<%BANNER%>

Isotropic versus Anisotropic Intracellular Electrical Properties of a Diffusion Tensor Magnetic Resonance Imaging (DT-MR...

Permanent Link: http://ufdc.ufl.edu/UFE0022339/00001

Material Information

Title: Isotropic versus Anisotropic Intracellular Electrical Properties of a Diffusion Tensor Magnetic Resonance Imaging (DT-MRI) Based Cardiac Model
Physical Description: 1 online resource (60 p.)
Language: english
Creator: Saaibi, Ana
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Myocardial fiber architecture largely determines the current pathways and current wavefront propagation in the heart. Disruption of this organization may give rise to severe chronic cardiac conditions related to electrical imbalances. In this study, underlying fiber direction information was obtained from DT-MRI data to predict intracellular electrical conductivity in cardiac tissue, and a finite element model of the heart was created to predict current pathways. Isotropic and anisotropic tissue properties were assigned to the heart tissue to compare the electrical behavior under different stimuli: (a) potential difference and (b) current point source stimulus. Significant differences in predicted current paths can be seen between isotropic and anisotropic cardiac models in response to both of the inputs. This DT-MRI modeling approach accounts for more realistic tissue properties that can more accurately predict the implications of myocardial infarction, which will be the focus of future studies. In addition, a preliminary time-dependent model was included to examine the implications of such simulations with a more realistic behavior.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ana Saaibi.
Thesis: Thesis (M.S.)--University of Florida, 2008.
Local: Adviser: Sarntinoranont, Malisa.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022339:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022339/00001

Material Information

Title: Isotropic versus Anisotropic Intracellular Electrical Properties of a Diffusion Tensor Magnetic Resonance Imaging (DT-MRI) Based Cardiac Model
Physical Description: 1 online resource (60 p.)
Language: english
Creator: Saaibi, Ana
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Myocardial fiber architecture largely determines the current pathways and current wavefront propagation in the heart. Disruption of this organization may give rise to severe chronic cardiac conditions related to electrical imbalances. In this study, underlying fiber direction information was obtained from DT-MRI data to predict intracellular electrical conductivity in cardiac tissue, and a finite element model of the heart was created to predict current pathways. Isotropic and anisotropic tissue properties were assigned to the heart tissue to compare the electrical behavior under different stimuli: (a) potential difference and (b) current point source stimulus. Significant differences in predicted current paths can be seen between isotropic and anisotropic cardiac models in response to both of the inputs. This DT-MRI modeling approach accounts for more realistic tissue properties that can more accurately predict the implications of myocardial infarction, which will be the focus of future studies. In addition, a preliminary time-dependent model was included to examine the implications of such simulations with a more realistic behavior.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ana Saaibi.
Thesis: Thesis (M.S.)--University of Florida, 2008.
Local: Adviser: Sarntinoranont, Malisa.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022339:00001


This item has the following downloads:


Full Text





ISOTROPIC VERSUS ANISOTROPIC INTRACELLULAR ELECTRICAL PROPERTIES OF
A DIFFUSION TENSOR MAGNETIC RESONANCE IMAGING (DT-MRI)-BASED
CARDIAC MODEL




















By

ANA MARIA SAAIBI


A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA

2008

































2008 Ana Maria Saaibi
































To Jose, Beatry & Silvia









ACKNOWLEDGMENTS

I would like to thank my family for their unconditional support throughout my career;

especially my mom and my dad for letting me dream while keeping me grounded, and my sister

for keeping me sane. Great appreciation goes to my advisor, Malisa Sarntinoranont for her

friendship, guidance and all the wonderful opportunities I received, and also to my committee

members Roger Tran-Son-Tay and John Forder. I would also like to thank my lab colleagues:

Jung Hwang Kim, Sung Lee, Greg Pishko and Xiaoming Chen for their help, advice and respect.

This project was supported in part by an appointment to the Research Participation

Program at the Center for Devices and Radiological Health administered by the Oak Ridge

Institute for Science and Education through and interagency agreement between the U.S.

Department of Energy and the U.S. Food and Drug Administration.









TABLE OF CONTENTS



A C K N O W L E D G M E N T S ..............................................................................................................4

L IST O F T A B L E S ......................................................................................................... ........ .. 7

LIST OF FIGURES ............................................. .. .......... ............ ...............8

A B S T R A C T .......................................................................................................... ..................... 10

CHAPTER

1 INTRODUCTION .................................. .. ........... ............................. 11

2 BACKGROUND ....................................... .. .......... ............................. 14

H e a rt D ise a se ................................................. ......... .. ............................................... ......... 14
External Defibrillators and Implantable Cardioverter Defibrillators................................ 15
H eart A natom y and Electrophysiology ............................................................. ................ 18
H heart A n ato m y ............................................................................................................... 18
E lectrop hy siology ............................................................................................................ 19
Overview of DT-MRI ................................................................... 20
M easuring Tissue Electrical Conductivity......................................................... ................ 21

3 M E T H O D S ........................................................................................................... ....... .. 24

Animal Preparation and MR Imagining .........................................................24
T issue e S egm entation .......................................................... ................................................. 2 5
Assigning Properties for Electrical Conductivity ..............................................................27
Finite E lem ent M odel of the H eart ......................................... ........................ ................ 28
Tw o-dim ensional M odel ..................... ................................................................. 28
T hree-dim ensional M odel .. ..................................................................... ................ 29
D efibrillation M odels ............. ................ ............................................... 32
Prelim inary Ishem ic M odel ................. .......................................................... 33
T im e-D dependent M odel .......................................... ......................... ................ 34

4 R E SU L T S ............................................................................................................. ........ .. 3 6

Two-dimensional Results of Validation Studies ...............................................................36
T hree-dim ensional R results ................................................................................ ................ 37
Voltage Input and Current Input Simulations ............... ....................................37
S en sitiv ity A n aly sis ................................................... .............................................. 3 9
Ischem ic T issue R esults............... .. .................. .................. ................ ............ .... ...... ........... 43
Tim e-D dependent M odel R results ..................................................................... ................ 45

5 D ISC U S SIO N ........................................................................................................ ....... .. 46









Previous Studies ........................................................... ...... ..................... 46
Interpretation of R results and A applications ........................................................ ................ 47
Future Work .............................................................................. 49

6 CONCLUSION................... ...................... ... .......... .............. ............... 51

APPENDIX

A DATA PROCESSING AND TISSUE SEGMENTATION SUBROUTINE .........................52

B TISSUE ABOVE THRESHOLD SUBROUTINE ............... .............. ..................... 55

L IS T O F R E F E R E N C E S ...............................................................................................................57

B IO G R A PH IC A L SK E T C H .............. ...................................................................... 60







































6









LIST OF TABLES


Table page

4-1. Summary of the magnitude of current paths at between two points at different
locations w ith the sam e seed point....................................... ...................... ................ 38









LIST OF FIGURES


Figure page

2-1. External defibrillator (image obtained from DRE Medical equipment 2008)............... 16

2-2. Implantable cardioverter-defibrillator (image obtained from Medtronic Inc. 2008)........ 16

2-3. Implantable cardioverter defibrillator and electrode placement (image obtained from
U.S. Department of Health and Human Services and NIH) ........................................17

2-4. Frontal view of a human heart. (image obtained from Mediplus Medical
Encyclopedia, ADAM) ..................................... ........ ...................19

2-5. Sequence schematic of the electrical propagation in the heart. A) location of nodes,
B) electrical activity starting at the SA node, C) depolarization wave going across the
atria, D) depolarization wave moving to the apex of the heart and E) depolarization
wave spreading upward from the apex (imaged obtained from Anatomy and
P hysiology, M arieb 2000).. ........................................................................ ................ 20

3-1. Experimental setup of isolated heart with the aorta cannulated. Left: perfused heart
with STH, right: replaced with PFC emulsion. (Figure courtesy of Min-Sin Hwang
Ph.D student, Biomedical Engineering Department, McKnight Brain Institute,
U n iv ersity o f F lo rid a) .......................................................... .............................................. 2 4

3-2. Tissue segmentation correspondence, (A) FA map of transverse image of the heart
showing left and right ventricle walls. (B) Tissue segmentation (red=non-tissue,
b lu e= h heart tissue e) ............................................................................................................... 2 6

3-3. Transverse image of the heart along the xy-plane, 2D isotropic model for validation
studies. (red=isotropic heart tissue, blue=non-tissue surroundings)...............................28

3-4. Mesh containing 24,840 brick elements corresponding to embedded ventricles and
surrounding media .......................................... ........................... 29

3-5. Transverse slice images of the isotropic electrical heart model in FEM software,
from apex= 1 to base=12 of the heart (blue=heart tissue with isotropic conductivity,
w hite= surrounding non-tissue) ......................................... ......................... ................ 30

3-6. Transverse slice images of anisotropic electrical heart model in FEM software, from
apex= 1 to base= 12 of the heart. Color contours provide directional information
(red=parallel to xy-plane, blue=perpendicular). .................................................. 31

3-7. External defibrillation model with anisotropic electrical conductivity properties ..........32

3-8. Implantable cardioverter-defibrillation model with anisotropic electrical conductivity
p rop erties....................................................................................................... . ....... .. 3 3









3-9. Transverse slice images of anisotropic electrical heart model in FEM software with
infarct region (dark blue), from apex=l to base=5 of the heart. (red=parallel to xy-
plane, blue= perpendicular) .. ...................................................................... ................ 34

4-1. Transverse cuts of the heart obtained from slice 7 of the imaging sequence. A) Fiber
orientation mapping using FLTView software. B) Electrical conductivity map in
COMSOL Multiphysics software, red=parallel to the xy-plane, blue=perpendicular.......36

4-2. Current direction for a potential difference between the apex and base within a
transverse cut of the heart 12 mm from the base. Arrows correspond to the current
d irectio n .......................................................................................................... ........ .. 3 7

4-3. Comparison between current streamlines in isotropic and anisotropic models.
A average from the sam e seed point ...................................... ...................... ................ 38

4-4. Slice image of a transverse cut of the heart with current path lines. A) Isotropic
model, red=uniform electrical conductivity. B) Anisotropic model,
blue=perperdicular to the xy-plane, red=parallel to xy-plane.......................................39

4-5. External defibrillation at low range. Normalized number of nodes of heart tissue
above 5 volts at different voltage inputs with a surface potential difference .................40

4-6. External defibrillation at high range. Normalized number of nodes of heart tissue
above 5 volts at different voltage inputs with a surface potential difference .................41

4-7. Implantable cardioversion at low-range. Normalized number of nodes of heart tissue
above 5 volts at different voltage with a point source input.........................................41

4-8. Implantable cardioversion at high range. Normalized number of nodes of heart tissue
above 5 volts at different voltage with a point source input.........................................42

4-9. Normalized number of nodes of heart tissue above 5 volts at different voltage inputs,
four-case com prison .......................................................................... .. .... .. ............... 43

4-10. Comparison between healthy and infarcted heart tissue above a 5 V threshold for a .......44

4-11. Sensitivity analysis of ischemic heart tissue with different electrical conductivity
v alu e s. .............................................................................................................. ....... .. 4 4









Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

ISOTROPIC VERSUS ANISOTROPIC INTRACELLULAR ELECTRICAL PROPERTIES OF
A DIFFUSION TENSOR MAGNETIC RESONANCE IMAGING (DT-MRI)-BASED
CARDIAC MODEL


By

Ana Maria Saaibi

August 2008

Chair: Malisa Sarntinoranont
Major: Mechanical and Aerospace Engineering

Myocardial fiber architecture largely determines the current pathways and current

wavefront propagation in the heart. Disruption of this organization may give rise to severe

chronic cardiac conditions related to electrical imbalances. In this study, underlying fiber

direction information was obtained from DT-MRI data to predict intracellular electrical

conductivity in cardiac tissue, and a finite element model of the heart was created to predict

current pathways. Isotropic and anisotropic tissue properties were assigned to the heart tissue to

compare the electrical behavior under different stimuli: (a) potential difference and (b) current

point source stimulus. Significant differences in predicted current paths can be seen between

isotropic and anisotropic cardiac models in response to both of the inputs. This DT-MRI

modeling approach accounts for more realistic tissue properties that can more accurately predict

the implications of myocardial infarction, which will be the focus of future studies. In addition, a

preliminary time-dependent model was included to examine the implications of such simulations

with a more realistic behavior.









CHAPTER 1
INTRODUCTION

Cardiac function is influenced by the three-dimensional organization of the myocardial

fibers. Cardiac fibers are arranged in a circumferential, longitudinal, and a sheet-like fashion,

forming counter-wound helices from the base to the apex of the heart.1 This fiber organization is

responsible for the delicate balance between mechanical and electrical functioning of the heart.

When electrical disruption of coordinated function occurs, this is associated with cardiac

arrhythmias which may lead to more serious conditions like ventricular fibrillation.2'3 In case of a

heart attack (myocardial infarction), blood supply to a section of the heart is interrupted, and this

oxygen shortage (ischemia) causes damage and possibly death of heart tissue. Injured heart tissue

conducts electricity slower than healthy heart tissue 4,5 and this difference in conduction velocity

often triggers a re-entry or a feedback loop. These re-entry waves and feedback loops are thought

to be the cause of many lethal cardiac arrhythmias.2

Previous studies have been conducted to reconstruct fiber architecture in the heart 6, but

this anatomical reconstruction is labor intensive and time consuming. Such studies consist of

perfusing, excising, and precisely cutting segments of the ventricular wall from which the

architecture is then carefully measured. Other methods to incorporate fiber orientation in heart

models have also been proposed 7,8. The method developed by Lorange et al.7 consisted of

nesting revolving ellipsoids from the endocardium to the epicardium, from 0 to 1200. Ellipsoid

dimensions were taken from a computer tomography scan of a frozen human heart. Models

developed by Vigmond and Leon 8 also included the rotating fiber anisotropy. They modeled

muscle fibers as a discrete cable network in a rectangular area. Within a plane, the fibers where

all parallel and there was a fixed clockwise rotation of fiber orientation between planes. The use

of this simplified geometry did not result in realistic echocardiograms (ECG). These methods









provide a solid foundation for the study of the structure-function relationship of myocardial

anisotropy, and provide approximations of the actual anatomical structure.

Recent studies used diffusion tensor magnetic resonance imaging (DT-MRI) to determine

the orientation of cardiac muscle fibers. 9,10,11 This technique yields an average diffusion tensor

for water in the tissue over an image voxel where the eigenvalues and eigenvectors determine the

magnitude and principal directions of diffusion rates, respectively. The principal direction

corresponds to the fiber orientation parallel to the long axis of the muscle fiber.4

Electrical conductivity is also a tensor, and it is predicted that electrical conductivity in

heart tissues is greatest along the cardiac muscle fiber direction.12 Cardiac fibers create a sheet-

like structure along the fiber direction. The next preferential direction for electrical conduction is

transverse to the fibers in the direction parallel with to the sheets. The least electrical conduction

occurs in the direction normal to the sheets.13

In this study, the underlying fiber direction information from the water diffusivity tensor

as measured by DT-MRI was used to predict intracellular electrical conductivity in cardiac

tissue. Based on this information an electrical finite element model of the heart was created.

Initially a two-dimensional model was created for validation purposes. Then, a steady-state

analysis in three dimensions was done and intracellular current paths were predicted in the left

ventricle. The current pathlines and voltage distribution was compared between models using

anisotropic and isotropic conductivity properties.

Such simulations are useful for predicting current propagation and current density

distribution patterns, as well as voltage distribution and sensitivity to electrical impulse location.

This may useful for establishing defibrillation threshold values and as well as optimizing

electrode placement. Ultimately, such models may be used to understand the consequences of









myocardial infarction on the electrical functioning of the heart and defibrillation characteristics.

Preliminary studies on a time-dependent model were prepared and analyzed against the steady-

state cases.

