a computational study of blood flow and oxygen transport during reperfusion and post-conditioning
DESCRIPTION
My Master's Thesis Completed at Rowan University, May 2009 - December 2010TRANSCRIPT
A COMPUTATIONAL STUDY OF BLOOD FLOW AND OXYGEN TRANSPORT
DURING REPERFUSION AND POST-CONDITIONING
by
Anthony Joseph La Barck, Jr.
A Thesis
Submitted in partial fulfillment of the requirements of the
Master of Science in Engineering Degree
of
The Graduate School
at
Rowan University
December 23, 2010
Thesis Chair: Thomas Lad Merrill, Ph.D.
©2010 Anthony Joseph La Barck, Jr.
ABSTRACT
Anthony Joseph La Barck, Jr.
A COMPUTATIONAL STUDY OF BLOOD FLOW AND OXYGEN TRANSPORT
DURING REPERFUSION AND POST-CONDITIONING
2009/10
Thomas L. Merrill, Ph.D.
Mechanical Engineering
Post-conditioning (PC) is a relatively new therapeutic strategy which aims at
limiting reperfusion injury after a heart attack. Although PC has shown to reduce the
amount of tissue death incurred by reperfusion injury, the biological mechanisms
involved in PC are not well understood. The goal of this thesis was to develop a
computational model which simulates the flow of blood through a coronary arterial
network and the resulting mass transport of oxygen in a capillary-tissue system during a
specific PC procedure. The computational model consisted of three sub-models: 1) a
model which simulates the flow of blood past a periodically inflating balloon according
to a specific PC procedure, 2) a model which simulates the flow of blood downstream in
a 1-D arterial network, and 3) a model which simulates the mass transport of oxygen
downstream in a capillary-tissue system. The PC algorithm tested in the model was based
on a procedure in a recent clinical trial. The results showed that PC significantly reduces
the concentration of oxygen in tissue, which may have a direct impact on reactive oxygen
species (ROS) production. The models in this paper lay the foundation for gaining a
better understanding behind the cardioprotective mechanisms of PC.
ii
ACKNOWLEDGEMENTS
I would like to thank my thesis advisor, Dr. Thomas L. Merrill, for his
encouragement, support, and insightful feedback throughout my graduate studies. I am
extremely grateful to him for introducing me into the field of biofluids and biotransport. I
would also like to thank my thesis committee members, Drs. Maria Tahamont, Smitesh
Bakrania, and Krishan Bhatia for their involvement and advice throughout my work. I
would also like to thank Rowan University for giving me the opportunity to pursue my
graduate studies, as well as providing me with the workstation used to solve all of the
computational models.
I cannot thank my parents, family, and friends enough for all of their
encouragement from the beginning of this work. To my parents: thank you for teaching
me to persevere, and to always strive for the best; and yes, I am finally done. To my
friends, thank you for all of the talks we have shared about my work, even though you
probably did not understand much of it.
iii
TABLE OF CONTENTS
ACKNOWLEDGEMENTS ................................................................................................ ii
LIST OF FIGURES ............................................................................................................ v
LIST OF TABLES ............................................................................................................. ix
CHAPTER 1 ....................................................................................................................... 1
1.1 Background ............................................................................................................ 1
1.2 Problem Statement ................................................................................................. 5
1.3 Organization of Thesis........................................................................................... 6
CHAPTER 2 ....................................................................................................................... 8
2.1 Steady Flow in a Straight Tube ............................................................................. 8
2.2 Pulsatile Flow in a Straight Tube: Womersley Solution ..................................... 10
2.3 Steady and Pulsatile Flow in Bifurcating Tubes ................................................. 15
2.4 Mass Transport of Oxygen in Capillaries and Tissues ........................................ 19
CHAPTER 3 ..................................................................................................................... 26
3.1 Finite Element Method – COMSOL ................................................................... 26
3.2 Numerical Integration and Fourier Operations – MATLAB ............................... 27
3.3 Computer Hardware ............................................................................................ 28
3.4 Breakdown of Models ......................................................................................... 28
3.5 Breakdown of Simulations .................................................................................. 30
3.6 Modeling Protocol ............................................................................................... 30
iv
3.7 Model 1 (1a & 1b) ............................................................................................... 32
3.8 Model 2 ................................................................................................................ 46
3.9 Model 3 ................................................................................................................ 52
CHAPTER 4 ..................................................................................................................... 61
4.1 Abstract ................................................................................................................ 61
4.2 Introduction ......................................................................................................... 62
4.3 Computational Methods ...................................................................................... 63
4.4 Results ................................................................................................................. 70
4.5 Discussion ............................................................................................................ 78
4.6 Conclusion ........................................................................................................... 81
CHAPTER 5 ..................................................................................................................... 83
5.1 Conclusions ......................................................................................................... 83
5.2 Future Work ......................................................................................................... 85
References ......................................................................................................................... 88
APPENDIX A ................................................................................................................... 93
APPENDIX B ................................................................................................................... 95
v
LIST OF FIGURES
Figure 1.1 - Flowchart of pathological events in reperfusion, and what role ROS plays in
it. ......................................................................................................................................... 2
Figure 1.2 - Angioplasty balloon catheter reopening an occluded vessel. 1.2a) fully
deflated balloon. 1.2b) partially inflated balloon. 1.2c) fully inflated balloon, block flow
through the vessel. During PC, the balloon will continue to inflate and deflate based on a
user-specified algorithm. Images shown are stills from a video of balloon angioplasty
animation3. .......................................................................................................................... 3
Figure 1.3 - Schematic of modeling pathway. .................................................................... 6
Figure 2.1 - 2-D schematic of fluid velocity in a straight tube of length L and radius R.
Flow is steady and fully-developed, meaning the fluid velocity profile is parabolic in the
tube and does not change over the length of the tube. ........................................................ 9
Figure 2.2 - Normalized cardiac waveform, based from waveform in He and Ku (1996).
........................................................................................................................................... 10
Figure 2.3 - Fourier series representation of cardiac waveform from Figure 2.2 for 2, 20,
and 60 harmonics. ............................................................................................................. 11
Figure 2.4 - Schematic of arterial bifurcations. The parent vessel is denoted as P; the two
daughter vessels D1 and D2 become new parent vessels for successive bifurcations...... 16
Figure 2.5 - Vessel numbering scheme and notation for an arterial network, as proposed
by Zamir37
, specific and general cases. ............................................................................. 19
Figure 2.6 - Schematic of vessels in systemic circulation (top) and topology of capillary
vessels (bottom). Images are taken from Dawson7 (top) and Doohan
8 (bottom) ............. 20
Figure 2.7 - Schematic of Krogh tissue Model. Rc is the radius of the capillary, Rt is the
radius of the surrounding tissue, and L is the length of the capillary. Blood within the
capillary flows from left to right. ...................................................................................... 21
Figure 2.8 - Hemoglobin oxygen-dissociation curve (ODC). The x-axis is the
concentration of oxygen dissolved in plasma (given in partial pressure of O2) and the
y-axis is the percent saturation of hemoglobin with oxygen31
. ......................................... 23
Figure 3.1 - Schematic of backstep problem used for numerical analysis validation. The
black arrows show the direction of flow. Lines within the fluid domain denote
streamlines. In the backstep problem, a recirculation zone forms immediately
downstream of the backstep. ............................................................................................. 27
Figure 3.2 - Schematic of modeling pathway. .................................................................. 31
vi
Figure 3.3 - Model 1a geometry. ...................................................................................... 33
Figure 3.4 - Model 1b geometry. ...................................................................................... 33
Figure 3.5 - Sectioning of balloon domain. Balloon is broken into two elliptical sections
and one cylindrical section................................................................................................ 35
Figure 3.6 - Diagram showing x-ranges for each balloon section. x1 is the lower limit on
the proximal balloon end, x2 is the upper limit on the proximal balloon end and the lower
limit on the middle balloon section, x3 is the upper limit on the middle balloon section
and the lower limit on the distal balloon end, and x4 is the upper limit on the distal
balloon end. ....................................................................................................................... 36
Figure 3.7 - Diagram representing the r-ranges for each balloon section. Each section has
the same lower radial limit r0 that is equal to the diameter of the catheter shaft. The upper
limits, which are a function of time, are r1(t), r2(t), and r3(t): the upper limit on the
proximal balloon end, the middle balloon section, and the distal balloon end, respectively.
........................................................................................................................................... 37
Figure 3.8 - Diagram representing the step inflation and deflation rate for one cycle of a
PC algorithm. Inflation took place at ton, while deflation occurred at toff. By using the
smoothing parameter of COMSOL‟s flc2hs function, the inflation and deflation step was
smoothed over an interval of 2*dt. This interval was centered over ton and toff. .............. 39
Figure 3.9 - Boundary conditions for Model 1a. .............................................................. 40
Figure 3.10 - Pulsatile flowrate waveform used in pulsatile flow simulations with and
without PC. Waveform shown is for one cardiac cycle. For steady flow simulations, the
time-averaged value of the pulsatile flowrate waveform for one cardiac cycle was used.40
Figure 3.11 - Boundary conditions for Model 1b. ............................................................ 41
Figure 3.12 - Plot of average velocity over entire computational domain vs. mesh size for
Model 1a. The final mesh used for all simulations contained 4,472 quadrilateral elements
(dashed circle in graph). .................................................................................................... 42
Figure 3.13 - Mesh for Model 1a. The bottom portion of the figure shows a close-up
section of the mesh............................................................................................................ 42
Figure 3.14 - Plot of average velocity vs. mesh size for Model 1a. The final mesh used for
all simulations contain 4,472 quadrilateral elements (dashed circle in graph). ................ 43
Figure 3.15 - Mesh for Model 1b. ..................................................................................... 43
vii
Figure 3.16 - Velocity field in Model 1a for the steady flow with PC simulation at three
different times: a) fully deflated balloon state, b) partially inflated balloon state, and c)
fully inflated balloon state. Some fluid velocity was seen in 3.15c, however its maximum
value was approximately 6 orders of magnitude smaller than the maximum velocity in
3.15a. The units for velocity in the color legend are in m/s. ............................................. 44
Figure 3.17 - Schematic of asymmetric bifurcation network showing the two „bounding‟
pathways: the path of the largest daughter branches and the path of the smaller daughter
branches. ........................................................................................................................... 47
Figure 3.18 - Coronary arterial tree for Model 2 simulations.. The area zoomed in shows
the site of the capillary vessel segment where the velocity will be recorded for each
simulation. ......................................................................................................................... 49
Figure 3.19 - Schematic of Model 3, showing boundary conditions used. ....................... 54
Figure 3.20 - Average partial pressure of oxygen vs. mesh size for Model 3. The partial
pressure varies with small element sizes, but remains relatively constant for mesh sizes
between 2,250 and 15,000 elements. The black dashed circle indicates the final mesh size
used for all simulations. .................................................................................................... 57
Figure 3.21 - Plot of final mesh used in simulations. Mesh is mapped with a total of
15,000 quadrilateral elements. ......................................................................................... 57
Figure 3.22 - Axial partial pressure of oxygen profiles in capillary and tissue for results
from Sharan et al. and results from a COMSOL model with same input parameters.
Oxygen partial pressure is recorded at two points: at r = 0 (centerline of capillary) and at r
= 10 (tissue radius). The radial (r) axial (z) distances are normalized by the radius of the
capillary (Rc). The error bars shown are ±5%. ................................................................. 59
Figure 3.23 - Radial partial pressure of oxygen profiles at various axial positions in
model for results from Sharan et al. and results from a COMSOL model with same input
parameters. The radial (r) and axial (z) distance is normalized by the radius of the
capillary (Rc). The error bars shown are ±5%. ................................................................. 60
Figure 4.1 - Schematic of Krogh tissue model. Rc is the radius of the capillary, Rt is the
radius of the surrounding tissue, and L is the length of the capillary. Blood within the
capillary flows from left to right. ...................................................................................... 63
Figure 4.2 - Pulsatile capillary blood velocity waveform used for pulsatile flow
simulations. Waveform has a period of T = 0.832s and is obtained by solving upstream
models with waveform taken from He and Ku17
. ............................................................. 66
Figure 4.3 – Schematic of model, showing boundary conditions used. ........................... 67
viii
Figure 4.4 - Average partial pressure of oxygen vs. mesh size for model. The
concentration varies with small element sizes, but remains relatively constant for mesh
sizes between 2,250 and 15,000 elements. The black dashed circle indicates the final
mesh size used for all simulations. ................................................................................... 68
Figure 4.5 - Plot of final mesh used in simulations. Mesh is mapped with a total of 15,000
quadrilateral elements. ...................................................................................................... 69
Figure 4.6 - Plot of axial partial pressure (PO2) of oxygen in capillary and tissue region
for steady flow without PC simulation. The model results in an outlet capillary PO2 of 40
mmHg, which is consistent with the venous PO2 value found in literature. The PO2
profiles are also in qualitative agreement with those given by Sharan et al29
. ................. 71
Figure 4.7 - Surface plot of PO2 for the steady flow without PC simulation. The color
legend on the right has units of mmHg. The maximum PO2 observed was 95 mmHg (at
inlet), and the minimum PO2 was approximately 28 mmHg. The minimum PO2 is located
at the „lethal corner‟, denoted by the red dot in the figure. .............................................. 71
Figure 4.8 - Plot of time-averaged PO2 for steady flow with and without PC simulations
a) in the capillary and b) in the tissue. For both plots, the axial distance is normalized
with respect to the capillary length L. ............................................................................... 73
Figure 4.9 - Plot of tissue PO2 over time at space-averaged point (solid line) and lethal
corner (dashed line) for the steady flow with PC simulation. Time axis (x-axis) is
normalized with respect to the PC algorithm‟s cycle time, t_cycle = 60s. a) PO2 over
entire PC algorithm, b) PO2 during first balloon inflation (dotted rectangle on 4.9a), and
c) PO2 during first balloon deflation (dashed rectangle on 4.9a). Figure 4.9c shows a
time-lag between when the PC balloon is deflated and when oxygen reaches the space
average point and lethal corner. ........................................................................................ 76
Figure 4.10 - Plot of time-averaged partial pressure of oxygen for pulsatile flow with and
without PC simulations. a) Normalized axial length along centerline of capillary. b)
Normalized axial length along tissue wall. In both plots, the axial length is normalized
with respect to the length of capillary. For comparative purposes, steady flow simulations
were included in the plots. ................................................................................................ 78
ix
LIST OF TABLES
Table 2.1 - Number of harmonics used in describing pressure gradient for each
simulation. ......................................................................................................................... 12
Table 3.1 - Dimensions for Model 1a geometry. .............................................................. 33
Table 3.2 - Dimensions for Model 1b geometry. .............................................................. 33
Table 3.3 - Recorded pressure drop for each simulation of Model 1b.. ........................... 45
Table 3.4 - Parameters for Model 2. ................................................................................. 49
Table 3.5 - Recorded capillary blood velocities in Model 2 for each simulation. ............ 50
Table 3.6 - Dimensions for Model 3 geometry. ................................................................ 52
Table 3.7 - Parameters used in Model 3. .......................................................................... 54
Table 3.8 - Mathematical description of boundary conditions for Model 3. .................... 55
Table 3.9 - Parameters used in model by Sharan et al.29
.................................................. 58
Table 4.1 - Dimensions for model geometry. ................................................................... 64
Table 4.2 - Parameters used in model. .............................................................................. 66
Table A.1 - COMSOL solver settings and solution times for Model 1a. . ....................... 93
Table A.2 - COMSOL solver settings and solution times for Model 1a. . ....................... 93
Table A.3 - COMSOL solver settings and solution times for Model 1a. . ....................... 94
1
CHAPTER 1
INTRODUCTION
1.1 Background
1.1.1 Ischemia-Reperfusion Injury
Heart disease is the leading cause of death in the United States, killing at least one
person every 38 seconds 23. One type of heart disease is ischemic heart disease, which is a
result of restricted coronary blood flow likely due to an atherosclerotic plaque blockage
of an artery. To prevent oxygen-starved heart tissue downstream of the blockage from
dying, clinicians use a balloon angioplasty catheter to unblock the clogged artery and
restore blood flow. This procedure is known as a percutaneous transluminal coronary
angioplasty (PTCA). When blood flow is restored to heart tissue, this is known as
reperfusion 32.
