and t ≠ 0. 
Fig. 6. PWV against alcohol concentration
Fig. 7. Comparison of Morgan-Kiely equation, equation (11) and measured values of PWV against
alcohol concentration
18
References
Atabek HB (1968) Wave Propagation through a Viscous Fluid Contained in a Tethered, Initially Stressed,
Orthotropic Elastic Tube. Biophys J. May; 8(5): 626–649.
Atabek HB, Lew HS (1966) Wave Propagation through a Viscous Incompressible Fluid Contained in an
Initially Stressed Elastic Tube. Biophys J. Jul; 6(4): 481–503.
Demiray H, Akgün G (1997) Wave propagation in a viscous fluid contained in a prestressed viscoelastic
thin tube. International Journal of Engineering Science, 35(10–11): 1065-1079
Dinenno FA, Jones PP, Seals DR & Tanaka H (2000) Age-associated arterial wall thickening is related to
elevations in sympathetic activity in healthy humans. Am J Physiol Heart Circ Physiol 278, H1205–H1210.
Eigenbrodt ML, Bursac Z, Rose KM, Couper DJ, Tracy RE, Evans GW, Brancati FL & Mehta JL (2006)
Common carotid arterial interadventitial distance (diameter) as an indicator of the damaging effects of age
and atherosclerosis, a cross-sectional study of the Atherosclerosis Risk in Community Cohort Limited
Access Data (ARICLAD), 1987–89. Cardiovasc Ultrasound 4, 1.
Engelen L, Ferreira I, Stehouwer CD, Boutouyrie P, Laurent S & Reference Values for Arterial
Measurements C (2013) Reference intervals for common carotid intima-media thickness measured with
echotracking: relation with risk factors. Eur Heart J 34, 2368–2380.
Feng J, Khir A (2010) Determination of wave speed and wave separation in the arteries using diameter and
velocity. Journal of Biomechanics. 43,455–462
Green DJ, Swart A, Exterkate A, Naylor LH, Black MA, Cable NT & Thijssen DH (2010) Impact of age,
sex and exercise on brachial and popliteal artery remodelling in humans. Atherosclerosis 210, 525–530.
Heijden van der Spek JJ, Staessen JA, Fagard RH, Hoeks AP, 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.
Hypertension 35, 637–642.
Homma S, Hirose N, Ishida H, Ishii T & Araki G (2001) Carotid plaque and intima-media thickness
assessed by b-mode ultrasonography in subjects ranging from young adults to centenarians. Stroke 32, 830–
835.
Khir A, Parker K (2002) Measurements of wave speed and reflected waves in elastic tubes and bifurcations.
Journal of Biomechanics. 35,775–783
Khir AW, Swalen MJP, Parker KH (2007) The simultaneous determination of wave speed and the arrival
time of reflected waves in arteries. Med Biol Eng Comput 45,1201–1210.
Korteweg D J (1878) Ueber die Fortpflanzungsgeschwindigkeit des Schalles in elastischen Röhren.
https://doi.org/10.1002/andp.18782411206
Lazović B, Mazić S, Zikich D, Žikić D (2015) The mathematical model of the radial artery blood pressure
waveform through monitoring of the age-related changes. Wave Motion 56:14–21
Li Y, Ashraf W, Khir A (2011) Experimental validation of non-invasive and fluid density independent
methods for the determination of local wave speed and arrival time of reflected wave. Journal of
Biomechanics. 44,1393–1399
Moens A I (1877) Over de voortplantingssnelheid van den pols (Ph.D. thesis). Leiden: S.C. Van Doesburgh.
19
Moens A I (1878) Die Pulskurve. Leiden: E.J. Brill
Morgan GW, Kiely JP (1954) Wave Propagation in a Viscous Liquid Contained in a Flexible Tube. The
Journal of the Acoustical Society of America 26, 323
Munckhof van den I, Scholten R, Cable NT, Hopman MT, Green DJ & Thijssen DH (2012) Impact of age
and sex on carotid and peripheral arterial wall thickness in humans. Acta Physiol (Oxf) 206, 220–228.
Papageorgiou Gl, Jones NB (1987a) Physical modeling of the arterial wall. Part 1: testing of tubes of various
materials. J Biomed Eng 9,153- 156.
Papageorgiou Gl, Jones NB (1987b) Physical modeling of the arterial wall. Part 2: Simulation of the nonlinear elasticity of the arterial wall. J Biomed Eng.9, 216–221.
Sandgren T, Sonesson B, Ahlgren AR & Lanne T (1998) Factors predicting the diameter of the popliteal
artery in healthy humans. J Vasc Surg 28, 284–289.
Sandgren T, Sonesson B, Ahlgren R & Lanne T (1999) The diameter of the common femoral artery in
healthy human: influence of sex, age, and body size. J Vasc Surg 29, 503–510.
Schmidt-Trucksass A, Grathwohl D, Schmid A, Boragk R, Upmeier C, Keul J & Huonker M (1999)
Structural, functional, and hemodynamic changes of the common carotid artery with age in male subjects.
Arterioscler Thromb Vasc Biol 19, 1091–1097
Stojadinovic B, Tenne T, Zikich D, Rajković N, Milošević N, Lazović B, Žikić D (2015) Effect of viscosity
on the wave propagation. Journal of Biomechanics. 48(15): 3969-3974
Tanaka H, Dinenno FA,Monahan KD, DeSouza CA & Seals DR (2001) Carotid artery wall hypertrophy
with age is related to local systolic blood pressure in healthy men. Arterioscler Thromb Vasc Biol 21, 82–
87
Westerhof N, Elzinga G, Sipkema P (1972) Forward and backward waves in the arterial system. Cardiovasc
Res 6,648–656.
Womersley R (1955) Method for the calculation of velocity, rate of flow and viscous drag in arteries when
the pressure gradient is known. J Physiol. Mar 28; 127(3): 553–563.
Womersley R (1957) An elastic tube theory of pulse transmission and oscillatory flow in mammalian
arteries. Wright Air Development Center (Technical Report WADC-TR)
Young T (1808) “Hydraulic investigations, subservient to an intended Croonian lecture on the motion of
the blood.” Philosophical Transactions of the Royal Society of London 98, 164-186 [Errata in Ref (26), p.
31].
Žikić D, Stojadinović B and Nestorović Z (2019) Biophysical modeling of wave propagation phenomena:
experimental determination of pulse wave velocity in viscous fluid-filled elastic tubes in a gravitation field
Eur Biophys J 48:407-411
20
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