Initially, a brief background is given on the topics necessary for a better understanding of

the issues addressed. The topics of the background overview are: heart diseases, devices to

overcome such diseased states, basic heart physiology, reconstruction of anatomy using DT-MRI

and techniques used to measure electrical conductivity. Then, a detailed step-by-step description

is presented on the creation of the computational heart model and the subsequent simulations.

Following this description, the results of the simulations are analyzed. The implications of the

results are provided along with an exploration of future studies. Lastly, conclusions are drawn

and the results obtained are summarized.









CHAPTER 2
BACKGROUND

Heart Disease

Heart attacks are the leading cause of death in developing countries. In the United States

alone, more than 10,000,000 are living with some form of heart disease. Every year,

approximately 1,200,000 people suffer a coronary attack every year, and about 40% of them die

as a result of this episode.14 This suggests that approximately every 65 seconds, a person in

America dies of a coronary event.

Heart attack or myocardial infarction, is a medical condition that occurs when an

insufficient blood supply reaches a certain region of the heart. This insufficient blood supply

event is referred to as an ischemic episode. This insufficient blood supply or oxygen shortage

produces damage or death to the cardiac tissue. This damaged tissue area is referred to as an

ischemic area. When the ischemic area is small and does not compromise the electrical system of

the heart, the likelihood that the patient will survive is very high. If the ischemic area is large,

and a region of infarcted or dead tissue arises, then myocardial arrhythmias occur.

Cardiac arrhythmia makes reference to any cardiac condition that involves an abnormal

electrical activity in the heart. These range from non-dangerous arrhythmias to severe

arrhythmias such as ventricular fibrillation. Some examples of cardiac arrythmias are:

tachycardia, bradycardia and fibrillation. When the cardiac rhythm exceeds 100 beats per minute

when at rest, this is classified as tachycardia. Extreme tachycardia makes the heart ventricles

contract rapidly and therefore they do not completely fill with blood in every blood cycle and

often leads to death.

On the other hand, when the cardiac rhythm is under 60 beats per minute, the condition is

called bradycardia. In extreme bradycardia, the heart pumps the correct amount of blood, but so









sparsely that the quantity of oxygen that gets to the organs and tissues is not enough to properly

oxygenate them. Cardiac fibrillation is present when there is an uncoordinated contraction of

either the atria or the ventricles in the heart. Atrial fibrillation is more common than ventricular

fibrillation. It often tends to become a chronic condition and leads to a small increase in the risk

of death. On the other hand, ventricular fibrillation is severely dangerous and depending on the

duration of the episode it often leads to a sudden death. These conditions are not diseases per se,

but are often a reflection of underlying cardiac tissue damage.

Nevertheless, one out of three people that have a heart attack die before they can receive

any type of medical treatment. Due to the frequency of sudden deaths, the development and

improvement of 'resuscitation' techniques for treating cardiac arrhythmias is very important.

Devices like external defibrillators and implantable cardioverter defibrillators need to be

optimized as well as the drug therapies which follow these episodes.

External Defibrillators and Implantable Cardioverter Defibrillators

In general, defibrillators work by delivering an electrical impulse to the heart that

simultaneously affects the majority of myocardial tissue cells and induces their simultaneous

depolarization. After a successful defibrillation, the heart resets its electrical cycle reactivating

the normal mechanical contractions starting with the atria and later with the ventricles. The

success of the defibrillation depends on the patient's condition as well as the amount of

myocardium that reaches a certain voltage gradient threshold.

External defibrillation and electric cardioversion defibrillation are therapies that deliver an

electrical shock to normalize abnormal rhythmic beatings of the heart. They are effective when

used promptly after a cardiac episode. External defibrillation is used when a patient is

experiencing ventricular fibrillation or ventricular tachycardia without a pulse. These two

episodes are lethal if there is no intervention.


























Figure 2-1. External defibrillator (image obtained from DRE Medical equipment 2008)

Electrical cardioversion is employed in order to revert any type of arrythmia except for that

mentioned above. The electric impulse sent is synchronized with the heart's electrical activity

and it can be administered either urgently during an extreme situation or selectively using

implantable cardioverter defibrillators (ICD).


Figure 2-2. Implantable cardioverter-defibrillator (image obtained from Medtronic Inc. 2008)

External defibrillators deliver the impulse through externally placed paddles. Paddle sizes

have a wide range, but common dimensions are 6 to 12 cm diameter circular paddles. These

paddles are placed directly on the thorax on the patient's skin. There is a slight paddle-


Y









placement dependence using these defibrillators as reported by Karlon et al. 15; but an overall

positioning near the heart region obtains the desired results. These devices deliver a wide range

of voltages, some deliver voltages within the high-voltage range from 700 V to 2000 V 16, while

others deliver voltages within the low-voltage range from 100 V to 500 V.15

Internal cardioverter-defibrillators deliver the electrical impulse on the heart's

endocardium. Usually, they are used in patients with recurrent arrhythmias or a chronic electrical

disruption condition. Unlike external defibrillators, their implantation requires a surgical

intervention. Most have two electrodes that are placed in the right atrium towards the base of the

heart and in the left ventricle towards the apex of the heart. Since the electrical impulse is

delivered directly on to the heart tissue, the magnitude of the shock required is smaller than that

required when using external defibrillators.



Sf Electrodes inserted
N into vein leading
... .. .- ,reafl






Electrodes Implantable
Sin heart- deflbrillalor
I Inserted
Rihat under skin
Right atrium and ventricle



Figure 2-3. Implantable cardioverter defibrillator and electrode placement (image obtained from
U.S. Department of Health and Human Services and NIH)

The dimensions of the electrodes that deliver the shock in ICDs are much smaller than the

external defibrillator paddles. Their dimensions are small so that a transvenous implantation is

possible, therefore the electrode placement within the heart has to be more precise.









Heart Anatomy and Electrophysiology


Heart Anatomy

The heart is the principal organ in the circulatory system. It is a striated muscle that acts as

a pump to synchronously circulate blood and nutrients through the whole body. It is located

slightly offset to the left in the middle of the thorax and it is surrounded by the pericardium. The

pericardium is responsible for its insulation and lubrication, preventing it from wear during its

normal mechanical functions. The heart consists of three main layers: the endocardium (the

innermost layer), the myocardium and the epicardium (the outermost layer). The endocardium is

a connective tissue membrane that covers inside of the heart and has direct contact with the

blood. The myocardium is the layer consisting of cardiac muscle fibers that are responsible for

the pumping mechanism. These cardiac muscle fibers are arranged in a circumferential,

longitudinal, and a sheet-like fashion, forming counter-wound helices from the base to the apex

of the heart.1 Surrounding the heart is the epicardium, (a part of the pericardium) responsible for

producing the pericardial fluid used to lubricate the layers of the heart during its mechanical

motion.

The heart has four main cavities, right and left atrium, and right and left ventricle (Figure

2-4). The right atrium receives deoxygenated blood from the body, pumps it to the left ventricle

which then pumps it to the lungs. Due to the low resistance faced when circulating blood to the

lungs, the right side of the heart does not have to exert the same amount of force as does the left

side.

The left atrium receives oxygenated blood from the lungs and pushes it to the left ventricle

through the mitral valve. The left ventricle then sends oxygenated blood to the whole body

through the aortic artery.










Interior of heart


Superior
vena cava Aorta
Pulmonary
artery

Left
Right atrium


ventricle
Right
ventricle



Figure 2-4. Frontal view of a human heart. (image obtained from Mediplus Medical
Encyclopedia, ADAM)

Electrophysiology

Cardiac muscle is self-excitatory; it does not require an external stimulus to trigger its

contractile functioning. Rhythmic contractions occur spontaneously, and the pace of these

contractions is regulated by the sinoatrial (SA) node. It is located in the superior wall of the right

atrium (Figure 2-5). The SA node produces an action potential, which is caused by the electric

depolarization of its membrane due to a concentration imbalance of Na and K ions. The current

produced by the SA node propagates slowly throughout the atria. Then, it passes to the ventricles

through the atrioventricular (AV) node which is located at the junction of both ventricles.

Depolarization occurs more rapidly from the AV node across the ventricles and travels towards

the apex of the heart. From the AV node, the depolarization wave propagates to the Bundle of

His; then upward from the apex of the heart to the extremity branches of the Bundle of His.

These fibers distribute the current to the ventricles through their terminal branches called

Purkinje fibers.











0& T Ik I I





A B C D E


Figure 2-5. Sequence schematic of the electrical propagation in the heart. A) location of nodes,
B) electrical activity starting at the SA node, C) depolarization wave going across the
atria, D) depolarization wave moving to the apex of the heart and E) depolarization
wave spreading upward from the apex (imaged obtained from Anatomy and
Physiology, Marieb 2000)

Overview of DT-MRI

Cardiac function is influenced by the three-dimensional organization of the myocardial

fibers. This fiber organization is responsible for the delicate balance between mechanical and

electrical functioning of the heart. Disruptions in normal fiber architecture are present in cardiac

conditions such as cardiac ischemia and ventricular hyperthrophy 17. Alterations to this fiber

organization may induce abnormal electrical propagations which may lead to severe arrhythmic

disruptions. In addition, devices like implantable cardioverter and external defibrillators rely on

accurately delivering the correct amount of charge to a specific location and exciting the correct

amount of mass percentage of the heart. Such tasks are influenced by the underlying fiber

architecture of myocardial fibers. Therefore there is an overall need for quantifying fiber

arrangement.

Recent studies used diffusion tensor magnetic resonance imaging (DT-MRI) to determine

the orientation of cardiac muscle fibers.9' 10, 11 This technique yields an average diffusion tensor

for water in the tissue over an image voxel, where the eigenvalues and eigenvectors determine

the magnitude and principal directions of rate of diffusion, respectively.









In DT-MRI, a linearly varying pulsed magnetic field gradient is applied to the tissue. Two

pulses in the same direction but opposite magnitude are applied. A reduction in signal happens

due to the movement of the protons during this time interval, and it can be related to the amount

of water diffusion through the following equation


A = e-b*D (2-1)


where A is the signal attenuation, D is the diffusion coefficient and b is a factor that

characterizes the gradient's shape, amplitude and timing.18 In anisotropic diffusion, the

coefficient D is a symmetric rank-2 tensor. This tensor characterizes water molecule mobility

along three axes that correspond to the MR machine's axes. Therefore, in order to properly

obtain the diffusion tensor one needs to take into account the tissue's local coordinates.

Measuring Tissue Electrical Conductivity

When characterizing electrical properties of tissue, capacitive and resistive elements need

to be specified. These two parameters vary with frequency 19 but at the frequencies of present

interest in the current study the effects of frequency dependence were disregarded. A more

detailed explanation of this assumption will be given in the following paragraph.

Electrical conductivity, G, (equal to 1/r where r is electrical ressistivity), and

permittivity, e, are needed to describe the electrical properties of tissue. These properties are

commonly measured using the four-electrode technique.20 When using this technique, for an

alternating current of frequency,f the ratio of voltage, V, and current, I, is proportional to

specific impedance, Z. This is a complex quantity and can be written as


= Z (2-2)
I
where









1
Z = (2-3)
(G + j2;rfrE)

In this equation, er is the permittivity of space which is a constant, 8.854*10-12 F/m. The phase

difference between current and voltage can be obtained by 21


2;rf EE
= tan ( )
G (2-4)

except for low frequencies, G>>2neer. Therefore, the magnitude of the ratio between voltage and

current is approximately proportional to the tissue resistivity (1/electrical conductivity). Tissue

resistivity is more commonly called, specific impedance.

The heart contracts by passing ionic current inside the muscle, therefore activating

rhythmic contractions that circulate blood throughout the body. This ionic current is generated by

the transport of Na+ and K- ions through a semi-permeable membrane. Ions move through small

cellular membrane gates which can either be open during an excitation state or closed at a resting

state. This ionic movement generates an action potential. 22

Action potential propagation is a complex electrochemical phenomenon of ionic imbalance,

but it can be described using the FitzHugh-Nagumo equations.23 This model is a two-

dimensional simplification of the Hodkin-Huxley model which models a spike generation in a

neuronal axon. There are three characteristics that describe the behavior of excitable media, i.e.

neurons and muscle cells: rest cell membrane potential, threshold for opening and closing ionic

gates in the membrane, and the diffusive spreading of the electrical signals.

The following equations are a general form of this model


-l = Au + (a- uI )(uI -1)u u2
at (2-5)
% /u2 2 )
at









Here ul is an action potential that activates the media, u2 is the gate regulator variable that

inhibits the system, a gives the threshold for excitation, F is the excitability of the system, and

0,y, and 6 are the parameters that describe the resting state and dynamics of the system.

Previous studies have been done where cardiac electrical propagation is described using

these equations. 19 Filipini et al. found a strong correspondence between their model and with the

electrical behavior of cardiac cells studied in vitro. However, since mechanical contractions were

not introduced in this model, they aim to propose a model that includes them in future studies.

This mathematical model describes the depolarization process of the cellular membrane that

characterizes wave propagation in nervous and cardiac tissue. If the external stimulus exceeds a

certain threshold value, the system initiates a wave propagation across the excitable media. If no

such threshold is reached, no propagation occurs.









CHAPTER 3
METHODS

Animal Preparation and MR Imagining

Min Hwang, a Ph.D. student in the John Forder laboratory, was responsible for the animal

experiments and the MR imaging procedure which was conducted at the McKnight Brain

Institute at the University of Florida.

DT-MRI cardiac data was obtained from an exsanguinated white male rabbit. Rabbit

surgery was conducted in accordance with the NIH guidelines on the use of animals in research

and the regulations of the Animal Care and Use Committee of the University of Florida. An

isolated and later arrested heart was used for this experiment because it maintains structural

integrity of the vasculature. A New Zealand White male rabbit was be anesthetized using a

mixture of ketamine/xylazine (40mg/kg: 10mg/kg, i.m.) followed by heparin (1000 U/kg, i.v.)

and was later exsanguinated.











Figure 3-1. Experimental setup of isolated heart with the aorta cannulated. Left: perfused heart
with STH, right: replaced with PFC emulsion. (Figure courtesy of Min-Sin Hwang
Ph.D student, Biomedical Engineering Department, McKnight Brain Institute,
University of Florida)

The excised heart was placed in a bath of cold cardiopledgic solution (40C). The heart

was transferred to a Langendorff apparatus and perfused retrogradely. An initial perfusion period

of 10 minutes washed the red blood cells out of the vascular space, permitted the heart to contract

normally, and the aortic valve to remain intact. A thin (1 mm-OD) polyethylene tube was









inserted in the left ventricle (LV) serving as a vent to avoid excess hydrostatic pressure

accumulation and distension of the left ventricle from Thebesian flow. Due to the sensitivity of

diffusion weighted images to motion, the heart was arrested prior to imaging by switching

perfusate to a modified St. Thomas' Hospital cardioplegic solution (STH).

MR imaging experiments were performed on an 11.1 T/ 40 cm clear bore magnet

(Magnex Instrument Inc. UK, Bruker Instrument console) with a loop-gap coil (32 mm diameter,

40 mm height) dual tuned to 1H/19F resonances. The temperature in the magnet was 28 290C.

Proton diffusion weighted images of the arrested rabbit heart with the cardiopledgic solution

were acquired by applying the gradients to give diffusion sensitizing factors (b values) of 80,

160, 250, 350, 460, 580, 710, 850, and 1000, in 6 directions with a standard spin echo pulse

sequence. Imaging parameters were TR = 1.5 s, TE = 29 ms, one average for all scans using A =

16.5 ms, 6 = 5.5 ms. Thus, a total of 55 scans were obtained per slice of 2 mm thickness each

with in-plane resolution of 0.5 x 0.5 mm2 and data matrix of 80 x 80. MR images were

processed with standard processing functions (Fourier transformation) and diffusion-weighted

images were fit to a rank-2 tensor model of tissue diffusion.