Although reperfusion may seem beneficial in saving heart tissue, its effects are
paradoxical. Oxygenated blood that is rapidly restored to ischemic heart tissue during
reperfusion can actually cause more damage than ischemic heart disease alone. When this
occurs, it is known as ischemia-reperfusion (I/R) injury (Fig. 1.1). The damaging effects
of I/R injury can account for up to 50% of the total infarct (region of dead heart tissue)
size36.
I/R injury can be thought of as a „cascade‟ of pathological events that ultimately
lead to irreversible heart tissue damage. Before reperfusion, ischemic heart tissue is
undergoing anaerobic metabolism. Chemical species that are produced from this process
2
react with the restoring oxygen in blood during reperfusion. As a result, an abundant
amount of reactive oxygen species (ROS) are produced. ROS play a critical role in the
pathogenesis of I/R injury. ROS are highly unstable and cytotoxic molecules that:
react with intracellular proteins that promote apoptosis and necrosis2.
cause cellular membrane damage leading to intracellular calcium overload. An
overload of calcium can permanently damage the cell‟s mitochondria and hence
render the cell useless32.
Figure 1.1 - Flowchart of pathological events in reperfusion, and what role ROS plays in it.
Observations of I/R injury have led clinicians and researchers to develop new procedures
to limit its damaging effects. One strategy, known as therapeutic hypothermia, involves
cooling the blood returning to ischemic tissue and reducing inflammatory response.
3
Another strategy focuses on mechanically altering reperfusion, called post-conditioning
(PC). This thesis will explore the biophysics of the PC procedure, specifically from a
fluid mechanics and chemical transport perspective.
1.1.2 Post-conditioning
Post-conditioning (PC) is a relatively new medical procedure which targets
minimizing heart tissue damage during reperfusion. PC is performed by applying brief
cycles of ischemia during the early minutes of reperfusion. One way this procedure is
performed is with the same PTCA balloon catheter used to open an occluded vessel after
a heart attack. The balloon attached to the end of the angioplasty catheter will remain in
place and periodically inflate (ischemia) and deflate (reperfusion) according to a user-
specified algorithm (Fig. 1.2). The result is a „stuttered‟ form of reperfusion rather than a
rapid continuous restoration of blood flow33.
1.2a 1.2b 1.2c
Figure 1.2 - Angioplasty balloon catheter reopening an occluded vessel. 1.2a) fully deflated balloon.
1.2b) partially inflated balloon. 1.2c) fully inflated balloon, block flow through the vessel. During PC,
the balloon will continue to inflate and deflate based on a user-specified algorithm. Images shown are
stills from a video of balloon angioplasty animation3.
In 2003, Zhao et al. first demonstrated that PC was cardioprotective in the left
anterior descending artery of a canine. Their experimental algorithm consisted of 60
minutes of prolonged ischemia, followed by 3 cycles of 30 seconds ischemia and 30
seconds reperfusion, followed by 3 hours of normal reperfusion. They found that the area
Balloon Vessel wall Plaque
4
of necrotic tissue (An) – to – tissue area at risk (Ar) ratio (An/Ar) during PC was reduced
by 48% when compared to the control group40. In 2008, Gao et al. conducted a series of
PC procedures with varying algorithms on the common carotid artery of Sprague-Dawley
rats12. They observed that the effectiveness of PC in preventing tissue damage depends on
the number of cycles used in the algorithm, and the duration of each ischemia and
reperfusion stage. Grandfeldt et al. 2009 provided a summary of PC studies in vitro and
in vivo for a variety of species and a variety of PC algorithms. The results showed that
infarct size was reduced by up to 50% in canines and up to 25% rats13.
Clinical trials of PC in humans have just begun to surface. A study done by Dr.
Tarek Helmy at the University of Cincinnati involved performing a PC algorithm on
patients with acute myocardial infarction (AMI). The algorithm consisted of 3 to 4 cycles
of 30 seconds ischemia followed by 30 seconds reperfusion18. This study was completed
in May 2010 and its results are unknown.
Studies have shown that the generation of ROS is attenuated during PC. Research
suggests this is due to the reduction in oxygen delivery to tissue2. While the prevention of
an abundant generation of ROS is beneficial to saving heart tissue, researchers have also
found that small amounts of ROS produced during the reperfusion cycles of PC help to
confer cardioprotection13, 39. Additionally, PC has been shown to reduce intracellular
calcium overload, as well as activate survival cellular proteins2, 16. These observations
demonstrate the difficulty in analyzing the impact of one biological reaction during PC,
since PC induces various reactions simultaneously.
5
1.2 Problem Statement
Although studies have shown that PC can significantly reduce infarct size, a
precise explanation regarding the mechanisms involved in its cardioprotection remains
elusive. To our knowledge, no study has investigated the link between the fluid dynamics
of blood and the mass transport of oxygen during a PC procedure.
The goal of this thesis is to develop a computational model that simulates the fluid
dynamics of blood flow during a PC procedure and the resulting mass transfer of oxygen
in the capillaries of the heart. The computational model will be broken down into 3
separate models (Fig. 1.3) that consist of:
Model 1 – a simulation of blood flow past a periodically inflating and deflating
angioplasty balloon catheter in the human left anterior descending artery.
Model 2 – a simulation of blood flow through the entire coronary arterial network
downstream of Model 1.
Model 3 – a Krogh tissue model of a single coronary capillary that simulates the
convection and diffusion of oxygen through a capillary and into the surrounding
tissue.
6
1.3 Organization of Thesis
This chapter (Chapter I) contains background information on ischemia-
reperfusion injury and post-conditioning. It also provides the problem statement and
specific objectives. Chapter II covers the biophysics for blood flow in straight tubes and
bifurcations, as well as the physics of mass transport of oxygen in the capillaries. Chapter
III discusses the computational methods used to implement the biophysics and how the
models were solved. Chapter IV is entitled „Journal Article‟ and provides a draft of a
future publication. Chapter IV presents the key results of the computational models.
Model 1a Model 1b
Model 2
Model 3 Outlet from Model 2
Outlet from Model 1b
Outlet from Model 1a
Figure 1.3 - Schematic of modeling pathway.
7
Chapter V concludes the thesis, providing a discussion of the results and
recommendations for future work.
8
CHAPTER 2
BIOPHYSICS BACKGROUND
This chapter will discuss the mathematical background needed to understand how
the computational models are solved. It will discuss the equations that govern steady and
pulsatile fluid flow through straight tubes and arterial bifurcations. The chapter will end
with the equations that govern the mass transport of oxygen in a capillary tissue system.
2.1 Steady Flow in a Straight Tube
Initial computational studies consisted of modeling steady flow in each model.
Fluid flow through a straight tube is governed by the Navier-Stokes and Conservation of
Mass differential equations35:
( )v
v v p v ft
(2.1)
0v (2.2)
where is the fluid density, v is the fluid velocity vector, p is the fluid pressure, is
the fluid viscosity, and f represents body forces acting on the fluid. These equations are
used to solve for the velocity of a fluid at a point in space and time. The symbol is the
del operator, and can be written in three dimensions for cylindrical coordinates as:
1 ˆˆ ˆr zr r z
(2.3)
9
where r, , and z are the directions in the cylindrical coordinate system and r , , z are
the unit vectors associated with their respective direction.
For steady, fully developed flow in a straight tube whose axis is aligned with the
z-direction (Fig. 2.1), flow will only be changing in the r-direction, so the θ- and z-terms
of the del operators are dropped. Since we are only dealing with steady flow, the transient
velocity term on the left side of Eq. (2.1) can also be dropped. Assuming no body forces
on the fluid are present, Eq. (2.1) is simplified to:
2
2
1dp w w
dz r r r
(2.4)
where w is the z-direction velocity of the fluid. The pressure across the length of a
straight tube will decrease linearly, so dp
dz, the pressure gradient, is a constant and can be
rewritten as p
L
, where L is the length of the tube. Equation (2.4) can now be integrated
over the radius of the tube to obtain the velocity of the fluid as function of tube radius.
The boundary conditions used to integrate Eq. (2.4) are as follows: 1) the velocity at the
center of the tube is at its maximum value, thus 0w
r
at r = 0, and 2) the velocity must
R r
z
L
Figure 2.1 - 2-D schematic of fluid velocity in a straight tube of length L and
radius R. Flow is steady and fully-developed, meaning the fluid velocity profile is
parabolic in the tube and does not change over the length of the tube.
10
satisfy the no-slip condition at the wall, thus w = 0 at r = R. Equation (2.5) below shows
the final equation for a steady velocity profile in a straight tube:
2 2
2( ) 1
4
p R rw r
L R
(2.5)
where R is the radius of the tube. By integrating over the cross-sectional area of the tube,
the equation for the flowrate Q through the tube can be found:
4
8
pRQ
L
(2.6)
2.2 Pulsatile Flow in a Straight Tube: Womersley Solution
Blood flow through the heart is pulsatile in nature. Pulsatile flow refers to a fluid
where its flowrate is changing over time in a periodic fashion (Fig. 2.2).
Figure 2.2 - Normalized cardiac waveform, based from waveform in He and Ku (1996).
In contrast to the previous section, the pressure gradient in pulsatile flow will be a
periodic function of time. However, the cardiac pressure gradient waveform is not a
simple function of time, as we can observe from the flowrate waveform depicted in Fig.
11
(2.2), assuming the pressure and flowrate are related. Although there is no defined
function for it, we can express the pressure gradient as a Fourier series expansion:
0 1 1 2 2cos sin cos 2 sin 2
cos sinn n
dpA A t B t A t B t
L
A n t B n t
(2.7)
To understand Eq. (2.7), it is easier to think of the Fourier series expansion of the
pressure gradient as a breakdown of the complex pressure gradient into simple sine and
cosine functions, or „harmonics‟. When summed together, the harmonics will
approximately resemble the original pressure gradient waveform. Figure (2.3) shows a
plot of the pressure gradient over time for different Fourier series expansions with a
varying number of harmonics.
Figure 2.3 - Fourier series representation of cardiac waveform from Figure 2.2 for 2, 20, and 60
harmonics.
12
From Fig. (2.3), we observe that increasing the number of harmonics gives a
better approximation of the original pressure gradient waveform. However, too many
harmonics (60 in the plot) can result in undesirable oscillations of the waveform.
In this thesis, the number of harmonics used to describe the pressure gradient
waveform for each simulation differed (Table 2.1). This is because of the period (T, see
below) differences in each waveform. For example, in simulations with pulsatile flow
without PC, the period T of the waveform was approximately 1s. On the other hand,
simulations of steady and pulsatile flow with PC had T = 210s, since the pressure
gradient over the entire PC procedure was treated as a periodic waveform; doing this
allowed for easier implementation of a transient velocity into Model 3. Steady flow is not
pulsatile; however, when PC is applied to it, it becomes periodic in time, enabling it to be
expressed in a Fourier series expansion.
Table 2.1 - Number of harmonics used in describing pressure gradient for each simulation. Simulation Number of Harmonics
Steady flow without PC 0 (steady-state problem)
Steady flow with PC 100
Pulsatile flow without PC 50
Pulsatile flow with PC 5000
In Eq. (2.7), ω is the fundamental frequency of the waveform, and can also be
written as 2
T
, where T is the period, in seconds, of the waveform. The coefficients A0,
An, and Bn are written as:
00
0
0
1( )
2( )cos
2( )sin
T
T
n
T
n
A f t dtT
A f t n t dtT
B f t n t dtT
(2.8)
13
where n is the harmonic number and f(t) is the original waveform function. Since there is
no explicit function for the waveform, f(t) must be presented in tabular form as a set of
data points over time, rather than analytical form37.
Thankfully, Fast-Fourier Transform (FFT) computer programs have been
developed which can take in data points of a complex waveform over one period and
output the value of the coefficients in Eq. (2.8). For example, pressure measurements
over time taken for a particular flow can be directly implemented into an FFT program.
The program will then output a number of Fourier series coefficients that is equal to the
number of data points that were put into the program. For instance, a dataset for the
pressure gradient over time with 50 data points will have 50 Fourier series coefficients.
For the reader interested in seeing how this is done by hand, Zamir37 provides an
excellent description of the process.
Equation (2.7) gives the Fourier series expansion of the pressure gradient in
trigonometric form. A much easier way to write this is in complex exponential form:
0
Ren
i t
n
dpa e
L
(2.9)
where i is the imaginary number and
n n na A B i (2.10
written in complex number form. The pulsatile fluid velocity is solved for by calculating
the fluid velocity for each harmonic component of the pressure gradient waveform.
Substituting a harmonic of the pressure gradient waveform into Eq. (2.4) and adding in
the transient velocity term from Eq. (2.1), the simplified Navier-Stokes equation
governing pulsatile flow in a straight tube becomes:
14
2
2
1i tn n nn
w w wa e
t r r r
(2.11)
where the subscript n on each velocity term demonstrates that the velocity being
computed is only for one harmonic of the pressure gradient waveform. A characteristic
solution38 to Eq. (2.11) is:
( , ) ( ) in t
n nw r t f r e (2.12)
where fn(r) is the time-independent part of the velocity. When this is substituted into Eq.
(2.11), the time-dependent parts of each term cancel, leaving:
2
2
( ) 1 ( )( ) n n
n n
f r f rin f r a
r r r
(2.13)
Equation (2.13) is an ordinary differential equation that depends only on the r-
direction and does not depend on time, t. This demonstrates that the time and space
dimensions are treated separately when solving for pulsatile flow.
Equation (2.13) also takes on the form of a zero-order Bessel differential
equation15 where its solution is a zero-order Bessel equation represented as:
0
0
( )( ) 1
( )
nn
a J rf r
i n J R
(2.14)
where J0 is a zero-order Bessel function of the first kind15 and λ is: 3i n
(2.15)
Equations (2.14) & (2.15) can be substituted into Eq. (2.12) to obtain the pulsatile
velocity for one harmonic of the pressure gradient waveform:
0
0
( )( , ) Re 1
( )
in tnn
a J rw r t e
i n J R
(2.16)
15
Summing all of the harmonic components of the velocity yields the Womersley solution
for the fluid velocity profile in pulsatile flow38:
0
1
( , ) ( ) ( , )n
nw r t w r w r t (2.17)
where w0(r) is the time-averaged velocity profile of the pulsatile waveform. Generally,
this value is equal to the velocity obtained for steady flow. If we integrate Eq. (2.16) over
the cross-sectional area of the tube, we can obtain the pulsatile flowrate for one harmonic
of the pressure gradient waveform:
4
1
2
0
2 ( )( ) Re 1
( )
in tnn
a R J rQ t e
i n RJ R
(2.18)
where Ω is the Womersley number37:
R
(2.19)
The Womersley number is a dimensionless value which characterizes pulsatile
flow similarly to how the Reynolds number characterizes laminar and turbulent flows.
Each harmonic of the flowrate waveform can be summed to obtain the Womersley
solution for pulsatile flowrate:
0
1
( ) ( )n
nQ t Q Q t (2.20)
where Q0 is the time-averaged value of the pulsatile flowrate waveform. Similar to
pulsatile velocity, this value is equal to the flowrate calculated in steady flow.
2.3 Steady and Pulsatile Flow in Bifurcating Tubes
Bifurcations, or branches, are the most common type of geometry seen in the
human vascular system. A bifurcation occurs when a parent vessel divides, or bifurcates,
into two daughter vessels(Fig. 2.4). As the bifurcations of arteries progress from larger to
16
smaller vessels, the daughter branches become parent branches for subsequent
bifurcations, as can be seen in Fig. (2.4). For a single bifurcation, the parent vessel P will
always be the largest of the three vessels. Assuming that the bifurcation is not symmetric
(the daughter branches do not have the same radii), there will be a large and a small
daughter branch, denoted as D1 and D2, respectively. θ1 and θ2 are the angles between P
and D1, and between P and D2, respectively.
