in-vitro experimental validation of finite element...
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IN-VITRO EXPERIMENTAL VALIDATION OF FINITE ELEMENT ANALYSIS OF BLOOD FLOW AND VESSEL WALL DYNAMICS
A DISSERTATION
SUBMITTED TO THE DEPARTMENT OF BIOENGINEERING
AND THE COMMITTEE ON GRADUATE STUDIES
OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
Ethan Oblivion Kung
November 2010
http://creativecommons.org/licenses/by-nc/3.0/us/
This dissertation is online at: http://purl.stanford.edu/hj038sy7972
© 2011 by Ethan Oblivion Kung. All Rights Reserved.
Re-distributed by Stanford University under license with the author.
This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.
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I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Charles Taylor, Primary Adviser
I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Michael McConnell
I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Christopher Zarins
Approved for the Stanford University Committee on Graduate Studies.
Patricia J. Gumport, Vice Provost Graduate Education
This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.
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Abstract
Biomechanical forces such as hemodynamic parameters and stress and strain in
blood vessel walls have significant effects on the initiation and development of
cardiovascular diseases, as well as on the operations of implantable medical devices.
Computational fluid dynamics is an emerging powerful numerical tool capable of
providing fine temporal and spatial resolutions, and much versatility, in the
quantifications of these cardiovascular biomechanical forces. The overall goal of this
research is to develop tools and methods for conducting in-vitro experiments, and to
acquire experimental data for the validation of the computational methods.
We first developed a physical Windkessel module which can provide realistic
vascular impedances at the outlets of flow phantoms in order to enable in-vitro
experiments that mimic in-vivo conditions. We also defined a corresponding analytical
model of the Windkessel module, and showed that upon proper characterization, the
analytical model can accurately predict the pressure and flow relationships produced by
the physical Windkessel module. The precise analytical model can then be prescribed as
a boundary condition for the finite element domain, resulting in a direct parallel between
the computational description of the physical model and the physical reality.
We then performed validation of the numerical method using the Windkessel
module, and a rigid, two outlet, patient-derived abdominal aortic aneurysm phantom
under resting and light exercise flow and pressure conditions. Physiological pressures
within the phantom, and flow waveforms through the two phantom outlets were achieved.
The numerical simulations predicted the flow split between the outlets to within 2%
accuracy of experimental measurements. Velocity pattern comparisons between
simulation and experiment also showed very favorable results, especially in the case of
whole-cycle averaged analysis.
Finally, we performed validation of the numerical method incorporating
deformable vessel walls, using two compliant flow phantoms under physiological flow,
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pressure, and deformation conditions. The compliant phantoms mimicked a patent
thoracic aorta, and one with an 84% coarctation (by area). The accuracy of numerical
predictions for flow and pressure normalized pulse amplitudes at various locations down
the length of the deformable vessels was within 10% of experimental measurements. The
accurate prediction of wave propagation behaviors indicated a realistic representation of
the vessel wall motion.
Future work could include further in-vitro validation incorporating deformable
phantoms with more complex geometries. The experimental techniques we have
developed can also be used in direct in-vitro evaluations of medical devices.
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Acknowledgments
The work I am presenting in this thesis would not have been possible without the
help of many individuals, and the process certainly would not have been as pleasant
without many others. I would like to first acknowledge my advisor, Prof. Charles Taylor,
who was the primary reason why I stayed at Stanford to continue with the PhD program
after completing my master’s degree. I can remember that during the few hiccups there
had been with the projects in the lab, no matter how uncertain situations got, Charley had
always silently handled the up and downs of the politics and selflessly looked out and
cared for me throughout my time here. His optimism and audacious personality have
made him both great fun to work/travel with, and a pillar to look to during the chaos of
the storms. I would also like to thank my reading committee members Dr. Christopher
Zarins and Dr. Michael McConnell. Dr. Zarins has been a stable presence during my
graduate school career. He has always been kind, encouraging, and very approachable
despite being one of the top vascular surgeons in the country. Dr. McConnell has
provided much valuable advice and input to my project, and has also been eager to
provide assistance when I needed. Finally, I would like to thank Prof. Gerald Pollack
from the University of Washington, who was a mentor to me during my early years after
college graduation and helped solidify my interest in Bioengineering.
I cannot imagine how I could have completed this work without the help of my
lab mate Dr. Andrea Les, who helped run every single one of my MR experiments,
meticulously edited my papers, and tirelessly taught me everything about the simulation
software I needed to use. Among everything miscellaneous, Andrea had simply been the
go-to person whenever I needed help. Together with Sandra Rodriguez, who served as
the MR tech for most of my experiments, they were excellent company to be with during
the scans and I will not forget the hilarious moments we had in the control room. I also
have to thank Anne Sawyer, Dr. Gary Glover, and Dr. Marc Alley for providing general
assistance with the scanning at the Lucas Center. I had received tremendous help from
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Dr. Chris Elkins with flow system setups and consultations regarding any general in-vitro
experimental concerns. In addition to being a total expert when it comes to setting up
hardware, Chris also had provided many valuable insights and ideas regarding my
experiments when I needed them. Lakhbir Johal at the Thermo Sciences machine shop
played a vital role in constructing the parts for my experiments. Several occasions he had
commuted in on weekends and holidays in order to work on the parts I needed so that I
could have them in time. I have appreciated his dedication, as well as his friendship.
Francisco Medina and Dr. Ryan Wicker from the UTEP Keck Center also had provided
much help in coordinating the construction of the in-vitro experiment phantoms.
Many current and previous members of the Taylor lab have provided assistance
towards various aspects of my research. Dr. Alberto Figueroa provided much
consultation and troubleshooting regarding the deformable simulations. Dr. Ryan Spilker
and Dr. Hyun Jin Kim provided the necessary tools for prescribing the boundary
conditions in my simulations. Nan Xiao is a programming guru who solved many coding
issues for me. Dr. Joy Ku and Dr. Mary Draney paved the path for my thesis research
with their own. I have also appreciated all of the lab members for their friendships, and
for constructing a relaxed, friendly, and enjoyable work atmosphere. I have particularly
enjoyed the fun times of TAing with Kelly Suh, Dr. Tina Morrison, Sanaz Saatchi, and
Rashmi Raghu, and the assimilation into the Persian culture by Hedi Razavi and Sanaz.
As part of the first incoming class of the Bioengineering department, I have
thoroughly enjoyed being in the department since its very beginning. I would like to
thank all of the Bioengineering faculties and staff members for making the department
what it is, and especially Olgalydia Urbano-Winegar for meticulously caring for each and
every one of the students like her own child. I would also like to thank the students in the
department for their friendships. In particular, Jerrod Schwartz, Julia Chen, Adam
Grossman, and Douglas and Sara Jones, have been untiring comrades getting through the
program together since the first year, and Anderson Nnewihe has been a fun music
performance buddy for many departmental events.
My time at Stanford has been some of the best years of my life, and a large part of
it has to be attributed to the Reformed University Fellowship, the Inter Varsity Graduate
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Fellowship, and the Chi Alpha Christian Fellowship at Stanford. David Jones, Pete
Sommer, and Glenn Davis, who are the pastors leading each of these groups have been
crucial mentors to me and taught me a tremendous amount about life beyond academic
and career pursuits. My friends from these groups have also walked with me through the
years. We have played together, laughed together, lamented together, or, sat together
with nothing to do. They have cared for and supported me, and they also have challenged
me and held me accountable to being a decent human being, all making me into a more
complete person. There are too many of these amazing friends to name, but in particular,
I would like to thank Alan Asbeck, Katie Turner, James Chu, Rob Majors, Edmond Chiu,
Eric Chu, Allie and Adam Leeper, Beth and Allen Jameson, Hannah and Mickey Sheu,
David Sangokoya, and Chrissy Guerra for their friendships over the years. I would also
like to thank Ellen Abell, Kimmy Wu, Julia Jang, Kassa Betre, and Yi Gu for our regular
get-togethers and the sharing of our lives. Finally, I would like to acknowledge my best
friend Cory Combs and my partner in crime David Williams for the fun times and for
their unfailing presence through the up and downs of life.
Lastly, and most importantly, I shall attempt to express my gratitude that I really
cannot express in words towards my family. They have provided the most supportive,
loving, and caring environment to nourish me in ways more than I could ever asked for.
My mom and dad have sacrificed their entire lives for me, always utterly disregarding
their own well-being and never wanting to ask for anything in return. They continue to
support my choice at every crossroad, rejoice with me at every success, and silently stand
by me every time I am too busy to talk to them. Together with my grandfather and
grandmother, they have always been proud of me at every phase of my life, providing the
motivation and hope for the future. I would especially like to thank my grandmother for
raising me up since I was a child, and always being a source of comfort till this day. My
cousins Jen Weng and Albert Lin have also been important parts of my life, and their
parents have always cared for me as their own child. Practically, I have four sets of
parents, and two siblings, all of whom have shaped me into who I am today. I am
incredibly thankful for the love and support they all have shown me.
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My research was supported by the National Institutes of Health (Grants P50
HL083800, P41 RR09784, and U54 GM072970) and the National Science Foundation
(0205741, and CNS-0619926 for computer resources). Over the course of my graduate
studies I had also received two scholarship awards from the Natural Sciences and
Engineering Research Council of Canada.
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Table of Contents
Chapter 1 : Introduction .............................................................................................1
1.1 Biomechanics and Vascular Disease ...............................................................1
1.2 Computational Fluid Dynamics .......................................................................1
1.3 In-vitro Validation ...........................................................................................3
1.4 Thesis Outline & Organization ........................................................................4
Chapter 2 : Technical Background ............................................................................5
2.1 Magnetic Resonance Imaging .........................................................................5
2.1.1 Nuclear Magnetic Resonance ..........................................................5
2.1.2 Received Signal ...............................................................................6
2.1.3 Longitudinal Spatial Localization ....................................................7
2.1.4 Transverse Spatial Localization .......................................................9
2.1.5 Phase Contrast Velocity Imaging ...................................................10
2.1.6 Cine Phase Contrast Pulsatile Velocity Imaging ...........................12
2.1.7 Field of View, Resolution, and Signal to Noise Ratio ...................13
2.2 Computational Fluid Dynamics .....................................................................14
2.2.1 Governing Equations .....................................................................14
2.2.2 Boundary Conditions & The Weak Form Equations .....................15
2.2.3 Finite Element Analysis .................................................................17
2.2.4 Incorporating Vessel Wall Deformability ......................................17
Chapter 3 : Development of a Physical Windkessel Module to Re-Create In-
Vivo Vascular Flow Impedance for In-Vitro Experiments ....................................19
3.1 Abstract ..........................................................................................................19
3.2 Introduction ...................................................................................................20
3.3 Methods .........................................................................................................21
3.3.1 Determining Target Windkessel Component Values ....................21
3.3.2 Flow Resistance Module ................................................................23
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3.3.3 Flow Capacitance Module .............................................................28
3.3.4 Flow Inductance .............................................................................29
3.3.5 Assembled Windkessel Module & Corresponding Analytical
Model .............................................................................................30
3.4 Experimental Testing & Data Analysis .........................................................31
3.4.1 Resistance Module .........................................................................31
3.4.2 Assembled Windkessel Module .....................................................33
3.5 Results & Discussions ...................................................................................34
3.5.1 Resistance Module .........................................................................34
3.5.2 Assembled Windkessel Module .....................................................36
3.6 Conclusion .....................................................................................................41
Chapter 4 : In-Vitro Validation of Finite Element Model of Abdominal
Aortic Aneurysm Hemodynamics Incorporating Realistic Outlet Boundary
Conditions ...................................................................................................................43
4.1 Abstract ..........................................................................................................43
4.2 Introduction ...................................................................................................44
4.3 Methods .........................................................................................................45
4.3.1 Anatomical Model Construction ....................................................45
4.3.2 Outlet Boundary Condition ............................................................46
4.3.3 In-Vitro Experiment .......................................................................50
4.3.4 In-Silico Simulation .......................................................................52
4.4 Results ...........................................................................................................54
4.4.1 PCMRI Flow Measurements at Different Slice Locations ............54
4.4.2 PCMRI vs. Flow Probe Measured Inlet Flow Waveforms ............55
4.4.3 Measured vs. Simulated Flow & Pressure Waveforms .................56
4.4.4 PCMRI vs. FEA Instantaneous Time Point Velocity Patterns ......57
4.4.5 PCMRI vs. FEA Whole-cycle Averaged Velocity Patterns ..........60
4.5 Discussion ......................................................................................................61
Chapter 5 : In Vitro Validation of Finite Element Analysis of Blood Flow in
Deformable Models ....................................................................................................65
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5.1 Abstract ..........................................................................................................65
5.2 Introduction ...................................................................................................66
5.3 Methods .........................................................................................................67
5.3.1 Physical Flow Phantom Construction & Characterization ............67
5.3.2 Outlet Boundary Condition ............................................................69
5.3.3 In-vitro Experiment ........................................................................72
5.3.4 In-silico Simulation ........................................................................74
5.4 Results ...........................................................................................................78
5.4.1 Flow & Pressure Waveforms in the Straight Phantom ..................78
5.4.2 Flow & Pressure Waveforms in the Stenotic Phantom ..................80
5.4.3 Impedance Modulus & Phase at the Inlet / Outlet .........................81
5.4.4 Through-Plane Velocity Patterns ...................................................83
5.5 Discussion ......................................................................................................85
Chapter 6 : Conclusion .............................................................................................89
6.1 Summary & Contributions ............................................................................89
6.2 Future Work ...................................................................................................91
6.2.1 In-vitro Validation .........................................................................91
6.2.2 In-vivo Validation ..........................................................................92
6.2.3 Direct Medical Device Evaluation .................................................93
Chapter 7 : References ..............................................................................................94
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List of Tables
Table 3.1 Estimated Resistance Values (and Numbers of Capillary Tubes) Resulting
From Various Combinations of Conduit Diameter (Maximum Laminar
Flow Rate), and Capillary Tube Size ...............................................................27
Table 3.2 Theoretical and Experimental Windkessel Component Values for the
Thoracic-Aortic and Renal Impedance Modules .............................................37
Table 4.1 List of Resistance Modules Used in the Experiment ........................................48
Table 4.2 Outlet BC Windkessel Component Values Prescribed to the FEA
Simulations ......................................................................................................53
Table 5.1 Theoretical and Experimental Windkessel Component Values for the
Straight and Stenotic Phantom Experiments ....................................................71
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List of Figures
Figure 2.1 Spatially varying resonant frequencies are achieved by the application of a
linearly varying gradient. Courtesy of Mary T. Draney. ...................................8
Figure 2.2 Under an applied linear gradient Gz, an RF excitation pulse with
bandwidth ∆ω would only excite the spins with resonant frequencies that
fall within ωo± ∆ω/2, corresponding to spatial locations ±∆z/2. Courtesy
of Mary T. Draney..............................................................................................8
Figure 2.3 The 2DFT Cartesian grid in k-space. Frequency encode gradients (Gx) in
the x direction are used to traverse through each line, and phase encode
gradients (Gy) in the y direction are used to step through the lines to fill
the grid. Courtesy of Mary T. Draney. ............................................................10
Figure 2.4 The data acquisition process for 3-component cine phase contrast velocity
imaging. One line of k-space at a particular Ky value is acquired per
cardiac cycle, until all of the phase encodes (PE) are completed. The
repetitions within each cardiac cycle are used to construct different time
frames retrospectively. Typically 24 time frames are constructed per
cardiac cycle. The NEX is the number of excitations to average together.
Courtesy of Mary T. Draney. ...........................................................................13
Figure 3.1 A basic three-element Windkessel model for component value estimation
purpose .............................................................................................................22
Figure 3.2 a) Maximum Laminar Flow Rate v.s. Number of Parallel Channels for
Various Resistance Values. b) Resistance v.s. Number of Parallel
Channels for Various Standard Capillary Tube Inside Diameters (ID) ...........25
Figure 3.3 a) Capillary Tube Resistance Module Construction b) Switchable
Resistance Setup ..............................................................................................28
Figure 3.4 a) Capacitance Module Construction b) Capacitor Inlet Contour ................29
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Figure 3.5 a) Assembled Impedance Module. b) Final Analytical Model of
Impedance Module ...........................................................................................31
Figure 3.6 Resistance Module Steady Flow Testing Setup ..............................................32
Figure 3.7 Impedance Module Pulsatile Flow Testing Setup ...........................................34
Figure 3.8 Resistance v.s. Flow Rate for Resistance Module with Theoretical
Resistance of a) 500 Barye*s/cm3, b) 6700 Barye*s/cm3, and a Partially
Closed Ball Valve ............................................................................................36
Figure 3.9 Comparisons Between Measured (solid lines) and Calculated (dots)
Pressure & Flow Waveforms for the Thoracic-Aortic Impedance Module
Under Four Different Flow Conditions............................................................39
Figure 3.10 Comparisons Between Measured (solid lines) and Calculated (dots)
Pressure & Flow Waveforms for the Renal Impedance Module Under Four
Different Flow Conditions ...............................................................................40
Figure 3.11 Comparisons Between Theoretical and Experimental Flow Impedance
Modulus and Phase for the a) Thoracic-aortic, and b) Renal, Impedance
Module .............................................................................................................41
Figure 4.1 Anatomical Phantom Model. a) MR Imaging data from an AAA patient.
b) 3D computer model constructed based on patient imaging data. c)
Physical phantom constructed from 3D computer model. ...............................46
Figure 4.2 In-vitro Experiment Flow System Setup Diagram ..........................................47
Figure 4.3 a) The physical Windkessel module assembly and the corresponding
analytical representation b) The resistance module c) The capacitance
module..............................................................................................................47
Figure 4.4 PCMRI Measured Flow Through the Abdominal Aorta at Different Slice
Locations for a) Resting condition, and b) Light exercise condition ..............54
Figure 4.5 PCMRI vs. Ultrasonic Flow Probe Measured Total Inlet Flow for a)
Resting condition, and b) Light exercise condition ........................................55
Figure 4.6 Measured In-vitro (Solid lines) vs. Simulated In-silico (dashed lines)
Pressure & Flow Waveforms for a) Resting condition, and b) Light
exercise condition ............................................................................................56
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Figure 4.7 Resting Condition Flow Velocity Comparisons: Between MR
Measurements and FEA Results at the a) Diastole, b) Acceleration, c)
Systole, and d) Deceleration time point, at three different slice locations
(S1-S3). Colour map and arrows correspond to through-plane and in-
plane velocities, respectively. ..........................................................................58
Figure 4.8 Light Exercise Flow Velocity Comparisons: Between MR Measurements
and FEA Results at the a) Diastole, b) Acceleration, c) Systole, and d)
Deceleration time point, at three different slice locations (S1-S3). Colour
map and arrows correspond to through-plane and in-plane velocities,
respectively. .....................................................................................................59
Figure 4.9 Whole-cycle Averaged Flow Velocity Comparisons: Between MR
measurements and FEA results at three different slice locations (S1-S3)
for a) Resting condition and b) Light exercise condition. ...............................60
Figure 5.1 Vessel Outer Diameter Versus Static Pressure for the a) Straight Phantom,
and b) Stenotic Phantom ..................................................................................69
Figure 5.2 In-vitro Flow Experiment Setup Diagram .......................................................72
Figure 5.3 Pressure and Flow Velocity Measurement Locations in a) the Straight
Phantom, and b) the Stenotic Phantom. Green section is deformable.