Tissue Segmentation

Image segmentation can be defined as the division of a particular image into distinct

regions, each having different properties. In this project, image segmentation was implemented

on a voxel by voxel basis using a custom Matlab (Matlab v. 6.5.0, Mathworks) subroutine. The

imaged volume was segmented into heart tissue and non-tissue regions. Such segmentation

allowed us to correctly assign myocardial tissue properties and the desired properties for the

surrounding regions.

A large number of image segmentation techniques are available in literature;25 however

there is no particular method that can be applied to all images or accepted for all imaged









subjects. In general, cardiac tissue segmentation has been implemented based on characteristic

feature values, i.e. relative anisotropy or fractional anisotropy. These values are used in cardiac

tissue segmentation because they describe the level of microstructure organization of the tissue.

Since muscle tissue is highly organized, these parameters can accurately distinguish between

tissue and non-tissue.

Fractional anisotropy (FA) values provide a measure of the extent of tissue anisotropy .

These values were calculated from the DTI data using


FA [3- )2 +(_ )2+ P_)2
FA-
J 2-2 2g (3-1)


where 2 is the mean diffusivity (/3 tr(D) ) and4/, 22 and a are the principal eigenvalues of water

diffusivity. FA was used to distinguish between aligned cardiac tissue (FA=1) and isotropic air

(FA=0) surrounding the heart. Figure 3-2 shows the correspondence between an FA

visualization map and the implemented segmentation. For FA values greater than 0.2, the voxel

was characterized as non-tissue, for values less than 0.2 it was assigned electrical conductivity

properties of heart tissue.











A B

Figure 3-2. Tissue segmentation correspondence, (A) FA map of transverse image of the heart
showing left and right ventricle walls. (B) Tissue segmentation (red=non-tissue,
blue=heart tissue)









Assigning Properties for Electrical Conductivity

One should note that DTI measures the effective tensor of water diffusivity in tissue which

is sensitive to the underlying tissue structure and G measures intracellular conductivity

properties. A strong correlation is assumed between the eigenvectors of the water diffusion

tensor and the eigenvectors of the electrical conductivity tensor based on tissue microstructure in

order to assign electrical propagation directionality. This 'cross property' relationship has been

previously studied and assigned by Tuch et al.25

DTI data was processed to assign fiber orientation to the electrical conductivity tensor

along the longitudinal, transverse and normal directions of the tissue using a customized Matlab

subroutine. This subroutine scanned every point of the DTI data and calculated the eigenvalues

and eigenvectors at every location. After segmenting the tissue as mention above, the electrical

conductivity tensor components in the local coordinate system of the heart were assigned to each

node by sorting them in descending order and creating the matrix


g1 0 0
G= 0 g22 0 (3-2)
0 0 g33

where gii are the conductivity eigenvalues and were obtained from Eason et al.26 Electrical

conductivity was assigned values in the global coordinate system using


G '=PGP (3-3)


where P is the transformation matrix and the columns of P are equivalent to the eigenvectors of

the water diffusion tensor at each point. In this way, the principal directions of the water

diffusion tensor provided the fiber orientation and the direction of maximum intracellular

conductivity in the global coordinate system.









Finite Element Model of the Heart

The electrical conductivity tensor data was assigned on a voxel-by-voxel basis to the 3-

dimensional model and on a pixel-by-pixel basis to the 2-dimensional model within the

multiphysics software package, COMSOL (COMSOL Multiphysics v. 3.3, Stockholm, Sweden).

Two-dimensional Model

In this preliminary model, a rectangular area of 80 x 80 mm was selected and a

quadrilateral mesh was implemented in which each pixel was assigned to an element in the mesh

for a total of 6400 elements.

An isotropic model was first undertaken by selecting a particular transverse image of the

heart near the base. For this case, the electrical conductivity tensor matrix was reduced to a

single-non-zero scalar value of 0.6 S/m. A voltage difference of 0.1 V was applied between

opposing boundaries and electrical insulation to the remaining 2 edges of the model (Figure 3-3).




Electric
insulation












Electric
insulation


Figure 3-3. Transverse image of the heart along the xy-plane, 2D isotropic model for validation
studies. (red=isotropic heart tissue, blue=non-tissue surroundings).









This isotropic model was then compared to a similar model having anisotropic tissue

properties. In the anisotropic model, the spatially-varying electrical conductivity values from the

tensor transformation were used.

Three-dimensional Model

A rectangular volume corresponding to a truncated image array was created using 24,840

quadratic brick elements, with 212,877 degrees of freedom for the dependent variable, V, and

electrical properties calculated for each voxel. Each brick element corresponds to an image

voxel. To reduce computation time, the atria were disregarded and only the ventricles were

modeled when brick elements were used. When the total mass of the heart was required, the

modeling was done with ventricle and atria volumes.











Figure 3-4. Mesh containing 24,840 brick elements corresponding to embedded ventricles and
surrounding media.

The continuity equation for conductive DC media yields a general form of Ohm's law,

which for a static case states that


V- J =-V-(G'VV- JF)= 0 (3-4)


where Je is an externally generated current density, Jis the induced current density, Vis the

electric potential, and G'is the electrical conductivity. A current source term, Q, was included,

and the externally generated current density was eliminated. Therefore, the following generalized

equation was used









-V-(G'VV)= Q.


Heart tissue was modeled as having isotropic and anisotropic electrical conductivity

values according to fiber orientation. In literature, reported electrical conductivity values have a

wide range. For the isotropic case, an electrical conductivity of 0.28 S/m was used 15 throughout

the tissue. This value is an approximation obtained from conductivities measured along the fiber

direction, transverse to the fiber direction and blood vessel conductivity. For the anisotropic case,

electrical conductivities were taken to be 0.625 S/m, 0.236 S/m and 0.11 S/m along the

longitudinal, transverse and normal fiber directions, respectively. 13,26









I 2 3 4


Figure 3-5. Transverse slice images of the isotropic electrical heart model in FEM software, from
apex=1 to base=12 of the heart (blue=heart tissue with isotropic conductivity,
white=surrounding non-tissue)


(3-5)







































Figure 3-6. Transverse slice images of anisotropic electrical heart model in FEM software, from
apex=l to base=12 of the heart. Color contours provide directional information
(red=parallel to xy-plane, blue=perpendicular).

Boundary conditions for the voltage difference simulation assumed electrical insulation on

the lateral faces surrounding the heart, that is

n-J = 0 (3-6)

and a potential difference Vo between the base and the apex of the heart.

When a current point source was modeled, electric insulation was used on all the faces

surrounding the heart. A current point source was defined, namely Qo, and a ground (zero

potential) point was defined on the surface of the ventricles. This arrangement is important to

determine electrode placement in the heart when defibrillation systems need to be implanted.









Steady-state equations were solved and studies compared (a) an input potential difference

between the apex and the base of the heart, and (b) current point sources at different locations. In

addition, modeling studies were done (c) that resemble external heart defibrillation using

different voltage magnitudes, and (d) implantable cardioverter defibrillators by having a voltage

point source within the left ventricle wall with different voltage magnitudes.

Defibrillation Models

Two types of defibrillators were modeled, external and implantable cardioverter

defibrillators. For the external defibrillation simulation, a potential difference was applied

between opposing faces of the rectangular model; namely the faces corresponding to the anterior

and posterior planes of a human body. This paddle placement location is referred to as anterior-

posterior paddle placement (APR3). Neumann boundary conditions were assigned to the

remaining faces so that there was no current flowing out of the volume (Figure 3-7).



















Figure 3-7. External defibrillation model with anisotropic electrical conductivity properties.

Implantable cardioverter defibrillations were modeled as voltage point sources located (by

visual inspection) at the superior wall of the right atrium and on the lower wall of the right









ventricle. A potential difference between these points was applied and electrical insulation

boundary conditions were assigned to all the boundaries in the volume (Figure 3-8).








0







40


Figure 3-8. Implantable cardioverter-defibrillation model with anisotropic electrical conductivity
properties.

Preliminary Ishemic Model

Several studies have been conducted to characterize the remodeling of the myocardial

architecture that occurs after myocardial infarction.5'1 However, the electrical implications of

such remodeling are not clearly understood. Chen et al. suggested that infracted heart tissue

exhibits a 37% decrease in relative anisotropy. This value is not small enough to be considered

as totally isotropic as water, but it does not exhibit the same level of organization as a healthy

myocardium. Other studies suggest that according to the degree of myocardial ischemia, a mere

decrease of electrical conductivity results, therefore slowing down the impulse propagation. In

addition, these studies suggest a complete lack of propagation in the presence tissue necrosis.27

A preliminary model of an ischemic heart was created. To do so, a simplified ichemic

geometry was implemented by introducing a cone-shape volume within the posterior wall of the

left ventricle (Figure 3-9). The ischemic region represented approximately 8% of the total heart









tissue. A low electrical conductivity of 0.18 S/m was assigned to this region and the tissue was

assumed isotropic.13 Simulations where a potential difference input was applied were done in

order to compare to the healthy heart tissue model.





IN





Figure 3-9. Transverse slice images of anisotropic electrical heart model in FEM software with
infarct region (dark blue), from apex=l to base=5 of the heart. (red=parallel to xy-
plane, blue=perpendicular)

Time-Dependent Model

A time-dependent model was implemented using a modified version of the FitzHug-

Nagumo equations for excitable media, Equation 3-7.28 A fully anisotropic electrical

conductivity tensor G' was implemented which has not been previously analyzes.

The electrical conductivity tensor affects the speed at which the tissue is excited as well

as the speed at which the tissue depolarizes. Non-linear membrane kinetics were implemented by

using modified FitzHugh-Nagumo equations for excitable media

-ul = V. G'Vu, + cu1 (u1 a)(1 u)- c2u2
at (3-7)
=su
at

G'is the electrical conductivity tensor and as well as in the general form of the equations, u, is

an activation variable, u2 is an inhibitor, a sets the excitation threshold, P the excitability, and 3,y,

and 6 are the parameters that describe the resting state and dynamics of the system.28









The boundary conditions for this simulation assumed that no current was flowing into or

out of the control volume. Therefore, insulating Neumann boundary conditions were assigned to

every face of the volume surrounding the heart. Initial conditions characterize an initial uniform

potential of distribution of 1 V throughout a section of the model for the activation variable,

while the adjacent sections remain at zero. For the inhibitor variable, the adjacent sections have a

value of 0.3 V.

Values for F, a, 0, y and 6 were obtained from literature from an FEM model done by

Filippi et al. 29 to be 0.01, 0.1, 0.5, 1 and 0 respectively, which are standard values used in simple

FitzHugh-Nagumo models. Preliminary data from these models is presented.









CHAPTER 4
RESULTS

Two-dimensional Results of Validation Studies

2D heart model simulations were carried out solely for the purpose of visualizing if the

data was being properly obtained from DT-MRI data, transformed into the correct coordinates,

and corresponded to the expected underlying fiber tissue arrangement.














A B


Figure 4-1. Transverse cuts of the heart obtained from slice 7 of the imaging sequence. A) Fiber
orientation mapping using FLTView software. B) Electrical conductivity map in
COMSOL Multiphysics software, red=parallel to the xy-plane, blue=perpendicular.

The red areas in Figure 4-1 (B) indicate the regions of larger electrical conductivity, this

implies that along these areas the fibers are aligned parallel to the plane of the transverse cut.

That is, the fibers are aligned almost horizontally. This inference can be validated when

comparing it with the FLTView fiber orientation map (Figure 4-1 (A)). It can be seen that the

fibers in this particular image and region of interest (red regions) are oriented parallel to the

plane of the image. These results were also validated by comparing them to similar studies found

in literature where the fiber architecture of the heart was reconstructed by histological

measurements of the fiber angle orientation.6









Three-dimensional Results

Voltage Input and Current Input Simulations

Significant differences can be seen between isotropic and anisotropic cardiac models in

response to the potential difference input. Figure 4-2 illustrates the current paths taken when

traveling from the apex to the base of the heart. In the isotropic model, the current direction is

largely perpendicular to the transverse cut; while in the anisotropic model, current follows a

more helical path. In addition, it can be seen that the current tends to follow the fiber orientation

of the left ventricular wall. Fibers tend to lie in planes parallel to the epicardium, then rotate

counterclockwise over approximately 1100 with increasing depth from the epicardium to the

endocardium going through a horizontal alignment near the midwall.





















Isotropic Anisotropic

Figure 4-2. Current direction for a potential difference between the apex and base within a
transverse cut of the heart 12 mm from the base. Arrows correspond to the current
direction.

The magnitude of the paths of selected current streamlines between two points was

compared in both models. Significant differences in magnitude were obtained. Streamlines with










starting points at the base of the left ventricle at (37 mm ,43 mm ,0 mm), and at the apex of the

left ventricle wall at (39 mm, 30 mm, 10 mm) were calculated for current point source and

voltage difference input models. The results of the simulations are summarized in Table 4-1.


Table 4-1. Summary of the magnitude of current paths at between two points at different
locations with the same seed point.
Simulation Tissue prope Start Point End Point Avg Magnitude
input (x,y,z)(mm) (x,y,z) (mm) (mm)

Voltage Isotropic 37, 43, 0 38,46,16 32.6
Anisotropic 37, 43, 0 38,44,16 36.3

Current Isotropic 39,30,10 5,5,15 45
Anisotropic 39,30,10 7,7,15 91



The magnitude of the distance between the same starting point and an average of three

adjacent ending points of current paths was calculated in isotropic and anisotropic models.

Anisotropic streamlines were found to be significantly longer. A plot of the averaged

streamlines clearly delineates the differences between the current behavior in isotropic and

anisotropic models (Figure 4-3).










End Point


0 Isotropic case
5020 Anisotropic case



y-axis (mm) 0 X

Figure 4-3. Comparison between current streamlines in isotropic and anisotropic models.
Average from the same seed point.









The following figure illustrates the paths current takes when a current point source was

implemented as the input. This source was located in the exterior wall of the left ventricle. Path

lines emerge from the point source in a slightly straight fashion in the isotropic model. Although

the tissue has isotropic conductivity, the lines are not completely straight throughout the entire

volume. Such behavior is attributed to the boundary effects at the edges of tissue and non-tissue

regions.

On the other hand, current does not exhibit straight pathlines in the anisotropic model.

This again, reflects the importance of the underlying fiber structure in electrical conduction in

myocardial tissue.


















Isotropic Anisotropic


Figure 4-4. Slice image of a transverse cut of the heart with current path lines. A) Isotropic
model, red=uniform electrical conductivity. B) Anisotropic model,
blue=perperdicular to the xy-plane, red=parallel to xy-plane.

Sensitivity Analysis

By definition, a sensitivity analysis is the study of how the output of a model varies when

certain parameters in the model are changed. This concept was applied to this model by

simulating external defibrillation and internal cardioversion.









Different voltage values were applied to simulate external defibrillators. As an overall

trend, when the magnitude of input voltage was increased, the percentage of heart tissue nodes at

certain voltage increased as well. After running these simulations, a Matlab subroutine (appendix

b) was implemented in order to see how many tissue nodes had reached a certain voltage

threshold under the different stimuli. The tissue and non-tissue nodes were segmented according

to FA values in this case as well.


0.16
4- 0.14 -
o
5 0.12

0.08 -o- Isotropic
N -U-- Anisotropic
2 0.06
0.04
z 0.02


0 2 4 6 8 10 12
Voltage Input (V)


Figure 4-5. External defibrillation at low range. Normalized number of nodes of heart tissue
above 5 volts at different voltage inputs with a surface potential difference

In external defibrillation models, different potentials were applied across the anterior and

posterior faces of the heart for isotropic and anisotropic tissue properties. Significant differences

could be observed at lower voltage inputs, 3 V to 10 V (Figure 4-5). The voltage input and the

normalized number of nodes show an exponential relationship in the anisotropic case and a more

linear relationship in the isotropic case.

On the other hand, at higher voltage input values the differences between isotropic and

anisotropic responses are not as evident. From Figure 4-6, one can note that an approximately

linear relationship exists between the voltage input and the normalized number of nodes in the











range of 15 V to 35 V. Outside this range the behavior is not quite linear as mentioned


previously.