For a single bifurcation, the pressure drop along each vessel segment can be
calculated by realizing that the pressure at the end of the parent vessel is equal to the
pressure in the beginning of both daughter branches. To express this mathematically, we
can rewrite Equation (2.6) as a function of vessel length for each vessel segment37:
0 0 0 0 04
0
8( ) (0)p x p Q x
R
1 1 0 0 1 14
1
8( ) ( )p x p L Q x
R
(2.21)
2 2 0 0 2 24
2
8( ) ( )p x p L Q x
R
where p, x, L, µ, R, and Q are the pressure, axial distance along the branch, the length of
the branch, the viscosity of the fluid, the radius of the branch, and the flowrate through
the branch, respectively. The subscripts 0, 1, and 2 denote the parent vessel, the larger
θ1
θ2
P
D2
D1
Figure 2.4 - Schematic of arterial bifurcations. The parent vessel is denoted as P; the
two daughter vessels D1 and D2 become new parent vessels for successive bifurcations.
17
daughter branch vessel, and the smaller daughter branch vessel, respectively. If we
assume that the pressure drop in the parent vessel is known, we can normalize the
pressure drop through each vessel segment with respect to the parent vessel‟s pressure
drop:
0 0 0( )P X X
4
0 1 11 1 1
1 0 0
( ) 1R Q L
P X XR Q L
(2.22)
4
0 2 22 2 2
2 0 0
( ) 1R Q L
P X XR Q L
where X0, X1, and X2 are the length coordinates for each vessel segment normalized by
the vessel‟s respective length. Assuming that the vessel lengths are proportional to their
respective radii37 results in reducing the pressure drop equations for the daughter vessel
segments in Eq. (2.22):
3
0 11 1
1 0
( ) 1R Q
P X XR Q
3
0 22 2
2 0
( ) 1R Q
P X XR Q
2.23
It has been observed that within the vascular system a power-law relationship exists
between the vessel radii and its corresponding flowrate37, such that:
0 1 2 0 1 2Q Q Q R R R (2.24)
where γ is the power-law index. Implementing this relationship into Eq. (2.23):
3
1
1 1( ) 1 1P X X
18
31
2 2
1( ) 1P X X
(2.25)
The term α in Eq. (2.25) is the bifurcation index, which provides a relationship
between the radii of the daughter vessels. The bifurcation index is a ratio of D2 to D1, so
its value is always between 0 and 1. For symmetric bifurcations, the bifurcation index is
equal to 1. Equation (2.25) can now be used to calculate the pressure drop through each
daughter vessel segment; their corresponding flowrate is calculated using Eq. (2.6).
When dealing with more than one bifurcation, as in the human coronary arteries,
the subscripts used in Eq. (2.21-25) must be generalized to account for a larger number of
vessels with an arterial tree. Zamir37 discussed a vessel numbering scheme that involves
two coordinates: the first coordinate being the „level‟ of the vessel in the arterial tree and
the second coordinate being the „sequential‟ position of the vessel within that specific
level of the tree. The „level‟ position corresponds to the bifurcation generation that the
vessel segment is located within, while the „sequential‟ position gives the location of the
vessel within a specific level. The level and sequential position coordinates are denoted
by the ‘j’ and ‘k’ subscripts used for each vessel segment, respectively(Fig. 2.5). The first
vessel of the tree structure is denoted as level „0‟, and is known as the root segment. The
vessels at the end of the tree are known as terminal vessel segments.
19
Assuming that the pressure drop in the root vessel segment is known, this
numbering system proposed by Zamir along with Eq. (2.24-25) can be used to calculate
the pressure and flowrate distribution throughout an entire arterial network.
For pulsatile flow through branching tubes, the process for calculating the
pressure and flowrate distribution is similar to that described in Section 2.2 for a straight
tube. Each harmonic of the complex pressure gradient waveform is solved for separately,
for each vessel segment. The harmonics are then summed to obtain the total pressure
drop over time for each vessel segment. The flowrate distribution is calculated in the
same manner37.
2.4 Mass Transport of Oxygen in Capillaries and Tissues
There are over 16 billion capillaries in the human body6, and over 3 million in the heart
alone11. Furthermore, the capillaries of the heart do not consist solely of straight and
branching tubes; the capillaries have a net-like structure with a complex topology (Fig.
2.6). Modeling fluid flow and mass transport of oxygen in all of these capillaries would
require a significant amount of computing power and time. Based on these observations,
a simplification in the geometry must be made to perform any modeling.
0,1
1,2
2,2 2,1
1,1
j,k
j+1,2k j+1,2k-1
Figure 2.5 - Vessel numbering scheme and notation for an arterial network, as
proposed by Zamir37
, specific and general cases.
20
Figure 2.6 - Schematic of vessels in systemic circulation (top) and topology of capillary
vessels (bottom). Images are taken from Dawson7 (top) and Doohan
8 (bottom)
The Krogh tissue model is a widely used geometric model which provides a
starting point for studying the transport of biological species in the body10. First
developed by August Krogh in 1919 to analyze oxygen transport in skeletal muscle, the
model idealizes the capillary-tissue structure by simplifying the capillary to a cylinder
which is concentrically surrounded by a cylindrical tissue region. Although a highly
simplified representation of the capillaries, the Krogh tissue model provides valuable
21
insight into the distribution of oxygen in a tissue region that surrounds a capillary. Based
on this reasoning, this work focuses on analyzing the blood flow and oxygen transport
through a single Krogh tissue model (Fig. 2.7).
The geometry represented in Fig. (2.7) will be used for all subsequent models that
involve the mass transport of oxygen.
Oxygen is carried by the blood from the lungs to capillary beds in two forms10:
bound to hemoglobin in the red blood cells, and
dissolved in plasma
To account for both transport mechanisms, a mass balance over the proposed Krogh
tissue model must be done for both forms of oxygen. In general, a mass balance for a
finite volume follows the equation below:
in out generated
bb b b
dCC C C
dt (2.26)
where Cb represents the total concentration of oxygen. In words, the total change in the
concentration of oxygen over time must equal the concentration of oxygen into the
volume minus the concentration of oxygen leaving the volume, plus any oxygen that is
Blood In Blood Out
Tissue
Capillary
Rt
Rc
L
z
r
Figure 2.7 - Schematic of Krogh tissue Model. Rc is the radius of the capillary, Rt is the
radius of the surrounding tissue, and L is the length of the capillary. Blood within the
capillary flows from left to right.
22
generated within the volume. If we begin by doing a mass balance on the Krogh tissue
model for oxygen bound to hemoglobin only, we get:
'(2 ) 2 ' 2 ' 2HbOz z z
Cr r z r rV C r rV C R r r z
t
(2.27)
where C’ represents the concentration of oxygen bound to hemoglobin, V is the average
velocity of blood in the capillary, and RHbO is the volumetric production rate of
oxygenated hemoglobin. Blood is assumed to be in „plug flow‟, meaning that the velocity
profile is uniform rather than parabolic - like in larger arteries. Also, since oxygen is only
being carried by hemoglobin in Eq. (2.27), diffusive transport is nonexistent and oxygen
is only transported by convection due to bulk flow10. By dividing Eq. (2.27) by 2 r r z ,
we can obtain the mass balance of oxygen bound to hemoglobin in differential form:
' '
HbO
C CV R
t z
(2.28)
A similar mass balance can be done for the oxygen that is dissolved in plasma. In
addition to being convectively carried away by blood flow, the oxygen dissolved in
plasma is transported in the axial and radial directions by diffusion, thus:
(2 ) 2 2 ( 2 ) ( 2 )
( 2 ) ( 2 ) 2
z z zr r r
OXYGEN
z z z
C C Cr r z r rV C r rV C r z D r z D
t r r
C Cr r D r r D R r r z
z z
(2.29)
and in differential form:
2
2
1r z OXYGEN
C C C CV D r D R
t z r r r z
(2.30)
Total Concentration Convection Radial Diffusion
Axial Diffusion Reaction
23
where C is the concentration of oxygen dissolved in the plasma, and Dr and Dz are the
radial and axial diffusivities of oxygen in blood, respectively. The reaction term, ROXYGEN
is the volumetric production rate of oxygen in plasma. If the reaction between oxygen
and hemoglobin is assumed to be in equilibrium10, the reaction terms in Eq. (2.28) and
(2.30) are equal and opposite.
The concentrations of oxygen bound to hemoglobin and dissolved in plasma are
related through the hemoglobin oxygen-dissociation curve (ODC). The ODC is an
equilibrium curve that gives the fractional saturation of hemoglobin with oxygen for a
specific concentration of oxygen dissolved in plasma10. Figure (2.8) below is a depiction
of the ODC.
Figure 2.8 - Hemoglobin oxygen-dissociation curve (ODC). The x-axis is the concentration of oxygen
dissolved in plasma (given in partial pressure of O2) and the y-axis is the percent saturation of
hemoglobin with oxygen31
.
A mathematical relation of the ODC can be represented by the Hill Equation10:
24
50
50
1
n
n
CC
CC
(2.31)
where ψ is the percent saturation of hemoglobin with oxygen and C50 is concentration of
oxygen dissolved in plasma when hemoglobin is 50% saturated. The term n is known as
the Hill parameter10. The relation between C and C‟ are as follows:
' '
' '
C C C Cm
t t C t
C C C Cm
z z C z
(2.32)
where m is related to the derivative, or slope, of the ODC by:
''sat
C dm C
C dC
(2.33)
where C‟sat is the concentration of oxygen bound to hemoglobin when hemoglobin is
100% saturated. With this relation, Eqs. (2.28) and (2.30) can be combined to express the
concentration of oxygen dissolved in plasma taking into account the dissociation of
oxygen from hemoglobin:
2
2
1(1 ) (1 ) r z
C C C Cm V m D r D
t z r r r z
(2.34)
The transport of oxygen in the tissue region is governed only by diffusion, since the
convective effects of blood flow are not present. For oxygen in the tissue, the mass
balance equation becomes:
(2 ) ( 2 ) ( 2 )
( 2 ) ( 2 ) 2
T T TT T
r r r
T TT T
METABOLIC
z z z
C C Cr r z r z D r z D
t r r
C Cr r D r r D R r r z
z z
(2.35)
25
and in differential form:
2
2
1T T TT T
r z METABOLIC
C C CD D R
t r r r z
(2.36)
where CT is concentration of oxygen in the tissue, Dr
T and Dz
T are the radial and axial
diffusivities for oxygen in tissue, and RMETABOLIC is the metabolic consumption rate of
oxygen. Eqs. (2.34) and (2.36) represent the governing equations used to solve for the
concentration of oxygen in the capillary and tissue regions of the Krogh tissue model.
26
CHAPTER 3
COMPUTATIONAL METHODS
This chapter will discuss the computational methods used to solve the models
outlined in Chapter I. First, a step-by-step protocol will be presented. Second, each model
will be described in detail regarding geometry, governing equations, and boundary
conditions.
The terms „model‟ and „simulation‟ used throughout this chapter have two distinct
meanings. „Model‟ will refer the type of geometry being solved for (i.e., Model 1, Model
2, or Model 3), while „simulation‟ will refer the specific solution type of each model:
Steady flow without PC
Steady flow with PC
Pulsatile flow without PC
Pulsatile flow with PC
3.1 Finite Element Method – COMSOL
Models 1 and 3 used the finite element method (FEM) using the commercial
software package COMSOL®5. COMSOL is a finite element analysis software that has
the capability to solve a wide range of physics problems such as heat transfer, structural
mechanics, and fluid dynamics. It is also able to solve problems where the physics are
coupled. Using FEM, the partial differential equations (PDE‟s) that govern a particular
type of physics are approximated as a system of ordinary differential equations (ODE‟s)
by breaking down a continuous computational domain into a mesh of discrete elements.
27
This mesh can be refined to obtain a better approximation of the original PDE‟s and a
more accurate solution to the physics problem. The ODE‟s are then solved by integration
with appropriate boundary conditions.
COMSOL is a widely used modeling tool which is verified with experimental
data. A common benchmarking problem to check the accuracy of COMSOL‟s finite-
element method is the laminar flow over a backstep problem (Fig. 3.1). COMSOL‟s
ability to model experimental findings accurately was validated against observations for
laminar flow over a backstep while varying the Reynolds number between 70 and 412[1].
The results showed that the difference between the computational solutions and the
experimental data for the flow recirculation length downstream of the step (Fig. 3.1) was
between 0.9 and 9% over the specified range of Reynolds numbers1. Taking into account
uncertainties in experimental measurements, these observations demonstrate that
COMSOL can accurately model experimental findings with its finite element method.
3.2 Numerical Integration and Fourier Operations – MATLAB
MATLAB®24 is a computer program that can perform mathematical and
numerical operations in a high-level programming environment. Specifically, the
Backstep Recirculation zone
Recirculation length
Figure 3.1 - Schematic of backstep problem used for numerical analysis validation. The
black arrows show the direction of flow. Lines within the fluid domain denote
streamlines. In the backstep problem, a recirculation zone forms immediately
downstream of the backstep.
28
program can be used for numerical studies as well as to perform the FFT analysis as
described in Chapter 2. The programming can also be integrated into a variety of other
programs, including COMSOL.
Since Model 2 contains a large amount of vessels in which the flowrate and
resulting velocity need to be calculated, using the FEM to solve that model would be
computationally expensive. As a result, a program was written in MATLAB instead.
3.3 Computer Hardware
All simulations were performed on an HP Z400 Workstation with a 3.20 GHz
Intel Zeon Quad-Core processor and 23.9 GB of RAM running Windows XP
Professional 64-bit Edition.
3.4 Breakdown of Models
This section provides a brief description of the models used in this study. The reader is
referred to Fig. (3.2) for a visual of each model, as well as a depiction of the modeling
process.
Model 1
Model 1 represented a portion of the human LAD artery which contains the
angioplasty balloon catheter. We placed the catheter in the LAD because this is a
common site for atherosclerotic plaque buildup17. For each simulation, the outlet flowrate
was recorded.
One input requirement for Model 2 was that the pressure drop in the root vessel
segment must be known. To determine this pressure drop, we created an „intermediate‟
model in COMSOL, Model 1b. This intermediate model was a straight tube that had
dimensions equal to that of the root vessel segment for Model 2. The outlet flowrate from
29
Model 1 was used as the inlet flowrate for this model, and the resulting pressure drop
over the length of the model was recorded. Finally, the recorded pressure drop was
inputted into Model 2.
Model 2
Model 2 represented the coronary arterial network (Fig. 3.2). The model was
designed to mimic the coronary vasculature that is downstream of Model 1a. From a
physiological standpoint, Model 2 branches from Model 1a. The mathematical equations
that describe blood flow through branching tubes from Chapter II were used to develop
the MATLAB program. The arterial tree generated by MATLAB contained at least one
capillary vessel segment where its corresponding flowrate and average velocity was
recorded. The velocity was used as the inlet velocity condition for Model 3.
Based on the large number of vessel segments contained within the coronary
vasculature, a finite-element solution would have been prohibitively demanding from a
computational perspective. Furthermore, Model 2 would have required that we specify
pressure boundary conditions for every outlet in COMSOL. A pressure measurement for
every coronary vessel down to the capillary level is not available in literature, so our
modeling efforts focused on simplifying Model 2 to a 1-dimensional model.
Model 3
Model 3 was a Krogh tissue geometry that represented a coronary capillary and
surrounding tissue. The transport of oxygen from the capillary to the surrounding tissue
was studied. The outlet velocity recorded from Model 2 for each simulation was used as
the inlet velocity condition for this Model.
30
3.5 Breakdown of Simulations
The simulations that were ran for each model varied by type of flow condition:
Steady flow without applied PC procedure
Steady flow with applied PC procedure
Pulsatile flow without applied PC procedure
Pulsatile flow with applied PC procedure
Steady flow simulations were ran initially for simplicity and verification. Pulsatile flow
simulations were then studied for two reasons:
To add physiological realism to our models, and
To determine if the pulsatility of the blood flow significantly impacted the
transfer of oxygen in the capillaries.