Grey section is rigid. Dimensions are in centimeters .....................................74
Figure 5.4 Summary of Boundary Condition Prescriptions for the Numerical
Simulations ......................................................................................................77
Figure 5.5 Straight Phantom Simulated Versus Measured Flow & Pressure a)
Waveforms and b) Normalized Pulse Amplitudes, at Different Locations .....79
Figure 5.6 Stenotic Phantom Simulated Versus Measured Flow & Pressure a)
Waveforms and b) Normalized Pulse Amplitudes, at Different Locations .....81
Figure 5.7 Simulated Versus Measured Impedance Modulus and Phase at the Inlet
and Outlet for the a) Straight, and b) Stenotic Phantom Experiment ..............82
Figure 5.8 Through-Plane Velocity Pattern Comparisons at the L2 Location for the
Straight Phantom Experiment at Four Different Time Points: Diastole,
Acceleration, Systole, and Deceleration ..........................................................84
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Figure 5.9 Through-Plane Velocity Pattern Comparisons at the a) L2, and b) L3
Location for the Stenotic Phantom Experiment at Four Different Time
Points: Diastole, Acceleration, Systole, and Deceleration ...............................85
Equation Chapter (Next) Section 1
Chapter 1: Introduction
1.1 Biomechanics and Vascular Disease
Biomechanical forces directly influence short-term and long-term responses of the
cardiovascular system. Hemodynamic parameters such as the three-dimensional blood
flow and pressure fields, as well as the stress and strain in blood vessels, have direct
effects on the initiation and development of cardiovascular diseases such as
atherosclerosis and aneurysms.1-5 For example, Glagov et al. found quantitative
evidences that low blood flow velocities and shear stresses can promote atherosclerosis
progression.2 The hemodynamic forces within blood vessels also directly affect
biological adaptations of vessel diameter, wall thickness, and endothelial growth.6-11
Knowledge of how in-vivo forces and tissue motions interact with implantable medical
devices is also essential for understanding and predicting device behaviors after
implantation. For example, the compliance interactions between a prosthetic bypass graft
and its adjacent native arteries has been hypothesized to be a factor in graft failure.12
Medical imaging has been used to investigate vessel strain13 and blood flow
hemodynamics,14 but is a diagnostic method limited to the conditions under which the
measurements were made and cannot be used to predict disease progression or outcomes
of medical, interventional or surgical therapy.
1.2 Computational Fluid Dynamics
Computational fluid dynamics (CFD) can be used to quantify 3-dimensional
blood flow velocities and pressures by employing a numerical technique to solve the
Navior-Stokes and continuity equations in an arbitrary 3D mesh.15 The analysis can
further include the deformability of vessel walls by incorporating the solid domain effects
on the fluid velocities at the vessel wall interface.16 The geometry of the 3D numerical
domain can often be extracted directly from clinical computed tomography or magnetic
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resonance imaging (MRI) data, enabling patient-specific quantification of hemodynamics
and tissue motion. In addition, ultrasound and MRI can provide patient-specific flow
data which can then be used together with pressure measurements to enable inlet and
outlet boundary condition prescriptions.
The CFD methods enable the study of hemodynamics and tissue motion with
temporal and spatial resolutions many times finer than what is possible with any imaging
modality. CFD also requires minimal patient involvement compared to direct in-vivo
quantification using imaging and probing methods. These advantages have made image-
based CFD a practical alternative for quantifying vessel strains and hemodynamic
conditions in the study of disease mechanisms15,17,18 and the design and evaluation of
medical devices.19-21 The ease of applying variations in geometry and flow conditions in
the computational domain also motivates the use of CFD in the planning and prediction
of surgical procedures.22,23
It is computationally infeasible to build 3D models of the entire vascular tree
containing billions of blood vessels. In the CFD analysis, the computational domain is
typically broken up into an upstream numerical domain where three-dimensional flow
and pressure are calculated using a numerical method (ie. the finite element method), and
a downstream analytical domain where the physical properties of downstream vasculature
not modeled in the numerical domain are represented with a simple lower-ordered model.
The downstream domain is coupled to the numerical domain as a boundary condition,
which dramatically influences the flow and pressure computation results in the numerical
domain.24 A multi-domain method that couples simple impedance, lumped-parameter, or
1-D models at the boundaries of the numerical domain24-30 has been shown to enable
highly physiological flow and pressure fields within the domain.24
Much work remains to validate CFD methods against experimental data. The
process of validation involves creating a direct parallel between a physical experimental
setup and a computational simulation, and then comparing the computation results to
measurements made in the physical system to show that the numerical method faithfully
captures the behaviors of the physical system.
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1.3 In-vitro Validation
An in-vitro validation approach enables a validation study to be done on a fully
characterized physical system with known parameters, and under a highly controlled
operational environment. It also allows for the ability to acquire high quality imaging
and other experimental data. In-vivo validation, on the other hand, while having been
previously attempted,31,32 is limited in the level of precision possible due to current
imaging constraints and the inherent variability and unknowns within an in-vivo system.
For example, there is currently no known direct relationship between the prescription of
an outlet boundary condition and the anatomy of the corresponding downstream
vasculature, thus an in-vivo validation approach would require the prescription to be done
through empirical methods. In general, it makes logical and practical sense to use the in-
vitro approach as the first stepping stone in the process of building a solid foundation for
CFD validations.
Previous in-vitro validation studies, due to the lack of adequate boundary
conditions in the physical setup, have not obtained realistic flow division and pressures to
accurately replicate in-vivo conditions. In addition, without appropriate outflow
boundary conditions to produce physiological pressures required to achieve realistic
deformations in a compliant model, previous in-vitro validations have been performed
mostly in rigid models. Experimental setups have prevalently implemented simple zero
pressure boundary conditions, which involve connections from phantom outlets directly
into a fluid reservoir.33-37 In the computational domain, the commonly prescribed flow
velocity boundary condition38-40 cannot be practically carried over to patient-based
computations, since obtaining flow velocity data for all of the outlets of a major blood
vessel in a patient is currently not feasible with the imaging capabilities available. Lastly,
many previous studies also employed steady flow,41-45 while others used relatively simple
phantom geometries and low flow rates.15,34,35,46-49
The most immediate need in the advancement of in-vitro CFD validation is the
creation of a physical impedance module that can produce physiological pressure under
physiological flow, and possesses parameters that can be directly prescribed to an
analytical model. Using such a module as the boundary condition, we would then be able
4
to perform validation using a realistic phantom geometry under realistic flows and
pressures. There is also a need for validation studies that incorporate the deformability of
the vessel wall, which has much influence on the flow and pressure inside a vessel, and is
required to capture wave propagation phenomena down the length of a vessel.
1.4 Thesis Outline & Organization
The research presented in this thesis describes the methods and techniques we
developed for the acquisition of experimental data that can be used as standards for CFD
validations, and presents the results of the validation studies we performed using this
data. Chapter 2 provides a general overview of magnetic resonance imaging, which is
the imaging modality we used for the in-vitro flow measurements, and a general
description of the CFD method we used for the in-silico simulations. Chapters 3-5 were
written as independent manuscripts for submission and thus contain some repeated
methods and background sections. Chapter 3 presents the design and manufacturing
methods for constructing a physical Windkessel module to provide realistic and
predictable vascular impedances, and defines a corresponding analytical model that can
be prescribed in the computational domain. Chapter 4 presents the validation work
performed on a rigid, two outlet, patient-derived abdominal aortic aneurysm geometry
under two different physiological flow and pressure conditions (resting and light
exercise). Chapter 5 presents the validation work performed on two deformable, single-
outlet geometries mimicking the descending thoracic aorta, one patent and one stenotic,
and both under physiological flow, pressure, and wall deformation conditions. Chapter 6
provides a summary and conclusion of the research, and discusses related future work.
5
Equation Section 2
Chapter 2: Technical Background
2.1 Magnetic Resonance Imaging
The techniques of magnetic resonance imaging (MRI) are based on manipulating
nuclear “spins” exhibited by atoms with an odd number of protons or neutrons. These
nuclear spins possess angular momentum and can be thought of as spinning charged
spheres that produce a magnetic moment. The combination of gradient magnetic fields
superposed on a static magnetic field, and radio-frequency excitations together can be
applied to align the magnetic moments and localize signals to produce images. We
present the fundamental physics and technical considerations of MRI in this section. The
contents of this section are primarily drawn from texts by Bushberg50 and Nishimura.51
2.1.1 Nuclear Magnetic Resonance
Under no external magnetic field, the magnetic spins of nuclei have no specific
spatial orientations and possess zero net magnetization. The application of a static
magnetic field provides a slightly lower energy state for the spins (parallel spin) when
they align with the external field, and thus results in a small (in the range of a few parts
per million) net magnetization in the longitudinal direction that is proportional to the
strength of the external field. The spins precess about the axis of the external field at a
frequency defined by the Larmor equation:
0o Bω γ= (2.1)
where γ is the gyromagnetic ratio specific to each type of atom, and 0B is the external
magnetic field strength.
6
For medical imaging, hydrogen atoms are typically the excitation targets due to
their abundance in living tissue. The gyromagnetic ratio specific to hydrogen is:
42.6 /2
MHz Tγπ= (2.2)
The precession frequency of hydrogen is 63.86MHz under a magnetic field
strength of 1.5T.
Under equilibrium, the net magnetization is static and entirely in the longitudinal
direction (Mz=Mo). In order to detect a signal from the net magnetization with a receiver
coil, a radiofrequency (RF) pulse B1 at the Larmor frequency is applied to cause the net
magnetization to precess about the Bo field, creating a time-varying transverse net
magnetization (Mxy). An appropriate duration of B1 (a 90 degree pulse) can excite the
longitudinal magnetization entirely into the transverse direction, where Mz=0 and
Mxy=Mo, and signal detection by the receiver coil is maximal. After such an excitation,
the process of the net magnetization returning to the equilibrium state where Mxy=0 and
Mz=Mo is governed by time constants T1 (recovery of the longitudinal magnetization)
and T2 (decay of the transverse magnetization):
( ) 10 (1 )
tT
zM t M e−
= − (2.3)
( ) 2(0)tT
xy xyM t M e−
= (2.4)
where Mxy(0) is the maximum transverse magnetization at time zero immediately after
the excitation pulse. The hydrogen atoms bound in different molecular structures have
different T1 and T2 due to differences in electron shielding, and thus image contrast can
be achieved through manipulating pulse sequence parameters to accentuate T1 and T2
weighting as desired.
2.1.2 Received Signal
The Bloch equation describes the time-dependent magnetization vector M:
0
2 1
(x y zM i M j )M M kddt T T
γ+ −
= × − −M M B (2.5)
7
where B is a time-varying magnetic field composed of a static field Bo, the excitation RF
field B1, and various time-varying gradient fields G that are used for spatial localization
and specialized sequences such as phase contrast imaging. The first term in the equation
describes the precession of the magnetization vector about the static field, the second
term in the equation describes the decay of the transverse magnetization, and the third
term in the equation describes the recovery of the longitudinal magnetization.
The receiver coil detects a signal that contains contributions from all of the
precessing transverse magnetization in the image volume. The detected signal S(t) is
described by:
( ) ( )( , ) , , ,xy xyvol zyx
s t M t dV M x y z t dxdydz= =∫ ∫∫∫r (2.6)
where r=(x,y,z). Using the solution to the Bloch equation, the detected signal is:
(2.7) ( ) ( )( )
02/ ( ), , ,0
t
o
i G rdiw tt T r
xyzyx
s t M x y z e e e dxdydzγ τ τ−
−∫
= ∫∫∫i
In the case of selective slice excitation (no z dependence), T2>>t, and defining
m(x,y) to be , the demodulated signal eventually reduces to:
(2.8) ( ) ( )( ) ( )
0 0
( ) ( )
,
t t
x yi G d x i G d y
yx
s t m x y e e dxdyγ τ τ γ τ τ− −∫ ∫
= ∫∫
2.1.3 Longitudinal Spatial Localization
Since the Larmor equation (Equation 2.1) states that the resonant frequency is a
function of the external magnetic field, to achieve selective slice excitation (a
requirement for Equation 2.8), a linear gradient field is applied in the longitudinal
direction to cause the magnetizations at different z locations to precess at different
Larmor frequencies as shown in Figure 2.1. All of the spins in the same z location
possess a particular Larmor frequency proportional to the magnetic field strength at the
particular z location, forming a “slice”. While the linear gradient is being applied, a
8
band-limited RF excitation pulse would only excite the spins within its target splice, as
shown in Figure 2.2.
Figure 2.1 Spatially varying resonant frequencies are achieved by the application of a linearly varying gradient. Courtesy of Mary T. Draney.
Figure 2.2 Under an applied linear gradient Gz, an RF excitation pulse with bandwidth ∆ω would only excite the spins with resonant frequencies that fall within ωo± ∆ω/2, corresponding to spatial locations ±∆z/2. Courtesy of Mary T. Draney.
9
2.1.4 Transverse Spatial Localization
If we define:
( ) ( )02
t
x xk t G dγ τ τπ
= ∫ (2.9)
( ) ( )02
t
y yk t G dγ τ τπ
= ∫ (2.10)
The signal equation in Equation 2.8 can be expressed as:
(2.11) ( ) ( ) ( ) ( )2, x yi k t x k t y
yx
s t m x y e dxdyπ ⎡ ⎤− +⎣ ⎦= ∫∫
We see that at any time t the received signal is precisely the Fourier Transform,
M(kx, ky), of m(x,y). The spatial frequencies (kx, ky) in the Fourier Transform are
defined by the time integrals of the applied gradient waveforms Gx(t) and Gy(t). We
express this as:
( ) ( ) ( ) ( )0 0
, ,2 2
t t
x y x ys t k k G d G dγ γτ τ τπ π
⎛ ⎞= = ⎜ ⎟
⎝ ⎠∫ ∫M M τ (2.12)
The spatial frequency domain is known as the “k-space”. During MR data
acquisition, the acquired signals fill information in k-space, and a 2-dimensional Fourier
Transform of the k-space information would then return an image in the physical space.
The data points near the center of k-space contain the low spatial frequency information
of the image, which mostly corresponds to image signal intensity. The data points near
the periphery of k-space contain the high spatial frequency information, which
corresponds to edge details of the image.
The most common strategy to acquire data in k-space is the two-dimensional
Fourier Transform imaging (2DFT), where a Cartesian grid of k-space data is filled by
acquiring lines at different values of Ky. Since the spatial frequencies kx and ky are
dependent on the time integral of the gradient fields in the x and y directions after each
excitation pulse, the k-space can be traversed by turning on the gradients in the x and y
direction for various lengths of time and at different levels of magnitudes. In 2DFT, the
phase encode gradient in the y direction is turned on until the desired value of Ky is
10
reached, and then the frequency encode gradient in the x direction is used to traverse
through Kx while data is acquired for the line of k-space at the particular value of Ky.
The repetition time (TR) determines the rate which the lines of k-space at different values
of Ky are acquired. Lines of k-space at different phase encode Ky values are acquired
until the Cartesian grid is filled, as shown in Figure 2.3.
Figure 2.3 The 2DFT Cartesian grid in k-space. Frequency encode gradients (Gx) in the x direction are used to traverse through each line, and phase encode gradients (Gy) in the y direction are used to step through the lines to fill the grid. Courtesy of Mary T. Draney.
2.1.5 Phase Contrast Velocity Imaging
The movement of an object within a gradient field creates a phase shift in the spin
that can be measured and quantified to provide velocity information. The phase shift of a
spin is dependent on its position r, and the gradient field G, as described in the equation:
( ) ( )0
tdφ γ τ τ= ∫ G ri τ (2.13)
11
The position as a function of time can be expressed as a Taylor series expansion:
( ) 21 ...2o r aτ τ τ= + + +r r v (2.14)
where ro is the position at time 0, v is the velocity, and a is the acceleration. Consider
velocity in the “x” direction, if assuming acceleration and higher order terms are
insignificant, the phase equation in 2.13 can be written as:
( ) ( )00 0
t t
x x xx G d v G dφ γ τ τ τ τ⎛ ⎞
== +⎜⎝ ⎠∫ ∫ τ ⎟ (2.15)
When a bipolar gradient waveform with zero area is used, the first term in the
equation is zero, and the phase shift becomes dependent only on the velocity:
( )( )0
TE
xv G t tdφ γ= ∫ t (2.16)
where TE is the echo delay time.
The term phase “contrast” comes from subtracting two acquisitions that are
identical except for the polarity of the gradient waveform, to obtain the phase difference.
The reason for using the phase difference is to remove unknown phase shifts at different
spatial locations due to the static magnetic field inhomogeneities. When the two
acquisitions are subtracted together, the unknown phase shifts are canceled out, resulting
in the static spins exhibiting no net phase shift, and only the moving spins exhibiting a
net phase difference described by:
( )1v Mφ γΔ = Δ (2.17)
where ( )1 0
TE
xM G t tdt= ∫ and 1 12M MΔ =
The velocity encoding (Venc) of an acquisition is defined as the velocity value that
results in a phase difference of 180 degrees:
( )1/encv π γ= ΔM (2.18)
The Venc of an acquisition is the upper limit which velocities can be measured
without aliasing. The gradient waveform can be adjusted to affect the amount of phase
12
difference resulting from a given velocity, and hence be used to set the Venc of an
acquisition. From equations 2.17 and 2.18, the velocity is then:
encvv φπ
⎛ ⎞= Δ⎜ ⎟⎝ ⎠
(2.19)
2.1.6 Cine Phase Contrast Pulsatile Velocity Imaging
In order to image time-varying pulsatile flow, all of the velocity imaging done in
this research uses a cine phase contrast imaging method which reconstructs a single cycle
of velocity data containing multiple time frames by acquiring data over many cycles.52 In
this method, data is continuously acquired at the same phase encode level during each
cardiac cycle. When a trigger signal is detected indicating the next cardiac cycle, the
phase encode level is changed and the next line in k-space is acquired. During each
cardiac cycle, the time of each repetition relative to the trigger signals is noted, and this
information is used to retrospectively interpolate time frames from the recorded data. In
other words, the final resulting image of a particular time frame is constructed from data
collected through different cardiac cycles, but at the same time point within each cardiac
cycle. Figure 2.4 shows a schematic diagram of this data acquisition process. The
retrospective interpolation assumes that the cardiac cycles are identical, and thus the
actual uniformity of cardiac cycles in the imaging subject is important for acquiring
accurate data.