1.2

S1
0o
-6 0.8

E 0.6
.4 --- Isotropic
0.4
[--I-M-Anisotropic
0.2
E
0
z 0
0 20 40 60 80 100 120

Voltage Input (V)

Figure 4-6. External defibrillation at high range. Normalized number of nodes of heart tissue
above 5 volts at different voltage inputs with a surface potential difference

A similar trend can be seen when modeling implantable cardioverter defibrillators. In


internal defibrillation models, a point source voltage difference was applied between the right


atrium and right ventricle. At low voltages (3 V to 10 V), there is a significant difference


between the behavior of anisotropic and isotropic tissue. The percentage of nodes that reach a 5


V threshold in anisotropic tissue increases radically when the voltage input reaches


approximately 9 V; while in the isotropic tissue model, a more linear behavior is observed.

0 09

0
C 0 07
S 006

E -005- -Anisotropic
C 004 -Isotropic
S 003-
0 002

0 0_1--------_-----_ -- ---

0 2 4 6 8 10 12
Point Source Voltage Input (V)


Figure 4-7. Implantable cardioversion at low-range. Normalized number of nodes of heart tissue
above 5 volts at different voltage with a point source input.










Significant differences can be observed between isotropic and anisotropic tissue models

(Figure 4-8) when a voltage point source was modeled (internal cardiversion-defibrillation). In

comparison to the results obtained when a surface potential difference was modeled at high-

range voltages, the behavior under isotropic and anisotropic tissue properties is clearly different.

There is a faster increase of the percentage of nodes above the defined threshold in the

anisotropic tissue model than in the isotropic model.


1.2



0
C 0.8
46

0.6 +- Anisotropic
-- Isotropic
0.4
M 0.2
E
Z 0 -
5 6 7 8 9 10 20 30 40 100
Point Source Voltage Input (V)


Figure 4-8. Implantable cardioversion at high range. Normalized number of nodes of heart tissue
above 5 volts at different voltage with a point source input.

The differences seen at the high-range voltage input are more important when modeling

point sources than when modeling surface potential difference. Better visualization of the results

is seen on the following graph (Figure 4-9). Eighty percent of anisotropic tissue reaches a voltage

threshold of 5 V at approximately 18 V of voltage point source input. When the tissue is

isotropic with the same input, eighty percent of its nodes reach above the threshold at

approximately 35 V of input. On the other hand, differences are not significant between

isotropic point source and isotropic face models. Nevertheless, it can be observed that in the

isotropic model, there is a faster increment of number of nodes than in the anisotropic model.










1.2
U)
0





Anisotropic point
0.4 Isotropic face -

ca 02
o4-/




v 04 -u-Anisotropic face
z 0 .... .....
0 5 10 15 20 25 30 35 40 45
Voltage Input (V)


Figure 4-9. Normalized number of nodes of heart tissue above 5 volts at different voltage inputs,
four-case comparison.

Ischemic Tissue Results

The ischemic model was compared to the healthy model with anisotropic tissue properties.

As it can be inferred from Figure 4-10, the response difference between the healthy and the

ischemic models is not major. Although healthy myocardium reached the voltage threshold at a

lower voltage input, it does not exhibit a significant difference from the ischemic heart. Eighty

percent of healthy heart tissue is excited above 5 V with a voltage input of approximately 28 V.

On the other hand, 80 % of heart tissue in the ischemic model surpasses such threshold at

approximately 35 V. However, it should be pointed out that depending on the severity of the

ischemia, the volume of infarcted heart tissue increases, and so does the disruption of electrical

function.

A sensitivity analysis was done were different isotropic electrical conductivity values were

assigned to the ischemic tissue region. They were done in order to better understand the how

these values impacted the results. Electrical conductivity values of 0.35 S/m, 0.2 S/m, 0.18 S/m











and 0.15 S/m were assigned, and the normalized number of tissue nodes stimulated above a 5 V


threshold results are summarized in Figure 4-11.


1.2 -


1


0.8


S 0.6


0.4


0.2


60
Voltage Input (V)


100 120


Figure 4-10. Comparison between healthy and infarcted heart tissue above a 5 V threshold for a


Voltage Input (V)


Figure 4-11. Sensitivity analysis of ischemic heart tissue with different electrical conductivity
values.


-U- Healthy face
potential input

-Ml-- Ischemic face
potential input









Time-Dependent Model Results

Qualitative information was obtained from the time-dependent simulations. Although still

in a developmental stage, differences could be observed between isotropic and anisotropic

responses to the FitzHug-Nagumo equations. Isotropic tissue was excited uniformly throughout

the tissue and the wavefront propagation of action potential was nearly symmetric. On the other

hand, anisotropic tissue presented varying shapes of wavefront propagation that are still under

analysis.









CHAPTER 5
DISCUSSION

Fiber architecture obtained from water diffusivity (DT-MRI) was used to predict

electrical conductivity in cardiac tissue and an intracellular electrical finite element model of the

heart was created. An isotropic model was also created in order to compare the paths taken by

currents under different stimulations conditions. In this case, fiber orientation was disregarded

and a uniform conductivity was assumed. Significant differences were seen between anisotropic

and isotropic model current paths lines. Streamlines in the isotropic model follow the shortest

path between two points, while in the anisotropic model they follow paths that reflect the

underlying muscle fiber orientation. Current follows the higher conductivity direction when

traveling between two points, and delineated the rotating organization of the fibers from the

epicardium to the endocardium in the left ventricular wall.

Previous Studies

Other studies have modeled the significance of fiber architecture in electrical propagation

in cardiac tissue. Knisley et al. 30 examined the role of spatial variation of voltage gradients on

the transmembrane voltage changes in rabbit hearts. They explored the voltages using a

bidomain computer model. They incorporated 2D fiber orientation and approximated the

orientation further away from the area of interest. In comparison, our study incorporates a high-

resolution 3D fiber architecture and the corresponding electrical conductivities in the appropriate

directions.

Wei et al. 31 compared isotropic and anisotropic computer heart models in body surface

electrocardiograms. Their model incorporated fiber arrangement by rotating fiber architecture

counterclockwise from the epicardial layer to the endocardial layer a total of 900. They modeled

transient electrical conduction and saw no significant differences in surface ECGs between









models. Their study incorporated both the fiber architecture and the action potential propagation

but only as an approximation. Its aim was to analyze the differences that could be detected in

surface ECGs.

Interpretation of Results and Applications

The presented model attempts to specifically describe the behavior of current patterns and

predict the percent of tissue stimulated when different stimuli are implemented. The modeling

approach is also able to account for more realistic tissue properties that can more accurately

predict the implications of an electrical imbalance which will be the focus of future studies.

This computational model may be useful for optimizing electrode placement and also for

predicting defibrillation thresholds that minimize damage of tissue. Simulation results suggest a

minimum and a maximum voltage range that a subject may undertake for successful

defibrillation while not suffering permanent damages. At the time of implantation of the

cardioverter defibrillators, safety-threshold testing is conducted.32 Ventricular fibrillation is

induced at the time of implantation to test whether or not the arrhythmia is terminated. Such

testing may cause permanent damages to the tissue and developing technologies to avoid such

injury are potentially beneficial.

There exists a critical mass hypothesis stating that a way to end an episode of ventricular

fibrillation is by electrically exciting a critical percentage mass of the heart.33 The exact amount

of mass that needs to be electrically activated is unclear, but estimates have established a range

of 75% to 100% of the myocardial tissue.34 In addition, it has been speculated that raising a

critical mass of myocardium above 5 V/cm will defibrillate the heart.34 According to this theory,

our developed modeling approach may predict DFTs for implantable cardioverter defibrillators

before implantation. When modeling point source inputs in anisotropic tissue, our rabbit heart

model (Figure 4-8) roughly shows a voltage range for successful defibrillation of 17 to 20 V.









This is significantly different if the tissue is assumed isotropic. In this case a voltage ranging

from 30 V to 45 V would successfully defibrillate the tissue.

Simulation results obtained when a potential difference input was applied, mimic an

external defibrillation. In an attempt to include the resistivity of the torso, without imbedding the

heart model into a whole torso model, the resistivity of non-tissue surrounding the heart was very

high (le-6 S/m). With this assumption, we could see that when the heart was modeled as

anisotropic, the voltage range that would defibrillate the heart was larger than when the heart was

stimulated using a point source. In a rabbit heart model, voltages ranging from 45 V to 100 V

would excite 90% of the mass above a threshold of 5 V. Standard external defibrillation voltage

thresholds for human hearts range from 200 V to 1000 V depending on the weight and diseased

condition of the patient.16 Compared to this range, our results do not seem to correspond, but one

should note that the heart DT-MRI data used in our model was obtained from the heart of a

rabbit which is smaller than for a human and may not directly apply to values obtained in human

studies.

Analyzing the results of the infarcted myocardium model, several observations can also be

made. Although the percentage of heart tissue stimulated in this case did not significantly differ

from that of healthy tissue, it did exhibit a different voltage distribution. Areas around the

infarcted region had increased voltage values compared to the rest of the heart tissue. This could

be attributed to boundary effects between healthy and unhealthy regions. This behavior be of

consequence due to the unorganized current propagation inside the infarcted region creating

regions of current recirculation affecting the potential distribution around the edges of the infarct.

Nevertheless, healthy and unhealthy myocardium exhibited a comparable percentage of tissue

excitation for the different voltage inputs.









Future Work

Previous DT-MRI studies have found that infarcted myocardium exhibits an increase in the

magnitude of water diffusivity. 17,35 Future work will use DT-MRI-based models to account for

regions of tissue damage to predict electrical propagation imbalance. Such models will be used

to analyze various infarction scenarios and determine possible implications in the mechanical

functioning of the heart.

The developed models may also be used to understand the implications of large external

electrical fields on myocardial conduction. To implement this approach one may start modeling a

magnetostatic case. When modeling electric behavior of biological tissue at very low

frequencies, a quasistatic approximation is valid. The induced electric field can be written in

terms of the magnetic vector potential A and the electric scalar potential q0 as 36

OA
E =-- V0. (5-1)
at

The tissue volume is assumed a conductive medium following the general form of Ohm's Law

J = o-E (5-2)

where J is the current density and c is the spatially varying conductivity tensor obtained from

DTI data. In a quasistatic approximation, the divergence of the current density Jis zero, so we

have

-V-(oVP)= 0. (5-3)

Combining equations (5-1)-(5-3), we obtain


-V --A7 V -.(oV) = 0 (5-4)


Also, the constitutive equation for magnetic fields needs to be included. For biological tissues,

the relative permeability is approximately 1, therefore









B= ,uH (5-5)

where B is the magnetic flux density or magnetic field, u/0 is the relative magnetic permeability

and His the magnetic field strength.

These equations aim to predict the effects that externally occurring electric and magnetic

fields have on the electrical behavior of the heart. In addition, they may be useful to define a

near-field electromagnetics standard in the presence of external electromagnetic forces. This

field may be characterized by observing at what distance electrodes need to be from the heart, so

that the effects of anisotropy can be ignored. Such analysis will be implemented in future studies.









CHAPTER 6
CONCLUSIONS

The overall goal of this project was to realistically model cardiac electrical anisotropy

and run sensitivity analyzes for different input and boundary conditions. Such simulations were

primarily done based on a steady-state case and preliminary studies were done on a time-

dependent model. This project also included simulations of infarcted myocardium based on

reported characteristics of such tissue. As a result, the general objectives of this project were

achieved together with the possibility for expansion in many directions.

Although an accurate heart geometry was used, there are certain limitations to the model

that need to be addressed. Heart tissue consists of different kinds of cells, i.e. Purkinje fibers, SA

node cells etc. These cells have different electrical characteristics that were not taken into

account.37 The tissue was assumed anisotropic throughout but with the same electrical excitation

characteristics. This clearly affects the propagation patterns in heart tissue, but these issues will

be addressed in future studies.









APPENDIX A
DATA PROCESSING AND TISSUE SEGMENTATION SUBROUTINE

test3a cuts atria.m

%Get eigenvalues and eigenvectors from .fit file DTI data
%Output the G tensor
%and separate files containing the anisotropic matrix values
%Organizes these values in rows of 80 columns (or the size of the image) in
%the x-direction

%Cuts atria
%
DT=openFLT('dti.fit');
eigen vec=fopen('eigen vec.txt','w'); %Eigenvector file
eigen val=fopen('eigen val.txt','w'); %Eigenvalues
Gtensor=fopen('Gtensor.txt', 'w'); %Conductivity tensor
el l=fopen('el 1 .txt','w');
el2=fopen('el2.txt','w');
el3=fopen('el3.txt','w');
e22=fopen('e22.txt','w');
e23=fopen('e23.txt','w');
e33=fopen('e33.txt','w');
sur=fopen('surface.txt','w');

%Electrical Conductivity
gl=0.625; % (S/m) Parallel to myofibers
gt=0.236; % (S/m) Transverse to myofibers but in the same plane
gn=0.1087; % (S/m) Normal to the layer

G=zeros(3,3); %Initialize matrices
g=zeros(3,3); %
D=zeros(80,80);
E 11 =zeros(1040,80);
E12=zeros(1040,80);
E13=zeros(1040,80);
E22=zeros(1040,80);
E23=zeros(1040,80);
E33=zeros(1040,80);

for k=1:5 %Number of slices
forj=14:68 %Size of the region of interest where the image is
for i=25:70 %Size of the region of interest where the image is
if ((i-40)A2+(j-40)A2<=33A2)
[v,l]=eig(matr(DT,i,j,k));%Function that gets eigenvalues and eigenvectors
trace=(l(1,1)+1(2,2)+1(3,3))/3;









FA=(sqrt(3 *((1(1,1)-trace)A2+(1(2,2)-trace)A2+(l(3,3)-
trace)A2)))/(sqrt(2*(l(1,1)A2+1(2,2)A2+1(3,3)A2)));

if (FA<0.2)
G(1,1)=0.0001;
G(1,2)=0.0001;
G(2,2)=0.0001;

elseif (FA>=0.2)%Sorts and assigns values
diag=[l(1,1),1(2,2),1(3,3)];
[lam,idxMax]=max(diag);
[lam,idxMin]=min(diag);
signEv=sign(diag);
if idxMax==1
if idxMin==2
g(1,1)=gl;
g(2,2)=gt;
g(3,3)=gn;
elseif idxMin==3
g(1,1)=gl;
g(2,2)=gn;
g(3,3)=gt;
end

elseif idxMax==2
if idxMin==1
g(1,1)=gt;
g(2,2)=gl;
g(3,3)=gn;
elseif idxMin==3
g(1,1)=gt;
g(2,2)=gl;
g(3,3)=gn;
end

elseif idxMax==3
if idxMin==2
g(1,1)=gt;
g(2,2)=gn;
g(3,3)=gl;
elseif idxMin==l
g(1,1)=gn;
g(2,2)=gt;
g(3,3)=gl;
end










end
G=v*g*v';
end

end

if (k==6)
if (FA>=0.2)
aa=l;
else
aa=2;
end
D(i,j)=aa;
%fprintf(sur,'%d\t %d\t %+15.6e\n',ij,aa);
end

E1l(j0+80*(k-1),i)=G(1,1);
E12(j+80*(k-1),i)=G(l,2);
E13(j+80*(k-1),i)=G(1,3);
E220(j+80*(k-1),i)=G(2,2);
E23(j+80*(k-1),i)=G(2,3);
E33(j+80*(k-1),i)=G(3,3);

fprintf(el l,'%+15.6e\t',G(1,1));
fprintf(el2,'%+15.6e\t',G(1,2));
fprintf(el3,'%+15.6e\t',G(1,3));
fprintf(e22,'%+15.6e\t',G(2,2));
fprintf(e23,'%+15.6e\t',G(2,3));
fprintf(e33,'%+15.6e\t',G(3,3));

end
end
end


fclose('all');









APPENDIX B
TISSUE ABOVE THRESHOLD SUBROUTINE

tiss nontiss.m

%Program that evaluates nodal points

clear all
clc
format long;


fidEvec=fopen('infa-aniso-point5v.txt');
Det=fopen('conductivity-point-infa.txt');


tissue=0;
tissuel=0;
i=l;
for r= 1:70920


[Evec, cnt] =fscanf(fidEvec,'%25e %25e %25e %25e\n',[1,4]);

El=i;
E4=Evec(4);

vol{El}=E4;


[Idx2, cnt]= fscanf(Det,'%25e', [1,3]);
[Evec2, cnt] =fscanf(Det,'%25e\n',[1,1]);


El l=i;
E41=Evec2(1);


con{E11}=E41;
i=i+l;

end



for q= 1:70920









if ((0.65>con{q}) & (con{q}>0.18))
tissue=tissue+1;
if(vol{q}>=0.005)
tissuel=tissuel+l;
end
end

end

fclose('all')









LIST OF REFERENCES


1. Streeter DD, Ross J, Patel DJ, Spotnitz HM, Sonneblick EH. Fiber orientation in canine
left ventricle during diastole and systole. Circulation Research, 1969. 24(3): p. 339-347

2. Tusscher K, Hren R, Panfilov A. Organization of ventricular fibrillation in the human
heart. Circulation Research, 2007. 100(12): p. E87-E101.