3.6 Modeling Protocol
Figure (3.2) shows schematically how post-conditioning fluid mechanics and
mass transfer were solved. A step-by-step protocol of the modeling process is described
below.
31
Model Protocol
1. Solve Model 1a.
a. For simulations without the applied post-conditioning procedure, record
the outlet flowrate and proceed to step 2.
b. For simulations with the applied post-conditioning procedure, solve Model
1a without the applied PC procedure and record the pressure at the inlet.
Then re-solve Model 1a with this pressure condition at the inlet. Record
the outlet flowrate.
Model 1a Model 1b
Model 2
Model 3 Outlet from Model 2
Outlet from Model 1b
Outlet from Model 1a
Figure 3.2 - Schematic of modeling pathway.
32
i. This is necessary because a flowrate condition cannot be
prescribed at the inlet when the balloon fully occludes the vessel.
2. Solve Model 1b.
a. Use recorded outlet flowrate from Model 1a as inlet flowrate condition for
model.
b. Record pressure at inlet for each simulation. Since the pressure is set to
zero at the outlet, this is actually the pressure drop through the vessel
segment.
3. Solve Model 2.
a. Use recorded pressure drop from Model 1b as pressure drop for root vessel
segment of model.
b. Run program and solve for the flowrate distribution. Record the capillary
outlet velocity.
4. Solve Model 3
a. Use recorded capillary outlet velocity as inlet velocity for capillary
domain.
b. Run model; analyze concentration of oxygen in capillary and tissue
regions for each simulation.
The next sections will describe each of the models used in detail.
3.7 Model 1 (1a & 1b)
Geometry
Figures (3.3) and (3.4) show the geometries used for Models 1a and 1b,
respectively. Model 1a is a 2-dimensional axial-symmetric straight tube geometry that
33
represents a section of the LAD artery containing a catheter shaft (Fig 3.3 cross-hatch).
Model 1b is also a 2-dimensional axial symmetric geometry, but without the catheter
shaft. Model 1b acts as a downstream section of the LAD artery from Model 1a. Tables
(3.1) and (3.2) provide the dimensions for Models 1a and 1b, respectively. The diameter
of the LAD artery is taken from literature6; the diameter of the catheter shaft is measured
from an actual balloon angioplasty catheter. The lengths of Model 1a and 1b have been
chosen such that their cumulative length does not exceed the typical length of an LAD
artery6.
Table 3.1 - Dimensions for Model 1a geometry.
Parameter Name Value Description
L_LAD 0.030 m Length of LAD artery
R_LAD 1.588x10-3
m Radius of LAD artery
L_CATH 0.025 m Length of catheter shaft
R_CATH 4.953x10-4
m Radius of catheter shaft
Table 3.2 - Dimensions for Model 1b geometry.
Parameter Name Value Description
R_LAD 1.588x10-3
m Radius of downstream LAD artery
L_LAD-ROOT 0.016 m Length of downstream LAD artery
L_CATH
L_LAD
R_LAD
R_CATH
Figure 3.3 - Model 1a geometry.
L_LAD-ROOT
R_LAD
Figure 3.4 - Model 1b geometry.
34
Governing Equations
The equations governing blood flow through Models 1a and 1b are the Navier-
Stokes and Conservation of Mass equations described in Section 2.1. For both models,
we assumed blood as a Newtonian fluid and treated the vessel walls as rigid.
Blood can be modeled as a Newtonian fluid if the diameters of the vessels under
study are much larger than the diameter of a red blood cell (7 μm). Waite and Fine34 state
that if vessels have a diameter less than 1mm, the viscosity of blood is no longer constant
and will behave as a non-Newtonian fluid. The diameter of Model 1a and 1b is 3.176
mm, and therefore the Newtonian fluid assumption is valid.
The LAD in the heart is elastic and will move in response to changes in fluid
pressure and/or muscle contractions. Although studies have shown that the elasticity of
vessels can increase wall shear stress (WSS) magnitude by 50% with a 2-10% variation
in vessel diameter27, Zamir demonstrated that the pulsatile flow rate and velocity profiles
in a straight elastic tube were not significantly different than those for a rigid tube37.
Based on these observations, Models 1a and 1b were assumed rigid.
Balloon Modeling
Our method to model the moving balloon domain in Model 1a was an infinite
viscosity method. Computational fluid dynamics (CFD) models have used this approach
to simulate an oscillating valve pin4. In the models that include a PC procedure, the fluid
viscosity term in Eq. 2.1 was changed from a normal value to a value several orders of
magnitude greater, simulating a solid region. Unlike typical fluid-structure interaction
(FSI) models, a solid domain that represents the moving obstacle in the flow field is not
35
needed. This approach simplifies the model significantly by eliminating the need to
couple fluid mechanics equations with solid mechanics equations.
As seen in Fig. (3.3), the time-dependent balloon domain was not a simple shape.
For this reason, the balloon was broken down into three main sections to simplify the
infinite viscosity method. Two elliptical sections for the ends of the balloon and one
cylindrical section for the middle of the balloon were created (Fig. 3.5).
Figure 3.5 - Sectioning of balloon domain. Balloon is broken into two elliptical sections and one
cylindrical section.
To describe the motion of the balloon during inflation and deflation, an
expression was written for each section of the balloon. The expression for each section is
a step function with a specific argument such that the output was 1 in the balloon section
and 0 elsewhere. Eqs. (3.1) through (3.3) show the expressions written for each balloon
section:
1 1 1
2 2 2
3 3 3
balloon xrange rrange
balloon xrange rrange
balloon xrange rrange
(3.1-3)
36
where xrange represents the length of each balloon section along the x-direction, and
rrange is the radial distance of each balloon section from the catheter shaft. The
subscripts 1, 2, and 3 are for the proximal balloon end, middle balloon section, and distal
balloon end, respectively. For each simulation, xrange is a constant, and can be written as
a product of step functions that will equal 1 if and only if an x value is between the
specified limits on the balloon length (Fig. 3.6). Eqs. (3.4-6) are the functions of xrange
for each balloon section.
Figure 3.6 - Diagram showing x-ranges for each balloon section. x1 is the lower limit on the proximal
balloon end, x2 is the upper limit on the proximal balloon end and the lower limit on the middle
balloon section, x3 is the upper limit on the middle balloon section and the lower limit on the distal
balloon end, and x4 is the upper limit on the distal balloon end.
1 1 2
2 2 3
3 3 4
2 ( , ) 2 ( , )
2 ( , ) 2 ( , )
2 ( , ) 2 ( , )
xrange flc hs x x scale flc hs x x scale
xrange flc hs x x scale flc hs x x scale
xrange flc hs x x scale flc hs x x scale
(3.4-6)
In Eqs. (3.4-6), flc2hs is COMSOL‟s smoothed Heaviside step function with a
continuous second derivative without overshoot. In the function, the first argument (
or xn nx x x ) is a logical expression given by the user. It is evaluated as 1 when the
expression is greater than 0 (true), and will be 0 otherwise (false). For example, when
x1<x<x2 in xrange1, both step functions in xrange1 are „true‟ and will return a 1, stating
x
r
37
that the certain x-value is within the proximal balloon section‟s x-range. The second
argument scale is a user-defined value that tells flc2hs to smooth the step transition of the
logical expression between –(scale) and scale. This smoothing prevents COMSOL from
divergent solutions. In our modeling, the smoothed transition needed to be precise so a
clear distinction between the balloon wall and the surrounding fluid was made. For this
reason, scale is set to 1*10-10
.
Since the balloon inflated and deflated in the radial direction, rrange for each
section was time-dependent (Fig. 3.7). Equations (3.7-9) below were the functions of
rrange for each balloon section.
Figure 3.7 - Diagram representing the r-ranges for each balloon section. Each section has the same
lower radial limit r0 that is equal to the diameter of the catheter shaft. The upper limits, which are a
function of time, are r1(t), r2(t), and r3(t): the upper limit on the proximal balloon end, the middle
balloon section, and the distal balloon end, respectively.
1 0 1
2 0 2
3 0 3
2 ( , ) 2 ( ( ) , )
2 ( , ) 2 ( ( ) , )
2 ( , ) 2 ( ( ) , )
rrange flc hs r r scale flc hs r t r scale
rrange flc hs r r scale flc hs r t r scale
rrange flc hs r r scale flc hs r t r scale
(3.7-9)
In Eqs. (3.7) through (3.9), the step functions returned a 1 when r is greater than
r0 and less than r1, r2, or r3. The time dependent functions r1, r2, r3 were represented by
Eqs. (3.10-12):
r1(t)
x
r
r2(t)
r3(t)
r0
38
2
21 2
2 1
2
2
33 2
4 3
( )( ) ( ) 1
( )
( ) ( )
( )( ) ( ) 1
( )
x xr t R t
x x
r t R t
x xr t R t
x x
(3.7.10-12)
where r1 and r3 were derived from the equation for an ellipsoid, and R(t) is the varying
distance of the balloon wall from the catheter shaft wall. R(t) is written as:
1 1( ) [ 2 ( , ) 2 ( , ) 2 ( , ) 2 ( , )]on off onk offkR t d flc hs t t dt flc hs t t dt flc hs t t dt flc hs t t dt
(3.13)
where d is the distance from the catheter shaft wall to the vessel wall. The bracketed part
of Eq. (3.13) represented the user-specified post-conditioning (PC) algorithm with k
reperfusion/occlusion cycles. As time progresses in the model, the balloon inflated when t
becomes greater than ton, and deflated when t is greater than toff. The inflation and
deflation rates were specified by the step function‟s smoothing parameter dt, where the
total time for the balloon to fully inflate or deflate is 2*dt. Our models have dt = 0.5
seconds, allowing a realistic 1-second inflation/deflation rate (Fig. 3.8).
39
Figure 3.8 - Diagram representing the step inflation and deflation rate for one cycle of a PC
algorithm. Inflation took place at ton, while deflation occurred at toff. By using the smoothing
parameter of COMSOL’s flc2hs function, the inflation and deflation step was smoothed over an
interval of 2*dt. This interval was centered over ton and toff.
By substituting Eqs. (3.4) through (3.13) back into Eqs. (3.1) through (3.3), an
expression for the viscosity becomes:
0 inf 1 inf 2 inf 3balloon balloon balloon (3.14)
where µ0 is the fluid viscosity, 3.600 x 10-3
Pa-s, and µinf is a user-defined large viscosity,
defined as 1.000 x 104 Pa*s. By applying Eq. (3.14) to the fluid subdomain, a pulsating
balloon region can be created with this infinite viscosity approach.
Boundary and Initial Conditions
Figure (3.9) describes the boundary conditions for each simulation that were
implemented on Model 1a. A symmetry condition was imposed at the centerline of the
vessel because the geometry is 2-D axial-symmetric; the walls of the vessel and catheter
shaft had a „No-Slip‟ condition which constrained the velocity to zero along them; at the
outlet, a zero pressure condition was imposed. For steady flow simulations, the inlet was
specified with a flowrate that was the time-averaged value for one cardiac cycle of the
pulsatile flowrate waveform used in pulsatile flow simulations.
0
1
2*dt 2*dt
ton toff
40
The pulsatile flowrate waveform used in our simulations was based off of the
waveform used by He and Ku17 for the left coronary artery (LCA). To obtain our flowrate
waveform for the LAD artery, He and Ku‟s waveform was multiplied by 35.6%,
representing a physiologic flow division from the LCA to the LAD and left circumflex
artery (LCX) of approximately 36%:64% (LAD:LCX)17. The flowrate waveform used in
our simulations is depicted in Fig. (3.10).
Figure 3.10 - Pulsatile flowrate waveform used in pulsatile flow simulations with and without PC.
Waveform shown is for one cardiac cycle. For steady flow simulations, the time-averaged value of the
pulsatile flowrate waveform for one cardiac cycle was used.
Figure (3.11) describes the boundary conditions implemented on Model 1b for
each simulation. The conditions at the wall, outlet, and centerline of the vessel were
Wall – no slip
Inlet
steady or pulsatile flowrate Outlet
zero pressure Axial symmetry
Figure 3.9 - Boundary conditions for Model 1a.
41
similar to those implemented for Model 1a. At the inlet, the recorded outlet flowrates
from Model 1a were used as the inlet flowrate condition.
The initial conditions for each transient simulation (steady flow with PC, pulsatile
flow without and with PC) were given by the solution to the steady flow without PC
simulation. This was done by storing the steady flow without PC solution and specifying
its velocity and pressure fields over the computational domain as initial values for the
transient simulations.
Mesh Statistics
A mesh convergence analysis was performed on Model 1a. This analysis
determined the number of mesh elements required in the computational domain so that
the solution becomes independent of the mesh size. The convergence study was
performed on the steady flow without PC simulation only; the same geometry and mesh
were used for all other simulations.
Inlet
(outlet from Model 1a)
Outlet
zero pressure
Wall – no slip
Axial symmetry
Figure 3.11 - Boundary conditions for Model 1b.
42
Figure 3.12 - Plot of average velocity over entire computational domain vs. mesh size for Model 1a.
The final mesh used for all simulations contained 4,472 quadrilateral elements (dashed circle in
graph).
The convergence criterion for the mesh in Model 1a was set so convergence was
achieved when the average velocity over the computational domain changed by less than
0.2% with an increase in mesh size. The criterion was satisfied with a mesh element size
of 4,470 quadrilateral elements (Figure 3.12). Figure (3.13) below shows the final mesh
used for Model 1a. Table (A.1) in Appendix A provides the solver settings used in
COMSOL, as well as the solution times for each simulation of Model 1a.
Figure 3.13 - Mesh for Model 1a. The bottom portion of the figure shows a close-up section of the
mesh.
43
A mesh convergence analysis was also performed on Model 1b. The convergence
study was performed on the steady flow without PC simulation only; the same geometry
and mesh were used for all other simulations. Using the same criteria as in Model 1a,
mesh convergence was achieved with a quadrilateral mesh size of 2,762 elements (Fig.
3.14). Figure (3.15) below shows the final mesh used for Model 1b. Table (A.2) in
Appendix A provides the solver settings used in COMSOL, as well as the solution times
for each simulation of Model 1b.
Figure 3.14 - Plot of average velocity vs. mesh size for Model 1a. The final mesh used for all
simulations contain 4,472 quadrilateral elements (dashed circle in graph).
Figure 3.15 - Mesh for Model 1b.
Results from Models 1a and 1b are briefly summarized below. Figure (3.16)
below shows the velocity field in Model 1a for the steady flow with PC simulation at
three different times: a fully deflated balloon state, a partially inflated balloon state, and a
fully inflated balloon state. The figure demonstrates that the no-slip condition is still
44
valid, even when using the infinite viscosity method to represent a solid pulsating
balloon.
Figure 3.16 - Velocity field in Model 1a for the steady flow with PC simulation at three different
times: a) fully deflated balloon state, b) partially inflated balloon state, and c) fully inflated balloon
state. Some fluid velocity was seen in 3.15c, however its maximum value was approximately 6 orders
of magnitude smaller than the maximum velocity in 3.15a. The units for velocity in the color legend
are in m/s.
Table (3.3) provides the pressure drop recorded in Model 1b for each simulation.
Two plots are shown for the pulsatile flow with PC simulation (full scale and zoomed) to
show the pressure drop waveform over time. For simulations that include the PC
procedure, the graphs show that the pressure drop goes to zero as the balloon fully
inflates and occludes the vessel upstream during the ischemic (balloon inflation) stages of
PC.
3.16a
3.16b
3.16c
45
Table 3.3 - Recorded pressure drop for each simulation of Model 1b.