The total scan time of a cine phase contrast sequence is:
1
total PET n NEXHR
= ⋅ ⋅ (2.20)
where nPE is the number of phase encodes (ie. The number of lines in k-space), NEX is
the number of excitations to average together, and HR is the heart rate.
13
Figure 2.4 The data acquisition process for 3-component cine phase contrast velocity imaging. One line of k-space at a particular Ky value is acquired per cardiac cycle, until all of the phase encodes (PE) are completed. The repetitions within each cardiac cycle are used to construct different time frames retrospectively. Typically 24 time frames are constructed per cardiac cycle. The NEX is the number of excitations to average together. Courtesy of Mary T. Draney.
2.1.7 Field of View, Resolution, and Signal to Noise Ratio
The field of view of an acquisition is determined by the sample spacing in the k-
space grid, and the spatial resolution is dependent on the value of the highest spatial
frequency included in the k-space grid. More narrowly spaced samples in the k-space
grid results in a larger field of view, and the inclusion of higher values of spatial
frequency results in finer spatial resolution. Increasing the field of view without
sacrificing spatial resolution, or vice versa, both require more k-space samples to be
acquired and hence increased scan time.
The signal to noise ratio is dependent upon the scan time, spatial resolution, and
main magnetic field strength:
oSNR B x y z AcquisitionTime∝ Δ Δ Δ (2.21)
14
In order to improve SNR without sacrificing spatial resolution or increasing scan
time, small receive-only coil are commonly used to only pick up signal from regions of
interest, but not noises from outside of the regions of interest.
For phase contrast velocity imaging, the velocity precision is given by:
2 encv
vSNR
σπ
⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠
(2.22)
The SNR of the velocity measurement is:
2v
enc
vSNR SNRv
π ⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
(2.23)
2.2 Computational Fluid Dynamics
In this section we present the general methodology of the finite element method
as applied to computational fluid dynamics.53-55 Starting with the governing equations
and the initial & boundary conditions, the strong form equations {S} are written into a
weak form {W}, which can then be approximated into the Galerkin form {G}, and
eventually written as a matrix form {M} to be solved numerically. An approximation
occurs in the transition from the weak form into the Galerkin form where the continuous
solution space is approximated into a finite-dimensional subset of that space.
2.2.1 Governing Equations
The strong form of equations is made up of an Initial Boundary Value Problem.
In an arbitrarily defined numerical domain Ω with boundary Γ where
, the velocity and pressure of a fluid can be ; i w h i w hΓ = Γ Γ Γ Γ Γ Γ =∅∪ ∪ ∩ ∩
15
represented by the Navier-Stokes and continuity equations for the conservation of
momentum and mass of a Newtonian incompressible fluid:
, ( ) in
( ) 012 with (2
t
T
u u u p div fdiv u
D D u u
ρ ρ τ
τ μ
⎫+ ⋅∇ = −∇ + + ⎪
)
Ω⎬= ⎪⎭
= = ∇ +∇
(2.24)
The initial condition of the problem is:
0( ,0) ( )u x u x x= ∈Ω (2.25)
At the inlet boundary , a Dirichlet condition is prescribed on the velocity field: iΓ
( , ) ( , )iniu x t u x t x= ∈Γ (2.26)
At the vessel wall boundary wΓ , for the rigid wall case, a no slip condition is prescribed:
( , ) 0 wu x t x= ∈Γ (2.27)
At the outflow boundary , a Neumann condition is prescribed: hΓ
( ) [ ] ( ) nt x,t pI n h x,t xτ h= − + = ∈Γ (2.28)
In the above equations, ( )x y zu u ,u ,u= is fluid velocity, p is pressure, μ is
viscosity, ρ is blood density, and f is the body force per unit volume. h in equation
2.28 is the traction vector imposed on hΓ .
2.2.2 Boundary Conditions & The Weak Form Equations
In order to write the strong form into the weak form, we define the trial solution
spaces U and P for velocity and pressure, and the respective weighting functions W and
Q as:
16
{ }{ }{ }{ }
1
1
1
( , ) ( ) , [0, ], ( , ) on , ( , ) 0 on
( , ) ( ) , [0, ], ( , ) 0 on , ( , ) 0 on
( , ) ( ), [0, ]
sd
sd
n ini w
ni w
u u t H t T u t u u t
w w t H t T w t w t
p p t H t T
q q p
= ⋅ ∈ Ω ∈ ⋅ = Γ ⋅ = Γ
= ⋅ ∈ Ω ∈ ⋅ = Γ ⋅ = Γ
= ⋅ ∈ Ω ∈
= ∈Q
W
U
P (2.29)
where H1 is the Sobolev space of functions with square-integrable values and first
derivatives in Ω , and nsd is the number of spatial dimensions.
We multiply the strong form equations by the weighting functions and integrate
over the domain, then apply the boundary and initial conditions, to obtain the weak form:
( ) ( ){ }, :
( ) 0h
tw u u u f w pI dx q udx
w pI nds qu nds
ρ ρ τ
τΩ Ω
Γ Γ
⋅ + ⋅∇ − +∇ − + − ∇ ⋅
− ⋅ − + ⋅ + ⋅ =
∫ ∫∫ ∫
(2.30)
We can incorporate the Coupled-Multidomain method developed by Vignon-
Clementel et al.24 to decompose the domain into an upstream numerical domain, and a
downstream reduced-order analytical domain. This allows the use of simpler analytical
models requiring less computational power to provide more general descriptions of the
downstream domain. The solution is also decomposed into a component defined within
the numerical domain, and another component defined within the analytical domain. The
continuity of momentum and mass fluxes is then enforced across the interface, which
allows for the simple analytical models capturing downstream behaviors to be coupled
into the numerical domain. The final weak form resulting from the Couple-Multidomain
method is:
( ) ( ) ( )
( )( )( )( )
,ˆ ˆ
ˆ
ˆ
ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ:
ˆ ˆ ˆ ˆˆ ˆ,
ˆ ˆ ˆ ˆˆ ˆ ˆ , 0
h
B
B
t
m m
c c
w u u u f w pI dx w pI nds
w u p nds q udx
qu nds q u p nds
ρ ρ τ τΩ Γ
Γ Ω
Γ Γ
⋅ + ⋅∇ − +∇ − + − ⋅ − + ⋅
− ⋅ + ⋅ − ∇ ⋅ +
⋅ + + ⋅ =
∫ ∫
∫ ∫
∫ ∫
M H
M H
(2.31)
where Mm, Mc, Hm and Hc are approximations defined by the analytical model chosen to represent the downstream domain.
17
2.2.3 Finite Element Analysis
A stabilized finite-element method can be used to transform the weak form into
the matrix form to be solved numerically.53,56 The matrix form of the equation is:
Kd F= (2.32)
where K is the matrix of equations, d is the unknown vector to be solved, and F is the
known vector based on the boundary and initial conditions. In a non-linear system, an
iterative process is necessary to find d. The process involves evaluating the matrix
system with an estimated solution, and finding the “residual”, which is the difference
between the matrix evaluation results and the target vector F. If the residual is not
acceptable, a new set of solution is tried and the process is repeated until a set of solution
is found that returns a reasonable residual. Various strategies exist for the solution
estimation and residual calculation.
2.2.4 Incorporating Vessel Wall Deformability
For deformable wall simulations, we incorporate a coupled momentum method
developed by Figueroa et al.,16 which adopts a linearized kinematics formulation for the
solid domain, and allows a fixed fluid mesh and nonzero velocities at the fluid-solid
interface. The no-slip Dirichlet boundary condition is removed from , and replaced
with a traction Neumann boundary condition determined by the motion of the vessel wall
as calculated using elastodynamics equations under thin-wall and membrane
approximations. The end result is that the effects of wall motion are embedded into the
fluid equations simply as additional terms for the mesh nodes on the fluid-solid interface,
leading to minimal increases in implementation complexity and computational efforts
compared to the rigid wall formulations.
wΓ
The strong form of the equations describing the vessel wall motion on a surface
domain sΩ can be presented by classic elastodynamics equations:
( ) ( ), , , 0,s s s suu b x tρ σ=∇⋅ + ∈Ω × T (2.33)
18
The initial conditions are:
( ) ( )( ) ( )
0
0, ,
, 0 ,
,0 ,
s
st t
u x u x x
u x u x x
= ∈Ω
= ∈Ω (2.34)
The boundary conditions at the inlet siΓ and outlet s
hΓ are:
( )
( ), , (0, )
, , (0, )
s si
s s sn h
u g x t T
t n h x tσ
= ∈Γ ×
= = ∈Γ × T (2.35)
In the above equations, sρ is the density of the vessel wall, u is the displacement
field, sσ is the vessel wall stress tensor, sb is the body force, ( )x0,tu and are the
initial conditions for velocity and displacement, respectively.
( )0u x
sg and sh are the essential
and natural boundary conditions prescribed on siΓ and s
hΓ .
The elastodynamics equations describing the wall motion can be written as
additional terms in the weak form of the fluid equations, replacing the terms
corresponding to the vessel wall boundary condition. This results in a single weak form
equation for the fluid domain:
( ) ( ){ }
( ){ }
,
,
0 :
( )
:h
s h
t
s s st
w u u u f w pI dx q udx
w pI nds qu nds
w u w u ds w h dl qu nds
ρ ρ τ
τ
ζ ρ σ ζ
Ω Ω
Γ Γ
Γ ∂Γ
= ⋅ + ⋅∇ − +∇ − + − ∇ ⋅
− ⋅ − + ⋅ + ⋅
+ ⋅ +∇ − ⋅ + ⋅
∫ ∫∫ ∫∫ ∫ ∫
sΓ
(2.36)
The terms on the third line of equation 2.36 are the additional terms resulting
from the vessel wall motion added to the rigid wall case. ζ is the vessel wall thickness.
Equation Section 3
19
Chapter 3: Development of a Physical
Windkessel Module to Re-Create In-Vivo
Vascular Flow Impedance for In-Vitro
Experiments
3.1 Abstract
Purpose: To create and characterize a physical Windkessel module that can provide
realistic and predictable vascular impedances for in-vitro flow experiments used for
computational fluid dynamics validation, and other investigations of the cardiovascular
system and medical devices.
Methods: We developed practical design and manufacturing methods for constructing
flow resistance and capacitance units. Using these units we assembled a Windkessel
impedance module and defined its corresponding analytical model incorporating an
inductance to account for fluid momentum. We tested various resistance units and
Windkessel modules using a flow system, and compared experimental measurements to
analytical predictions of pressure, flow, and impedance.
Results: The resistance modules exhibited stable resistance values over wide ranges of
flow rates. The resistance value variations of any particular resistor are typically within
5% across the range of flow that it is expected to accommodate under physiologic flow
conditions. In the Windkessel impedance modules, the measured flow and pressure
waveforms agreed very favorably with the analytical calculations for four different flow
conditions used to test each module. The shapes and magnitudes of the impedance
modulus and phase agree well between experiment and theoretical values, and also with
those measured in-vivo in previous studies.
20
Conclusions: The Windkessel impedance module we developed can be used as a
practical tool to provide realistic vascular impedance for in-vitro cardiovascular studies.
Upon proper characterization of the impedance module, its analytical model can
accurately predict its measured behavior under different flow conditions.
3.2 Introduction
Computational fluid dynamics (CFD) is a powerful tool for quantifying
hemodynamic forces in the cardiovascular system. In CFD simulations, realistic outflow
boundary conditions are necessary to represent physical properties of the downstream
vasculature not modeled in the numerical domain, and to produce physiologic levels of
pressure.24 While various types of boundary condition implementations exist,29,57-60
previous studies show that impedance-based boundary condition is the preferred
approach for coupling wave reflections from the downstream vasculature into the
numerical domain,24 and that simple lumped-parameter model representations can
provide realistic impedances similar to those provided by a more complicated method
employing a distributed parameter model.57 The Windkessel model, due to its simplicity
and ability to provide physiologically realistic impedances,61-64 is a practical method of
prescribing suitable boundary conditions to the numerical domain in CFD simulations.
The Windkessel model is represented as a circuit containing lumped elements of
resistance, capacitance, and inductance. Although these elements are more generally
interpreted in an electrical system, there is a direct analogy between the governing
equations of an electric circuit and those of a fluid system, where the fluid pressure, the
fluid volume, and the volumetric flow rate directly parallels voltage, electrical charge,
and electrical current, respectively. For example, the relationship between voltage and
current related by electrical resistance as described by the equation V=IR, can be directly
modified into P=QR to describe the relationship between pressure and flow rate related
by the fluid resistance.
When used to mimic vascular impedances, associations exist between the lumped
component values in a Windkessel model and in-vivo physiological parameters. The
resistance and inductance values are associated with the density and viscosity of blood,
21
and with the geometry and architecture of the vasculature which are functions of both the
anatomy and the vascular tone. The capacitance value is most affected by the physical
properties and the vascular tone of the large arteries. Since the properties of blood, the
blood vessel anatomy and physical properties, and the vascular tone do not vary
significantly within the time frames of a cardiac cycle, it is the general practice to
implement an analytical Windkessel model with fixed component values.
In order to validate CFD against experimental data, methods must be developed to
reliably construct a physical model of the Windkessel boundary condition such that there
is a direct parallel between the experimental setup and the CFD simulation. In this paper
we present the theories, principles, practical design considerations, and manufacturing
processes for physically constructing the resistance and capacitance components of a
Windkessel impedance module. These methods enable the construction of Windkessel
components with values that are predictable and constant throughout their operating
ranges. We also present an analytical model that describes the physical Windkessel
module, and incorporates an inductance to account for fluid momentum. We
manufactured several resistance units and tested them independently in a flow loop to
verify their operations. Windkessel modules that mimic the thoracic-aortic and renal
impedances were then assembled and tested under physiologic pulsatile flow conditions,
and experimental measurements were compared to analytical predictions of pressure and
flow.
3.3 Methods
3.3.1 Determining Target Windkessel Component Values
Target values to aim for in the design and construction of the Windkessel
components must first be determined. The component value estimation may be
performed using a basic three element Windkessel model consisting of a proximal
resistor (Rp), a capacitor (C), and a distal resistor (Rd) as shown in Figure 3.1. The target
component values are those that would result in the desired pressure and flow
relationship reflecting the particular vascular impedance to be mimicked. For a periodic
flow condition, the pressure and flow is related by the equation in the frequency domain:
22
( ) ( ) ( )P Q Zω ω ω= (3.1)
where ω is the angular frequency, Q is the volumetric flow rate, and Z is the impedance
of the three-element Windkessel circuit:
( ) 1
dp
d
RZ Rj CR
ωω
= ++
(3.2)
In previous reports, blood flow waveforms at various locations in the vascular tree
have been obtained with imaging modalities such as ultrasound or phase-contrast
magnetic resonance imaging,65-67 and pressure waveforms have been obtained with
pressure cuffs or arterial catheters.68 Using the available in-vivo flow and pressure
waveform data, together with Equations 3.1 and 3.2, an iterative process can be
performed to find the target Windkessel component values for mimicking the in-vivo
vascular impedance at a specific location. We begin by using the flow data and initial
guesses of the component values as input parameter into Equations 3.1 and 3.2 to
calculate a resulting pressure waveform. The component values can then be adjusted
with the goal of matching the calculated pressure to the in-vivo measured pressure
waveform. For any given input flow, the total resistance (sum of Rp and Rd) can be
adjusted to vertically shift the calculated pressure waveform, and the ratio of Rp/Rd as
well as the capacitance can be adjusted to modulate the shape and pulse amplitude of the
calculated pressure waveform. Once we determine the component values which give the
desired pressure and flow relationship, we then consider them the target values in the
design and construction of the components.
Figure 3.1 A basic three-element Windkessel model for component value estimation purpose
23
3.3.2 Flow Resistance Module
Theory and Construction Principles
In Poiseuille’s solution for laminar flow in a straight cylinder, the relationship
between the pressure drop across the cylinder (∆P) and volumetric flow rate (Q) is:
4
8P l Qrμ
πΔ = (3.3)
The flow resistance defined as R=∆P/Q is then:
4
8 lRrμ
π= (3.4)
where μ is the dynamic viscosity of the fluid, l is the length of the cylinder, and r is the
radius of the cylinder.
Equation 3.3 holds true in a laminar flow condition, where the resistance is
constant and independent of flow rate. In turbulent flow, however, the additional energy
loss leads to the pressure drop across the flow channel becoming proportional to the flow
rate squared (∆P α Q2), implying that the total effective resistance as defined by R=∆P/Q
is proportional to the flow rate (R α Q). Since our goal is to create a constant resistance
that is independent of flow rate, it is thus important to avoid turbulence and maintain
laminar flow. An approximate condition for laminar flow in a circular cylinder is the
satisfaction of the following equation for Reynolds number:
Re 1200 v r Qrυ πυ
= = < (3.5)
where ν is flow velocity, r is the radius of the flow conduit, and υ is the kinematic
viscosity of the fluid.
Equation 3.4 shows that with a single cylindrical channel of a given length, a high
flow resistance can be achieved by drastically decreasing the cylinder radius. According
to Equation 3.5, however, decreasing the radius means that the flow conduit can only
accommodate a lower flow rate while maintaining laminar flow. For physiological
ranges of flows and impedances, it is generally the case that an Rp made from a single
flow channel of a reasonable length would not be able to accommodate the required
24
amount of flow. For example, the typical infra-renal aortic impedance results in an Rp of
approximately 500 Barye*s/cm3, and the peak flow at that anatomical location is
approximately 100 cc/s. Using a single cylindrical channel of length 10cm, and a fluid
kinematic viscosity of 0.04 g/cm*s, the radius of such a resistor would be 0.22cm.
Equation 3.5 indicates that the maximum flow rate this resistor can accommodate in
laminar flow condition is 33 cc/s, much less than the peak flow that will flow through it.
We present mathematically how such a problem can be overcome by using a large
number of small channels in parallel, which simultaneously allows for high resistance
and laminar flow at high flow rates. Consider “N” number of parallel flow channels with
radius “r”. We define:
A – combined cross sectional area of all channels
Q – combined volumetric flow through all channels
Qchan – volumetric flow rate through each channel
Re – Reynolds number
Rchan – resistance of each channel
Rtotal – combined resistance of all the parallel channels
The following two equations describe the geometry and resistances of the flow channels:
2rAN
π= (3.6)
Nchan
totalRR = (3.7)
From Equations 3.4, 3.5 and 3.6, we obtain the following proportionalities:
4
1rchanR ∝ (3.8)
ReQrA∝ (3.9)
1
Nr ∝ (3.10)
25
Substituting Equations 3.8 and 3.6 into 3.7
2
1ArtotalR ∝ (3.11)
Substituting Equation 3.9 into 3.11
3
ReQrtotalR ∝ (3.12)
Substituting Equation 3.10 into 3.12 and re-arranging, we finally have
3
2
Re totalQ R N∝ (3.13)
Equation 3.13 indicates that in order to achieve a high resistance at a high flow
rate, while maintaining a low Reynolds number, a large number of parallel channels is
required. Figure 3.2a is an illustration that shows the relationship between N and the
maximum laminar flow rate for various values of Rtotal.