3. Salama G, Choi R. Imaging ventricular fibrillation. Journal of Electrocardiology, 2007.
40(6): p. S56-S61.

4. Hsu EW, Muzikant AL, Matulevicius, Penland RC. Magnetic resonance myocardial
fiber-orientation mapping with direct histological correlation. American Journal of
Physiology, 1998. 274(5 PART 2): p. H1627-H1634.

5. Hsu EW, Xue R, Holmes A, Forder JR. Delayed reduction of tissue water diffusion after
myocardial ischemia. American Journal of Physiology-Heart and Circulatory
Physiology, 1998. 275(2): p. H697-H702.

6. Nielsen P, LeGrice I, Smaill BH, Hunter PJ. Mathematical-model of geometry and
fibrous structure of the heart. American Journal of Physiology, 1991. 260(4): p. H1365-
H1378.

7. Lorange M, Gulrajani M. A computer heart model incorporating anisotropic propagation
Model construction and simulation of normal activation. Journal of Electrocardiology,
1993. 26(4): p. 245-261.

8. Vigmond EJ, Leon L. Effect of fiber rotation on the initiation of re-entry in cardiac tissue.
Medical & Biological Engineering & Computing, 2001. 39(4): p. 455-464.

9. Scollan D, Holmes A, Zhang J, Wnslow RL. Reconstruction of cardiac ventricular
geometry and fiber orientation using magnetic resonance imaging. Annals of Biomedical
Engineering, 2000. 28(8): p. 934-944.

10. Winslow R, Scollan DF, Greenstein JL, Yung CK, Baumgartner W, Bhanot G, Gresh L,
Rogowitz E. Mapping, modeling, and visual exploration of structure-function
relationships in the heart. Ibm Systems Journal, 2001. 40(2): p. 342-359.

11. Basser P, Mattiello J, Lebihan D, Estimation of the effective self diffusion tensor from
the NMR-spin echo. Journal of Magnetic Resonance Series B, 1994. 103(3): p. 247-254.

12. Steendijk P, Mur G, Van Der Velde E, Baan J. The 4-electrode resistivity technique in
anisotropic media theoretical-analysis and applications on myocardial tissue in-vivo.
IEEE Transactions on Biomedical Engineering, 1993. 40(11): p. 1138-1148.









13. Hooks D, Trew M, Caldweel B, Sands G, LeGrice I, Smaill B. Laminar arrangement of
ventricular myocytes influences electrical behavior of the heart. Circulation Research,
2007. 101: p. E103-E112.

14. .Heart Attack and Angina Statistics. American Heart Assosiation (2003).

15. Karlon W, Eisenberg S, Lehr J. Effects of paddle placement and size on defibrillation
current distribution: a three dimensional finite element model. IEEE Transactions on
Biomedical Engineering, 1993. 40(3): p. 246-255.

16. Sobel, Rachel K. A Shocking Story: Handy Defibrillators. US News & World Report.
28 September 1998

17. Chen JJ, Song SK, Liu W, McLean M, Allen JS, Tan J. Remodeling of cardiac fiber
structure after infarction in rats quantified with diffusion tensor MRI. American Journal
of Physiology-Heart and Circulatory Physiology, 2003. 285(3): p. H946-H954.

18. Le Bihan D, Denis MD, Mangin JF, Poupon C, Clark CA. Diffusion tensor imaging:
concepts and applications. Journal of Magnetic Resonance aging. 2001. 13: 543-546.


19. Tuch DS, Wedeen VJ, Dale AM, George JS. Conductivity tensor mapping of the human
brain using diffusion tensor MRI. Proceedings of the National Academy of Sciences of
the USA, 2001. 98(20): p. 11697-11701.

20. Plonsey R, Barr R. The four-electrode resistivity technique as applied to cardiac muscle.
IEEE Transactions on Biomedical Engineering 1982. 29(7):p. 541-546

21. Fallert M, Mirotznik M, Downing, S. Myocardial electrical impedance mapping of
ischemic sheep hearts and healing aneurysms. Circulation, 1993. 87: 199-207.

22. .Mathews G, Cellular Physiology of Nerve and Muscle Fourth Edition, 2003.

23. FitzHugh R. Impulses and physiological states in theoretical models of nerve membrane.
Biophysical Journal, 1961. vol 1: 445-466.


24. Filippi S, Cherubini C. Multiphysics models of biological systems. Exerptfrom the
Proceedings of COMSOL Users Conference 2006.

25. Pham D, Xu C, Prince J. Current methods in medical image segmentation. Annual review
Of Biomedical Engineering 2000. Vol. 2: 315-337.

26. Eason J, Schmidth J, Dabasinskas A, Siekas G, Aguel F, Trayanova N. Influence of
anisotropy on local and global measures of potential gradient in computer models of
defibrillation. Annals of Biomedical Engineering, 1998. 26(5): p. 840-849.










27. Liu F, Xia L, Zhang X. Analysis of the influence of the electrical asynchrony on regional
mechanics of the infarcted left ventricle using electromechanical heart models. JSME
International Journal 2003. Vol (46): 1.

28. Usyk T, LeGrice I, McCulloch A. Computational model of three-dimensional cardiac
electromechanics. Computing and Visualization in Science 2002. 4 (4): p.249-257.

29. Filippi S, Cherubini C. Multiphysics models of biological systems. Excerpt from the
Proceedings of COMSOL Users Conference 2006

30. Knisley SB, Trayanova N, Aguel F. Roles of electric field and fiber structure in cardiac
electric stimulation. Biophysical Journal, 1999. 77(3): p. 1404-1417.

31. Wei DM, Okazaki 0, Harumi K, Harasawa E, Hosaka H. Comparative simulation of
excitation and body surface electrocardiogram with isotropic and anisotropic computer
heart models. IEEE Transactions on Biomedical Engineering, 1995. 42(4): p. 343-357.

32. Russo A, Sauer W, Gerstenfeld EP, Hsia HH, Lin D. Defibrillation threshold testing: is it
really necessary at the time of implantable cardioverter-defibrillator insertion?. Heart
Rhythm 2005. 2(5) : p. 456-461.

33. Mower M, Mirowski M, Spear JF, Moore EN. Patterns of ventricular activity during
catheter defibrillation. Circulation 1974. 44:858-861.

34. Blanchanard S, Ideker R. The process of defibrillation in implantable cardioverter-
defibrillators, edited by N.A.M.I. Estes, A. Manoli and P. Wang. New York: Marcel
Dekker 1994, p. 1-27.

35. Scollan D, Holmes A, Winslow R, Forder, J. Histological validation of myocardial
microstructure obtained from diffusion tensor magnetic resonance imaging. Ame
Physiological Society 1998., H2308-H2317.

36. Norbury J. Classical electrodynamics for undergraduates. University of Wisconsin 1997.
p.63-89.

37. Tsalinkakis D, Zhang H, and Fotiadis, D. Phase response characteristics of sinoatrial
node cells. Computers in Biology and Medicine, 2007. Vol 27 (1): 8-10.









BIOGRAPHICAL SKETCH

Ana Maria Saaibi was born in 1983 in Bucaramanga, Colombia, and in the fall of 2001 she

received her high-school diploma from Colegio Panamericano in her home town. In the spring

2002, she began her engineering and her collegiate tennis career at Tulane University in New

Orleans, Louisiana where she double majored in mechanical engineering and mathematics. She

received her Bachelor of Science in Engineering degree in the fall of 2005. In 2006 after starting

graduate school in the Biomedical Engineering department at Tulane University, she transferred

to the Mechanical and Aerospace Engineering department at University of Florida. She will

receive her Master of Science degree in mechanical engineering with a minor in biomedical

engineering from the University of Florida in August 2008.





PAGE 1

1 ISOTROPIC VERSUS ANISOTROPIC INTR ACELLULAR ELECTRICAL PROPERTIES OF A DIFFUSION TENSOR MAGNETIC RE SONANCE IMAGING (DT-MRI)-BASED CARDIAC MODEL By ANA MARIA SAAIBI A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2008

PAGE 2

2 2008 Ana Maria Saaibi

PAGE 3

3 To Jose, Beatry & Silvia

PAGE 4

4 ACKNOWLEDGMENTS I would like to thank my family for th eir unconditional support throughout my career; especially my mom and my dad for letting me dream while keeping me grounded, and my sister for keeping me sane. Great appreciation goes to my advisor, Malisa Sarntinoranont for her friendship, guidance and all the wonderful opportun ities I received, and also to my committee members Roger Tran-Son-Tay and John Forder. I w ould also like to thank my lab colleagues: Jung Hwang Kim, Sung Lee, Greg Pishko and Xiaomi ng Chen for their help, advice and respect. This project was supported in part by an appointment to the Research Participation Program at the Center for De vices and Radiological Health administered by the Oak Ridge Institute for Science and Education through a nd interagency agreement between the U.S. Department of Energy and the U. S. Food and Drug Administration.

PAGE 5

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........7 LIST OF FIGURES................................................................................................................ .........8 ABSTRACT....................................................................................................................... ............10 CHAPTER 1 INTRODUCTION..................................................................................................................11 2 BACKGROUND................................................................................................................... .14 Heart Disease.................................................................................................................. ........14 External Defibrillators and Implan table Cardioverter Defibrillators......................................15 Heart Anatomy and Electrophysiology..................................................................................18 Heart Anatomy................................................................................................................18 Electrophysiology............................................................................................................19 Overview of DT-MRI.............................................................................................................20 Measuring Tissue Elec trical Conductivity..............................................................................21 3 METHODS...................................................................................................................... .......24 Animal Preparation and MR Imagining.................................................................................24 Tissue Segmentation............................................................................................................ ...25 Assigning Properties for Electrical Conductivity...................................................................27 Finite Element Model of the Heart.........................................................................................28 Two-dimensional Model.................................................................................................28 Three-dimensional Model...............................................................................................29 Defibrillation Models...............................................................................................32 Preliminary Ishemic Model......................................................................................33 Time-Dependent Model...........................................................................................34 4 RESULTS...................................................................................................................... .........36 Two-dimensional Results of Validation Studies....................................................................36 Three-dimensional Results.....................................................................................................37 Voltage Input and Current Input Simulations.................................................................37 Sensitivity Analysis.........................................................................................................39 Ischemic Tissue Results........................................................................................................ ..43 Time-Dependent Model Results.............................................................................................45 5 DISCUSSION................................................................................................................... ......46

PAGE 6

6 Previous Studies............................................................................................................... .......46 Interpretation of Results and Applications.............................................................................47 Future Work.................................................................................................................... ........49 6 CONCLUSION................................................................................................................... ....51 APPENDIX A DATA PROCESSING AND TISSUE SE GMENTATION SUBROUTINE.........................52 B TISSUE ABOVE THRESHOLD SUBROUTINE.................................................................55 LIST OF REFERENCES............................................................................................................. ..57 BIOGRAPHICAL SKETCH.........................................................................................................60

PAGE 7

7 LIST OF TABLES Table page 4-1. Summary of the magnitude of current paths at between two points at different locations with the same seed point.....................................................................................38

PAGE 8

8 LIST OF FIGURES Figure page 2-1. External defibrillator (image obt ained from DRE Medical equipment 2008)..................16 2-2. Implantable cardioverterdefibrillator (image obtained from Medtronic Inc. 2008)........16 2-3. Implantable cardioverter defibrillator and electrode placement (image obtained from U.S. Department of Health and Human Services and NIH)..............................................17 2-4. Frontal view of a human heart. (image obtained from Mediplus Medical Encyclopedia, ADAM)......................................................................................................19 2-5. Sequence schematic of the electrical pr opagation in the heart. A) location of nodes, B) electrical activity starting at the SA node, C) depolarization wave going across the atria, D) depolarization wave moving to th e apex of the heart and E) depolarization wave spreading upward from the apex (imaged obtained from Anatomy and Physiology, Marieb 2000)..................................................................................................20 3-1. Experimental setup of isolat ed heart with the aorta cannulated. Lef t: perfused heart with STH, right: replaced with PFC em ulsion. (Figure courtesy of Min-Sin Hwang Ph.D student, Biomedical Engineering De partment, McKnight Brain Institute, University of Florida)........................................................................................................24 3-2. Tissue segmentation correspondence, (A) FA map of transverse image of the heart showing left and right ventricle walls. (B) Tissue segmenta tion (red=non-tissue, blue=heart tissue)............................................................................................................. ..26 3-3. Transverse image of the heart along the xy-plane, 2D isotropi c model for validation studies. (red=isotropic heart tissu e, blue=non-tissue surroundings)..................................28 3-4. Mesh containing 24,840 brick elements corresponding to embedded ventricles and surrounding media.............................................................................................................29 3-5. Transverse slice images of the isotro pic electrical heart model in FEM software, from apex=1 to base=12 of the heart (b lue=heart tissue with isotropic conductivity, white=surrounding non-tissue)..........................................................................................30 3-6. Transverse slice images of anisotropic electrical heart model in FEM software, from apex=1 to base=12 of the heart. Color contours provide directional information (red=parallel to xy-plane, blue=perpendicular).................................................................31 3-7. External defibrillation model with anis otropic electrical cond uctivity properties.............32 3-8. Implantable cardioverterdefibrillation model with anis otropic electric al conductivity properties..................................................................................................................... .......33

PAGE 9

9 3-9. Transverse slice images of anisotropic electrical heart model in FEM software with infarct region (dark blue), from apex=1 to base=5 of the heart. (red=parallel to xyplane, blue=perpendicular)................................................................................................34 4-1. Transverse cuts of the heart obtained fr om slice 7 of the imaging sequence. A) Fiber orientation mapping using F LTView software. B) Elect rical conductivity map in COMSOL Multiphysics software, red=paralle l to the xy-plane, blue=perpendicular.......36 4-2. Current direction for a potential diff erence between the apex and base within a transverse cut of the heart 12 mm from the base. Arrows corr espond to the current direction ............................................................................................................................37 4-3. Comparison between current streamlines in isotropic and anisotropic models. Average from the same seed point.....................................................................................38 4-4. Slice image of a transver se cut of the heart with curre nt path lines. A) Isotropic model, red=uniform electrical condu ctivity. B) Anisotropic model, blue=perperdicular to the xy-pl ane, red=parallel to xy-plane............................................39 4-5. External defibrillation at low range. No rmalized number of nodes of heart tissue above 5 volts at different voltage inpu ts with a surface potential difference....................40 4-6. External defibrillation at high range. Normalized numb er of nodes of heart tissue above 5 volts at different voltage inpu ts with a surface potential difference....................41 4-7. Implantable cardioversion at low-range. Normalized number of nodes of heart tissue above 5 volts at different volta ge with a point source input..............................................41 4-8. Implantable cardioversion at high range. Normalized number of nodes of heart tissue above 5 volts at different volta ge with a point source input..............................................42 4-9. Normalized number of nodes of heart tis sue above 5 volts at di fferent voltage inputs, four-case comparison.........................................................................................................43 4-10. Comparison between healthy and infarcte d heart tissue above a 5 V threshold for a.......44 4-11. Sensitivity analysis of ischemic heart tissue with different electrical conductivity values......................................................................................................................... ........44

PAGE 10

10 Abstract of Thesis Presen ted to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science ISOTROPIC VERSUS ANISOTROPIC INTR ACELLULAR ELECTRICAL PROPERTIES OF A DIFFUSION TENSOR MAGNETIC RE SONANCE IMAGING (DT-MRI)-BASED CARDIAC MODEL By Ana Maria Saaibi August 2008 Chair: Malisa Sarntinoranont Major: Mechanical and Aerospace Engineering Myocardial fiber architecture largely dete rmines the current pathways and current wavefront propagation in the hear t. Disruption of this organiza tion may give rise to severe chronic cardiac conditions related to electri cal imbalances. In this study, underlying fiber direction information was obtained from DT-MRI data to predict intracellular electrical conductivity in cardiac tissue, and a finite elemen t model of the heart was created to predict current pathways. Isotropic and an isotropic tissue proper ties were assigned to the heart tissue to compare the electrical behavior under different stimuli: (a) potential difference and (b) current point source stimulus. Significant differences in predicted current path s can be seen between isotropic and anisotropic cardiac models in re sponse to both of the inputs. This DT-MRI modeling approach accounts for more realistic tis sue properties that can more accurately predict the implications of myocardial infarction, which w ill be the focus of future studies. In addition, a preliminary time-dependent model was included to examine the implications of such simulations with a more realistic behavior.