Simulation Recorded pressure drop
Steady flow without
PC 22.91 Pa
Pulsatile flow
without PC
Steady flow with PC
Pulsatile flow with
PC
46
Pulsatile flow with
PC – close up
3.8 Model 2
Model 2 represents the coronary arterial network that branches off from the LAD
(Model 1b) and terminates at the coronary capillary level. As stated earlier, the geometry
and the flowrate through each vessel of the network is computed using a program
developed in MATLAB (see Appendix B) which follows the bifurcation theory for
branching tubes, as explained in Chapter 2. The assumptions used for Model 2 are
Newtonian blood flow, rigid vessels, and one-dimensional blood flow.
In Model 2, the diameter of the vessel segments in the arterial tree become small
enough where non-Newtonian effects may impact vessel flowrate. Huo and Kassab19
showed that a constant viscosity underestimates the flowrate in smaller vessels by
approximately one order of magnitude when compared with a viscosity that is a function
of diameter. However, the ultimate goal of Model 2 was to solely obtain a physiological
outlet capillary velocity, and not analyze the entire flowrate distribution through the
model. The bifurcation index and power law index parameters of Model 2 were varied to
achieve this; thus the effect of non-Newtonian viscosity was not considered.
47
The goal for the MATLAB program was to generate a coronary network that
would begin at a root vessel segment diameter of 3.176 mm (average LAD dia. in
human6) and contain at least one capillary vessel segment with a diameter of 6.75 µm6 to
determine an average capillary blood velocity.
To develop the program in MATLAB, we needed to apply bifurcation theory
from Chapter 2. Recall that for an asymmetric bifurcation, there will always be a larger
and a smaller daughter branch that will bifurcate from the parent branch. For a network of
asymmetric bifurcations, there are many pathways that can lead to the capillary vessels
from the root vessel segment; however the two „bounding‟ pathways are either the one
that follows the largest daughter branches or the one that follows the smallest daughter
branches (Fig. 3.17)37. By following the pathway with the smallest daughter branches, we
can reach a capillary vessel segment from the root vessel segment with the least amount
of bifurcations and vessels in between.
To determine the number of levels in the arterial tree, a relationship must be made
between the root vessel segment and the capillary vessel segment. In Chapter 2, Eq. 2.24
showed that a power-law exists between the vessel radii in a single bifurcation. This
Small daughter branches
Large daughter branches
Figure 3.17 - Schematic of asymmetric bifurcation network showing the two
‘bounding’ pathways: the path of the largest daughter branches and the path of
the smaller daughter branches.
48
equation can be rewritten to form a general relationship between a parent vessel and a
small daughter branch vessel:
1
0
2
1R
R
(3.15)
where R0 and R2 are the radii of the parent vessel and smaller daughter branch vessel,
respectively, α is the bifurcation index, and γ is the power law index. This relationship
can be extended from a single bifurcation to a network of bifurcations. If we follow the
pathway of small daughter branches to the capillary vessel segment, the ratio of root
vessel segment diameter to capillary vessel segment diameter becomes:
Nroot
cap
R
R (3.16)
where Rroot and Rcap are the radii of the root vessel segment and capillary vessel segment
respectively, and N is the number of bifurcation levels in the arterial network. Using the
diameter values stated above for the root vessel segment and capillary vessel segment, we
determined root
cap
R
R. An initial λ value was calculated with a power-law index and
bifurcation index value based off of literature37. To determine N, the logarithm of root
cap
R
R
with base λ was calculated. Since N must be a whole number, its value is rounded to the
nearest integer; λ is then adjusted to account for this rounding using Eq. (3.16). With this
relationship in place, the geometric structure of an entire coronary network with a
constant α and γ can be created such that the terminal vessel segment along the small
daughter branch pathway has the desired capillary diameter.
49
In the MATLAB program, the bifurcation index α and power law index γ were
varied to achieve a physiological capillary velocity14 of approximately 0.55 mm/s with an
inlet pressure drop recorded from Model 1b. Both indices varied such that λ remained the
same value; this was necessary since λ dictated the overall structure of the arterial tree.
Table (3.4) below gives the final parameters used in the program to create the arterial
network depicted in Fig. (3.18).
Table 3.4 - Parameters for Model 2.
Parameter Value Description
γ 2.880 Power-law index
α 0.6902 Bifurcation index
N 14 Number of bifurcation levels
Rroot 1.588x10-3
m Radius of root vessel segment
Rcap 3.375x10-6
m Radius of capillary
Total Number of Vessels 16,383 -
Figure 3.18 - Coronary arterial tree for Model 2 simulations.. The area zoomed in shows the site of
the capillary vessel segment where the velocity will be recorded for each simulation.
Capillary vessel segment
50
Table (3.5) below summarizes the capillary blood velocity recorded for each
simulation of Model 2. For the steady flow without PC simulation, the recorded outlet
velocity was 0.557 mm/s. This is within the range of capillary velocities computed in
Kassab et al.‟s mathematical model of coronary capillary blood flow, which is based off
of anatomic and elasticity data of a coronary capillary network22. All other simulations
had this average velocity as well, due to the initial condition imposed on Models 1a and
1b. All of the simulations except for the steady flow without PC simulation had small
oscillations in the capillary velocity over time. These oscillations are due to Gibbs
Phenomenon, which is a typical behavior for Fourier series waveforms where the original
function or dataset has discontinuities or „jumps‟15. The oscillations are damped by
including more terms in the Fourier series expansion of the inlet pressure gradient
waveform; however, as stated in Chapter 2, too many terms (harmonics) in the expansion
can produce undesirable oscillations.
Table 3.5 - Recorded capillary blood velocities in Model 2 for each simulation.
Simulation Recorded capillary blood velocity
Steady flow
without PC 0.5570 mm/s
Pulsatile flow
without PC
51
Steady flow
with PC
Pulsatile flow
with PC
Pulsatile flow
with PC – close
up
52
For the pulsatile flow simulations the velocity within the capillary varied from
approximately -1.100 mm/s to 2.250 mm/s (minimum to maximum). This range of values
was slightly wider than the range of values determined in the mathematical model of
blood flow in a coronary capillary created by Fibich et al.9. However, their model
included elasticity of capillaries; this will give rise to lower velocities in the axial
direction since blood flow will be in the radial and axial directions with compliant
vessels37. Based on this observation, we believed that the pulsatile flow simulations
performed in Model 2 were reasonable.
3.9 Model 3
Geometry
Model 3 is a Krogh tissue geometry that represented a coronary capillary and
surrounding tissue region. The geometry used for Model 3 is that shown in Fig. (2.6).
Table (3.6) provides the dimensions of the geometry.
Table 3.6 - Dimensions for Model 3 geometry.
Dimension Value Description
Rc 3.375x10-6
m Radius of capillary
Rt 2.000x10-5
m Radius of Krogh tissue
L 5.500x10-4
m Length of capillary
The radius and length of the capillary were chosen based on values given in
literature6, 11, while the radius of the Krogh tissue was calculated so that the outlet partial
pressure of oxygen in the capillary was approximately 40 mmHg, the partial pressure of
oxygen in blood returning to the heart10, 14. This value was in the physiological range of
coronary intercapillary distances of 2.56 x 10-5
m ± 7.9 x 10-6
m, reported by Karch et
al.21.
53
Governing Equations
The governing equations used to solve for the concentration of oxygen in the
capillary and tissue are Eqs. (2.34) and (2.36), respectively:
2
2
1(1 ) (1 ) r z
C C C Cm V m D r D
t z r r r z
1T T
T
r METABOLIC
C CD R
t r r r
The assumptions made in the governing equations to solve Model 3 are as
follows:
The Krogh tissue model assumes the capillary is a straight tube. Realistically, the
capillaries can vary in shape between straight, T-shaped, H-shaped, and/or hairpin
shaped11.
The Krogh tissue model assumes that the tissue region is being oxygenated only
by a single capillary unit. The impact of other capillaries on oxygen transport in
the tissue region has been neglected.
Axial diffusion in the tissue is neglected. The length of the Krogh tissue is much
greater than its radius, so this assumption is reasonable.
Velocity through the capillary is assumed to be in „plug‟ flow, due to the presence
of red blood cells29. The result is a uniform rather than a parabolic velocity profile
seen in larger arteries.
Metabolic consumption of oxygen is the only reaction taking place in the tissue;
all other reactions are neglected. The metabolic consumption rate of oxygen is
capillary
tissue
54
assumed constant (zero-order) and does not vary with the local concentration of
oxygen (first-order).
The parameters used to solve the governing equations are described in Table (3.7).
All values provided were determined from existing literature29. The term H is the
solubility coefficient of oxygen in blood; this coefficient was used to describe the
concentration of oxygen in terms of partial pressure (mmHg).
Table 3.7 - Parameters used in Model 3.
Parameter Value Literature Source
Dr 1.12 x 10-9
m2/s [29]
Dz 1.12 x 10-9
m2/s [29]
DT
r 1.7 x 10-9
m2/s [29]
V See Table 3.8.1 -
RMETABOLIC 0.0705 mol/m3-s [14]
H 1.527 x 10-3
mol/m3-mmHg [29]
C’sat 9.100 mol/m3
[29]
n 2.647 [29]
Boundary and Initial Conditions
To solve the governing mass transfer Eqs. (2.34) and (2.36), boundary conditions
are required. Figure (3.19) shows the boundary conditions used in simulations after
simplifying the Krogh tissue model shown in Fig. (2.6) to a 2-D axial-symmetric
geometry. Table (3.8) provides a mathematical description of these conditions.
Tissue
Capillary 3c - Symmetry
3t - No Flux
1t - No Flux 2t - No Flux
1c - Concentration 2c - No Flux
r
z
Figure 3.19 - Schematic of Model 3, showing boundary conditions used.
55
Table 3.8 - Mathematical description of boundary conditions for Model 3.
Capillary Tissue
Boundary Condition Equation Boundary Condition Equation
1c Inlet
Concentration
0,z
inC C 1t No Flux
0,z
0TC
z
2c No Flux ,z L 0C
z
2t No Flux
,z L
0TC
z
3c Symmetry 0,r 0C
r
3t No Flux
,tr R
0TC
r
In the capillary region, the inlet was prescribed with a constant partial pressure of
oxygen of 95 mmHg. This value represents the concentration of oxygen in arterial blood
that perfuses muscle tissue10. At the outlet of the capillary, oxygen is carried out by blood
flow only, so the diffusion flux in the axial direction is zero. At the axis of symmetry, the
geometry is symmetrical about the centerline of the capillary, giving rise to boundary 3c.
In the tissue region, no-flux boundaries are imposed on the left and right sides of
the tissue. Similar to boundary condition 2c, this condition implies that oxygen cannot
leave the tissue by axial diffusion. At the tissue radius wall, the flux in the radial direction
was set to zero. The condition applied here states that the oxygen profile in the tissue
region between two capillaries is symmetric.
The initial conditions for Model 3 were set similarly to how the initial conditions
for Models 1a and 1b were set. The initial conditions for each transient simulation (steady
flow with PC, pulsatile flow with and without PC) are given by the solution to the steady
flow without PC simulation. This was done by storing the steady flow without PC
56
solution and specifying its concentration field over the computational domain as initial
values for the transient simulations.
Mesh Statistics
Before the transient model was solved for, a mesh independence study was
performed to minimize the discretization error over the computational domain. The test
was done by solving the steady flow without PC simulation for a variety of mesh sizes.
Model 3 solutions were considered to be mesh independent when the average
concentration of oxygen over the entire domain did not significantly change with an
increase in the number of mesh elements. A convergence criterion was arbitrarily set to
when the difference in the average partial pressure of oxygen between mesh sizes was
less than 0.05%. Figure (3.20) shows a plot of the average partial pressure of oxygen vs.
the number of mesh elements. It can be seen from this plot that there is little difference in
the partial pressure of oxygen between 2,250 and 15,000 elements. For each simulation
we have chosen a mapped mesh with 15,000 quadrilateral elements to ensure that the
solution is mesh independent even when the flow field changes due to pulsatile flow
and/or balloon movement.
57
Figure 3.20 - Average partial pressure of oxygen vs. mesh size for Model 3. The partial pressure
varies with small element sizes, but remains relatively constant for mesh sizes between 2,250 and
15,000 elements. The black dashed circle indicates the final mesh size used for all simulations.
Figure 3.21 below shows the final mesh used for Model 3. Table (A.3) in
Appendix A gives the solver settings used in COMSOL, as well as the solution times for
each simulation of Model 3.
Figure 3.21 - Plot of final mesh used in simulations. Mesh is mapped with a total of 15,000
quadrilateral elements.
Model Validation
Before Model 3 simulations were run, an accuracy check was performed to build
confidence in our modeling. This was done by running a computational model that exists
in published literature which is similar to Model 3. Once a direct comparison to this
58
relevant computational model was made, the parameters of that model were changed to
fit our specific simulations for Model 3. Although we could no longer make a direct
comparison to the model from literature, by previously running that relevant
computational model, we were able to build sufficient confidence in our accuracy of
subsequent Model 3 simulations.
Sharan et al. have developed a finite-element model of oxygen transport in a
systemic capillary using the Krogh tissue model29. The overall geometry used by Sharan
et al. is similar to the geometry for Model 3, however, the physical dimensions are
different. Furthermore, the metabolic oxygen consumption rate used varies from that
stated in Table (3.7). Table (3.9) gives the parameter values used by Sharan et al. in their
computational study29.
Table 3.9 - Parameters used in model by Sharan et al.29
Parameter Value
Dr 1.12 x 10-5
cm2/s
Dz 1.12 x 10-5
cm2/s
DT
r 1.7 x 10-5
cm2/s
Rc 3.25 x 10-4
cm
Rt 3.25 x 10-3
cm
L 60*Rc cm
V 0.03 cm/s
RMETABOLIC 3.72 x 10-8
mol/cm3-s
H 1.527 x 10-9
mol/cm3-mmHg
C‟sat 9.1 x 10-6
mol/cm3
n 2.6472
Figure (3.22) shows the axial partial pressure of oxygen profiles obtained by
Sharan et al. for the capillary and tissue regions. On the same plot, the results from
COMSOL using the same input parameters as Sharan et al. is given. For both data sets,
the axial and radial lengths are normalized by the radius of the capillary.
59
Figure 3.22 - Axial partial pressure of oxygen profiles in capillary and tissue for results from Sharan
et al. and results from a COMSOL model with same input parameters. Oxygen partial pressure is
recorded at two points: at r = 0 (centerline of capillary) and at r = 10 (tissue radius). The radial (r)
axial (z) distances are normalized by the radius of the capillary (Rc). The error bars shown are ±5%.
Figure (3.23) shows the radial partial pressure of oxygen profiles obtained by
Sharan et al. at various axial positions in the Krogh tissue model. On the same plot, the
results from COMSOL using the same input parameters as Sharan et al. is given. For both
data sets, the radial and axial lengths are normalized by the radius of the capillary.
60
Figure 3.23 - Radial partial pressure of oxygen profiles at various axial positions in model for results
from Sharan et al. and results from a COMSOL model with same input parameters. The radial (r)
and axial (z) distance is normalized by the radius of the capillary (Rc). The error bars shown are
±5%.
From Figs. (3.22) and (3.23) we see that the results from COMSOL have
reasonably reproduced the numerical findings given by Sharan et al29. Based on this
conclusion, we feel that our modeling approach for simulating oxygen transport in a
Krogh tissue model using COMSOL is accurate.
The next chapter entitled „Journal Article‟ will provide a draft for a potential
publication which will summarize the key results from Model 3 simulations.
61
CHAPTER 4
JOURNAL ARTICLE
This chapter provides a draft journal article arising from this thesis work. The
chapter is organized and written as it would be presented in a publication; the reader who
has read the previous thesis chapters may go directly to Sections 4.4 and 4.5 which give
the results and discussion of Model 3, respectively. Upon completion of a final draft, the
author plans to submit this chapter to the Annals of Biomedical Engineering Journal.
The title of the article will be “Tissue oxygen transfer during post-conditioning.” The
references for the journal article are included with other references for the thesis.