Figure 3.2 a) Maximum Laminar Flow Rate v.s. Number of Parallel Channels for Various Resistance Values. b) Resistance v.s. Number of Parallel Channels for Various Standard Capillary Tube Inside Diameters (ID)
Calculated using: Fluid dynamic viscosity = 0.046 g/cm*s, Capillary Tube Length = 10 cm
26
Practical Design and Construction Methods
To assemble a large number of small parallel channels in a practical and robust
way, we placed thin-walled glass capillary tubes (Sutter Instrument, CA) inside a
plexiglass cylinder as shown in Figure 3.3a. We applied a small amount of silicone
rubber adhesive sealant (RTV 102, GE Silicones, NY) in between the capillary tubes
around their middle section to adhere the tubes to one another, and to block fluid
passageways through the gaps in between the tubes. We then applied a small amount of
epoxy (5 Minute Epoxy, Devcon, MA) between the plexiglass surface and the bundle of
capillary tubes to secure the capillary tubes inside the plexiglass cylinder.
The theoretical resistance of the resistance module is given by:
4
8 lRNrμ
π= (3.14)
where μ is the dynamic viscosity of the working fluid, l is the length of the capillary
tubes, r is the inside radius of each individual capillary tube, and N is the total number of
capillary tubes in parallel.69
For a standard capillary tube length of 10cm, Figure 3.2b shows the relationship
between the number of tubes and the resulting resistance for various standard capillary
tube sizes that can be readily purchased.
Using the same principle of parallel channels, Figure 3.3b shows a method for
creating a switchable resistance module where the resistance value can be changed during
an experiment. Multiple resistance modules can be placed in parallel, with control valves
that open and close to add in or remove parallel resistor(s) in order to decrease or increase
the effective total resistance.
The resistance module must be connected to tubing at each end. It is important to
ensure that laminar flow is maintained throughout the connection tubing, and that
diameter changes at the connection junctions are minimized to avoid the creation of
turbulence. We constructed Table 3.1 to aid the design process of choosing an
appropriate combination of a standard capillary tube size and connection tubing size,
such that the resistance module can connect smoothly to its inlet and outlet tubing, and
that the connection tubing itself can also accommodate the maximum flow rate required.
27
The maximum laminar flow for any particular flow conduit diameter can be calculated
from Equation 3.5, and is listed below each conduit diameter in the table. Note that the
Reynolds number within the capillary tubes is much lower than that in the connection
tubing (due to the smaller diameter of the capillary tubes), thus the critical factor in
maintaining laminar flow is the connection tubing diameter. From Table 3.1, the optimal
capillary tube size for constructing the resistance module is determined by identifying a
resistance value that is close to the desired target value, in combination with a conduit
diameter that can accommodate the maximum expected flow. Once the capillary tube
size is determined, a circle packing algorithm70 can then be used to determine the precise
plexiglass cylinder diameter required to house the specific number of capillary tubes
needed for obtaining the desired resistance. Upon completing the actual construction of
the resistance module, we manually count the number of capillary tubes in the plexiglass
cylinder, and use the resulting count, together with the measured dynamic viscosity of the
working fluid and Equation 3.14, to determine the theoretical resistance of the module.
Capillary Tubes *OD / ID (mm)
Estimated Resistance(Barye*s/cm3) & (Number of Capillary Tubes) for Conduit Diameters & (Maximum Laminar Flow Rates):
1” (200cc/s)
3/4” (150 cc/s)
5/8” (125 cc/s)
1/2” (100 cc/s)
3/8” (75 cc/s)
1/4” (50 cc/s)
2 / 1.56 231 (137) 410 (77) 591 (54) 923 (34) 1641 (19) 3693 (9) 1.5 / 1.1 525 (244) 934 (137) 1345 (95) 2101 (61) 3735 (34) 8404 (15) 1.2 / 0.9 750 (381) 1334 (214) 1920 (149) 3000 (95) 5334 (54) 12002 (24) 1 / 0.78 923 (548) 1641 (308) 2364 (214) 3693 (137) 6566 (77) 14773 (34) 1 / 0.75 1080 (548) 1920 (308) 2765 (214) 4321 (137) 7681 (77) 17282 (34)
Table 3.1 Estimated Resistance Values (and Numbers of Capillary Tubes) Resulting From Various Combinations of Conduit Diameter (Maximum Laminar Flow Rate), and Capillary Tube Size
Calculated using: Fluid dynamic viscosity = 0.046 g/cm*s Fluid density = 1.1 g/mL Capillary Tube Length = 10 cm Circle packing density = 0.85 by area *OD/ID stands for Outside Diameter / Inside Diameter
28
/ P
Figure 3.3 a) Capillary Tube Resistance Module Construction b) Switchable Resistance Setup
3.3.3 Flow Capacitance Module
The capacitance of a fluid system is define as C=∆V/∆P where ∆V and ∆P are the
changes in volume and pressure. In a closed system at constant temperature, an ideal gas
exhibits the behavior PV=(P+∆P)(V-∆V), where P and V are the reference pressure and
volume. The capacitance of a pocket of air is then:
( )aC V V= −Δ (3.15)
We constructed the capacitance module with a plexiglass box that can trap a
precise amount of air, which acts as a capacitance in the system (Figure 3.4a). Equation
3.15 indicates that, as fluid enters the capacitor and compresses the air, the capacitance of
the module would decrease. For small changes in volume relative to the reference
volume, however, a reasonably constant capacitance can be maintained. As fluid enters
and exits the box, the vertical level of the fluid in the box rises and falls slightly. The
varying fluid level contributes to an additional capacitance that is in series with the
capacitance due to air compression. The pressure change in the fluid due to the varying
fluid level under the effects of gravity and fluid mass is:
/P g h g V Aρ ρΔ = Δ = Δ (3.16)
where ρ is the fluid density, g is the gravitational constant, and A is the area of the
fluid/air interface (assuming a column of fluid with constant cross-sectional area). The
capacitance due to the varying fluid level is then:
29
/ ( )vC A gρ= (3.17)
Since Cv is in series with Ca the overall capacitance “C” can be approximated by Ca alone
if Cv>>Ca:
1v a
1 1For C C : ( ) ~a va
a v a v
C CCC C C C
−>> = + =+
C (3.18)
In the actual construction of the capacitance module, we designed the box to be
large enough so that the approximation in Equation 3.18 is true. We also designed a
smooth contour for the inlet of the capacitance module (Figure 3.4b) in order to minimize
flow turbulences and thus avoid parasitic resistances. In addition, two access ports are
included at the top of the capacitance module for air volume modulation and pressure
measurements, and a graduated scale on the sidewalls for air volume measurement
(Figure 3.4a).
Figure 3.4 a) Capacitance Module Construction b) Capacitor Inlet Contour
3.3.4 Flow Inductance
The flow inductance is an inherent parameter of a fluid system resulting from the
fluid mass. It describes how a force, manifest as a pressure differential, is required to
accelerate a body of fluid. The inductance in a fluid system creates a pressure drop in
response to a change in flow as described by the equation:
dQP Ldt
Δ = (3.19)
30
where L is the inductance value.
Consider a volume of fluid with density “ρ” and mass “m” inside a cylinder with
cross-sectional area “A” and length “l”. The acceleration of the fluid can be expressed
as:
( )Qd Aadt
= (3.20)
The force “F” required to accelerate the fluid mass, and applied to the cross-
sectional area results in a pressure differential:
/P F AΔ = (3.21)
Substituting Newton’s second law and Equation 3.20 into Equation 3.21:
2
dQm dtPA
Δ = (3.22)
Equations 3.19 and 3.22 together indicate that L = m / A2. Since “m” is related to
the density and volume of the fluid, we can express the inductance as:
/L l Aρ= (3.23)
Equation 3.23 allows direct calculation of the inductance value of a fluid body
from the fluid density and the geometry of the flow conduit.
3.3.5 Assembled Windkessel Module & Corresponding Analytical Model
We assembled the Windkessel impedance module by putting together two
resistors and one capacitor as shown in Figure 3.5a. Note that in such a physical setup
the reference pressure of the capacitor is the initial pressure within the capacitor when the
system is in no-flow equilibrium, and thus the capacitor is considered to be connected to
the “ground”. In the analytical model, inductive effects of the fluid body is taken into
account62 and the impedance module is represented as an LRCR circuit as shown in
Figure 3.5b. Note that even though there is an inductance associated with the
downstream resistance Rd, since the flow through Rd is typically nearly constant, the
31
presence of the inductance is transparent to the operation of the impedance unit.
Incorporating only the upstream inductance in the analytical model is sufficient to fully
capture the behavior of the physical impedance module.
Figure 3.5 a) Assembled Impedance Module. b) Final Analytical Model of Impedance Module
3.4 Experimental Testing & Data Analysis
3.4.1 Resistance Module
We tested the operation of the resistance modules with a setup depicted in Figure
3.6. We used a 1/12 horse-power, 3100RPM, steady flow pump (Model 3-MD-HC, Little
Giant Pump Co., OK) to drive flow through the resistance module. The working fluid in
the flow system was a 40% glycerol solution with a dynamic viscosity similar to that of
blood. For data acquisition, we used an ultrasonic transit-time flow probe to monitor the
flow through the system. We placed the externally clamped flow probe (8PXL,
Transonic Systems, NY) around a short section of Tygon tubing R3603, and sent the
signals from the probe into a flowmeter (TS410, Transonic Systems, NY). For pressure
measurements, we inserted catheter pressure transducers (“Mikro-Tip” SPC-350, Millar
Instruments, Huston, TX) into the flow conduit immediately upstream and downstream of
the resistance module to capture instantaneous pressure readings, and obtain the pressure
drop across the resistor. We sent the signals from each catheter pressure transducer into
a pressure control unit (TCB-600, Millar Instruments, TX) which produces an electrical
output of 0.5V per 100mmHg of pressure. We recorded the data from the flow meter and
32
the pressure control units at a sample rate of 2kHz using a data acquisition unit (USB-
6259, National Instruments, Austin, TX) and a LabVIEW program (LabVIEW v.8,
National Instruments, Austin, TX). We averaged 8000 samples of flow and pressure
(effectively, 4 seconds of flow and pressure) to obtain each data point. We then divided
the measured pressure drop across the resistor by the measured volumetric flow rate
through the resistor to obtain the resistance value.
The flow control for the steady pump consisted of a LabVIEW program that
directed the data acquisition unit to send a voltage to an isolation amplifier (AD210,
Analog Devices, MA), which then produced the same control voltage to feed into a
variable frequency drive (Stratus, Control Resources Inc., MA) that drove the flow pump
to produce different constant flow rates through the flow loop. The purpose of including
the isolation amplifier in the signal chain was to electronically de-couple the high-power
operation of the variable frequency pump drive from the data acquisition unit to avoid
signal interference.
In addition to the resistance modules, we also tested the resistance of a partially
closed ball valve, which has commonly been used as a method to produce flow resistance
in previous literatures.37,71 We adjusted the relative resistance of the ball valve by
adjusting the proportion that the valve was closed.
Figure 3.6 Resistance Module Steady Flow Testing Setup
33
3.4.2 Assembled Windkessel Module
We tested the assembled Windkessel impedance modules using a setup depicted
in Figure 3.7. A custom-built, computer-controlled pulsatile pump in parallel with a
steady flow pump produced physiological-level, pulsatile, and cyclic flow waveforms
into the Windkessel module. Two ultrasonic transit-time flow probes (8PXL & 6PXL,
Transonic Systems, NY) were used to monitor the volumetric flows through Rp and Rd.
For pressure measurements, we inserted catheter pressure transducers into the flow
conduits and into the capacitor chamber to capture the pressure waveforms at three points
in the circuit. The flow and pressure data were recorded at a sample rate of 96 samples
per second. We averaged approximately 50 cycles of flow and pressure data to obtain
one representative cycle of flow and pressure waveforms. We used the pressures
measured at P3 as the ground reference, and subtracted it from the pressures measured at
P1 and P2, to obtain the true pressure waveforms at P1 and P2.
We tested two impedance modules, one mimicking the in-vivo thoracic-aortic
impedance, and the other mimicking the in-vivo renal impedance, using four different
input flow waveforms approximately simulating physiological flows for each module.
We included input flow waveforms with different periods, as well as considerably
different shapes, to investigate the impedance behavior of each module across a wide
range of flow conditions.
The impedance of the analytical Windkessel circuit in Figure 3.7 can be
represented by the equation:
( ) 1
dp
d
RZ j L Rj CR
ω ωω
= + ++
(3.24)
By prescribing the measured input flow waveform, and the values of the lumped
components, we calculated the theoretical pressure waveform at P1 using Equations 3.1
and 3.24. We then calculated the theoretical pressure waveform at P2 and the flow
waveform Qd using the equation ∆P=QR.
34
Figure 3.7 Impedance Module Pulsatile Flow Testing Setup
3.5 Results & Discussions
3.5.1 Resistance Module
Figure 3.8 presents results of resistance verses flow rate for two of the resistance
modules, and for a partially closed ball valve. In Figure 3.8a, the theoretical resistance of
the resistance module is 500 Barye*s/cm3. The measured resistance is very close to the
expected theoretical value, and the resistance module exhibits relatively constant
resistance values over the range of flow rates tested. The variation in the resistance value
between flow rates of 20 cc/s and 100 cc/s is approximately 5%. The ball valve on the
other hand, exhibits a resistance that varies linearly with the flow rate. Figure 3.8b shows
results of a resistance module with theoretical resistance of 6700 Barye*s/cm3, and a ball
valve adjusted to produce a higher flow resistance. We see similar results at this higher
regime of resistance values. The value variation of the resistance module between flow
rates of 20 cc/s and 60 cc/s is approximately 7%. Note that a resistance unit with
resistance in the higher regime typically only needs to accommodate relatively low flows
in its actual operation. If placed within a Windkessel module under physiologic flows,
the expected maximum flow through such a resistor in Figure 3.8b would be
approximately 30 cc/s. All of the other resistor modules we have tested (but not shown
35
here) also exhibited similar behaviors of relatively constant resistance values over the
range of flow rates they are expected to accommodate. The resistance of the ball valve
showing a linear dependence on flow rate, and extrapolated value of zero at zero flow,
suggests that it is a result of turbulence alone as discussed in section 3.3.2. For the
resistance modules, the slight increases in the resistance value with flow rate suggest that
there is a small amount of turbulence present in the modules.
The resistance variations at the low flow regions are likely due to measurement
imprecision, but not due to the actual resistance change or instability in the resistance
module. Very low flows and small pressure drops across the resistor result in low signal-
to-noise for both the ultrasonic flow probe and the pressure transducers, and hence
difficulties in obtaining precise measurements. Fortunately, the fact that the pressure
drop across the resistor is insignificant during very low flows, means that the resistance
value also have minimal impact during that period. The accuracy of experimental
confirmation of resistance values during the very low flow regions is thus of minimal
importance.
At a fixed flow rate, we found that the resistance value of a resistance module
may decrease over time by up to 5%. The decrease may be due to trapped air bubbles
being purged out of the capillary tubes over time with flow (since the presence of air
bubbles in the tubes would obstruct the fluid passage and result in elevated resistance).
This source of resistance variation can be minimized with careful removal of air from the
flow system during setup to minimize the amount of air that would be trapped in the
resistor during operation.
36
Figure 3.8 Resistance v.s. Flow Rate for Resistance Module with Theoretical Resistance of a) 500 Barye*s/cm3, b) 6700 Barye*s/cm3, and a Partially Closed Ball Valve
3.5.2 Assembled Windkessel Module
Table 3.2 shows the theoretical Windkessel component values as calculated from
their physical constructions, where the values of L calculated from the geometry of the
physical system as described in section 3.3.4, the values of resistances calculated from
their construction details as described in section 3.3.2, and the values of C calculated
from the operating pressure and air volume in the capacitors as described in section 3.3.3.
The experimental component values in Table 3.2, unless otherwise noted, were
determined from the experimentally measured pressure and flow data, using a method
similar to that described in section 3.3.1. For the thoracic-aortic impedance module, the
inductance and resistances behaved as theoretically predicted, where the observed
capacitance in the actual experiment was larger than the theoretical expectation. For the
renal impedance module, we experimentally determined the resistance values from steady
flow tests of the impedance module, and found that the actual resistances were about 15%
less than theoretical. The inductance value, on the other hand, was higher than
theoretical. The capacitance value was consistent with the theoretical prediction. The
differences between the theoretical and experimental component values may be attributed
to variations in the physical construction of the components, as well as to the connection
parts in between the components.
37
Thoracic-Aortic Renal Theoretical Experimental Theoretical Experimental
L (Barye*s2/cm3) 7 7 16 26 Rp (Barye*s/cm3)
245 245 3050 2522
C (cm3/Barye)
2.3 e-4 4.0 e-4 1.3 e-4 1.3 e-4
Rd (Barye*s/cm3)
4046 4046 5944 5221
Table 3.2 Theoretical and Experimental Windkessel Component Values for the Thoracic-Aortic and Renal Impedance Modules
We prescribed the experimental component values from Table 3.2 in the
analytical calculations of pressure and flow. Figure 3.9 shows pressure and flow
comparisons between experimental measurements and analytical calculations for the
impedance module mimicking the in-vivo aortic impedance at the thoracic level. For all
of the four different input flow waveforms tested, the measured pressure waveforms at P1
and P2 (as denoted in Figure 3.7), and the flow waveform through Rd, all agree extremely
well with the analytical calculations in their shapes, phases, and magnitudes. The
maximum difference between measured and calculated pressure (P1 & P2) and flow (Qd)
is 6% and 8%, respectively. Note that two different cyclic periods (1 second and 0.75
second) were included in the test and analysis, and the impedance module performed
predictably under flow conditions with both period lengths. Figure 3.10 shows similar
results for the impedance module mimicking the renal impedance. There is the same
excellent match between experimental measurements and analytical calculations of
pressure and flow waveforms for all of the four different flow conditions tested. The
maximum difference between measured and calculated pressure (P1 & P2) and flow (Qd)
is 8% and 15%, respectively.
In Figures 3.9 and 3.10, the flow waveforms show that much of the pulsatility in
the input flow is absorbed by the capacitor, and the flow through the downstream resistor
is fairly constant. This implies that for any given input flow waveform, the proximal
resistor Rp needs to be able to accommodate the peak flow of the input waveform, where
38
the downstream resistor Rd only needs to accommodate approximately the averaged flow
of the input waveform.