PAGE 11

11 CHAPTER 1 INTRODUCTION Cardiac function is influenced by the threedimensional organization of the myocardial fibers. Cardiac fibers are arrang ed in a circumferential, longitu dinal, and a sheet-like fashion, forming counter-wound helices from the base to the apex of the heart.1 This fiber organization is responsible for the delicate bala nce between mechanical and electr ical functioning of the heart. When electrical disruption of coordinated functi on occurs, this is asso ciated with cardiac arrhythmias which may lead to more seri ous conditions like vent ricular fibrillation.2,3 In case of a heart attack (myocardial infarcti on), blood supply to a section of th e heart is interrupted, and this oxygen shortage (ischemia) causes damage and possi bly death of heart tiss ue. Injured heart tissue conducts electricity slower than healthy heart tissue 4,5 and this difference in conduction velocity often triggers a re-entry or a feedback loop. Th ese re-entry waves and feedback loops are thought to be the cause of many le thal cardiac arrhythmias.2 Previous studies have been conducted to reconstruct fiber architecture in the heart 6, but this anatomical reconstruction is labor intens ive and time consuming. Such studies consist of perfusing, excising, and precisely cutting segments of the ve ntricular wall from which the architecture is then carefully measured. Other me thods to incorporate fiber orientation in heart models have also been proposed 7,8. The method developed by Lorange et al .7 consisted of nesting revolving ellipsoids from the endocardium to the epicardium, from 0o to 120o. Ellipsoid dimensions were taken from a computer tomogr aphy scan of a frozen human heart. Models developed by Vigmond and Leon 8 also included the rotating fiber anisotropy. They modeled muscle fibers as a discrete cable network in a r ectangular area. Within a plane, the fibers where all parallel and there was a fixed clockwise rota tion of fiber orientation between planes. The use of this simplified geometry did not result in realistic echocardiograms (ECG). These methods

PAGE 12

12 provide a solid foundation for the study of the st ructure-function relatio nship of myocardial anisotropy, and provide approximations of the actual anatomical structure. Recent studies used diffusion tensor magnetic resonance imaging (DT-MRI) to determine the orientation of cardiac muscle fibers. 9,10,11 This technique yields an average diffusion tensor for water in the tissue over an image voxel wher e the eigenvalues and eigenvectors determine the magnitude and principal directions of diffusion rates, respectively. The principal direction corresponds to the fiber orientation parallel to the long axis of the muscle fiber.4 Electrical conductivity is also a tensor, and it is predic ted that electric al conductivity in heart tissues is greatest along th e cardiac muscle fiber direction.12 Cardiac fibers create a sheetlike structure along the fiber direc tion. The next preferential direc tion for electrical conduction is transverse to the fibers in the direction parallel with to the sh eets. The least electrical conduction occurs in the direction normal to the sheets.13 In this study, the underlying fiber direction in formation from the water diffusivity tensor as measured by DT-MRI was used to predict in tracellular electrical c onductivity in cardiac tissue. Based on this information an electrical finite element model of the heart was created. Initially a two-dimensional model was created for validation purposes. Then, a steady-state analysis in three dimensions was done and intracellular current paths were predicted in the left ventricle. The current pathlines and voltage di stribution was compared between models using anisotropic and isotropic conductivity properties. Such simulations are useful for predicti ng current propagation and current density distribution patterns, as well as voltage distribution and sensitivity to electrical impulse location. This may useful for establishing defibrillation threshold values and as well as optimizing electrode placement. Ultimately, such models may be used to understand the consequences of

PAGE 13

13 myocardial infarction on the electri cal functioning of the heart and defibrillation characteristics. Preliminary studies on a time-dependent model we re prepared and analyzed against the steadystate cases. Initially, a brief background is given on the topics necessary for a better understanding of the issues addressed. The topics of the backgr ound overview are: heart diseases, devices to overcome such diseased states, basic heart phys iology, reconstruction of anatomy using DT-MRI and techniques used to measure electrical cond uctivity. Then, a detailed st ep-by-step description is presented on the creation of the computational heart model and the subsequent simulations. Following this description, the results of the simu lations are analyzed. The implications of the results are provided along with an exploration of future studies. Lastl y, conclusions are drawn and the results obtained are summarized.

PAGE 14

14 CHAPTER 2 BACKGROUND Heart Disease Heart attacks are the leading cause of death in developing countries. In the United States alone, more than 10,000,000 are living with some form of heart disease. Every year, approximately 1,200,000 people suffer a coronary atta ck every year, and about 40% of them die as a result of this episode.14 This suggests that approxima tely every 65 seconds, a person in America dies of a coronary event. Heart attack or myocardial infarction, is a medical condi tion that occurs when an insufficient blood supply reaches a certain region of the heart. This insufficient blood supply event is referred to as an ischemic episode. This insufficient blood s upply or oxygen shortage produces damage or death to the cardiac tissue. Th is damaged tissue area is referred to as an ischemic area. When the ischemic area is small and does not compromise th e electrical system of the heart, the likelihood that the pa tient will survive is very high. If the ischemic area is large, and a region of infarcted or dead tissue arises, then myocardial arrhythmias occur. Cardiac arrhythmia makes refe rence to any cardi ac condition that involves an abnormal electrical activity in the hear t. These range from non-dangerous arrhythmias to severe arrhythmias such as ventricu lar fibrillation. Some examples of cardiac arrythmias are: tachycardia, bradycardia and fibrillation. When the cardiac rhythm exceeds 100 beats per minute when at rest, this is classified as tachycardi a. Extreme tachycardia makes the heart ventricles contract rapidly and therefore they do not co mpletely fill with blood in every blood cycle and often leads to death. On the other hand, when the cardiac rhythm is under 60 beats per minute, the condition is called bradycardia. In extreme bradycardia, the heart pumps th e correct amount of blood, but so

PAGE 15

15 sparsely that the quantity of oxygen that gets to the organs and tissues is not enough to properly oxygenate them. Cardiac fibrillati on is present when there is an uncoordinated contraction of either the atria or the ventricles in the heart. Atrial fibrillation is more common than ventricular fibrillation. It often tends to become a chronic co ndition and leads to a small increase in the risk of death. On the other hand, ventricular fibrilla tion is severely dangerous and depending on the duration of the episode it often leads to a sudden death. These c onditions are not diseases per se, but are often a reflection of unde rlying cardiac tissue damage. Nevertheless, one out of three people that have a heart attack die before they can receive any type of medical treatment. Due to the fr equency of sudden deaths, the development and improvement of resuscitation techniques for tr eating cardiac arrhythmia s is very important. Devices like external defibri llators and implantable cardiovert er defibrillators need to be optimized as well as the drug therapies which follow these episodes. External Defibrillators and Implantable Cardioverter Defibrillators In general, defibrillators work by deliveri ng an electrical impulse to the heart that simultaneously affects the majority of myocardi al tissue cells and indu ces their simultaneous depolarization. After a successful defibrillation, th e heart resets its elect rical cycle reactivating the normal mechanical contractions starting with the atria and later w ith the ventricles. The success of the defibrillation de pends on the patients conditio n as well as the amount of myocardium that reaches a certain voltage gradient threshold. External defibrillation and electric cardioversio n defibrillation are ther apies that deliver an electrical shock to normalize abnor mal rhythmic beatings of the heart. They are effective when used promptly after a cardiac ep isode. External defibrillation is used when a patient is experiencing ventricular fibril lation or ventricular tachycar dia without a pulse. These two episodes are lethal if there is no intervention.

PAGE 16

16 Figure 2-1. External defibrillator (image obtained from DRE Medical equipment 2008) Electrical cardioversion is employed in order to revert any type of arrythmia except for that mentioned above. The electric impulse sent is sync hronized with the heart s electrical activity and it can be administered either urgently dur ing an extreme situation or selectively using implantable cardioverter defibrillators (ICD). Figure 2-2. Implantable cardiover ter-defibrillator (image obtai ned from Medtronic Inc. 2008) External defibrillators deliver the impulse th rough externally placed paddles. Paddle sizes have a wide range, but common dimensions are 6 to 12 cm di ameter circular paddles. These paddles are placed directly on the thorax on th e patients skin. There is a slight paddle-

PAGE 17

17 placement dependence using these defibrillators as reported by Karlon et al 15; but an overall positioning near the heart region obtains the desire d results. These devices deliver a wide range of voltages, some deliver voltages within th e high-voltage range from 700 V to 2000 V 16, while others deliver voltages w ithin the low-voltage range from 100 V to 500 V.15 Internal cardioverter-defibrillators deli ver the electrical impulse on the hearts endocardium. Usually, they are used in patients w ith recurrent arrhythmias or a chronic electrical disruption condition. Unlike extern al defibrillators, their impl antation requires a surgical intervention. Most have two electrodes that are pl aced in the right atrium towards the base of the heart and in the left ventricle towards the apex of the heart. Since the electrical impulse is delivered directly on to the heart tissue, the magnitude of the shoc k required is smaller than that required when using external defibrillators. Figure 2-3. Implantable cardioverter defibrillato r and electrode placement (image obtained from U.S. Department of Health and Human Services and NIH) The dimensions of the electrodes that deliver the shock in ICDs are much smaller than the external defibrillator paddles. Their dimensions are small so th at a transvenous implantation is possible, therefore the electrode placement within the heart has to be more precise.

PAGE 18

18 Heart Anatomy and Electrophysiology Heart Anatomy The heart is the principal organ in the circulatory system. It is a striated muscle that acts as a pump to synchronously circulate blood and nutri ents through the whol e body. It is located slightly offset to the left in the middle of the thorax and it is surrounded by the pericardium. The pericardium is responsible for its insulation a nd lubrication, preventing it from wear during its normal mechanical functions. The heart consists of three main layers: the endocardium (the innermost layer), the myocardium and the epicardi um (the outermost layer). The endocardium is a connective tissue membrane that covers inside of the heart and has direct contact with the blood. The myocardium is the laye r consisting of cardiac muscle fi bers that are responsible for the pumping mechanism. These cardiac muscle fi bers are arranged in a circumferential, longitudinal, and a sheet-like fashion, forming c ounter-wound helices from the base to the apex of the heart.1 Surrounding the heart is the epicardium, (a part of the pericardium) responsible for producing the pericardial fluid used to lubricate the layers of the heart during its mechanical motion. The heart has four main cavities, right and le ft atrium, and right and left ventricle (Figure 2-4). The right atrium receives deoxygenated blo od from the body, pumps it to the left ventricle which then pumps it to the lungs. Due to the low resistance faced when circulating blood to the lungs, the right side of the heart does not have to exert the same amount of force as does the left side. The left atrium receives oxygenated blood from th e lungs and pushes it to the left ventricle through the mitral valve. The left ventricl e then sends oxygenated blood to the whole body through the aortic artery.

PAGE 19

19 Figure 2-4. Frontal view of a human heart. (image obtai ned from Mediplus Medical Encyclopedia, ADAM) Electrophysiology Cardiac muscle is self-excitatory; it does not require an external stimulus to trigger its contractile functioning. Rhythmic contractions occur spontaneously, and the pace of these contractions is regulated by the sino atrial (SA) node. It is located in the superior wall of the right atrium (Figure 2-5). The SA node produces an ac tion potential, which is caused by the electric depolarization of its membrane due to a concentr ation imbalance of Na and K ions. The current produced by the SA node propagates slowly throughout the atria. Then, it passes to the ventricles through the atrioventricular (AV) node which is located at th e junction of both ventricles. Depolarization occurs more rapidly from the AV node across the ventricles and travels towards the apex of the heart. From the AV node, the depolarization wave propa gates to the Bundle of His; then upward from the apex of the heart to the extremity branches of the Bundle of His. These fibers distribute the curr ent to the ventricles through th eir terminal branches called Purkinje fibers.

PAGE 20

20 Figure 2-5. Sequence schematic of the electrical propagation in the heart. A) location of nodes, B) electrical activity starting at the SA node, C) depolarization wave going across the atria, D) depolarization wave moving to th e apex of the heart and E) depolarization wave spreading upward from the apex (imaged obtained from Anatomy and Physiology, Marieb 2000) Overview of DT-MRI Cardiac function is influenced by the threedimensional organization of the myocardial fibers. This fiber organization is responsible for the delicate balance between mechanical and electrical functioning of the heart. Disruptions in normal fiber architecture are present in cardiac conditions such as cardiac isch emia and ventricular hyperthrophy 17. Alterations to this fiber organization may induce abnormal el ectrical propagations which may lead to severe arrhythmic disruptions. In addition, devices like implantable car dioverter and external defibrillators rely on accurately delivering the correct amount of charge to a specific location and exciting the correct amount of mass percentage of the heart. Such tasks are in fluenced by the underlying fiber architecture of myocardial fibers. Therefore there is an overall need for quantifying fiber arrangement. Recent studies used diffusion tensor magnetic resonance imaging (DT-MRI) to determine the orientation of cardiac muscle fibers.9, 10, 11 This technique yields an average diffusion tensor for water in the tissue over an image voxel, wh ere the eigenvalues and eigenvectors determine the magnitude and principal directions of rate of diffusion, respectively.