4.1 Abstract
Post-conditioning (PC) is a relatively new therapeutic strategy which aims at
limiting reperfusion injury after a heart attack. Although PC has shown to reduce the
amount of tissue death incurred by reperfusion injury, the biological mechanisms
involved in PC are not well understood. The goal of this study was to develop a
computational model to simulate the steady and pulsatile flow of blood and the resulting
mass transport of oxygen in a capillary-tissue system during a specific PC procedure. The
equations governing the mass transport of oxygen were solved using the finite-element
method. The effects of oxygen dissociation from hemoglobin were also considered. The
PC algorithm tested in the model was based off of a recent clinical trial. The model in this
study shows that PC significantly reduces the amount of oxygen delivered to tissue,
which may significantly impact biological reactions that occur during PC, such as the
production of reactive oxygen species (ROS). This study also showed that oxygen is
62
depleted and restored rapidly during the ischemic and reperfusion phases, respectively,
indicating that the design of the reperfusion phases of a PC algorithm should be further
investigated. The model in this paper lays the foundation for gaining a better
understanding behind the cardioprotective mechanisms of PC.
4.2 Introduction
Heart disease is the leading cause of death in the United States, killing at least one
person every 38 seconds23. Ischemic heart disease occurs when coronary blood flow to
the heart is reduced, limiting the amount of oxygen and nutrients the heart receives which
causes tissue death. Fortunately clinicians have found many therapies aimed at reducing
tissue death from ischemic heart disease, one of which is a percutaneous transluminal
coronary angioplasty (PTCA). However, when blood flow is restored after the procedure,
the rapid reperfusion from sudden balloon deflation can cause further injury to the
oxygen-starved heart tissue, leading to increased cell injury and cell death. Studies in
animal models with ischemic heart disease have shown that reperfusion injury may
account for up to 50% of the final infarct size36. Recent therapeutic strategies such as
post-conditioning (PC) have been shown to reduce the amount of reperfusion injury on
the heart by applying brief periods of ischemia during the early moments of reperfusion.
This procedure intermittently occludes blood flow during reperfusion by periodically
inflating and deflating an angioplasty balloon according to a specific algorithm39. Zhao et
al. was one of the first to show that PC reduced reperfusion injury in a canine model by
applying 3 cycles of 30 seconds of reperfusion followed by 30 seconds of ischemia (re-
occlusion) at the onset of reperfusion. Their study demonstrated that PC reduced tissue
AN/AAR (area of necrosis/area at risk) by 48%40.
63
Several reviews on experimental studies revealed that PC reduces the amount of
reactive oxygen species (ROS) generated from ischemia/reperfusion (I/R) injury2, 16.
Investigators have observed that the reintroduction of oxygen back to ischemic tissue
creates a „burst‟ production of ROS, which is believed to play a critical role in cell death2.
However, the exact underlying mechanisms by which PC protects the myocardium from
lethal reperfusion injury are not well understood2, 13, 16. Furthermore, an optimal PC
algorithm that minimizes tissue death has yet to be determined. To the best of our
knowledge, no reference has investigated the relationship between blood fluid dynamics
and species mass transfer in the coronary capillaries during PC.
To gain a better understanding of PC‟s cardioprotective effects, this study
investigated the link between blood fluid dynamics and oxygen transport in the coronary
capillaries during PC, and how disrupting flow at the capillary level changes oxygen
transport behavior.
4.3 Computational Methods
The model in this study uses a Krogh tissue geometry that will be representative
of a coronary capillary and surrounding tissue (Fig. 4.1). Table (4.1) provides the
dimensions of the geometry.
Blood In Blood Out
Tissue
Capillary
Rt
Rc
L
z
r
Figure 4.1 - Schematic of Krogh tissue model. Rc is the radius of the capillary, Rt is the radius of the
surrounding tissue, and L is the length of the capillary. Blood within the capillary flows from left to
right.
64
Table 4.1 - Dimensions for model geometry.
Dimension Value Description
Rc 3.375x10-6
m Radius of capillary
Rt 2.000x10-5
m Radius of Krogh tissue
L 5.500x10-4
m Length of capillary
The radius and length of the capillary were chosen based on values given in
literature11, 14, while the radius of the Krogh tissue was calculated so that the outlet partial
pressure of oxygen in the capillary was approximately 40 mmHg14. This value was in the
physiological range of coronary intercapillary distances of 2.56 x 10-5
m ± 7.9 x 10-6
m,
reported by Karch et al.21.
Governing Equations
The governing equations used to solve for the concentration of oxygen in the
capillary and tissue are Eqs. (4.1) and (4.2), respectively:
2
2
1(1 ) (1 ) r z
C C C Cm V m D r D
t z r r r z
(4.1)
1T T
T
r METABOLIC
C CD R
t r r r
(4.2)
where C is the concentration of oxygen dissolved in the plasma, Dr and Dz are the radial
and axial diffusivity values of oxygen in blood, respectively, CT is the tissue oxygen
concentration, DrT and Dz
T are the tissue radial and axial diffusivity values for oxygen,
respectively, V is the velocity of the blood in the capillary that depends on the PC
algorithm, and RMETABOLIC is the volumetric metabolic consumption rate of oxygen. The
term m is related to the derivative, or slope, of the oxygen dissociation curve (ODC) by:
''sat
C dm C
C dC
(4.3)
65
where C'sat is the concentration of oxygen bound to hemoglobin when hemoglobin is
100% saturated. The ODC is an equilibrium curve that gives the fractional saturation of
hemoglobin with oxygen for a specific concentration of oxygen dissolved in plasma. A
mathematical relation of the ODC can be represented by the Hill Equation:
50
50
1
n
n
CC
CC
(4.4)
where ψ is the percent saturation of hemoglobin with oxygen, C50 is concentration of
oxygen dissolved in plasma at which hemoglobin is 50% saturated, and n is the Hill
parameter.
The assumptions made in the governing equations to solve the model are as follows:
Axial diffusion in the tissue is neglected. The length of the Krogh tissue is much
greater than its radius.
Velocity through the capillary is assumed to be in „plug‟ flow, due to the presence
of red blood cells10. The result is a uniform rather than parabolic velocity profile,
like that seen in larger arteries.
Metabolic consumption of oxygen is the only reaction taking place in the tissue;
all other reactions are neglected. The metabolic consumption rate of oxygen is
assumed constant (zero-order) and does not vary with the local concentration of
oxygen (first-order).
The parameters needed to solve the governing equations are described in Table (4.2).
All values provided were determined from existing literature. The term H is the solubility
66
coefficient of oxygen in blood; this coefficient is used to describe the concentration of
oxygen in terms of partial pressure (mmHg).
Table 4.2 - Parameters used in model.
Parameter Value Literature Source
Dr 1.12 x 10-9
m2/s [29]
Dz 1.12 x 10-9
m2/s [29]
DT
r 1.7 x 10-9
m2/s [29]
V 5.57 x 10-4
m/s [22]
RMETABOLIC 0.0705 mol/m3-s [14]
H 1.527 x 10-3
mol/m3-mmHg [29]
C’sat 9.100 mol/m3
[29]
n 2.647 [29]
In addition to steady flow, the impact of pulsatile flow on the transport of oxygen
was also examined in this study. Figure (4.2) below shows the pulsatile blood velocity
waveform used for all pulsatile flow simulations. The velocity waveform was calculated
from computational models which simulated the blood flow upstream of the capillary
under pulsatile flow conditions with and without PC applied to the flow. The cardiac
cycle time for the pulsatile flowrate waveform was T = 0.832 seconds17.
Figure 4.2 - Pulsatile capillary blood velocity waveform used for pulsatile flow simulations.
Waveform has a period of T = 0.832s and is obtained by solving upstream models with waveform
taken from He and Ku17
.
67
To solve the governing mass transfer Eqs. (4.1) and (4.2), boundary conditions
were required. Figure (4.3) shows the boundary conditions used in simulations after
simplifying the Krogh tissue model shown in Fig. (4.1) to a 2-D axial-symmetric
geometry.
In the capillary region, the inlet was prescribed with a constant partial pressure of
oxygen of 95 mmHg. This value represented the partial pressure of oxygen in arterial
blood that perfuses muscle tissue10, 14. At the outlet of the capillary, oxygen was carried
out by blood flow only, so the diffusion flux in the axial direction was zero. At the axis of
symmetry, the geometry was symmetrical about the centerline of the capillary, giving rise
to boundary 3c.
In the tissue region, no-flux boundaries were imposed on the left and right sides
of the tissue. Similar to boundary condition 2c, this condition implied that oxygen cannot
leave the tissue by axial diffusion. At the tissue radius wall, the flux in the radial direction
was set to zero. The condition applied here stated that the oxygen profile in the tissue
region between two capillaries was symmetric.
The initial conditions for each transient simulation (steady flow with PC, pulsatile
flow without and with PC) are given by the solution to the steady flow without PC
simulation. This is done by storing the steady flow without PC solution and specifying its
3c - Symmetry
3t - No Flux
1t - No Flux 2t - No Flux
1c - Concentration 2c - No Flux
r
z
Tissue
Capillary
Figure 4.3 – Schematic of model, showing boundary conditions used.
68
concentration field over the computational domain as initial values for the transient
simulations.
Before the transient model was solved for, a mesh independence study was
performed to minimize the discretization error over the computational domain. The test
was done by solving the steady flow without PC simulation for a variety of mesh sizes. A
convergence criterion was arbitrarily set to when the difference in the average partial
pressure of oxygen between increasing mesh sizes was less than 0.05%. Figure (4.3)
shows a plot of the average partial pressure of oxygen vs. the number of mesh elements.
It can be seen from this plot that there is little difference in the partial pressure of oxygen
between 2,250 and 15,000 elements. For subsequent simulations we have chosen a
mapped mesh with 15,000 quadrilateral elements to ensure that the solution is mesh
independent. Figure (4.5) below shows the final mesh used for the model.
Figure 4.4 - Average partial pressure of oxygen vs. mesh size for model. The concentration varies
with small element sizes, but remains relatively constant for mesh sizes between 2,250 and 15,000
elements. The black dashed circle indicates the final mesh size used for all simulations.
69
Figure 4.5 - Plot of final mesh used in simulations. Mesh is mapped with a total of 15,000
quadrilateral elements.
All mass transfer equations were solved using the commercial software package
COMSOL (version 4.0a). COMSOL uses the finite element method to solve the partial
differential mass transfer equations. With this method, the partial differential equations
are approximated by a system of ordinary differential equations when are then solved
using the Newton-Raphson Method. COMSOL‟s ability to model experimental findings
accurately was validated against observations for laminar flow over a backstep while
varying the Reynolds number between 70 and 412[1]. The results showed that the
difference in the computational solutions and experimental data for the flow recirculation
length downstream of the step over the range of Reynolds numbers was between 0.9 and
9%. Taking into account uncertainties in experimental measurements, these observations
demonstrate that COMSOL can accurately model experimental findings with its finite
element method.
All simulations were performed on an HP Z400 Workstation with a 3.20 GHz
Intel Zeon Quad-Core processor and 23.9 GB of RAM running Windows XP
Professional 64-bit Edition.
70
4.4 Results
Before simulations were ran, an accuracy check was performed to build
confidence in our modeling. This was done by solving a finite-element model developed
by Sharan et al. which simulated the transport of oxygen in a Krogh tissue model29. We
found that our solutions were in good agreement with the results from their simulations
(±5%).
Initial simulations were performed under steady flow conditions without an
imposed PC algorithm. Figure (4.6) shows the partial pressure of oxygen (PO2) as a
function of axial distance in both the capillary (solid line) and tissue (dashed line)
regions. The axial distance (x-axis) was normalized with respect to the capillary length L.
The capillary PO2 was recorded at the centerline of the capillary (axis of symmetry in
geometry), while the tissue PO2 was recorded at the tissue radius Rt. Figure (4.6) shows
that at the outlet of the capillary, the PO2 in the capillary is approximately 40 mmHg,
which is consistent with published literature10, 14. In both the capillary and tissue regions,
the PO2 drops rapidly within the first 25% of the capillary length, which is consistent
with other computational models on mass transport of oxygen in capillary-tissue
systems20, 28, 29.
71
Figure 4.6 - Plot of axial partial pressure (PO2) of oxygen in capillary and tissue region for steady
flow without PC simulation. The model results in an outlet capillary PO2 of 40 mmHg, which is
consistent with the venous PO2 value found in literature. The PO2 profiles are also in qualitative
agreement with those given by Sharan et al29
.
Figure 4.7 - Surface plot of PO2 for the steady flow without PC simulation. The color legend on the
right has units of mmHg. The maximum PO2 observed was 95 mmHg (at inlet), and the minimum
PO2 was approximately 28 mmHg. The minimum PO2 is located at the ‘lethal corner’, denoted by the
red dot in the figure.
tissue
capillary
72
Figure (4.7) is a surface plot of the capillary and tissue PO2 for the steady flow
without PC simulation. As expected, the maximum PO2 of 95 mmHg is seen at the inlet
of the capillary. The minimum PO2 of approximately 28 mmHg was observed at the
„lethal corner‟ (red dot), a term often used to describe the region of a Krogh tissue model
that would first become oxygen-starved under a high metabolic rate31.
Although the boundary conditions of this model stated that axial diffusion in the
tissue region was ignored, Fig. (4.7) shows some axial dependence on the PO2
distribution near the capillary-tissue interface. This is due to the blood flow that exists in
the capillary and the continuity boundary condition which must be satisfied at the
interface.
Figure (4.8) shows the time-averaged axial PO2 for steady flow simulations with
and without PC in the capillary (4.8a) and tissue (4.8b) regions. To obtain the time-
averaged PO2 for each region, the PO2 at each specific time step were added together and
then divided by the total number of time steps in the solution time. This calculation was
not necessary for the steady flow without PC simulation since it was a steady state
problem. Similar to Fig. (4.6), the axial distance (x-axis) was normalized with respect to
the capillary length L. The capillary PO2 was recorded at the centerline of the capillary
(axis of symmetry in geometry), while the tissue PO2 was recorded at the tissue radius Rt.
The simulations of steady flow with PC showed a significant decrease in the PO2 in both
the capillary and tissue when compared to the steady flow without PC simulation. From a
qualitative standpoint, the PO2 profiles for steady flow with PC simulations had a similar
shape to those in simulations without PC.
73
4.8a
4.8b
Figure 4.8 - Plot of time-averaged PO2 for steady flow with and without PC simulations a) in the
capillary and b) in the tissue. For both plots, the axial distance is normalized with respect to the
capillary length L.
74
Post-conditioning with the specified algorithm showed a drop in the average PO2
of 22 mmHg and 19 mmHg in the capillary and tissue regions, respectively. Interestingly,
PC showed that the time-averaged PO2 was reduced by 26% to 44% in the capillary
region along the axial length of the capillary. Similarly, in the tissue region PO2 was
reduced by 30% to 46% along the tissue wall.
Figure (4.9a) shows the dynamic variation of the PO2 in the tissue region at two
points: the space-averaged point20 at r = Rc+Rt/2 and z = L, and the lethal corner31 at r = Rt
and z = L. The time-axis (x-axis) is normalized with respect the PC algorithm‟s cycle
time, t_cycle. For the PC algorithm used in this paper, t_cycle = 60s (30s reperfusion/30s
ischemia). In Fig. (4.9a), small oscillations can be seen at the starting points of the
reperfusion (deflation) phases, where the PO2 becomes negative. They were believed to
be insignificant, since their magnitude is less than 5% of the steady PO2 value. Figures
(4.9b) and (4.9c) show a close-up portion of Fig. (4.9) at the first balloon inflation stage
and first balloon deflation stage, respectively. In Figs. (4.9a) and (4.9b), the dotted
vertical line indicates the transition time between reperfusion (balloon deflated) and
ischemic phases (balloon inflated) of one PC cycle.