By subtracting Qd from Q, we can calculate the flow into the capacitor, which
then can be integrated to find the change in fluid volume inside the capacitor over each
cardiac cycle. From calculations of pressure and volume with Equation 3.15, we
confirmed that the variation of capacitance value due to the volume change over each
cardiac cycle is less than 3% from the reference value for both impedance modules.
Figure 3.11 shows the impedance modulus and phase as derived from the
analytical model, and as calculated from the four sets of experimental pressure and flow
data for each module. For both impedance modules, there is close agreement between the
theoretical impedance modulus & phase, and those determined from the experimental
data of all of the four different flow conditions. This further shows that the impedance
modules behave very consistently even when the flow conditions were changed. The
general shapes and magnitudes of the impedance modulus and phase also compare well
with those measured in-vivo in previous studies.61,62,64,69
39
Figure 3.9 Comparisons Between Measured (solid lines) and Calculated (dots) Pressure & Flow Waveforms for the Thoracic-Aortic Impedance Module Under Four Different Flow Conditions
Note that P1, P2, Qd, and Q are the pressures and flows in the impedance module as depicted in Figure 3.7
40
Figure 3.10 Comparisons Between Measured (solid lines) and Calculated (dots) Pressure & Flow Waveforms for the Renal Impedance Module Under Four Different Flow Conditions
Note that P1, P2, Qd, and Q are the pressures and flows in the impedance module as depicted in Figure 3.7
41
Figure 3.11 Comparisons Between Theoretical and Experimental Flow Impedance Modulus and Phase for the a) Thoracic-aortic, and b) Renal, Impedance Module
3.6 Conclusion
We showed that using the methods presented here, we can construct flow
resistance units with stable resistance values over wide ranges of flow rates. This is a
significant advancement from the common practice of using a partially closed valve to
create flow resistances. The resistance value of the units we constructed can both be
theoretically determined from construction details, and experimentally confirmed from
pressure and flow measurements. We further showed that the impedance module
assembled from individual resistor and capacitor components performs very consistently
42
across different flow conditions, and that the corresponding analytical model faithfully
captures the behavior of the physical system. When actually employing the physical
Windkessel module in other experimental applications, whenever possible, flow and
pressure data should be used to confirm or adjust the lumped component value
assignments in the corresponding analytical model. We have shown that upon proper
characterization of a particular impedance module, the analytical model can then
accurately predict its behavior under different flow conditions.
Compared to the Windkessel module previously presented by Westerhof et al.,69
the methods presented here offer simpler and more robust construction, and include
considerations for minimizing turbulence in order to minimize parasitic resistances and
resistance variations across different flow rates. The analytical model presented here also
includes the physical effects of inductance, offering a more complete description of the
physical system.
In conclusion, the Windkessel impedance module we developed can be used as a
practical tool for in-vitro cardiovascular studies. Implementing the Windkessel module
in a physical setup enables the experimental system to replicate realistic blood pressures
under physiologic flow conditions. The ability to construct in-vitro physical systems to
mimic in-vivo conditions can aid in the direct physical testing of implantable
cardiovascular medical devices such as stents and stent grafts, and enable reliable
measurements of how the in-vivo forces and tissue motions will interact with the devices.
In the area of CFD validation, well-characterized physical Windkessel modules
connected to the outlets of a physical phantom will allow prescriptions of the same outlet
boundary condition in the computational domain, such that the boundary condition
prescription in-silico is representative of the physical reality. Furthermore, the ability to
implement realistic impedances in-vitro enables experimental studies involving
deformable materials, where realistic pressures are absolutely essential for obtaining
proper fluid-solid interactions. These studies will be useful for investigating the pulsatile
motions of blood vessels, and wave propagations in the cardiovascular system. The work
presented here serves as a basis to contribute towards more rigorous cardiovascular in-
vitro experimental studies in the future.
43
Equation Section 4
Chapter 4: In-Vitro Validation of Finite
Element Model of Abdominal Aortic
Aneurysm Hemodynamics Incorporating
Realistic Outlet Boundary Conditions
4.1 Abstract
Purpose: To validate numerical simulations of flow and pressure in an abdominal aortic
aneurysm (AAA) using phase-contrast MRI (PCMRI), and an in-vitro phantom under
physiological flow and pressure conditions.
Materials and Methods: We constructed a 2-outlet physical flow phantom based on
patient imaging data of an AAA, and developed a physical Windkessel model to use as
outlet boundary conditions. We then acquired PCMRI data in the phantom while it
operated under conditions mimicking a resting and a light exercise physiological state.
Next, we performed in-silico numerical simulations, and compared experimentally
measured velocities, flows, and pressures in the in-vitro phantom to those computed in
the in-silico simulations.
Results: There was a high degree of agreement in all of the pressure and flow waveform
shapes and magnitudes between the experimental measurements and simulated results.
The average pressures and flow split difference between experiment and simulation were
all within 2%. Velocity patterns showed excellent agreement between experimental
measurements and simulated results, especially in the case of whole-cycle averaged
comparisons.
Conclusion: We demonstrated methods to perform in-vitro phantom experiments with
physiological flows and pressures, showing excellent agreement between numerically
44
simulated and experimentally measured velocity fields and pressure waveforms in a
complex, patient-specific AAA geometry.
4.2 Introduction
Hemodynamic parameters such as the three-dimensional blood flow and pressure
fields, as well as the stress and strain in blood vessels, have direct effects on the initiation
and development of cardiovascular diseases such as atherosclerosis and aneurysms.1-4,72
The hemodynamic forces within blood vessels also directly affect biological adaptation
of vessel diameter and wall thickness.7,10 The design and evaluation of implantable
medical devices such as stents and stent grafts require knowledge of how the in-vivo
forces and tissue motions will interact with the devices. Medical imaging can be used to
investigate these hemodynamic parameters, but with limited temporal and spatial
resolutions. As computing resources increase, image-based computational fluid
dynamics (CFD) methods are becoming powerful tools to quantify these hemodynamic
conditions. The ability of CFD techniques to finely resolve time and space enables the
study of disease mechanisms,15,17,18,73,74 and can aid in the design and evaluation of
medical devices.19-21,75 The ease of applying variations in geometry and flow conditions
in the computational domain also motivates the use of CFD in the planning and
prediction of surgical procedures.22,23 However, much work remains to provide proper
input parameters to CFD simulations, and to validate CFD methods against experimental
data. Phase contrast MRI (PCMRI) is a versatile in-vivo technique that can provide
velocity data to be used as input parameters at discrete locations, and to validate
numerical computations at other locations.
Previous In-vitro experiments have been performed to compare CFD computed
velocity fields with those measured with PCMRI. However, these prior studies did not
include realistic outflow boundary conditions (BC), which are required to represent
physical properties of downstream vasculature not modeled in the numerical domain, and
to produce physiologic levels of pressure. For example, previous in-vitro CFD validation
studies have implemented simple zero pressure BC in the physical setup (phantom outlets
connect directly into a fluid reservoir),33,36 and many also employed steady flow.42,45 In
45
the computational domain, flow velocity BC were commonly prescribed,38,39 but this type
of BC is difficult to employ in patient-specific CFD studies, since obtaining flow velocity
data for all of the outlets of a major blood vessel in a patient is very difficult. In addition,
realistic patient-derived abdominal aortic aneurysm (AAA) geometry under pulsatile flow
has not been considered in in-vitro validation models. Complex or even turbulent flows
within AAA geometries have commonly been reported in literature,68,76 making patient-
derived AAA geometries more challenging targets for validation studies.
In this study, we present results from a complex AAA geometry under
physiological flows, and Windkessel BC (a practical BC prescription method in CFD
simulations to provide physiologically realistic impedances24,57,77), to achieve rigorous
validations of the numerical method. We built a 2-outlet physical model (flow phantom)
of a patient-specific AAA, and constructed physical analogs of a Windkessel lumped-
parameter model which we attached to the outlets of the flow phantom to provide
physiologically-realistic outlet flow impedances. We then used a 1.5T MRI scanner to
acquire PCMRI data in multiple 2D planes in the AAA phantom while it was under
pulsatile, and physiologically-realistic flow and pressure conditions. The use of PCMRI
in this study allows us to follow a similar protocol for future in-vivo validations. Next,
we performed in-silico numerical CFD simulations using a finite-element analysis (FEA)
technique, and prescribed outlet BC using analytical models of the Windkessel directly
corresponding to its physical construction. We then compared experimentally measured
velocities, flows, and pressures in the in-vitro AAA phantom to those computed in the in-
silico CFD simulations.
4.3 Methods
4.3.1 Anatomical Model Construction
From a gadolinium-enhanced MR angiography scan of an AAA patient (Figure
4.1a), we constructed a 3D computational anatomical model (Figure 4.1b) that includes
the AAA, and the renal and common iliac arteries.68 The anatomical domain extends
from about 4cm superior to the first renal branches to 4cm inferior to the aortic
bifurcation, and includes a few centimeters of the renal arteries. The final computational
46
model shown in Figure 4.1b includes extensions we added to the anatomical domain
vessels to accommodate the two-outlet connections to the rest of the flow loop. Finally,
we used this computational model to physically construct a rigid AAA phantom for the
in-vitro experiment using a stereolithography technique (Viper™ si2 stereolithography
machine, 3D Systems Corporation, Rock Hill, SC) and MR-compatible resin
(WaterShed® XC 11122, DSM Somos®, Elgin, IL) 78 (Figure 4.1c).
Figure 4.1 Anatomical Phantom Model. a) MR Imaging data from an AAA patient. b) 3D computer model constructed based on patient imaging data. c) Physical phantom constructed from 3D computer model.
4.3.2 Outlet Boundary Condition
We used a four-element Windkessel model consisting of an inductance (L),
proximal resistance (Rp), capacitance (C), and distal resistance (Rd), as outlet BC (Figure
4.2 & Figure 4.3a). Physically, we designed the Windkessel modules such that
physiologically realistic flows and pressures were achieved in the phantom, and that the
resistance and capacitance values of the lumped-parameters remained reasonably
constant over the whole operating range of the flow conditions.
47
Figure 4.2 In-vitro Experiment Flow System Setup Diagram
Figure 4.3 a) The physical Windkessel module assembly and the corresponding analytical representation b) The resistance module c) The capacitance module
48
Resistance
In order to obtain significant flow resistance while keeping the flow laminar
(which is required to obtain a resistance value that is independent of flow rate), we
constructed the resistance module by placing a large number of thin-walled glass
capillary tubes (Sutter Instrument, CA) in parallel with each other inside a plexiglass
cylinder (Figure 4.3b). Using Poiseuille’s law and the equation for parallel resistances,
the resistance of the module can be calculated by:
4
8Resistance lNrμ
π= (4.1)
where μ is the dynamic viscosity of the working fluid, l is the length of the capillary
tubes, r is the inside radius of each individual capillary tube, and N is the total number of
capillary tubes in parallel.69 Table 4.1 shows the list of resistance modules we used in the
experiment with their construction details and resulting resistances.
Resistance Module
Cylinder Diameter
Capillary Tubes
Length (cm)
Capillary Tubes ID/OD*
(mm)
# of Capillary
Tubes
Resistance (dynes-s/cm5)
Rp1 1” 10 1.1 / 1.5 233 549 Rd1-1** 3/8” 10 0.78 / 1 71 7132 Rd1-2** 0.71cm 5 1.1 / 1.5 16 4000 Rp2 5/8” 10 0.75 / 1 194 3053 Rd2 3/8” 10 0.9 / 1.2 48 5951
Table 4.1 List of Resistance Modules Used in the Experiment
Resistances values are calculated using a fluid viscosity of 0.0457 dynes-s/cm2 as measured from the working fluid.
* ID/OD stands for inside diameter/outside diameter **Rd1-1 and Rd1-2 are placed in parallel, and together act as the “Rd1” shown in Figure 4.2. A valve turns Rd1-2 on and off for the light exercise and resting flow conditions to change the effective resistance value of “Rd1”.
49
/ P
Capacitance
The capacitance of a fluid system is define as C=∆V/∆P where ∆V and ∆P are the
changes in volume and pressure. In a closed system at constant temperature, an ideal gas
exhibits the behavior PV=(P+∆P)(V-∆V), where P and V are the reference pressure and
volume. The capacitance of a column of air is then:
( )aC V V= −Δ (4.2)
For small changes in volume relative to the reference volume, a reasonably constant
capacitance can be obtained with an air column. We constructed the capacitance module
with a plexiglass box that can trap a precise amount of air, which acts as a capacitance in
the system (Figure 4.3c). The capacitance box has an inlet and an outlet, and as the fluid
enters and exits the box, the fluid level in the box rises and falls slightly. The varying
fluid level also contributes to a capacitance that is in series with the capacitance due to air
compression. The pressure change in the fluid due to the varying fluid level ∆h is:
/P g h g V Aρ ρΔ = Δ = Δ (4.3)
where ρ is the fluid density, g is the gravitational constant, and A is the area of the
fluid/air interface (assuming a constant cross-sectional area column of fluid). The
capacitance due to the varying fluid level is then:
/ ( )vC A gρ= (4.4)
Since Cv is in series with Ca, if Cv>>Ca the overall capacitance can be
approximated by Ca alone. In the actual construction of the capacitance module, we
designed the box to be large enough so that such approximation is true. We also
designed a smooth contour for the inlet of the module in order to minimize flow
turbulences and thus avoid parasitic resistances (Figure 4.3c).
Inductance
The flow inductance results from the fluid momentum, and can be calculated from
the geometry of the physical system. The flow inductance in a fluid conduit is:
/L l Aρ= (4.5)
where l and A are the length and the cross-sectional area of the conduit.
50
4.3.3 In-Vitro Experiment
Flow Loop Setup
We placed the flow phantom and the outlet impedance modules in a flow system
as shown in Figure 4.2. The experiment mimicked two different physiological
conditions: a resting condition, and a light exercise condition. We computed the resting,
supra-renal level aortic flow waveform by summing the infra-renal blood flow velocity
previously acquired by PCMRI in the same AAA patient, and published data of renal
flow.66 For the light exercise condition, we increased the average flow to two times that
of resting, increased the heart rate from 60 bpm to 80 bpm, and decreased the
downstream resistance of the aortic outlet Rd1 (by turning on a valve to allow flow
through Rd1-2 which is in parallel with Rd1-1) to mimic vasodilation of the lower
extremity vessels. This mimics a very mild exercise condition such as light walking. We
then used a custom-built, MR-compatible, and computer-controlled pulsatile pump,37 in
parallel with a 1/12 horse-power, 3100RPM, steady flow pump (Model 3-MD-HC, Little
Giant Pump Co., OK) with a ball valve attached at its outlet for flow rate control, to
physically reproduce the supra-renal flow waveforms as the input flow to the phantom.
At the inlet of the phantom we placed one meter of straight, rigid tubing, a honeycomb
flow straightener, and two pressure-stabilization grids, in order to provide sufficient
entrance conditioning to generate a stable and fully-developed Womersley flow profile at
the phantom inlet. The working fluid in the flow system was a 40% glycerol solution
with a dynamic viscosity similar to that of blood, and contained 0.5% Gadolinium.
Flow & Pressure Probe Measurements
We used an MR-compatible ultrasonic transit-time flow sensor to monitor the
total input flow to the phantom. We placed the externally clamped flow probe (8PXL,
Transonic Systems, NY) around a short section of Tygon tubing R3603 immediately
upstream of the one-meter flow conditioning rigid tubing, and sent the signals from the
probe into a flowmeter (TS410, Transonic Systems, NY) with its low pass filter setting at
160Hz. For pressure measurements, we inserted MR-compatible catheter pressure
51
transducers (“Mikro-Tip” SPC-350, Millar Instruments, Houston, TX) through small
ports on the sides of the phantom to capture pressure waveforms at the phantom outlets,
and also immediately downstream of the parallel impedance modules to record the
pressure to be used as reference ground pressure. Note that the pressure data presented in
the results section is relative to the reference ground pressure. We sent the signals from
each catheter pressure transducer into a pressure control unit (TCB-600, Millar
Instruments, TX) which produces an electrical output of 0.5V per 100mmHg of pressure.
We recorded the data from the flow meter and the pressure control units at a sample rate
of 96 Hz using a data acquisition unit (USB-6259, National Instruments, Austin, TX) and
a LabVIEW program (LabVIEW v.8, National Instruments, Austin, TX). We acquired
pressure and flow data intermittently (approximately every 0.5~1 hour) throughout the
experiment in between MR scans. We averaged approximately 50 cycles of flow rate and
pressure data to obtain one averaged cycle for each acquisition. We then averaged the
measurements from all the acquisition throughout the experiment to obtain one cycle of
flow rate and pressure data that represents each of the two flow conditions. The cycles of
flow rate and pressure measurements were stable both in between the cycles of one
acquisition, and between the different acquisitions throughout the experiment.
MRI
We acquired flow velocity data at different slice locations within the phantom
(Figure 4.2) using a cardiac-gated 2D 3-component cine PCMRI sequence in a 1.5T GE
MR scanner (Signa, GE Medical Systems, Waukesha, WI) and an 8-channel cardiac coil.
The slice locations represent the mid-aneurysm location for each lobe of the bilobed
aneurysm anatomy, and also a location directly downstream of the renal branches where
flow is likely to be complex. The imaging parameters were: 256x192 acquisition matrix
reconstructed to 256x256, 24x24 cm2 field of view, 5mm slice thickness, TR=~13 ms,
TE=~5.5 ms, 20 degree flip angle, and NEX=2. The through-plane velocity encoding
(Venc) for slices 1-3 were 100, 50, and 50 cm/s, respectively, for the resting flow
condition, and 100, 75, 50 cm/s, respectively, for the light exercise flow condition. The
in-plane Venc for slices 1-3 were 30, 30 and 50 cm/s, respectively, for the resting flow
52
condition, and 30, 40, and 50 cm/s, respectively, for the light exercise flow condition.
The LabVIEW program which controlled the pulsatile pump produced a trigger signal
that was converted by an electrocardiogram (ECG) simulator (Shelley Medical Imaging
Technologies, London, Ontario, Canada) into an ECG signal used by the MRI system for
gating, and 24 time points per cardiac cycle synchronized to this ECG signal were
reconstructed. The temporal resolution of the velocity data was two times the TR
(~26ms). We placed vitamin E capsules (Schiff Nutrition Group, Inc, Salt Lake City,
UT) as well as saline bags around the flow region of each slice to produce the reference
signals of stationary fluids, which we then used for baseline eddy current correction with
a linear correction algorithm in the analysis of the PCMRI data.