PAGE 21

21 In DT-MRI, a linearly varying pulsed magnetic fi eld gradient is applied to the tissue. Two pulses in the same direction but opposite magnitu de are applied. A reduc tion in signal happens due to the movement of the protons during this time interval, and it can be related to the amount of water diffusion through the following equation beDA (2-1) where A is the signal attenuation, D is the diffusion coefficient and b is a factor that characterizes the gradients shape, amplitude and timing.18 In anisotropic diffusion, the coefficient D is a symmetric rank-2 tensor. This tens or characterizes water molecule mobility along three axes that correspond to the MR mach ines axes. Therefore, in order to properly obtain the diffusion tensor one needs to take into account the tissues local coordinates. Measuring Tissue Electrical Conductivity When characterizing electrical properties of tissue, capacitive and resistive elements need to be specified. These two para meters vary with frequency 19 but at the freque ncies of present interest in the current study the effects of frequency dependence were disregarded. A more detailed explanation of this assumption will be given in the following paragraph. Electrical conductivity, G (equal to 1/r where r is el ectrical ressistivity), and permittivity, are needed to describe the electrical properties of tissue. These properties are commonly measured using the four-electrode technique.20 When using this technique, for an alternating current of frequency, f the ratio of voltage, V and current, I is proportional to specific impedance, Z This is a complex quantity and can be written as V Z I (2-2) where

PAGE 22

22 1 (2)rZ Gjf (2-3) In this equation, r is the permittivity of space which is a constant, 8.854*10-12 F/m. The phase difference between current and voltage can be obtained by 21 12 tan()rf G (2-4) except for low frequencies, G >>2 r. Therefore, the magnitude of the ratio between voltage and current is approximately proporti onal to the tissue resistivity (1 /electrical conduc tivity). Tissue resistivity is more commonly called, specific impedance. The heart contracts by passing ionic current inside the muscle, therefore activating rhythmic contractions that circ ulate blood throughout the body. This ionic current is generated by the transport of Na+ and Kions through a semi-permeable membrane. Ions move through small cellular membrane gates which can either be open during an excitation state or closed at a resting state. This ionic movement ge nerates an action potential. 22 Action potential propagation is a complex elec trochemical phenomenon of ionic imbalance, but it can be described using the FitzHugh-Nagumo equations.23 This model is a twodimensional simplification of the Hodkin-Huxley model which models a spike generation in a neuronal axon. There are three charac teristics that describe the beha vior of excitable media, i.e. neurons and muscle cells: rest cell membrane pot ential, threshold for opening and closing ionic gates in the membrane, and the diffusive spreading of the electrical signals. The following equations are a general form of this model 1 1112 2 12()(1) () u uuuuu t u uu t (2-5)

PAGE 23

23 Here u1 is an action potential that activates the media, u2 is the gate regulator variable that inhibits the system, gives the threshold for excitation, is the excitability of the system, and and are the parameters that describe the re sting state and dynamics of the system. Previous studies have been done where cardi ac electrical propagati on is described using these equations.19 Filipini et al found a strong correspondance betw een their model and with the electrical behavior of cardiac cells studied in vitro However, since mechanical contractions were not introduced in this model, they aim to propose a model that includes them in future studies. This mathematical model describes the depolari zation process of the cellular membrane that characterizes wave propagation in nervous and card iac tissue. If the exte rnal stimulus exceeds a certain threshold value, the system initiates a wave propagation across the excitable media. If no such threshold is reached, no propagation occurs.

PAGE 24

24 CHAPTER 3 METHODS Animal Preparation and MR Imagining Min Hwang, a Ph.D. student in the John Forder laboratory, was responsible for the animal experiments and the MR imaging procedure which was conducted at the McKnight Brain Institute at the University of Florida. DT-MRI cardiac data was obtained from an exsanguinated white male rabbit. Rabbit surgery was conducted in accordance with the NIH guidelines on the use of animals in research and the regulations of the Animal Care and Us e Committee of the University of Florida. An isolated and later arrested heart was used for this experiment because it maintains structural integrity of the vasculature. A New Zealand Wh ite male rabbit was be anesthetized using a mixture of ketamine/xylazine (40mg/kg: 10mg/ kg, i.m.) followed by heparin (1000 U/kg, i.v.) and was later exsanguinated. Figure 3-1. Experimental setup of isolated heart with the aorta cannulated Left: perfused heart with STH, right: replaced with PFC emul sion. (Figure courtesy of Min-Sin Hwang Ph.D student, Biomedical Engineering De partment, McKnight Brain Institute, University of Florida) The excised heart was placed in a bath of cold cardiopledgic solution (4C). The heart was transferred to a Langendorff apparatus and perf used retrogradely. An initial perfusion period of 10 minutes washed the red blood cells out of the vascular space, permited the heart to contract normally, and the aortic valve to remain intact. A thin (1 mm-OD) polyethylene tube was

PAGE 25

25 inserted in the left ventricle (LV) serving as a vent to avoid excess hydrostatic pressure accumulation and distension of the left ventricle fr om Thebesian flow. Due to the sensitivity of diffusion weighted images to motion, the hear t was arrested prior to imaging by switching perfusate to a modified St. Thomas Ho spital cardioplegic solution (STH). MR imaging experiments were performe d on an 11.1 T/ 40 cm clear bore magnet (Magnex Instrument Inc. UK, Bruker Instrument console) with a loop-ga p coil (32 mm diameter, 40 mm height) dual tuned to 1H/19F resonances. The temperature in the magnet was 28 29C. Proton diffusion weighted images of the arrested rabbit heart with the cardiopledgic solution were acquired by applying the gradients to give diffusion sensitizing factors (b values) of 80, 160, 250, 350, 460, 580, 710, 850, and 1000, in 6 directions with a standard spin echo pulse sequence. Imaging parameters were TR = 1.5 s, TE = 29 ms, one average for all scans using = 16.5 ms, = 5.5 ms. Thus, a total of 55 scans were obtained per slice of 2 mm thickness each with in-plane resolu tion of 0.5 0.5 mm2 and data matrix of 80 80. MR images were processed with standard processing functions (Fourier transformation) and diffusion-weighted images were fit to a rank-2 tensor model of tissue diffusion. Tissue Segmentation Image segmentation can be defined as the divi sion of a particular image into distinct regions, each having different properties. In th is project, image segmentation was implemented on a voxel by voxel basis using a custom Matlab (Matlab v. 6.5.0, Mathworks) subroutine. The imaged volume was segmented into heart tissu e and non-tissue regions. Such segmentation allowed us to correctly assign myocardial tissu e properties and the desired properties for the surrounding regions. A large number of image segmentation techniques are available in literature;25 however there is no particular method that can be appl ied to all images or accepted for all imaged

PAGE 26

26 subjects. In general, cardiac ti ssue segmentation has been implemented based on characteristic feature values, i.e. relative anisotropy or fract ional anisotropy. These values are used in cardiac tissue segmentation because they describe the leve l of microstructure organization of the tissue. Since muscle tissue is highly organized, these parameters can accurately distinguish between tissue and non-tissue. Fractional anisotropy (FA) values provide a meas ure of the extent of tissue anisotropy These values were calculated from the DTI data using 222 123 222 123()()() 3 2 FA (3-1) where is the mean diffusivity ( tr(D) ) and1 2 and 3 are the principal eigenvalues of water diffusivity. FA was used to distinguish between aligned cardiac tissue (F A=1) and isotropic air (FA=0) surrounding the heart. Figure 3-2 shows the correspondence between an FA visualization map and the implemented segmentati on. For FA values greater than 0.2, the voxel was characterized as non-tissue, for values less than 0.2 it was assigned electr ical conductivity properties of heart tissue. Figure 3-2. Tissue segmentation correspondence, (A ) FA map of transverse image of the heart showing left and right ventricle walls. (B) Tissue segmenta tion (red=non-tissue, blue=heart tissue)

PAGE 27

27 Assigning Properties for Electrical Conductivity One should note that DTI measures the effective tensor of water diffusivity in tissue which is sensitive to the underlying tissue structure and G measures intracellular conductivity properties. A strong correlation is assumed between the eigenvectors of the water diffusion tensor and the eigenvectors of th e electrical conductivity tensor based on tissue microstructure in order to assign electrical propaga tion directionality. This cross property relationship has been previously studied and assigned by Tuch et al.25 DTI data was processed to assign fiber orient ation to the electrical conductivity tensor along the longitudinal, transverse and normal directions of the ti ssue using a customized Matlab subroutine. This subroutine scanne d every point of the DTI data and calculated the eigenvalues and eigenvectors at every location. After segmenti ng the tissue as menti on above, the electrical conductivity tensor components in the local coordina te system of the heart were assigned to each node by sorting them in descendi ng order and creating the matrix 11 22 3300 00 00 g g g G (3-2) where gii are the conductivity eigenvalue s and were obtained from Eason et al.26 Electrical conductivity was assigned values in the globa l coordinate system using G'=PGPT (3-3) where P is the transformation matrix and the colu mns of P are equivalent to the eigenvectors of the water diffusion tensor at each point. In this way, the principal directions of the water diffusion tensor provided the fi ber orientation and the direct ion of maximum intracellular conductivity in the globa l coordinate system.

PAGE 28

28 Finite Element Model of the Heart The electrical conductivity tens or data was assigned on a voxel-by-voxel basis to the 3dimensional model and on a pixel-by-pixel ba sis to the 2-dimensional model within the multiphysics software package, COMSOL (COMSOL Multiphysics v. 3.3, Stockholm, Sweden). Two-dimensional Model In this preliminary model, a rectangula r area of 80 x 80 mm was selected and a quadrilateral mesh was implemented in which each pixel was assigned to an element in the mesh for a total of 6400 elements. An isotropic model was first undertaken by sel ecting a particular tran sverse image of the heart near the base. For this cas e, the electrical co nductivity tensor matrix was reduced to a single-non-zero scalar value of 0.6 S/m. A vo ltage difference of 0.1 V was applied between opposing boundaries and electrical insulation to the remaining 2 edges of the model (Figure 3-3). Figure 3-3. Transverse image of the heart along the xy-plane, 2D isotr opic model for validation studies. (red=isotropic heart tissu e, blue=non-tissue surroundings).

PAGE 29

29 This isotropic model was then compared to a similar model having anisotropic tissue properties. In the anisotropic model, the spatially -varying electrical conduc tivity values from the tensor transformation were used. Three-dimensional Model A rectangular volume corresponding to a trunc ated image array was created using 24,840 quadratic brick elements, with 212,877 degrees of freedom for the dependent variable, V, and electrical properties ca lculated for each voxel. Each bric k element corresponds to an image voxel. To reduce computation time, the atria we re disregarded and only the ventricles were modeled when brick elements were used. When the total mass of the heart was required, the modeling was done with vent ricle and atria volumes. Figure 3-4. Mesh containing 24, 840 brick elements corresponding to embedded ventricles and surrounding media. The continuity equation for conductive DC me dia yields a general form of Ohms law, which for a static case states that (')0eJGJ V (3-4) where J e is an externally ge nerated current density, J is the induced current density, V is the electric potential, and G is the electrical conductiv ity. A current source term, Q, was included, and the externally generated curre nt density was eliminated. Theref ore, the following generalized equation was used

PAGE 30

30 (') GQ V. (3-5) Heart tissue was modeled as having isotropi c and anisotropic el ectrical conductivity values according to fiber orientation. In literatu re, reported electrical c onductivity values have a wide range. For the isotropic case, an el ectrical conductivity of 0.28 S/m was used 15 throughout the tissue. This value is an approximation obta ined from conductivities measured along the fiber direction, transverse to the fi ber direction and blood vessel con ductivity. For the anisotropic case, electrical conductivities were taken to be 0.625 S/m, 0.236 S/m and 0.11 S/m along the longitudinal, transverse and norma l fiber directions, respectively. 13,26 Figure 3-5. Transverse slice images of the isotro pic electrical heart model in FEM software, from apex=1 to base=12 of the heart (blue= heart tissue with is otropic conductivity, white=surrounding non-tissue)

PAGE 31

31 Figure 3-6. Transverse slice images of anisotropi c electrical heart model in FEM software, from apex=1 to base=12 of the heart. Color contours provide directional information (red=parallel to xy-plane, blue=perpendicular). Boundary conditions for the voltage differen ce simulation assumed electrical insulation on the lateral faces surroundi ng the heart, that is 0 n J (3-6) and a potential difference Vo between the base and the apex of the heart. When a current point source was modeled, electric insulation was used on all the faces surrounding the heart. A current point source was defined, namely Qo, and a ground (zero potential) point was defined on the surface of the ventricles. This arrangement is important to determine electrode placement in the heart when de fibrillation systems need to be implanted.

PAGE 32

32 Steady-state equations were solved and studies compared (a) an input potential difference between the apex and the base of the heart, and (b) current point s ources at different locations. In addition, modeling studies were done (c) that resemble external heart defibrillation using different voltage magnitudes, and (d) implantabl e cardioverter defibrillators by having a voltage point source within the left ventricle wall with different voltage magnitudes. Defibrillation Models Two types of defibrillators were modeled, external and implantable cardioverter defibrillators. For the external defibrillati on simulation, a potential difference was applied between opposing faces of the rectangular model; namely the faces corresponding to the anterior and posterior planes of a human body. This paddle placement location is re ferred to as anteriorposterior paddle placement (APR3). Neumann boundary conditions were assigned to the remaining faces so that there was no current flowing out of the volume (Figure 3-7). Figure 3-7. External defibrillation model with an isotropic electrical conductivity properties. Implantable cardioverter defibrillations were modeled as voltage point sources located (by visual inspection) at the superior wall of th e right atrium and on the lower wall of the right

PAGE 33

33 ventricle. A potential difference between thes e points was applied a nd electrical insulation boundary conditions were assigned to all th e boundaries in the volume (Figure 3-8). Figure 3-8. Implantable cardiover ter-defibrillation model with an isotropic electr ical conductivity properties. Preliminary Ishemic Model Several studies have been conducted to char acterize the remodeling of the myocardial architecture that occurs after myocardial infarction.5,17 However, the electrical implications of such remodeling are not clearly understood. Chen et al suggested that infracted heart tissue exhibits a 37% decrease in relative anisotropy. Th is value is not small enough to be considered as totally isotropic as water, but it does not exhibit the same le vel of organization as a healthy myocardium. Other studies suggest that according to the degree of myocardial ischemia, a mere decrease of electrical conductivity results, ther efore slowing down the im pulse propagation. In addition, these studies suggest a complete lack of propagation in the presence tissue necrosis.27 A preliminary model of an ischemic heart was created. To do so, a simplified ichemic geometry was implemented by introducing a cone-s hape volume within the posterior wall of the left ventricle (Figure 3-9). The ischemic region represented approximately 8% of the total heart

PAGE 34

34 tissue. A low electrical conduc tivity of 0.18 S/m was assigned to this region and the tissue was assumed isotropic.13 Simulations where a potential differe nce input was applied were done in order to compare to the healthy heart tissue model. Figure 3-9. Transverse slice images of anisotropi c electrical heart model in FEM software with infarct region (dark blue), from apex=1 to base=5 of the heart. (red=parallel to xyplane, blue=perpendicular) Time-Dependent Model A time-dependent model was implemented us ing a modified version of the FitzHugNagumo equations for excita ble media, Equation 3-7.28 A fully anisotropic electrical conductivity tensor G was implemented which has not been previously analyzes. The electrical conductivity tensor affects the speed at which the tissue is excited as well as the speed at which the tissue depolarizes. N on-linear membrane kinetics were implemented by using modified FitzHugh-Nagumo equations for excitable media 1 1111122 2 1'()(1) () u Gucuuucu t u uu t (3-7) G is the electrical conductivity tensor and as well as in the general form of the equations, u1 is an activation variable, u2 is an inhibitor, sets the excitation threshold, the excitability, and and are the parameters that describe the resting state and dynam ics of the system.28

PAGE 35

35 The boundary conditions for this simulation a ssumed that no current was flowing into or out of the control volume. Therefore, insulati ng Neumann boundary conditions were assigned to every face of the volume surrounding the heart. In itial conditions characterize an initial uniform potential of distributi on of 1 V throughout a section of the model for the activation variable, while the adjacent sections remain at zero. For th e inhibitor variable, the adjacent sections have a value of 0.3 V. Values for and were obtained from literature from an FEM model done by Filippi et al. 29 to be 0.01, 0.1, 0.5, 1 and 0 respectively, which are standard values used in simple FitzHugh-Nagumo models. Preliminary data from these models is presented.