75
4.9a
4.9b
Reperfusion Phase Ischemic Phase
76
4.9c
Figure 4.9 - Plot of tissue PO2 over time at space-averaged point (solid line) and lethal corner (dashed
line) for the steady flow with PC simulation. Time axis (x-axis) is normalized with respect to the PC
algorithm’s cycle time, t_cycle = 60s. a) PO2 over entire PC algorithm, b) PO2 during first balloon
inflation (dotted rectangle on 4.9a), and c) PO2 during first balloon deflation (dashed rectangle on
4.9a). Figure 4.9c shows a time-lag between when the PC balloon is deflated and when oxygen
reaches the space average point and lethal corner.
The lethal corner PO2 was lower than that at the space-averaged point because the
lethal corner is farther away from the oxygen source (i.e., the capillary). As expected, the
PO2 in the tissue fell to zero during the ischemic phases of the PC algorithm, and returned
to its steady-state value during the reperfusion phases. From Fig. (4.9c) we see that there
was a time-lag between when the PC balloon is deflated (dashed line in Fig. 4.9c) and
when the PO2 in the tissue at the space-averaged point and lethal corner begin to restore
to their steady-state value; this is due to the time is takes for the blood in the capillary to
travel from the inlet to the outlet of the capillary (approximately 1s). The time it takes for
the PO2 in the tissue to fall to zero during the balloon inflation stages was relatively rapid
(3s) when compared to the total time of the ischemic (inflation) phase of one PC cycle
Reperfusion Phase Ischemic Phase
77
(30s). Figure (4.9c) shows that the „relief time‟, the time it takes for the PO2 in the tissue
to return to steady-state after the ischemic phase, is also relatively rapid (2.4s) compared
to the total time of the reperfusion (deflation) phase of one PC cycle (30s); our study
shows that this time is 8% of the total reperfusion phase time, which is consistent with
Hyman et al.20.
Pulsatile flow simulations without PC and with PC were performed to observe
any impact that pulsatile blood velocity may have on the transport of oxygen. Figure
(4.10) shows the time-averaged PO2 along the centerline of the capillary (Fig. 4.10a) and
along the tissue wall (Fig. 4.10b). When compared to steady flow simulations with and
without PC, it is clear from these plots that pulsatility of the capillary blood velocity did
not significantly impact the transport of oxygen; solutions were within 5% of each other.
4.10a
78
4.10b
Figure 4.10 - Plot of time-averaged partial pressure of oxygen for pulsatile flow with and without PC
simulations. a) Normalized axial length along centerline of capillary. b) Normalized axial length
along tissue wall. In both plots, the axial length is normalized with respect to the length of capillary.
For comparative purposes, steady flow simulations were included in the plots.
4.5 Discussion
The results from this study showed that the model could accurately predict
physiological values of PO2 in both a coronary capillary and surrounding tissue.
Assuming a PO2 of 95 mmHg at the capillary inlet, the model predicted an outlet
capillary PO2 of approximately 40 mmHg, which is consistent with published literature10,
14. The results from this study also show that the majority of oxygen transport in the
capillary occurs within the first 25% of the capillary length. This result is in good
qualitative agreement with findings by others who have modeled the steady-state
transport of oxygen in a Krogh tissue geometry10, 20, 28, 29, despite differences in the
79
parameters of the models such as capillary and tissue radius, capillary length, rate of
metabolic of consumption in tissue, and velocity of blood in the capillary.
The results from this study showed that with simulations of PC, the time it takes
for the PO2 to fall to zero at the lethal corner of the tissue region as a result of balloon
inflations is on the order of seconds. This observation was consistent with what was
found by Hyman et al.20, who studied the mass transfer of oxygen including with
occlusion periods in a Krogh tissue model. They found that the time for the PO2 to drop
to zero at the lethal corner following occlusion was 6.8 seconds. Our results estimated
that this time was twice as fast as given by Hyman et al.; however, the tissue radius and
metabolic rate used by Hyman et al. were 3.5 x 10-5
m and 2.2 x 10-3
mol/m3-s,
respectively. These variations in parameters could account for the difference in PO2
depletion times. The results from the current study also showed that the restoration of
PO2 from upstream balloon deflations was relatively rapid when compared to the duration
of the ischemic/reperfusion phases; the time it took for the PO2 to return to its steady-
state value was 8% of the duration of the ischemic phases. Hyman et al. simulated the
mass transport of oxygen in a Krogh tissue model subject to an occlusion time period and
included an „oxygen-debt‟ term which accounted for the production of lactic acid as a by-
product of anaerobic metabolism. They found that the time it took for oxygen to return to
its steady-state value after a period of occlusion was approximately 7% of the occlusion
time20. Despite notable differences in the model parameters and the occlusion times,
Hyman et al. studied occlusion times between 2 and 5 minutes, our results are in general
agreement with those of Hyman et al.
80
The present study showed that at the onset of the reperfusion phases of PC,
oxygen is rapidly restored to the surrounding tissue after a period of occlusion. When
oxygen is restored to oxygen-starved tissue, it reacts with other chemical species
produced as a result of anaerobic metabolism, generating various forms of reactive
oxygen species (ROS) which contribute to reperfusion injury2. Although PC has been
shown to significantly reduce the amount of ROS generated during reperfusion30, the
short „bursts‟ of oxygenated blood during the reperfusion cycles of PC have also been
shown to generate small amounts ROS that can trigger several protection mechanisms
which ultimately provide cardioprotection from reperfusion injury2, 16.
Furthermore, the rapid restoration of oxygen during the reperfusion (deflation)
phases of PC demonstrates that the design of PC algorithms is important to minimize
reperfusion injury and resulting heart tissue damage. The duration of the reperfusion
phases for each cycle, as well as the number of cycles in the algorithm, should be clearly
identified to assess whether a specific PC algorithm is effective in attenuating reperfusion
injury. In a recent animal study which performed a PC procedure on the common carotid
arteries of rats, a variety of PC algorithms were tested and showed varying results. For
instance, PC algorithms with 10 cycles of 10 seconds reperfusion/ 10 seconds occlusion
showed the most reduction in infarct size, while an algorithm with 3 cycles of the same
reperfusion/occlusion durations showed minimal benefit. Additionally, 3 cycles of 30
seconds reperfusion/ 10 seconds occlusion offered a significant reduction in infarct size,
while 10 cycles of the same reperfusion/occlusion durations showed no difference
between the control group12. A review done on several experiments which differed in PC
algorithms and species type showed that the percent reduction in infarct size when
81
compared to the control group varied between 2% and 74%13. The results described in
this paper and the observations from the literature stress the importance of gaining a
better understanding of the link between the fluid and mass transfer dynamics during PC.
4.6 Conclusion
This computational study used the finite-element method to simulate the effects of
altered blood flow in a capillary on the mass transfer of oxygen to a surrounding Krogh
tissue model during a specific PC procedure. Both steady flow and pulsatile flow
conditions in the capillary were observed. The mass transfer of oxygen with and without
PC was compared. The model‟s initial simulations without a PC procedure have shown to
produce physiological results which are consistent with published literature. It was
observed through our model that a PC procedure significantly reduces the amount of
oxygen that is delivered to the surrounding tissue; we speculate that this has a direct
influence on the amount of ROS generated. Furthermore, our computational model has
shown oxygen depletion and oxygen restoration at the onset of the ischemic phases and
reperfusion phases, respectively, to be relatively quick when compared to the duration of
the PC phases. The observations in this study lay the foundation in gaining a better
understanding in the cardioprotective mechanisms of PC.
The model described in this paper has limitations. First, the geometry of the
capillary and tissue were idealized as concentric cylinders in a Krogh tissue model. This
assumption simplifies the shape of the capillaries to a straight tube, whereas in the human
body this is not always the case. Furthermore, we assume that the tissue surrounding the
capillary is being supplied with oxygen only by that single capillary; in the body the
capillaries can have tortuous shapes and are not always evenly spaced apart, which may
82
impact the oxygen delivery to a specific tissue region. With the use of MRI and other
imaging processes, the geometry of the model described in this study could be modified
to an anatomically correct section of tissue which contains more than one capillary
vessel. A more realistic capillary-tissue system can provide valuable insight into the
overall spatial distribution of the concentration of oxygen for a section of tissue.
The model described in this paper uses a constant metabolic consumption rate of
oxygen, which is also known as zero-order metabolism. Future improvements to the
model could include first-order metabolism, where the metabolic consumption rate of
oxygen becomes dependent on the local concentration of oxygen.
The focus of this study was to develop a computational model that could simulate
the mass transfer of oxygen during a specific PC procedure; thus the number of PC
algorithms tested on the model was limited to one. Future studies using this model should
include studying an array of PC algorithms which vary in the number of cycles, the
duration of the ischemic/reperfusion phases, and the initiation time of the PC procedure
during reperfusion. Additionally, the model should include the dynamics of biochemical
species, such as a specific ROS produced from oxygen. By doing so, the model could be
validated with published experimental data that observed ROS levels30.
83
CHAPTER 5
CONCLUSIONS AND FUTURE WORK
This chapter will briefly recap the work done, the significance of the work, and
discuss the model limitations as well recommendations for improvements to the models
that can be made in future studies.
5.1 Conclusions
This thesis developed a computational model which simulated the fluid dynamics
of blood flow throughout an arterial network under steady and pulsatile flow conditions,
with and without a post-conditioning (PC) procedure applied to the flow. The resulting
mass transfer of oxygen downstream in a capillary-tissue system was also modeled. The
goal of this thesis was to provide a connection between the fluid dynamics of blood flow
and the mass transfer of oxygen as a means to understand the cardioprotective
mechanisms of PC.
Model 1 was a finite-element model which simulated the flow of blood around a
balloon catheter which pulsated according to a specific PC algorithm. Rather than
treating the balloon as a solid and performing complex fluid-structure interaction and
moving mesh simulations, the balloon was represented by applying a time-dependent
viscosity term to the area of the fluid domain where the balloon is located. This viscosity
term was orders of magnitude larger than the surrounding blood viscosity, thus portraying
the balloon as a solid region. Results from the simulations of this model showed that our
method to modeling the balloon as a highly viscous region was accurate in two main
ways: 1) the vessel containing the balloon became fully occluded and blood flow was
84
stopped as the balloon inflated, and 2) the no-slip condition was sufficiently satisfied
across the fluid-balloon interface. The method used to depict the balloon and its
corresponding movements eliminated the potential problems in numerical stability that
can arise during fluid-structure interaction models, specifically where the computational
mesh would reduce to a zero-thickness as the balloon inflates and occludes the vessel.
In Model 2, the flowrate and average velocity through a 1-D representation of the
coronary arterial network were calculated using bifurcation theory presented by Zamir37.
The goal of this model was to take the outlet flow and pressure from the upstream Model
1 and obtain a physiologic average outlet velocity of blood at a terminal capillary
segment, serving as a „link‟ between Model 1 and Model 3. The results from this model
show that with successful tuning of the bifurcation index and power law index
parameters, an accurate flowrate distribution can be achieved throughout the entire 1-D
network. More importantly, the model predicted a reasonable average capillary blood
velocity. The computational cost of Model 2 was significantly reduced by using a 1-D
representation of the coronary arterial network instead of a 3-D representation. The 3-D
representation of the network would not only require a large amount of computer
memory to solve for the flowrate and velocity of blood, but boundary conditions
(pressure or flowrate) for each terminal segment of the model must be known in advance.
Model 3 used the resulting capillary velocity from Model 2 to simulate the mass
transfer of oxygen in a Krogh tissue model during a PC procedure. The model‟s initial
simulations without a PC procedure produced PO2 distributions in the capillary and tissue
regions which are consistent with published literature. It was observed through our model
that a typical PC procedure can significantly reduce the amount of oxygen delivered to
85
surrounding tissue; we speculate that this has a direct influence on the amount of ROS
generated. Furthermore, Model 3 has shown that oxygen depletion and oxygen
restoration at the onset of the ischemic (inflation) phases and reperfusion (deflation)
phases, respectively, are relatively quick when compared to the duration of a particular
phase. These observations suggest that the design of PC algorithms should limit the
duration of reperfusion phases to prevent an increase in reperfusion injury during PC.
Taken as a whole, our modeling approach could perhaps be used to carefully fine tune PC
algorithms, dialing in specific inflation and deflation phase characteristics.
5.2 Future Work
Each of the models described in this thesis have some limitations. For instance, in
Model 1, the vessel was assumed to be rigid, and blood was assumed a Newtonian fluid.
Additionally, as the balloon deflated during the PC procedure, the fully deflated balloon
became flush with the wall of the catheter shaft. This assumed that the deflated balloon
had no impact on the flow field during the reperfusion phases of the PC algorithm.
Future studies utilizing Model 1 should incorporate the non-Newtonian fluid
characteristics of blood to probe the sensitivity of the model under PC flow. The model
should also include vessel elasticity to observe the interaction between wall motion and
balloon inflations. The equation describing the balloon motion in Chapter 3 could also be
modified so that the balloon could deflate to a specified thickness above the catheter
shaft. Observations could be made on how the deflated balloon impacts the flow field.
In Model 2, the geometry of the arterial network was created using 1-D
bifurcation theory which assumes a constant bifurcation index throughout the model. The
blood flowing through the model was also assumed as a Newtonian fluid. Although the
86
flowrate distribution in the network represented physiological values, the pressure drop
across the entire model (from root segment to capillary segment) was approximately 5
mmHg, which is much smaller than the 75 – 85 mmHg pressure drop seen in the body
from arteries to capillaries14. This was due to the specification of the bifurcation index
and power law index parameters, as well as the assumption of non-Newtonian blood.
Future studies with Model 2 should consider implementing a non-Newtonian viscosity
for blood into the program; Pries et al.26 have developed a viscosity model that is a
function of vessel diameter which has been implemented into a 1-D model of an arterial
network by Mittal et al.25, similar to Model 2 in this thesis. Their work showed that the
non-Newtonian effects of blood increase the fluid resistance in smaller vessels which will
decrease the blood velocity and flowrate.
The geometry of the capillary and tissue in Model 3 were idealized as concentric
cylinders in a Krogh tissue model. This assumption greatly simplified the shape of
capillaries to straight tubes, whereas in the human body this is not always the case.
Furthermore, we assumed that the tissue surrounding the capillary was being supplied
with oxygen only by that single capillary; in the body the capillaries can have tortuous
shapes and are not always evenly spaced apart. With the use of MRI and other imaging
processes, the geometry of the model described in this study could be modified to an
anatomically correct section of tissue which contains more than one capillary vessel. A
more realistic capillary-tissue system could provide insight into the overall spatial
distribution of the concentration of oxygen throughout a section of tissue.
The model described in this paper used a constant metabolic consumption rate of
oxygen, which is also known as zero-order metabolism. Future improvements to the
87
model could include first-order metabolism, where the metabolic consumption rate of
oxygen becomes dependent on the local concentration of oxygen.
The focus of this study was to develop a computational model that could simulate
the mass transfer of oxygen during a typical PC procedure; thus the number of PC
algorithms tested on the model was limited to one. Future studies using this model should
include studying an array of PC algorithms which vary in the number of cycles, the
duration of the ischemic/reperfusion phases, and the starting time for the PC procedure
after the start of normal reperfusion. Additionally, the model should include the reactions
of other biochemical species, such as a specific reactive oxygen species (ROS) produced
from oxygen during reperfusion injury. By doing so, the model could be validated with
published experimental data that observed ROS levels30.
88
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93
APPENDIX A
COMSOL Solver Settings and Solution Times
Table (A.1) - COMSOL solver settings and solution times for Model 1a.
Simulation Solver Type
Time-
stepping
Method
Relative
Tolerance
Absolute
Tolerance
Solution
Time (s)
Steady flow
w/o PC
Direct
(PARDISO) N/A 0.001 N/A 2
Steady flow
w/ PC
Direct
(PARDISO) BDF 0.050
0.005/10
(velocity/pressure) 257
Pulsatile
flow w/o
PC
Direct
(PARDISO) BDF 0.001
0.00001/10
(velocity/pressure) 221
Pulsatile
flow w/ PC
Direct
(PARDISO) BDF 0.001
0.005/10
(velocity/pressure) 19,440
Table (A.2) - COMSOL solver settings and solution times for Model 1b.