4.3.4 In-Silico Simulation
We performed the numerical simulation of flow and pressure using a custom
stabilized finite-element method to solve the incompressible Navier-Stokes equations,
assuming rigid walls and a Newtonian fluid with a density of 1.1 g/cm3 and dynamic
viscosity of 0.0457 dynes-s/cm2 as measured from the working fluid.79 We discretized
the 3D anatomical model into an isotropic finite-element mesh with a maximum edge
size of 0.1 cm and containing 3.2 million linear tetrahedral elements using commercial
mesh generation software (MeshSim, Simmetrix, Inc., NY). For each of the two flow
conditions, we first ran a steady flow simulation with the mean probe-measured flow
prescribed at the inlet, and the sum of the proximal and distal resistances in each of the
outlet Windkessel models as the resistance BC for each of the outlets. For the steady
flow simulation, we used a timestep size of 0.001 second and ran the simulation for 0.5
second, which was sufficient for the pressure inside the domain to stabilize. We then
used the pressure and velocity results from the steady simulation as the initial condition
for the pulsatile flow simulation. For the pulsatile flow simulation, we mapped the
inflow waveform (the averaged representative cycle of flow probe measurement) to the
inlet face using a Womersley velocity profile. For the BC at the 2 outlets, we prescribed
a Windkessel model with the lumped-parameter component values (Table 4.2) calculated
from their physical constructions. We used an augmented lagrangian method to constrain
53
the shapes of velocity profiles at the outlets to prevent divergence.80 This technique has
been shown to have very little effect to the flow and pressure calculations in the
numerical domain.80 We used a timestep size of 0.0004 second, and simulated 11 cardiac
cycles for each of the two flow conditions. The first cycle of the simulation result was
discarded, and the last 10 cycles where the pressures had stabilized were included in the
final analysis. The cycle-to-cycle variations in the velocity pattern of the numerical
simulation results indicate the presence of aperiodic features in the velocity pattern.
Since the PCMRI technique combines measurements acquired over multiple cycles into
one cycle of velocity data, we averaged the simulation results from corresponding time
points in the last 10 successive cycles to mimic the PCMRI data acquisition method.37
The aperiodic features in the velocity patterns tend to also be the higher spatial frequency
features which cannot be resolved by MR imaging due to its limited spatial resolution.
Cycle to cycle averaging of the simulation results removes these high spatial frequency
features, making the comparisons to MR data more meaningful.
Resting Condition Light Exercise Condition Aortic outlet Renal outlet Aortic outlet Renal outlet
L (dynes-s2/cm5) 7 16 7 16 Rp (dynes-s/cm5) 549 3053 549 3053 C (cm5/dynes) 0.0003253 0.0001644 0.0003222 0.0001590 Rd (dynes-s/cm5) 7132 5951 2563 5951
Table 4.2 Outlet BC Windkessel Component Values Prescribed to the FEA Simulations
54
4.4 Results
4.4.1 PCMRI Flow Measurements at Different Slice Locations
Figure 4.4 PCMRI Measured Flow Through the Abdominal Aorta at Different Slice Locations for a) Resting condition, and b) Light exercise condition
In the case of a rigid phantom and flow conservation, the total flow across any
arbitrary axial section of the phantom is theoretically identical at any instant in time.
Figure 4.4 shows the flow through the abdominal aorta as measured with PCMRI at the
three different slice locations, for the resting and light exercise conditions. We found that
the flows measured at the different locations agree extremely well. The average flow
measured at S1, S2, and S3 are 24.1, 24.0, and 24.5 cc/s respectively (2% maximum
difference) for the resting condition, and 68.4, 68.5, and 68.7 cc/s respectively (0.4%
maximum difference) for the light exercise condition. For the resting condition,
however, the flow waveform acquired from S1 was temporally shifted by one sampling
point compared to S2 and S3. In Figure 4.4, we have shifted the S1 waveform to align it
to the S2 and S3 waveforms.
55
4.4.2 PCMRI vs. Flow Probe Measured Inlet Flow Waveforms
Figure 4.5 PCMRI vs. Ultrasonic Flow Probe Measured Total Inlet Flow for a) Resting condition, and b) Light exercise condition
PCMRI flow measurements were validated against ultrasonic flow probe
measurements. Figure 4.5 shows flow comparisons between the PCMRI measurements
(the sum of the aortic and renal branches at S3), and the flow probe measurements at the
inlet of the phantom, for the resting and light exercise flow conditions. Figure 4.5 shows
very close agreement between the PCMRI and flow probe measured flow waveforms
both in their magnitudes and shapes. The shapes of the waveforms are nearly identical
even down to the small features of local peaks and troughs. Slight disagreement between
the first data point in the PCMRI measured flow and the flow probe data could be due to
the retrospective reconstruction of the PCMRI data. For the resting condition, the
average flows were 42 cc/s and 46 cc/s as measured by the flow probe and PCMRI
respectively. For the light exercise condition, the average flows were 88 cc/s and 96 cc/s
as measured by the flow probe and PCMRI respectively. For both flow conditions, the
averaged flow measured by PCMRI was 8-9% higher than that measured by the flow
probe.
56
4.4.3 Measured vs. Simulated Flow & Pressure Waveforms
Figure 4.6 Measured In-vitro (Solid lines) vs. Simulated In-silico (dashed lines) Pressure & Flow Waveforms for a) Resting condition, and b) Light exercise condition
Figure 4.6 shows comparisons between in-vitro, MR measured, and in-silico,
simulated flow waveforms, and comparisons between in-vitro, catheter probe measured,
and in-silico, simulated pressure waveforms, for each of the two flow conditions. We
used the PCMRI data at S3, which included both the aortic and the renal branch outlets,
to construct the aortic and renal flow waveforms. There is excellent agreement between
the experimental data and simulated results for the pressure and flow waveforms. The
pressure and flow waveforms are physiologic in shape and magnitude. Specifically, the
renal flow is always antegrade,66 but the abdominal aortic flow is retrograde during part
of the cardiac cycle in resting condition. The flow split ratios between the aortic and
renal branches are 54:46 (predicted by the simulation) and 53:47 (measured by MR) for
the resting condition, and 74:26 (predicted) and 72:28 (measured) for the light exercise
condition. The predicted and measured average pressures are 131 mmHg and 128 mmHg
respectively (2% difference) for the resting condition, and 153 mmHg and 159 mmHg
respectively (3.8% difference) for the light exercise condition.
57
4.4.4 PCMRI vs. FEA Instantaneous Time Point Velocity Patterns
Figures 4.7 and 4.8 show through-plane and in-plane velocity comparisons
between the cycle-to-cycle ensemble-averaged simulation results, and 3-component
PCMRI velocity data. For each of the flow conditions, the comparisons are made at four
different time points of the cardiac cycle (diastole, acceleration, systole, and
deceleration), and at three slice locations within the abdominal aorta (S1, S2, and S3).
For both flow conditions, we found good agreement between the simulation results and
PCMRI data in all of the comparisons at the systole and deceleration time points, and also
in the S3 location at all time points.
For the resting condition in Figure 4.7, flow velocities are very low at the diastole
time point, leading to low signal-to-noise in the MR data, and also poor agreement with
simulation results; however, diastolic velocities measured with PCMRI at the S3 location
surprisingly still agreed well with simulation under such a circumstance. Many in-plane
velocity comparisons at the S1 location showed patterns that seem to mismatch at first
glance, but contained features in common upon more careful examination. One such
example is deceleration at S1 under the resting condition, where the MR data show a
counter-clockwise vortex in the upper half of the slice and a clockwise vortex in the
bottom half, and the FEA results show a less prominent, but existing counter-clockwise
vortex at the top of the slice, and downward velocities along the bottom right wall
forming a clockwise vortex structure with the right-ward velocities near the center of the
slice.
For the light exercise condition in Figure 4.8, S2 at deceleration is an example of
how even the detailed flow features can be matched in the comparisons: Positive and
negative through-plane velocities along two opposite walls of the vessel are shown in
both the simulation and PCMRI results with similar magnitudes and shapes, and in-plane
velocities moving away from the high through-plane flow area creating two vortices
towards the center are also shown in both results. The less favorable comparisons
occurred at S1 and S2 at the acceleration time point, and also at S1 at the diastole time
point. For these same slices and time points, we also found significant cycle-to-cycle
velocity pattern variations in the simulation results.
58
Figure 4.7 Resting Condition Flow Velocity Comparisons: Between MR Measurements and FEA Results at the a) Diastole, b) Acceleration, c) Systole, and d) Deceleration time point, at three different slice locations (S1-S3). Colour map and arrows correspond to through-plane and in-plane velocities, respectively.
59
Figure 4.8 Light Exercise Flow Velocity Comparisons: Between MR Measurements and FEA Results at the a) Diastole, b) Acceleration, c) Systole, and d) Deceleration time point, at three different slice locations (S1-S3). Colour map and arrows correspond to through-plane and in-plane velocities, respectively.
60
4.4.5 PCMRI vs. FEA Whole-cycle Averaged Velocity Patterns
Figure 4.9 shows velocities averaged over all of the time points in the cardiac
cycle at the three different slice locations for each of the two flow conditions. There is
remarkable agreement between the PCMRI measurements and FEA results in all of the
slices for both flow conditions and for both the through-plane and in-plane velocities.
Even complex patterns of in-plane velocities at the S1 location match between the
simulation results and experimental data. For the resting condition, even though the
orientations of the S1 in-plane velocities are slightly different between the comparisons,
upon closer examination it is clear that both contain the clockwise and counter-clockwise
vortices in the lower left and upper right parts of the slice.
Figure 4.9 Whole-cycle Averaged Flow Velocity Comparisons: Between MR measurements and FEA results at three different slice locations (S1-S3) for a) Resting condition and b) Light exercise condition.
61
4.5 Discussion
The purpose of this study was to produce a set of in-vitro, high-quality
experimental data that can be used to compare against FEA results of flow and pressure
within a complex and realistic anatomy, and under physiological flow and pressure
conditions. We have built a highly controlled and characterized experimental setup
consisting of a precisely constructed, patient-derived flow phantom, well-defined BC
modules, and highly periodic cardiac cycles, in order to acquire such experimental data.
We used PCMRI, an imaging method that could be employed in-vivo. We also used a
two-outlet, patient-specific AAA anatomy under two physiological conditions – resting
and light exercise. We demonstrated that with the methods described in this paper, we
were able to construct and characterize lumped parameter impedance modules with
predictable component values and behaviors, which enabled us to accurately prescribe
corresponding analytical models at the outlet boundaries of the numerical simulations,
and predict the flows through the two outlets of the phantom and the pressures within the
phantom. In addition, the complex impedances provided by the impedance modules are
important in achieving flows through the aortic and renal outlets that are dramatically
different in their shapes and phases.
The close agreement between PCMRI volumetric flow measurements at different
slice locations demonstrated the precision of the PCMRI measurements. The close
agreement between the PCMRI and ultrasonic flow probe measured flow waveform
shapes showed that the PCMRI measurements had sufficient temporal resolution to
capture the various frequency components in the flow waveforms. A discrepancy of
<10% in the mean flow was both within the specified absolute accuracy of the ultrasonic
flow probe, as well as consistent with the anticipated accuracy of PCMRI flow
measurements.65,81
We found instantaneous flow velocity pattern agreements between MR and FEA
results in the complex and realistic AAA geometry, even in complex in-plane velocity
features. The S1 location exhibited more complex flow and cycle-to-cycle variations
compared to the other slice locations, possibly due to the fact that it was immediately
downstream of the renal branches, and because it had a small cross-sectional area
62
resulting in high flow velocity and high Reynold’s number. The S3 location, on the other
hand, exhibited less complex flow and aperiodicity due to its lower flow velocities. Since
PCMRI is acquired over multiple cardiac cycles, data accuracy depends heavily on
periodicity. Cycle-to-cycle variations in simulation results are likely due to flow
turbulence,68 and previous studies have shown that flow turbulence causes MR signal
loss82,83 which also contributes to measurement challenges. Large amounts of cycle-to-
cycle variations also require a large number of cycles in the FEA simulation to be
averaged in order to obtain a representative and stable ensemble average. Thus, the
effects of complex flow and aperiodicity are detrimental to both the MR measurements
and simulation result interpretations, leading to poorer comparisons. We indeed found
less favorable comparisons at the S1 location, and consistently good comparisons at the
S3 location as expected. Another interesting finding was that the presence of complex
flow and aperiodicity upstream did not seem to affect the downstream flow stability and
predictability, which is consistent with the findings of Les et al.68 At the diastolic time
point in the light exercise condition, even though there was significant aperiodicity in the
S1 location, stable and periodic flow resumed downstream at S2 and S3, where very good
agreements were obtained between the PCMRI and FEA results. At the acceleration time
point in the light exercise condition, there was also much aperiodicity at both S1 and S2,
and yet further downstream at S3, the flow features agreed well.
The velocity encoding (Venc) of the PCMRI acquisition is typically set to a value
that is able to capture the highest flow velocities during systole without aliasing.
However, such Venc setting may not be appropriate for obtaining high signal-to-noise
data during diastole if the flow velocities are near zero during that period of the cardiac
cycle. If one needs to obtain more accurate flow velocity data during diastole, a separate
acquisition with a lower Venc assignment may be called for. It is useful to note that the
volumetric flow measurements was not affected by the lower signal-to-noise ratio: the
MR measured volumetric flow waveform agreed closely with the flow probe measured
waveform even during the resting condition diastolic period when the flow was very low
and the MR measured velocity pattern appeared noisy.
63
When the flow velocities were averaged over all of the time points within the
cardiac cycle, the S1 comparison for the light exercise condition was slightly better than
that of the resting condition, likely due to the lack of backflow through the abdominal
aorta. During parts of the cardiac cycle under resting condition where backflow occurred
through the abdominal aorta, and continuous forward flow persisted through the renal
branch, complex and aperiodic flow at the nearby S1 location was likely to occur and
contribute to the slight differences in the comparison. The ability to predict whole-cycle
averaged flow velocities is clinically useful, as many studies have correlated parameters
such as mean blood flow and mean shear stress to blood vessel adaptation behavior and
disease progression.3,4,6,7 In general, the good comparisons of the whole-cycle averaged
velocity patterns we found at all of the three slice locations for both flow conditions bode
well for the application of CFD in cardiovascular studies.
In regards to the sources of errors in this study, we found that the segmentation
and baseline correction variabilities in processing the MR data seemed to have little
effect on the results. The geometric tolerance of the phantom manufacturing from the
CAD model could also influence results, although it has been shown that slight geometric
variations have minimal impact on simulated flow and pressure.37 Slight variations in the
cyclic pump output could affect the PCMRI data integrity, but in general we were able to
obtain highly periodic flows from the flow pumps. The values of the lumped parameter
components, especially of the resistance modules, also have a large impact on the
resultant pressure waveforms in the experiment. The slight differences between the
measured and simulated pressures could be possibly attributed to small differences
between the theoretical and the actual resistance values of the resistance modules. The
inlet flow prescription can affect FEA simulation results, and slight differences in the
numerical prescription of the input flow compared to that in the physical experiment
could introduce errors in the simulation results. The numerical simulation also assumed
fluid behaviors that must be precisely matched by the actual fluid used in the experiment.
The mesh size we used for this problem was likely sufficient to solve the problem domain
with reasonable fidelity, yet a larger mesh would certainly provide better mesh
convergence. Since each set of the PCMRI data was acquired over hundreds of cardiac
64
cycles, an ensemble average of the simulation results over a larger number of cycles
might also provide better comparisons to the MR data.
The lumped parameter BC modules we developed in this study can provide the
necessary tool to perform further in-vitro studies incorporating additional physiologically
realistic aspects, such as compliance of the blood vessel. The ability to reproduce
physiological pressures is absolutely essential for obtaining physiological deformation in
an in-vitro experimental setup, allowing the investigations of behaviors mimicking in-
vivo tissue motion and wave propagation. The effects on flow velocities and pressure due
to vessel wall motion can be captured computationally by prescribing at the fluid-solid
interface non-zero fluid velocities that are based on solid domain calculations.16 By
prescribing accurate vessel wall properties for the solid domain calculations, we expect
the computational simulation to be capable of predicting flow and pressure in a
deformable geometry.
In conclusion, we have demonstrated the capabilities of numerical simulations to
accurately predict flow and pressure in a complex AAA geometry, and we have
developed the necessary methods towards further validations of the numerical technique,
which could eventually show that the numerical method can be applied clinically to make
accurate predictions of pressure and flow in the vasculatures of a real patient.
Equation Section 5
65
Chapter 5: In Vitro Validation of Finite
Element Analysis of Blood Flow in
Deformable Models
5.1 Abstract
Purpose: To validate numerical simulations of flow and pressure incorporating
deformable walls using in-vitro flow phantoms under physiological flow and pressure
conditions.
Materials and Methods: We constructed two deformable flow phantoms mimicking a
normal and a restricted thoracic aorta, and developed a physical Windkessel model to use
as the outlet boundary condition. We then used magnetic resonance imaging, flow
probes, and pressure catheters to acquire flow and pressure data in the phantom while it
operated under physiological conditions. Next, in-silico numerical simulations were
performed, and velocities, flows, and pressures in the in-silico simulations were
compared to those measured in the in-vitro phantoms.
Results: The experimental measurements and simulated results of pressure and flow
waveform shapes and magnitudes compared favorably at all of the different measurement
locations in the two deformable phantoms. The average difference between measured
and simulated flow and pressure was approximately 3.5 cc/s (13% of mean) and 1.5
mmHg (1.8% of mean), respectively. Velocity patterns also showed good qualitative
agreement between experiment and simulation especially in regions with less complex
flow patterns.
Conclusion: We demonstrated the capabilities of numerical simulations incorporating
deformable walls to capture both the vessel wall motion and wave propagation by
accurately predicting the changes in the flow and pressure waveforms at various locations
down the length of the deformable flow phantoms.
66
5.2 Introduction
The stress and strain in blood vessels, as well as hemodynamic parameters such as
the three-dimensional blood flow and pressure fields, have direct effects on the initiation
and development of cardiovascular diseases such as atherosclerosis and aneurysms.2,84,85
Knowledge of how in-vivo forces and tissue motions interact with implantable medical
devices is also essential for understanding and predicting their behavior after
implantation. For example, compliance mismatch between a prosthetic bypass graft and
its adjacent native arteries has been hypothesized to lead to graft failure.12 Medical
imaging has been used to investigate vessel strain and blood flow hemodynamics, but
with limited temporal and spatial resolutions, and often with discomfort to the patients as
they are required to remain motionless for long periods during imaging sessions. Image-
based computational fluid dynamics (CFD) methods, due to their minimal patient
involvement and their ability to finely resolve time and space, have been a practical
alternative to quantifying vessel strains and hemodynamic conditions for studies of
disease mechanisms15,17,18 and the design and evaluation of medical devices.19-21 The
ease of applying variations in geometry and flow conditions in the computational domain
also motivates the use of CFD in the planning and prediction of surgical procedures.22,23
Previous studies of cardiovascular CFD included the use of rigid wall models,86 and
dynamically deforming models.87 Considering that vessel wall deformability often
influences flow velocities and pressures, and that wave propagation phenomena can only
be captured when considering wall deformation since blood behaves as an incompressible
fluid, it is advantageous to include blood vessel deformability in numerical simulations
whenever possible.
Much work remains to validate CFD methods against experimental data.
Previous in-vitro validation studies have been performed only for the rigid case, likely
due to the lack of realistic outflow boundary conditions, which are required to represent
physical properties of downstream vasculature and to produce physiologic levels of
pressure. For example, implementations of simple zero pressure boundary conditions in
the physical setup where phantom outlets connect directly into a fluid reservoir33,36,37
67
would not be able to provide the pressures required to achieve realistic deformations in a
compliant model.
In this study, we present results from two compliant phantoms under
physiological flows and Windkessel boundary conditions for the validation of the
numerical method incorporating wall deformability. The Windkessel boundary condition
is a practical boundary condition prescription method in CFD simulations that can
provide physiologically realistic impedances.24,57,77 We built a normal and a restricted
physical model (flow phantom) comparable in size to the descending thoracic aorta, and
constructed a physical analog of the Windkessel model to be attached to the outlets of
each flow phantom to provide physiologically-realistic outflow impedances. A 1.5T MRI
system was then used to acquire phase-contrast magnetic resonance imaging (PCMRI)
flow velocity data in multiple 2D planes in the phantoms while they were under pulsatile
physiologically-realistic flow and pressure conditions. The use of PCMRI in this study
enables us to follow a similar protocol for future in-vivo validation. Next, we performed
in-silico numerical CFD simulations incorporating a coupled momentum method for wall
deformability,16 and prescribed Windkessel boundary conditions that directly
corresponded to the physical setup. We then compared the flows, pressures, and velocity
patterns measured in the in-vitro phantoms to those computed in the in-silico CFD
simulations.
5.3 Methods
5.3.1 Physical Flow Phantom Construction & Characterization
We constructed two flow phantoms each containing a compliant vessel with an
unpressurized diameter of 2cm, length of 25 cm, and thickness of 0.08 cm. The
diameters and thicknesses of the vessels were selected such that physiological
diameters88,89 and strains90,91 mimicking the descending thoracic aorta would be achieved
under physiological operating pressures. One of the vessels was constructed to be a
simple straight cylinder, and the other to be a straight cylinder containing a stenosis of
diameter 0.88 cm at its center, which equivalents to an 84% area reduction relative to the
mean operating diameter of the phantom.
68
To fabricate a compliant vessel, we used a multi-step dip-spin coating technique
which entailed dipping an aluminum rod machined to the desired inner geometry of the
vessel into a silicone mixture.92,93 The important factors in obtaining the desired vessel
wall thickness were the silicone mixture viscosity, dipping withdrawal speed, rod
diameter, and number of dips. We set the silicone viscosity to 1500-2000 cp, dipped the
aluminum rod vertically into the silicone mixture, and withdrew it at a controlled speed
of 23.8 centimeters per minute. To obtain a uniform thickness circumferentially, we then
set the rod on a horizontal rotating fixture for 30 minutes while the silicone dried. The
entire process was repeated twice to obtain the desired thickness of 0.08 cm. Finally, the
rod was set to cure in a heat convection oven at 100 °C for 4 hours.
We connected the inlet and outlet of each compliant vessel to a rigid section of
flow conduit in order to allow for easy connection to the rest of the experimental flow
setup. We also glued the bottom edge of the vessel along a ridge of width 6.8mm using a
small amount of epoxy (5 Minute Epoxy, Devcon, MA) to mimic the in-vivo tethering of
arteries to surrounding tissues such as the spine. The rigid inlet and outlet sections,
together with the compliant vessel glued to the ridge, made up a “flow phantom.” For the
stenotic phantom, a rigid ring was placed around the stenosis to make this region of the
compliant vessel essentially “rigid”. This mimics the in-vivo material property of an
arterial stenosis comprised of a stiff plaque.
We performed static pressurization tests to characterize the phantom deformation
under different levels of pressures. We pressurized the phantoms using a syringe while
monitoring the internal pressure with a catheter pressure transducer (“Mikro-Tip” SPC-
350, Millar Instruments, Houston, TX). The outer diameter of the compliant vessels at
various pressurization levels was measured using a digital caliper (CD-6” CS, Mitutoyo
Corp., Kawasaki, Japan). Figure 5.1 shows the static pressurization characterization data
for the two phantoms. There was an approximately linear relationship between the
expansion of the compliant vessels and the increasing static pressures within the expected
operating pressure range.
69
Figure 5.1 Vessel Outer Diameter Versus Static Pressure for the a) Straight Phantom,
and b) Stenotic Phantom
We also characterized the viscoelastic properties of the silicone material using a
dynamic mechanical analyzer (DMA Q-800, TA Instruments, New Castle, DE). We took
small rectangular pieces (20 x 5 x 0.7mm) of the silicone material, and tested them using
a multi-frequency sweep controlled strain method. We applied a pre-load of 0.45N in
order to deform the material to an approximately 12% static strain, we then applied an
oscillating strain of 5% over a range of incremental frequencies from 0.10 – 6.0 Hz.
These settings approximately correspond to the operating conditions of the phantoms in
the flow experiments. When subjected to an oscillating strain at 1Hz, the storage moduli
(which represent the elastic behavior of the material) in all of the six different samples
tested were approximately 10% higher compared to when subjected to an oscillating
strain at 0.1Hz.
5.3.2 Outlet Boundary Condition
We used a four-element Windkessel model consisting of an inductance (L),
proximal resistance (Rp), capacitance (C), and distal resistance (Rd), at the outlet
boundary of the phantom (Figure 5.2). Physically, the Windkessel module was designed
such that physiologically realistic flows and pressures were achieved in the phantom, and
70
that the specific values of the resistance and capacitance components remained
reasonably constant over the operating range of each experiment.
We constructed the resistance module by placing a large number of thin-walled
glass capillary tubes (Sutter Instrument, CA) in parallel with each other inside a
plexiglass cylinder. Using Poiseuille’s law and the equation for parallel resistances, the
resistance value of the module is:69 Resistance=8 μ l/(Nπr4), where μ is the dynamic
viscosity of the working fluid, l is the length of the capillary tubes, N is the total number
of capillary tubes in parallel, and r is the inner radius of each individual capillary tube.
We used 129 capillary tubes with inner diameters of 0.156 cm, and 125 tubes with inner
diameters of 0.078 cm, for Rp and Rd, respectively. The theoretical resistance values
corresponding to the specific viscosity of the working fluid in each of the two phantom
experiments are listed in Table 5.1.
The capacitance of a fluid system is C=∆V/∆P where ∆V and ∆P are the changes
in volume and pressure. In a closed system at constant temperature, an ideal gas exhibits
the behavior PV=(P+∆P)(V-∆V), where P and V are the reference pressure and volume.
The capacitance of a pocket of air is then: C = (V - ∆V) / P. For small changes in
volume relative to the reference volume, a reasonably constant capacitance can be
obtained with an air pocket. We constructed the capacitance module with a plexiglass
box that can trap a precise amount of air to act as a capacitance in the system. The
capacitance box has an inlet and an outlet, and as the fluid enters and exits the box, the
fluid level in the box rises and falls slightly. The varying fluid level contributes to an
additional capacitance that is in series with the capacitance due to air compression. This
additional capacitance, if significantly larger than the air compression capacitance, has
minimal impact on the overall capacitance. The capacitance due to the varying fluid
level is proportional to the cross-sectional area of the box. In the actual construction of
the capacitance module, we made the box sufficiently large such that the capacitance
contribution from the varying fluid level can be disregarded. We also designed a smooth
contour for the inlet of the module in order to minimize flow turbulence and thus avoid
parasitic resistances. The air volume we used in the capacitor module was 210mL and
300mL at ground pressure (atmosphere pressure), for the straight and stenotic phantom
71
experiment, respectively. The theoretical capacitance at the average operating pressure
(relative to ground) of 100mmHg and 91mmHg for the straight and stenotic phantom
experiments, respectively, is listed in Table 5.1.
The flow inductance results from the fluid momentum, and can be calculated from
the geometry of the physical system. The flow inductance in a fluid conduit is: L = ρl/A,
where l and A are the length and the cross-sectional area of the conduit, respectively.
The theoretical inductance of the Windkessel module used in the phantom experiments is
listed in Table 5.1.
Straight Phantom Experiment Stenotic Phantom Experiment
Theoretical Experimental Theoretical Experimental
L (Barye s2 cm‐3) 7 7 7 7
Rp (Barye s cm‐3) 246 242 241 259
C (cm3 Barye‐1) 1.6 e‐4 1.3 e‐4 2.3 e‐4 1.9 e‐4
Rd (Barye s cm‐3) 4055 3990 3980 4279
Table 5.1 Theoretical and Experimental Windkessel Component Values for the Straight
and Stenotic Phantom Experiments
72
5.3.3 In-vitro Experiment
Figure 5.2 In-vitro Flow Experiment Setup Diagram
We performed two in-vitro experiments, one with each flow phantom. For each
experiment, the flow phantom and the outlet impedance module were placed in a flow
system as shown in Figure 5.2. To produce the input flow to the phantom, we used a
custom-built, MR-compatible, computer-controlled pulsatile pump to physically
reproduce flow waveforms similar to the descending aortic flow measured in patients
with aortic coarctations.94 Immediately upstream of the phantom we placed one meter of
straight, rigid tubing, a honeycomb flow straightener, and two pressure-stabilization
grids, in order to provide sufficient entrance conditioning to generate a stable and fully-
developed Womersley flow profile at the phantom inlet. The working fluid in the flow
system was a 40% glycerol solution with a dynamic viscosity similar to that of blood, and
contained 0.5% Gadolinium. The fluid viscosity for the straight and stenotic phantom
experiments was measured to be 0.0461 poise and 0.0452 poise, respectively.
We inserted MR-compatible catheter pressure transducers (“Mikro-Tip” SPC-350,
Millar Instruments, Houston, TX) through small ports on the sides of the phantom to
73
capture pressure waveforms at various locations within the phantom (Figure 5.3), and
also immediately downstream of the outlet impedance module. The signals from each
catheter pressure transducer were sent into a pressure control unit (TCB-600, Millar
Instruments, TX) which generates an electrical output of 0.5V per 100mmHg of pressure.
An MR-compatible ultrasonic transit-time flow sensor was used to monitor the total input
flow to the phantom. We placed the externally clamped flow probe (8PXL, Transonic
Systems, NY) around a short section of Tygon tubing R3603 immediately upstream of
the one-meter flow conditioning rigid tubing, and sent the signals from the probe into a
flowmeter (TS410, Transonic Systems, NY) with its low pass filter setting at 160Hz. The
data from the flow meter and the pressure control units was recorded at a sample rate of
96 samples per second using a data acquisition unit (USB-6259, National Instruments,
Austin, TX) and a LabVIEW program (LabVIEW v.8, National Instruments, Austin,
TX). For each data acquisition, we averaged approximately 50 cycles of flow and
pressure data to obtain one representative cycle of flow and pressure waveforms. The
flow and pressure waveforms were stable in between the cycles of each acquisition. We
used the pressures measured downstream of the outlet impedance module as the ground
reference, and subtracted it from all of the other pressure measurements to obtain the true
pressure waveforms relative to the ground pressure.
We acquired through-plane flow velocity data at different locations within the
phantoms (Figure 5.3) using a cardiac-gated 2D cine PCMRI sequence in a 1.5T GE MR
scanner (Signa, GE Medical Systems, Waukesha, WI) and an 8-channel cardiac coil.
The imaging parameters were: 256x192 acquisition matrix reconstructed to 256x256,
18x18 cm2 field of view, 5mm slice thickness, TR=11~14 ms, TE=5~7 ms, 20 degree flip
angle, and NEX=2. A velocity encoding gradient of 50 cm/s was used for all
measurements, except for the L2, L3, and L4 measurements of the stenotic phantom,
where the velocity encoding gradient was 100, 200, and 100 cm/s, respectively. The
LabVIEW program which controlled the pulsatile pump produced a trigger signal that
was converted by an electrocardiogram (ECG) simulator (Shelley Medical Imaging
Technologies, London, Ontario, Canada) into an ECG signal used by the MRI system for
gating, and 24 time points per cardiac cycle synchronized to this ECG signal were
74
reconstructed. The temporal resolution of the velocity data was double the TR (~26ms).
We placed vitamin E capsules (Schiff Nutrition Group, Inc, Salt Lake City, UT) as well
as saline bags around the flow region of each acquisition location to produce the
reference signals of stationary fluids, which were then used for baseline eddy current
correction with a linear correction algorithm in the analysis of the PCMRI data.
Figure 5.3 Pressure and Flow Velocity Measurement Locations in a) the Straight
Phantom, and b) the Stenotic Phantom. Green section is deformable. Grey section is
rigid. Dimensions are in centimeters
5.3.4 In-silico Simulation
We performed the numerical simulation of blood flow and pressure using a
custom stabilized finite-element method to solve the incompressible Navier-Stokes
equations.95 The deformability of the wall is incorporated by a coupled momentum
method developed by Figueroa et al., which adopts a linearized kinematics formulation
for the solid domain, and allows for a fixed fluid mesh and nonzero fluid velocities at the
fluid-solid interface16. The end result is that the effects of wall motion are embedded into
the fluid equations simply as additional terms defined on the fluid-solid interface, leading
to minimal increases in implementation complexity and computational efforts compared
to rigid wall formulations.
75
Due to the fixed fluid mesh and linearized wall mechanics implementation of the
coupled momentum method, we must use a mesh that most closely resembles the average
geometry of the phantom during its operation. Calculations of the solid domain behavior
require the definition of the elastic modulus of the vessel wall material. The data from
the static tests of the compliant phantoms shown in Figure 5.1 can be used to find the
radii of the compliant tubes at their respective average operating pressures, and the elastic
modulus of the silicone material.
An analytical equation describing the expansion of a pressurized circular,
cylindrical vessel made of an isotropic material and under small strain is:96
2
2 21.5 ( )
i o
o i
PR RER R RΔ
=− + Δ
(5.1)
where E is the elastic modulus of the material, Ri is the reference inner radius of the
vessel, Ro is the reference outer radius of the vessel, ∆P is the change in pressure, and ∆R
is the resulting change in radius.
Equation 5.1 describes a linear relationship between ∆R and ∆P. Ro is related to
Ri by the vessel wall thickness, which we assume remains unchanged over small strains.
Using the experimentally measured average operating pressure in the phantom, and
prescribing the values of E and Ri, a plot of theoretical diameter versus pressure graph
similar to Figure 5.1 can be generated. For each phantom, we modify the values of E and
Ri until the theoretical plot coincides with the linear best fit of the static pressurization
test results shown in Figure 5.1, and determine the operating radius of the phantom, as
well as the elastic modulus of the silicone material. The operating radius was determined
to be 1.13 cm at the operating pressure of 96 mmHg for the straight phantom, and 1.11
cm at the operating pressure of 89 mmHg for the stenotic phantom. The static elastic
modulus of both phantoms was 9.1x105 Pa. Using the result of the dynamic mechanical
analysis previously discussed in the phantom characterization section, the effective
elastic modulus of the silicone material at the fundamental operating frequency of the
experiments (1 Hz) was then 1.0x106 Pa.
At the outlet boundary, we prescribed a Windkessel model (Figure 5.4) with the
lumped-parameter component values determined from the experimentally measured flow
76
and pressure at the phantom outlet. Mass conservation dictates that the average flow
throughout different locations in the phantom is constant. For the analysis of Windkessel
component values, we offset the outlet PCMRI flow measurement such that the average
flow is equal to that measured by the flow probe, in order to exclude any effects of
background correction variations. We used the measured average flow and pressure
values at the phantom outlet to determine the total Windkessel resistance (Rp+Rd). The
experimentally determined total Windkessel resistance in the straight and stenotic
phantom experiment was 1.6% lower, and 7% higher, respectively, compared to
theoretical. We used this ratio to scale the theoretical Rp and Rd to obtain the
experimental values of the resistances. We then performed an analysis of the Windkessel
model impedance:
( ) 1
dp
d
RZ j L Rj CR
ω ωω
= + ++
(5.2)
to determine the value of the capacitance that will result in an impedance best reflecting
the measured pressure and flow relationship at the phantom outlet. We prescribed these
experimentally determined Windkessel component values (as listed in Table 5.1) in the
numerical simulations. For the stenotic phantom simulation, due to the presence of
turbulence in the domain, an augmented lagrangian method was used at the outlet to
constrain the shapes of velocity profiles to prevent divergence.97 This technique has been
shown to have very little effect to the flow and pressure calculations in the numerical
domain.97
We constructed computational 3D solid models from the physical construction
details and the operating radii of each phantom. Each 3D solid model was discretized
into an isotropic finite-element mesh with a maximum edge size of 0.1 cm using
commercial mesh generation software (MeshSim, Simmetrix, Inc., NY). For the stenotic
phantom mesh, we further refined a region of length 8cm distal to the stenosis using a
maximum edge size of 0.03 cm, followed by another region of 4 cm downstream
discretized using a maximum edge size of 0.06 cm. The straight phantom and stenotic
phantom mesh contain approximately 1.5 million, and 3.5 million linear tetrahedral
elements, respectively. Sections of the vessel wall boundary in the meshes were set to be
77
rigid or deformable according to the physical construction of each phantom (Figure 5.4).
We set the initial values of pressure and vessel wall distention in the mesh based on the
average pressure in the physical experiment and the vessel wall properties. We then
mapped the flow waveform measured by the flow probe to the inlet face of the
computational domain using a Womersley velocity profile (Figure 5.4). In the
simulations we used a time step size of 0.42 milliseconds which resulted in 2400 time
steps per cardiac cycle. For the straight phantom simulation, we simulated 5 cardiac
cycles and used the data from the last cycle where the pressures had stabilized in the final
analysis. For the stenotic phantom, due to the presence of cycle-to-cycle variations in the
velocity pattern, we simulated 14 cardiac cycles and used the ensemble-averaged data
from the last 10 cycles in the final analysis. Since the PCMRI technique combines
measurements acquired over multiple cycles into one cycle of velocity data, we
ensemble-averaged the stenotic phantom simulation results to mimic the PCMRI data
acquisition method.37
Figure 5.4 Summary of Boundary Condition Prescriptions for the Numerical
Simulations
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5.4 Results
5.4.1 Flow & Pressure Waveforms in the Straight Phantom
Figure 5.5 compares the in-silico and in-vitro flow and pressure waveforms and
normalized pulse amplitudes at various locations in the straight phantom. There is good
agreement between the simulated and measured waveform shapes and amplitudes for
both flow and pressure, throughout the various locations in the phantom. The average
difference between the measured and simulated flows is 3.7 cc/s, which is 12% of the
average flow (31 cc/s). The average difference between the measured and simulated
pressures is 2.0 mmHg, which is 2.4% of the average pressure (85 mmHg). In both the
simulation results and experimental measurements, we clearly observe progressive flow
waveform damping, as well as progressive pressure waveform pulse amplitude increase,
down the 25cm length of the deformable vessel. A significant pressure waveform shape
change occurs between the inlet and L1, which is the transition from the rigid section into
the deformable section. The most prominent pressure pulse amplitude increase also
occurs between the inlet and L1. The experimental measurements show that, of the
approximate 10% increase in the normalized pressure pulse amplitude from the inlet to
the outlet, roughly 7% occurred at the inlet and L1 transition. Lastly, both simulation and
measurement showed an approximately 50% decrease in the normalized flow pulse
amplitude between the inlet and the outlet of the phantom.