Simulation Solver Type
Time-
stepping
Method
Relative
Tolerance
Absolute
Tolerance
Solution
Time (s)
Steady flow
w/o PC
Direct
(PARDISO) N/A 0.001 N/A 2
Steady flow
w/ PC
Direct
(PARDISO) BDF 0.050
0.005/10
(velocity/pressure) 437
Pulsatile
flow w/o
PC
Direct
(PARDISO) BDF 0.001
0.00001/10
(velocity/pressure) 188
Pulsatile
flow w/ PC
Direct
(PARDISO) BDF 0.001
0.005/10
(velocity/pressure) 18,000
Table (A.3) - COMSOL solver settings and solution times for Model 3.
Simulation Solver Type
Time-
stepping
Method
Relative
Tolerance
Absolute
Tolerance
Solution
Time (s)
Steady flow
w/o PC
Direct
(PARDISO) N/A 0.001 N/A < 1
Steady flow
w/ PC
Direct
(PARDISO) BDF 0.050 0.005 9543
Pulsatile
flow w/o PC
Direct
(PARDISO) BDF 0.001 0.00001 11
Pulsatile
flow w/ PC
Direct
(PARDISO) BDF 0.001 0.005 13,027
95
APPENDIX B
MATLAB Programs for Model 2
Program used for steady flow without PC simulations
% This program will build an arterial network tree structure based on bifurcation theory. The program will then estimate pressure drop along
each path within the bifurcation. The results will be used to estimate
the flow distribution in the arterial network.
close all clear all clc
%---INPUT PARAMETERS---%
N = 14; % Number of levels in the tree structure
V = (2^N)-1; % Number of vessels in tree structure
alpha = .690195; % Bifurcation Index
gamma = 2.88; % Power Law Index
l = 0.016; % Length of Root Segment, m
d = 0.003176; % Diameter of Root Segment, m
mu = 0.0036; % Fluid Viscosity, Pa-s
%---CALCULATED VARIABLES---%
lambda_1 = (1+alpha^gamma)^(1/gamma);
% Relation between parent branch and large daughter branch radius
lambda_2 = ((1+alpha^gamma)/(alpha^gamma))^(1/gamma);
% Relation between parent branch and small daughter branch radius
theta_1 = acosd(((1+alpha^3)^(4/3)+1-(alpha^4))/(2*(1+alpha^3)^(2/3)));
% Angle made between parent branch and large daughter branch
theta_2 = acosd(((1+alpha^3)^(4/3)+(alpha^4)-
1)/(2*(alpha^2)*(1+alpha^3)^(2/3)));
% Angle made between parent branch and small daughter branch
% For ease of calculations, we need to number each vessel in the
segment, as well as give a numbering convention to each based on the
level and position of the vessel within that level.
%---VESSEL NUMBERING--% % Pre-allocation of arrays for computational speed num = zeros(1,V); % Vessel Number Array j = zeros(1,V); % Level Number Array k = zeros(1,V); % Position Number in Level Array L = zeros(1,V); % Length Array theta = zeros(1,V); % Angle Array x1 = zeros(1,V); % Proximal x-coordinate array x2 = zeros(1,V); % Distal x-coordinate array y1 = zeros(1,V); % Proximal y-coordinate array y2 = zeros(1,V); % Distal y-coordinate array lambda = zeros(1,V);
% Relation between vessel and parent vessel radius lambda_root = zeros(1,V);
% Relation between vessel and root vessel radius dp = zeros(1,V); % Pressure Drop in each vessel segment x3 = zeros(1,V); % Proximal x-coordinate array y3 = zeros(1,V); % Distal y-coordinate array x4 = zeros(1,V); % Proximal x-coordinate array y4 = zeros(1,V); % Distal y-coordinate array R = zeros(1,V); % Resistances in each vessel segment Q = zeros(1,V); % Flowrate in each vessel segment
num(1,1) = 1; j(1,1) = 0; k(1,1) = 1; L(1,1) = l; D(1,1) = d; theta(1,1) = 0; x1(1,1) = 0; y1(1,1) = 0; x2(1,1) = 0; y2(1,1) = -(L(1,1)); lambda(1,1) = 1; lambda_root(1,1) = lambda(1,1); dp(1,1) = 0.1718; % Relates to an inlet flowrate of 60 mL/min x3(1,1) = 0; y3(1,1) = 100; x4(1,1) = L(1,1); x3(1,1) = 0; y4(1,1) = y3(1,1)-dp(1,1);
for i = 2:1:V
i num(1,i) = i; % Numbering convention for numbering vessels, j(1,i) = fix(log2(i)); k(1,i) = num(1,i)-(2^(j(1,i)))+1;
if mod(num(1,i),2)==0
% If vessel number is even do below, if it is odd, do what is under
'else' statement.
L(1,i) = (L(1,i-(i/2))/lambda_1); %---LENGTH CALCULATION---% D(1,i) = (D(1,i-(i/2))/lambda_1); %---DIAMETER CALCULATION---% theta(1,i) = theta(1,i-(i/2))-theta_1;%---ANGLE CALCULATION---% x1(1,i) = x2(1,i-(i/2)); %---CALCULATION OF COORDINATES--% y1(1,i) = y2(1,i-(i/2)); x2(1,i) = x2(1,i-(i/2)) + L(1,i)*sind(theta(1,i)); y2(1,i) = y2(1,i-(i/2)) - L(1,i)*cosd(theta(1,i)); lambda(1,i) = lambda_1; lambda_root(1,i) = lambda(1,i)*lambda_root(1,i-(i/2)); dp(1,i) = dp(1,1)*(lambda_root(1,i))^(3-gamma); x3(1,i) = x4(1,i-(i/2)); y3(1,i) = y4(1,i-(i/2)); x4(1,i) = x4(1,i-(i/2)) + L(1,i); y4(1,i) = y4(1,i-(i/2)) - dp(1,i);
else
L(1,i) = (L(1,i-((i-1)/2)-1)/lambda_2); D(1,i) = (D(1,i-((i-1)/2)-1)/lambda_2); theta(1,i) = theta(i-((i-1)/2)-1)+theta_2; x1(1,i) = x2(i-((i-1)/2)-1); y1(1,i) = y2(i-((i-1)/2)-1); x2(1,i) = x2(i-((i-1)/2)-1) + L(i)*sind(theta(i)); y2(1,i) = y2(i-((i-1)/2)-1) - L(i)*cosd(theta(i)); lambda(1,i) = lambda_2; lambda_root(1,i) = lambda(1,i)*lambda_root(1,i-((i-1)/2)-1); dp(1,i) = dp(1,1)*(lambda_root(1,i))^(3-gamma); x3(1,i) = x4(1,i-((i-1)/2)-1); y3(1,i) = y4(1,i-((i-1)/2)-1); x4(1,i) = x4(1,i-((i-1)/2)-1) + L(1,i); y4(1,i) = y4(1,i-((i-1)/2)-1) - dp(1,i);
end
end
for i = 1:1:V
i R(1,i) = (128*mu*L(1,i))/(pi*(D(1,i)^4)); Q(1,i) = (dp(1,i)*133.32)/R(1,i);
end
figure(1) x = [x1(1:V);x2(1:V)]; %---PLOTTING GEOMETRIC NETWORK---% y = [y1(1:V);y2(1:V)]; line(x,y) hold on
figure(2) xa = [x3(1:V);x4(1:V)]; %---PLOTTING PRESSURE DISTRIBUTION---% ya = [y3(1:V);y4(1:V)]; line(xa,ya) hold on
% Outlet Capillary Velocity
D(V) Q(V) v_cap = Q(V)/(pi*((D(V)^2)/4))
Program used for Steady Flow with PC, and Pulsatile Flow with and without PC
simulations
% This program will build an arterial network tree structure based on bifurcation theory. The program will then estimate the pulsatile
pressure drop along each path within the bifurcation tree. The results will be used to estimate the pulsatile flow distribution in the
arterial network.
close all clear all clc
%---INPUT PARAMETERS---%
A = 14; % Number of levels in the tree structure V = (2^A)-1; % Number of vessels in tree structure alpha = 0.690195; % Bifurcation Index gamma = 2.88; % Power Law Index l = 0.016; % Length of Root Segment, m d = 0.003176; % Diameter of Root Segment, m mu = 0.0036; % Fluid Viscosity, Pa-s rho = 1060; % Fluid Density, kg/m^3 T = 210; % Period of one cardiac wave, s w = (2*pi)/T; % Heart rate, radians per second N = 26251;
% Number of data points in pressure drop given for root segment dt = T/(N-1); % Time step value t_N = 0:dt:T;
% Time domain for given pressure values in root segment t = 0:dt:T; % Times for flow waveform
%---CALCULATED VARIABLES---%
lambda_1 = (1+alpha^gamma)^(1/gamma);
% Relation between parent branch and large daughter branch radius lambda_2 = ((1+alpha^gamma)/(alpha^gamma))^(1/gamma);
% Relation between parent branch and small daughter branch radius theta_1 = acosd(((1+alpha^3)^(4/3)+1-(alpha^4))/(2*(1+alpha^3)^(2/3)));
% Angle made between parent branch and large daughter branch theta_2 = acosd(((1+alpha^3)^(4/3)+(alpha^4)-
1)/(2*(alpha^2)*(1+alpha^3)^(2/3)));
% Angle made between parent branch and small daughter branch
% For ease of calculations, we need to number each vessel in the
segment, as well as give a numbering convention to each based on the
level and position of the vessel within that level.
%---VESSEL NUMBERING--% % Pre-allocation of arrays for computational speed
num = zeros(1,V); % Vessel Number Array j = zeros(1,V); % Level Number Array k = zeros(1,V); % Position Number in Level Array
L = zeros(1,V); % Length Array theta = zeros(1,V); % Angle Array x1 = zeros(1,V); % Proximal x-coordinate array x2 = zeros(1,V); % Distal x-coordinate array y1 = zeros(1,V); % Proximal y-coordinate array y2 = zeros(1,V); % Distal y-coordinate array lambda = zeros(1,V);
% Relation between vessel and parent vessel radius lambda_root = zeros(1,V);
% Relation between vessel and root vessel radius dp = zeros(length(t),V); % Pressure Drop in each vessel segment x3 = zeros(1,V); % Proximal x-coordinate array y3 = zeros(1,V); % Distal y-coordinate array x4 = zeros(1,V); % Proximal x-coordinate array y4 = zeros(1,V); % Distal y-coordinate array R = zeros(1,V); % Resistances in each vessel segment Q = zeros(length(t),V); % Flowrate in each vessel segment Wom = zeros(1,V); % Womersley Number for each vessel segment
num(1,1) = 1; j(1,1) = 0; k(1,1) = 1; L(1,1) = l; D(1,1) = d; theta(1,1) = 0; x1(1,1) = 0; y1(1,1) = 0; x2(1,1) = 0; y2(1,1) = -(L(1,1)); lambda(1,1) = 1; lambda_root(1,1) = lambda(1,1); dp(:,1) = []; % INSERT INLET PRESSURE DATA AS A SINGLE COLUMN HERE.
PRESSURE IS COMPUTED FROM PRESSURE DROP OF MODEL 1b. %
x3(1,1) = 0; y3(1,1) = 0; x4(1,1) = L(1,1); x3(1,1) = 0; y4(1,1) = y3(1,1)-dp(1,1);
for i = 2:1:V
i num(1,i) = i; % Numbering convention for numbering vessels j(1,i) = fix(log2(i)); k(1,i) = num(1,i)-(2^(j(1,i)))+1;
if mod(num(1,i),2)==0
% If vessel number is even do below, if it is odd, do what is under
'else' statement.
L(1,i) = (L(1,i-(i/2))/lambda_1); %---LENGTH CALCULATION---% D(1,i) = (D(1,i-(i/2))/lambda_1); %---DIAMETER CALCULATION---% theta(1,i) = theta(1,i-(i/2))-theta_1;%---ANGLE CALCULATION---% % x1(1,i) = x2(1,i-(i/2)); %---CALCULATION OF COORDINATES--% % y1(1,i) = y2(1,i-(i/2));
% x2(1,i) = x2(1,i-(i/2)) + L(1,i)*sind(theta(1,i)); % y2(1,i) = y2(1,i-(i/2)) - L(1,i)*cosd(theta(1,i)); lambda(1,i) = lambda_1; lambda_root(1,i) = lambda(1,i)*lambda_root(1,i-(i/2)); dp(:,i) = dp(:,1)*(lambda_root(1,i))^(3-gamma); % x3(1,i) = x4(1,i-(i/2)); % y3(1,i) = y4(1,i-(i/2)); % x4(1,i) = x4(1,i-(i/2)) + L(1,i); % y4(1,i) = y4(1,i-(i/2)) - dp(1,i);
The coordinate calculations are commented out for a quicker run time of
the program.
else
L(1,i) = (L(1,i-((i-1)/2)-1)/lambda_2); D(1,i) = (D(1,i-((i-1)/2)-1)/lambda_2); theta(1,i) = theta(i-((i-1)/2)-1)+theta_2; % x1(1,i) = x2(i-((i-1)/2)-1); % y1(1,i) = y2(i-((i-1)/2)-1); % x2(1,i) = x2(i-((i-1)/2)-1) + L(i)*sind(theta(i)); % y2(1,i) = y2(i-((i-1)/2)-1) - L(i)*cosd(theta(i)); lambda(1,i) = lambda_2; lambda_root(1,i) = lambda(1,i)*lambda_root(1,i-((i-1)/2)-1); dp(:,i) = dp(:,1)*(lambda_root(1,i))^(3-gamma); % x3(1,i) = x4(1,i-((i-1)/2)-1); % y3(1,i) = y4(1,i-((i-1)/2)-1); % x4(1,i) = x4(1,i-((i-1)/2)-1) + 1; % y4(1,i) = y4(1,i-((i-1)/2)-1) - dp(1,i);
end
end
clear theta Q R x1 x2 x3 x4 y1 y2 y3 y4 lambda_root lambda
dp_fs = fft(dp(:,V)); dp_FS = dp_fs/N;
n = 5000; a0 = zeros(1); a0_flow = zeros(size(a0)); an = zeros(n,1); bn = zeros(size(an)); An = zeros(size(an)); Lambda = zeros(1,1); dp_n = zeros(n,length(t)); q = zeros(n, length(t)); pressure = zeros(length(t),1); flowrate = zeros(length(t),1);
for i = 16383
%Only capillary vessel segment pressure and flow are computed for% i
a0(1) = dp_FS(1); a0_flow(1) = (pi*a0(1)*(((D(1,i)/2)^4)/(8*mu*L(1,i)))); an(:,1) = 2*real(dp_FS(2:n+1,1)); bn(:,1) = -2*imag(dp_FS(2:n+1,1)); An(:,1) = an(:,1)-bn(:,1)*sqrt(-1); Wom(1,i) = sqrt((rho*w)/mu)*(D(1,i)/2); Lambda(1,i) = ((sqrt(-1)-1)/(sqrt(2)))*Wom(1,i); m = 1; for m = 1:n m dp_n(m,:) = An(m,1)*exp(sqrt(-1)*w*m*t); q(m,:) = ((pi*((D(1,i)/2)^4))/(sqrt(-
1)*mu*L(1,i)*(Wom(1,i)^2)))*(1-
((2*besselj(1,Lambda(1,i)))/(Lambda(1,i)*besselj(0,La
mbda(1,i)))))*dp_n(m,:);
end pressure(:,1) = a0(1) + real(sum(dp_n,1)); flowrate(:,1) = a0_flow(1) + real(sum(q,1)); end
v_cap = (flowrate(:,1)/(pi*((D(1,V)^2/4))));
% Average outlet velocity, m/s
realcap = real(v_cap)
figure('renderer','painters') % Plot of average capillary velocity
plot(t.',v_cap*1000) xlabel('Time (s)') ylabel('Velocity (mm/s)') title('Average Pulsatile Velocity in Capillary vs. Time') axis([0 60 -2 3])