79
Figure 5.5 Straight Phantom Simulated Versus Measured Flow & Pressure a)
Waveforms and b) Normalized Pulse Amplitudes, at Different Locations
80
5.4.2 Flow & Pressure Waveforms in the Stenotic Phantom
Figure 5.6 shows the comparisons of flow and pressure waveforms and
normalized pulse amplitudes at various locations in the stenotic phantom. The average
difference between the measured and simulated flow waveforms is 3.4 cc/s, which is 13%
of the average flow (26 cc/s). The average difference between the measured and
simulated pressure waveforms is 1.0 mmHg, which is 1.2% of the average pressure (82
mmHg). We observe similar trends in pressure and flow behavior down the length of the
stenotic phantom as those seen in the straight phantom. Between L2 and L3 of the
stenotic phantom (across the stenosis), there is no visible difference in the flow
waveforms, but there is a significant pressure drop. The drop in the peak pressure across
the stenosis is 3.8 mmHg in the simulation, and 4.8 mmHg in the experimental
measurement. The decrease in the normalized pressure pulse amplitude across the
stenosis is 8.1% and 11.6% in simulation and measurement, respectively. A significant
pressure waveform shape change occurred across the stenosis and is reflected in both
measurement and simulation. As in the straight phantom case, the most prominent
pressure pulse amplitude increase also occurs between the inlet and L1, which is the
transition from the rigid section into the deformable section. The experimental
measurements show that, of the 7% increase in pressure pulse amplitude between inlet
and L2, about 6% occurred between the inlet and L1 transition. Both simulation and
measurement show only an approximately 40% (compared to 50% in the straight
phantom) decrease in the flow pulse amplitude between the inlet and the outlet of the
stenotic phantom.
81
Figure 5.6 Stenotic Phantom Simulated Versus Measured Flow & Pressure a)
Waveforms and b) Normalized Pulse Amplitudes, at Different Locations
5.4.3 Impedance Modulus & Phase at the Inlet / Outlet
We compare the impedance modulus and phase at the phantom inlet and outlet
between simulation and measurement in Figure 5.7. In both phantom experiments, there
is agreement between the simulated and measured impedance across physiologically
82
relevant frequencies. For the impedance phase, the experimental measurements exhibited
more fluctuations at the higher frequency range compared to the simulated results. For
frequencies in the 1~3 Hz range, both the simulation and measurement show that the
impedance modulus increases from the inlet to the outlet. For both phantoms, the general
shapes and magnitudes of the impedance modulus and phase compare favorably with
those measured in-vivo in previous studies.61,64,69
Figure 5.7 Simulated Versus Measured Impedance Modulus and Phase at the Inlet and
Outlet for the a) Straight, and b) Stenotic Phantom Experiment
83
5.4.4 Through-Plane Velocity Patterns
We compare the simulated and measured through-plane velocity patterns at four
different time points in the cardiac cycle: diastole, acceleration, peak systole, and
deceleration. Figure 5.8 shows results for the L2 location in the straight phantom
experiment. There is excellent qualitative agreement between the simulated and
measured velocity pattern at all four time points. At acceleration and systole, there is
forward flow of similar magnitudes and shapes across the slice, and a visible layer of
decreased flow velocities near the vessel wall in both simulation and experiment. At
diastole and deceleration, both simulation and experiment showed forward flow near the
center, and a prominent region of backflow at the perimeter of the vessel. We generally
observed a sustained Womersley velocity profile throughout the different locations in the
straight phantom. Figure 5.9 shows flow velocity comparison results for the L2 and L3
locations in the stenotic phantom. In the L2 location, the pre-stenosis location in the
stenotic phantom, we found nearly identical results as those presented for the straight
phantom. In L3, the post-stenosis location, both simulation and measurement show a
circularly-shaped jet of high forward velocities near the vessel center, and backward
velocities (recirculation) around the vessel perimeter at the systole and deceleration time
points. While the high velocity jet in the simulation contains slight irregularities in its
shape, its size is comparable to that in the measured results.
84
Figure 5.8 Through-Plane Velocity Pattern Comparisons at the L2 Location for the
Straight Phantom Experiment at Four Different Time Points: Diastole, Acceleration,
Systole, and Deceleration
85
Figure 5.9 Through-Plane Velocity Pattern Comparisons at the a) L2, and b) L3
Location for the Stenotic Phantom Experiment at Four Different Time Points: Diastole,
Acceleration, Systole, and Deceleration
5.5 Discussion
Figures 5.5 and 5.6 show that physiological flows and pressures94,98 were
achieved in the experiments. The damping of the flow waveform down the length of the
phantom is the result of the flow being temporarily stored in the deformable tube, which
essentially acts as a capacitance in the system. Since there is a rigid section at the center
of the stenotic phantom, the total amount of deformable section is smaller compared to
86
that in the straight phantom. With less deformable volume to absorb the flow pulse, the
inlet waveform is better preserved and we indeed observed smaller flow waveform
damping between the inlet and outlet in the stenotic phantom. The accurate numerical
prediction of flow waveform shapes and magnitudes at different locations down the
length of the phantoms indicates that the calculated fluid velocities at the vessel wall
boundary accurately correspond to the physical vessel wall movement, faithfully
capturing the compliant behavior of the vessel. The prediction of the decrease in the flow
waveform peaks down the length of the phantom requires accurate calculations of the
vessel wall expansion in response to the increase in pressure, where additional fluid is
stored in the vessel during the systolic period. On the other hand, the prediction of the
gradual increase in flow waveform minimums requires accurate calculations of the elastic
behavior of the vessel wall, which releases the stored fluid during the diastolic period.
The flow waveform damping behavior observed in the simulations and experiments is
generally consistent with the actual blood flow behavior in-vivo, where the pulsatility
resulting from the pumping heart is damped out throughout the vasculature, and
eventually transformed into steady flow in the capillaries.
Since a significant portion of the cardiac cycle was diastole where the flow was
low or retrograde, the average flow rates were relatively low compared to systolic flow
rates. The average difference of approximately 3.6 cc/s between the measured and
simulated flow was 12~13% relative to the average flow rates, but was only under 5%
relative to the systolic flow rates, which were between 80~140 cc/s.
The progressive increase of pressure pulse amplitude down the length of the
phantom is also consistent with the in-vivo observation where the pulse pressure
progressively increases from the brachial artery downstream towards the radial artery.99
It has been generally believed that such phenomenon is attributed to the increased
stiffness of the downstream arteries. Our experimental and simulation results suggest
that wave propagation and reflection alone could contribute to a pressure pulse increase
under the condition of constant vessel stiffness.
Across the rigid and deformable junction where there is a mismatch in
characteristic impedances, the change in the pressure waveform shape and the prominent
87
increase in the pressure pulse amplitude could be attributed to wave reflections. Across a
stenosis, pressure waveform changes also occur due to energy losses in the post-stenosis
turbulent flow region. The numerical simulation accurately captured both the wave
reflections between the rigid and deformable sections, and the energy loss across a
stenosis, accurately predicting the changes in the pressure waveform at different locations
within the straight and stenotic phantoms.
The impedance modulus increase between the phantom inlet and outlet during the
1-3Hz frequency range reflects the capacitive effect of the deformable tube. Pulsatile
flow enters the inlet with relative ease due to the compliance in the deformable vessel
downstream, resulting in a lower impedance modulus. At the outlet of the phantom, there
is no deformable region downstream to manifest the effect of the lowered resistance to
pulsatile flow. This phenomenon is only prominent in the lower frequency range partly
because of the physical characteristics of the deformable tube dictating its response to
dynamic strain, and partly because of the low modulus values in the higher frequency
region making any differences difficult to observe. The increased prominence of outlet
impedance phase oscillations at the higher frequency region in the experimental
measurements could be due to the small high frequency component in the flow and
pressure waveforms, making the noise in the measurements relatively high.
The favorable comparison in velocity patterns between simulation and
measurement for the straight phantom is consistent with our expectation due to the trivial
geometry of the phantom. We also expect the pre-stenosis location in the stenotic
phantom to show similar results since complex flow originates from the stenosis and
propagates to the regions downstream. At the L3 location in the stenotic phantom, which
is immediately downstream of the stenosis where complex and recirculating flow occurs,
the simulation showed a smooth contour for the flow pattern right up to the time frame
immediately prior to peak systole. After which point the flow begins to decelerate and
diverge, resulting in slight irregularities in the shapes of the flow patterns in the
simulation results. We found that the irregularities in the flow pattern shapes were
correlated with mesh resolution, where a simulation computed on a finer mesh resulted in
fewer irregularities. In regions containing complex and diverging flow, it is thus
88
important to define a desired balance between flow pattern prediction accuracy and
computational cost.
The vessel wall motion in the numerical simulation is sensitive to the prescribed
thickness and elastic modulus of the vessel wall. Both the wall motion and the prescribed
vessel geometry affect the volumetric flow and the pressure changes down the length of
the vessel. Prescription of higher elastic modulus or smaller vessel diameter would result
in diminished wall motion and smaller flow waveform damping, and vice versa. The
method we developed to determine the relevant geometry and vessel wall properties
requires direct manipulations and observations of the vessel which are only possible in-
vitro. To apply the numerical simulation in an in-vivo setting, additional methods would
need to be developed to determine the equivalent values of vessel parameters for the
numerical model.
In conclusion, in this study we have produced a set of in-vitro, high-quality
experimental data that can be used to compare against CFD results of flow and pressure
within a compliant vessel under physiological conditions. The deformable CFD
simulation utilizing the coupled momentum method and a fixed fluid mesh was capable
of capturing realistic vascular flow and pressure behaviors. There were good predictions
of flow and pressure waveforms down the length of a straight and a stenotic deformable
phantom, indicating that the numerical simulation captured both the vessel wall motions
and wave reflections accurately. Due to the good comparisons in pressure and flow, the
impedance comparisons were also favorable. The simulated and measured flow and
pressure results were similar to those previously measured in-vivo. The numerical
simulation was able to track velocity patterns very well in regions with simple flow. In
regions containing more complex and diverging flow, a finer mesh resolution was
required for the simulation to capture the velocity patterns faithfully. The results
presented in this paper show promising potential for the numerical technique to make
accurate predictions of vascular tissue motion, and blood flow and pressure under the
influence of blood vessel compliance. This study provides the cornerstone for further
deformable validation studies involving more complex geometries, and in-vivo validation
studies that could ultimately support the use of CFD into clinical medicine.
89
Chapter 6: Conclusion
6.1 Summary & Contributions
In the research presented in this thesis, we developed methods for conducting the
necessary in-vitro experiments to acquire experimental data which we then used to
validate CFD simulations. We first developed methods to construct and characterize a
physical Windkessel module which can provide realistic vascular impedances to enable
in-vitro experimental conditions mimicking in-vivo conditions. The resistance units we
built as part of the Windkessel module generally exhibited flow resistance values that are
stable within 5% across the relevant flow ranges of the experiments. The Windkessel
unit as a whole behaved very consistently across different flow conditions. Upon proper
characterization of the physical Windkessel module, its corresponding analytical model
we defined can accurately predict the pressure and flow relationships produced by the
module. The physical Windkessel module can be connected to the outlet of a physical
phantom in flow experiments to produce physiologic pressures within the phantom.
Computationally, the analytical model of the Windkessel can be directly prescribed at the
outlet of the numerical domain, such that the boundary condition prescription in-silico is
representative of the physical reality. This was the first time that such direct parallels
between the computational and physical domain have been possible under physiological
pressure and flow conditions. The creation of the Windkessel module for reproducing
realistic vascular impedances opens up new realms for highly sophisticated in-vitro
experimental setups and computational validation studies.
We utilized the Windkessel module in an in-vitro experimental CFD validation
study involving a rigid, two outlet, and patient specific AAA phantom. We built a highly
controlled and characterized experimental setup to operate the AAA phantom and its
Windkessel outlet modules under two physiological conditions, and acquired high-
fidelity experimental pressure and PCMRI flow velocity data within the phantom.
Physiological pressure within the phantom and flow through the renal and aortic outlets
90
of the phantom were observed, with the measured pressure and flow split being within
5% of numerical predictions, demonstrating the impressive performance of the
Windkessel impedance module in a CFD validation application. Together with the
precisely constructed AAA phantom, accurate prescriptions of the physical reality in the
numerical domain enabled us to demonstrate excellent numerical predictions of flow
velocity patterns within the complex AAA geometry of the phantom.
We then proceeded to conduct an in-vitro validation study involving compliant
physical phantoms, and CFD simulations incorporating vessel wall deformability. Using
a Windkessel impedance module at the phantom outlet, we were able to achieve
physiological pressures in the compliant phantoms, which was essential for achieving
proper wall deformations that would result in realistic wave propagation. This was the
first time a deformable in-vitro CFD validation study has been reported under
physiological pressure and flow. We acquired high-fidelity pressure and PCMRI flow
velocity data at various locations down the length of two deformable phantoms, one
mimicking a patent healthy thoracic aorta, and one mimicking a stenotic thoracic aorta.
The experimental data enabled us to demonstrate the capability of the numerical
simulations incorporating wall deformability to predict flow and pressure waveform
changes down the length of the compliant phantoms, implying also the accurate
predictions of the dynamic motions of the vessel wall.
Over the course of this research, we have not only developed a powerful tool to
enable realistic and precise boundary condition prescriptions in the physical and
computational domain, we have also utilized this tool to produce two sets of
comprehensive, high quality experimental data that could be used by researchers in the
field as standards to validate numerical methods involving rigid or deformable
geometries. Using the same data, we performed CFD validation studies to show the
promising potentials of the numerical methods to accurately predict pressure, flow, and
tissue motion in the cardiovascular system, drawing a big step towards applying image-
based CFD methods into clinical use.
91
6.2 Future Work
6.2.1 In-vitro Validation
Upon having performed studies in a complex rigid phantom, and in relatively
simple deformable phantoms, the next step in in-vitro validation studies is to employ
deformable phantoms of more complex geometries. The first challenge that must be
overcome in such an effort is the construction of a uniform, non-symmetrical compliant
vessel. The current construction method for the compliant vessel relies on spinning to
obtain a uniform wall thickness across the vessel, meaning that the technique is only
applicable to radially symmetrical geometries. An approach that enables the construction
of uniform and non-symmetric compliant phantoms involves utilizing a dissolvable inner
mold, and a hollow outer mold, and filling the gap in between them with the deformable
silicone material. However, practical challenges that must be overcome include the
precise alignment of the inner and outer molds, and the proper filling of the silicone into
the gap, especially when small wall thicknesses are required for obtaining realistic strains
under physiological pressures. Thus far the dipping technique we have used for the
compliant phantom construction remains the most feasible and practical approach for
producing phantoms with the proper behaviors for in-vitro experiments under
physiological conditions. It is possible to conduct validation studies using non-radially-
symmetrical phantoms constructed with the dipping technique, which will have non-
uniform wall thicknesses, but methods must be developed to precisely characterize the
varying wall thicknesses across the phantom, and to prescribe the same varying
thicknesses in the numerical simulation.
Another incremental step in increasing the realism of in-vitro validation studies is
the incorporation of additional outlets in the phantom. We have shown that in a two
outlet phantom, very realistic, and drastically different flow waveforms can be directed
through each outlet by prescribing appropriate outlet impedances with the Windkessel
modules. However, large arteries in-vivo often have many more branches drawing
different flow waveforms, thus the incorporation of more of those branches in an in-vitro
validation study can further investigate the robustness of the CFD method in an
increasingly more realistic and complex scenario.
92
6.2.2 In-vivo Validation
One of the final steps before being able to apply CFD methods clinically is to
perform validation against in-vivo experimental data. A prominent challenge of in-vivo
validation is the prescription of outlet boundary conditions in the numerical domain.
Current patient-based CFD attempts have utilized arbitrary fitting approaches for
prescribing outlet boundary conditions in order to obtain the expected physiological
pressure and flow.68,100 To perform a rigorous validation study, methods should be
developed to directly relate physiological parameters to an equivalent Windkessel model
or impedance.
For in-vivo validation of CFD methods incorporating deformable walls, a
significant challenge is the prescription of vessel wall properties for the solid domain
calculations. Methods must be developed to characterize the material properties of blood
vessels in-vivo using non-invasive or minimally-invasive techniques. In addition, the
numerical method typically incorporates a simplified solid mechanics model for the
vessel wall, assuming homogeneous material properties and linearly elastic behaviors. It
is thus also important to develop methods to determine the equivalent values of vessel
wall thickness and elastic modulus appropriate for the simplified model in the
computational domain, drawing direct parallels between in-vivo measurements and in-
silico parameter prescriptions. The surrounding tissues around the blood vessel have
restrictive or damping effects on the vessel wall motion, which also must be considered
in the numerical simulation. Methods yet to be developed are needed to meaningfully
represent the effects of external tissues through either modifying the vessel wall
properties to encapsulate these effects, or adding appropriate external forces in the solid
domain calculations. These methods must also include meaningful ways of drawing
parallels between the measured in-vivo parameters and the corresponding in-silico
representations.
In order to apply CFD methods into clinical use, the same challenges as
mentioned here for in-vivo validation exist. Progress towards in-vivo CFD validations is
indeed also direct advancements towards clinical applications of CFD.
93
6.2.3 Direct Medical Device Evaluation
The experimental setup we have developed over the course of this research can be
readily used in the direct experimental testing of cardiovascular medical devices. Stents
and stent grafts can be deployed in deformable phantoms and studied under physiological
flow and vessel wall deformation. Device migration and fatigue behaviors can then be
directly observed and quantified. The durability and accuracies of implantable sensors
can be rigorously evaluated in-vitro under physiological pressure, flow, and tissue motion
conditions. There is much potential for establishing in-vitro testing protocols for various
cardiovascular medical devices to evaluate both their safety and efficacies using the
experimental techniques presented in this thesis.
In conclusion, the developments and findings resulting from this thesis research
motivate exciting future investigations that could contribute further to the advancement
of the cardiovascular biomedical engineering field.
94
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