PAGE 36

36 CHAPTER 4 RESULTS Two-dimensional Results of Validation Studies 2D heart model simulations were carried out solely for the purpose of visualizing if the data was being properly obtained from DT-MRI data, transformed into the correct coordinates, and corresponded to the expected underlying fiber tissue arrangement. Figure 4-1. Transverse cuts of the heart obtained from slice 7 of the imaging sequence. A) Fiber orientation mapping using F LTView software. B) Elect rical conductivity map in COMSOL Multiphysics software, red=paralle l to the xy-plane, blue=perpendicular. The red areas in Figure 4-1 (B) indicate the regions of larger electrical conductivity, this implies that along these areas the fibers are aligned parallel to th e plane of the tr ansverse cut. That is, the fibers are aligned almost horizon tally. This inference can be validated when comparing it with the FLTView fiber orientation map (Figure 4-1 (A)). It can be seen that the fibers in this particular image and region of inte rest (red regions) are oriented parallel to the plane of the image. These results were also va lidated by comparing them to similar studies found in literature where the fiber architecture of the heart was r econstructed by histological measurements of the fiber angle orientation.6

PAGE 37

37 Three-dimensional Results Voltage Input and Cur rent Input Simulations Significant differences can be seen between is otropic and anisotropic cardiac models in response to the potential differe nce input. Figure 4-2 illustrates the current paths taken when traveling from the apex to the base of the heart. In the isotropic model, the current direction is largely perpendicular to the transverse cut; whil e in the anisotropic model, current follows a more helical path. In addition, it can be seen that the current tends to follow the fiber orientation of the left ventricular wall. Fibers tend to lie in planes parallel to the epicardium, then rotate counterclockwise over approximately 110 with in creasing depth from the epicardium to the endocardium going through a horizontal alignment near the midwall. Figure 4-2. Current direction fo r a potential difference between the apex and base within a transverse cut of the heart 12 mm from the base. Arrows corr espond to the current direction. The magnitude of the paths of selected current streamlines between two points was compared in both models. Significant differences in magnitude were obtained. Streamlines with

PAGE 38

38 starting points at the base of th e left ventricle at ( 37 mm ,43 mm ,0 mm), and at the apex of the left ventricle wall at (39 mm, 30 mm, 10 mm) were calcul ated for current point source and voltage difference input models. Th e results of the simulations are summarized in Table 4-1. Table 4-1. Summary of the magnitude of curre nt paths at between two points at different locations with the same seed point. Simulation input Tissue property Start Point (x,y,z)( mm) End Point Avg (x,y,z) (mm) Magnitude (mm) Isotropic 37, 43, 0 38,46,16 32.6 Voltage Anisotropic 37, 43, 0 38,44,16 36.3 Isotropic 39,30,10 5,5,15 45 Current Anisotropic 39,30,10 7,7,15 91 The magnitude of the distance between the sa me starting point and an average of three adjacent ending points of current paths was calcu lated in isotropic and anisotropic models. Anisotropic streamlines were found to be signi ficantly longer. A plot of the averaged streamlines clearly delineates the differences be tween the current behavior in isotropic and anisotropic models (Figure 4-3). Figure 4-3. Comparison between current stream lines in isotropic and anisotropic models. Average from the same seed point.

PAGE 39

39 The following figure illustrates the paths cu rrent takes when a current point source was implemented as the input. This source was located in the exterior wall of the left ventricle. Path lines emerge from the point sour ce in a slightly straight fashio n in the isotropic model. Although the tissue has isotropic conductivity, the lines are not complete ly straight throughout the entire volume. Such behavior is attributed to the bound ary effects at the edges of tissue and non-tissue regions. On the other hand, current does not exhibit st raight pathlines in the anisotropic model. This again, reflects the importan ce of the underlying fiber structur e in electrical conduction in myocardial tissue. Figure 4-4. Slice image of a tran sverse cut of the heart with current path lines. A) Isotropic model, red=uniform electrical condu ctivity. B) Anisotropic model, blue=perperdicular to the xy-pl ane, red=parallel to xy-plane. Sensitivity Analysis By definition, a sensitivity analysis is the study of how the output of a model varies when certain parameters in the model are changed. This concept was applied to this model by simulating external defibrillati on and internal cardioversion.

PAGE 40

40 Different voltage values were applied to simu late external defibrillators. As an overall trend, when the magnitude of input voltage was in creased, the percentage of heart tissue nodes at certain voltage increased as well. After running these simulations, a Matl ab subroutine (appendix b) was implemented in order to see how ma ny tissue nodes had reached a certain voltage threshold under the different stimuli. The tissue and non-tissue nodes were segmented according to FA values in this case as well. Figure 4-5. External defibrilla tion at low range. Normalized nu mber of nodes of heart tissue above 5 volts at different voltage inpu ts with a surface potential difference In external defibrillation mode ls, different potentials were applied across the anterior and posterior faces of the heart for isotropic and an isotropic tissue properties. Significant differences could be observed at lower voltage inputs, 3 V to 10 V (Figure 4-5). Th e voltage input and the normalized number of nodes show an exponential relationship in the anisotropic case and a more linear relationship in the isotropic case. On the other hand, at higher voltage input values the differences between isotropic and anisotropic responses are not as evident. From Figure 4-6, one can note that an approximately linear relationship exists between the voltage i nput and the normalized number of nodes in the 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 024681012 Voltage Input (V)Normalized number of nodes (%) Isotropic Anisotropic

PAGE 41

41 range of 15 V to 35 V. Outside this range th e behavior is not quite linear as mentioned previously. Figure 4-6. External defibrilla tion at high range. Normalized nu mber of nodes of heart tissue above 5 volts at different voltage inpu ts with a surface potential difference A similar trend can be seen when modeling implantable cardioverter defibrillators. In internal defibrillation models, a point source voltage difference was applied between the right atrium and right ventricle. At low voltages (3 V to 10 V), there is a significant difference between the behavior of anisotr opic and isotropic tissue. The per centage of nodes that reach a 5 V threshold in anisotropic tissue increases radically when the voltage input reaches approximately 9 V; while in the isotropic tissue model, a more linear behavior is observed. 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 024681012Point Source Voltage Input (V)Normalized number of nodes (%) Anisotropic Isotropic Figure 4-7. Implantable cardiover sion at low-range. Normalized number of nodes of heart tissue above 5 volts at different volta ge with a point source input. 0 0.2 0.4 0.6 0.8 1 1.2 020406080100120Voltage Input (V)Normalized number of nodes (%) Isotropic Anisotropic

PAGE 42

42 Significant differences can be observed between isotropic an d anisotropic tissue models (Figure 4-8) when a voltage point source was mo deled (internal cardivers ion-defibrillation). In comparison to the results obtained when a surf ace potential difference was modeled at highrange voltages, the behavior under isotropic and anisotropic tissue properties is clearly different. There is a faster increase of the percentage of nodes above the defined threshold in the anisotropic tissue model than in the isotropic model. 0 0.2 0.4 0.6 0.8 1 1.2 5678910203040100 Point Source Voltage Input (V)Normalized number of nodes (% ) Anisotropic Isotropic Figure 4-8. Implantable cardiover sion at high range. Normalized number of nodes of heart tissue above 5 volts at different volta ge with a point source input. The differences seen at the high-range volta ge input are more important when modeling point sources than when modeling surface potential difference. Better visualization of the results is seen on the following graph (Figure 4-9). Eighty percent of anisotropic tissue reaches a voltage threshold of 5 V at approximately 18 V of vo ltage point source input. When the tissue is isotropic with the same input, eighty percen t of its nodes reach above the threshold at approximately 35 V of input. On the other hand, differences are not significant between isotropic point source and isotropic face models. Nevertheless, it can be observed that in the isotropic model, there is a faster increment of number of nodes than in the anisotropic model.

PAGE 43

43 0 0.2 0.4 0.6 0.8 1 1.2 051015202530354045Voltage Input (V)Normalized number of nodes (%) Anisotropic point Isotropic face Isotropic point Anisotropic face Figure 4-9. Normalized number of nodes of heart tissue above 5 vo lts at different voltage inputs, four-case comparison. Ischemic Tissue Results The ischemic model was compared to the hea lthy model with anisotr opic tissue properties. As it can be inferred from Figure 4-10, the re sponse difference between the healthy and the ischemic models is not major. Although healthy myocardium reached the voltage threshold at a lower voltage input, it does not ex hibit a significant difference from the ischemic heart. Eighty percent of healthy heart tissue is excited above 5 V with a vol tage input of approximately 28 V. On the other hand, 80 % of heart tissue in th e ischemic model surpasses such threshold at approximately 35 V. However, it should be point ed out that depending on the severity of the ischemia, the volume of infarcted heart tissue in creases, and so does the disruption of electrical function. A sensitivity analysis was done were different is otropic electrical conduc tivity values were assigned to the ischemic tissue region. They we re done in order to be tter understa nd the how these values impacted the result s. Electrical conductivity values of 0.35 S/m, 0.2 S/m, 0.18 S/m

PAGE 44

44 and 0.15 S/m were assigned, and the normalized nu mber of tissue nodes stimulated above a 5 V threshold results are su mmarized in Figure 4-11. 0 0.2 0.4 0.6 0.8 1 1.2 020406080100120Voltage Input (V)Normalized number of nodes (%) Healthy face potential input Ischemic face potential input Figure 4-10. Comparison between hea lthy and infarcted heart tissue above a 5 V threshold for a -0.2 0 0.2 0.4 0.6 0.8 1 1.2 020406080100120 Voltage Input (V)Normalized number of nodes (%) Infarct face potential input (0.35 S/m) Infarct face potential input (0.2 S/m) Infarct face potential input (0.18 S/m) Infarct face potential input (0.15 S/m) Figure 4-11. Sensitivity analysis of ischemic heart tissue with different electrical conductivity values.

PAGE 45

45 Time-Dependent Model Results Qualitative information was obtained from the time-dependent simulations. Although still in a developmental stage, differences could be observed between isotropic and anisotropic responses to the FitzHug-Nagum o equations. Isotropic tissue wa s excited uniformly throughout the tissue and the wavefront propagation of action potential was nearly symmetric. On the other hand, anisotropic tissue presente d varying shapes of wavefront propagation that are still under analysis.

PAGE 46

46 CHAPTER 5 DISCUSSION Fiber architecture obtained from water diffusivity (DT-MRI) was used to predict electrical conductivity in cardiac tissue and an in tracellular electri cal finite element model of the heart was created. An isotropic model was also created in order to compare the paths taken by currents under different stimulati ons conditions. In this case, fi ber orientation was disregarded and a uniform conductivity was assumed. Significan t differences were seen between anisotropic and isotropic model current paths lines. Streamlines in the isotropic model follow the shortest path between two points, while in the anisotr opic model they follow paths that reflect the underlying muscle fiber orientat ion. Current follows the high er conductivity direction when traveling between two points, and delineated th e rotating organization of the fibers from the epicardium to the endocardium in the left ventricular wall. Previous Studies Other studies have modeled the significance of fiber architectur e in electrical propagation in cardiac tissue. Knisley et al.30 examined the role of spatial variation of voltage gradients on the transmembrane voltage changes in rabbit hearts. They explored the voltages using a bidomain computer model. They incorporated 2D fiber orientation and approximated the orientation further away from th e area of interest. In comparison, our study incorporates a highresolution 3D fiber architecture a nd the corresponding electrical c onductivities in the appropriate directions. Wei et al. 31 compared isotropic and anisotropic computer heart models in body surface electrocardiograms. Their model incorporated fiber arrangement by rota ting fiber architecture counterclockwise from the epicardial layer to th e endocardial layer a total of 90. They modeled transient electrical conduction and saw no si gnificant differences in surface ECGs between

PAGE 47

47 models. Their study incorporated both the fiber architecture and the action potential propagation but only as an approximation. Its aim was to analyze the differences that could be detected in surface ECGs. Interpretation of Results and Applications The presented model attempts to specifically de scribe the behavior of current patterns and predict the percent of tissue stimulated when different stimuli are implemented. The modeling approach is also able to account for more real istic tissue properties that can more accurately predict the implications of an electrical imbalance which will be the focus of future studies. This computational model may be useful for optimizing electrode placement and also for predicting defibrillation thresholds that minimize damage of tissue. Simulation results suggest a minimum and a maximum voltage range that a subject may undertake for successful defibrillation while not suffering permanent da mages. At the time of implantation of the cardioverter defibrillators, safe ty-threshold testing is conducted.32 Ventricular fibrillation is induced at the time of implantation to test whet her or not the arrhythmia is terminated. Such testing may cause permanent damages to the tissu e and developing technol ogies to avoid such injury are potentially beneficial. There exists a critical mass hypothesis stating th at a way to end an ep isode of ventricular fibrillation is by electrically exciting a critical percentage mass of the heart.33 The exact amount of mass that needs to be electri cally activated is unclear, but es timates have established a range of 75% to 100% of the myocardial tissue.34 In addition, it has been speculated that raising a critical mass of myocardium above 5 V/cm will defibrillate the heart.34 According to this theory, our developed modeling approach may predict DFTs for implantable cardioverter defibrillators before implantation. When modeling point source inputs in anisotropic tissue, our rabbit heart model (Figure 4-8) roughly shows a voltage rang e for successful defibrillation of 17 to 20 V.

PAGE 48

48 This is significantly different if the tissue is assumed isotropic. In this case a voltage ranging from 30 V to 45 V would successfully defibrillate the tissue. Simulation results obtained when a potential difference input was applied, mimic an external defibrillation. In an attempt to include the resistivity of the to rso, without imbedding the heart model into a whole torso model, the resis tivity of non-tissue surrounding the heart was very high (1e-6 S/m). With this assumption, we could see that when the heart was modeled as anisotropic, the voltage range that would defibrillate the heart was larger than when the heart was stimulated using a point source. In a rabbit heart model, vo ltages ranging from 45 V to 100 V would excite 90% of the mass above a threshold of 5 V. Standard external defibrillation voltage thresholds for human hearts range from 200 V to 1000 V depending on the weight and diseased condition of the patient.16 Compared to this range, our results do not seem to correspond, but one should note that the heart DT-MRI data used in our model was obtained from the heart of a rabbit which is smaller than for a human and may not directly apply to values obtained in human studies. Analyzing the results of the infarcted myocardi um model, several observations can also be made. Although the percentage of he art tissue stimulated in this case did not significantly differ from that of healthy tissue, it did exhibit a different voltage distribution. Areas around the infarcted region had increased voltage values compar ed to the rest of the heart tissue. This could be attributed to boundary effects between healt hy and unhealthy regions. This behavior be of consequence due to the unorganized current pr opagation inside the in farcted region creating regions of current recirculation affecting the potential distribution around the edges of the infarct. Nevertheless, healthy and unhealt hy myocardium exhibited a comp arable percentage of tissue excitation for the different voltage inputs.

PAGE 49

49 Future Work Previous DT-MRI studies have found that infarc ted myocardium exhibits an increase in the magnitude of water diffusivity.17, 35 Future work will use DT-MRI-b ased models to account for regions of tissue damage to predict electrical prop agation imbalance. Such models will be used to analyze various infarction scenarios and dete rmine possible implications in the mechanical functioning of the heart. The developed models may also be used to understand the implications of large external electrical fields on myocardial conduction. To implement this approach one may start modeling a magnetostatic case. When modeling electric be havior of biological tissue at very low frequencies, a quasistatic approxi mation is valid. The induced elec tric field can be written in terms of the magnetic vector potential A and the electric scalar potential as 36 A E t (5-1) The tissue volume is assumed a conductive medium following the general form of Ohms Law J E (5-2) where J is the current density and is the spatially varying co nductivity tensor obtained from DTI data. In a quasistatic approximation, the divergence of the current density J is zero, so we have ()0 (5-3) Combining equations (5-1)-(5-3), we obtain ()0A t (5-4) Also, the constitutive equation for magnetic fields needs to be included. For biological tissues, the relative permeability is approximately 1, therefore

PAGE 50

50 0BH (5-5) where B is the magnetic flux density or magnetic field, 0 is the relative magnetic permeability and H is the magnetic field strength. These equations aim to predic t the effects that externally occurring electric and magnetic fields have on the electrical behavior of the hear t. In addition, they may be useful to define a near-field electromagnetics sta ndard in the presence of external electromagnetic forces. This field may be characterized by observing at what dist ance electrodes need to be from the heart, so that the effects of anisotropy can be ignored. Such analysis will be implemented in future studies.

PAGE 51

51 CHAPTER 6 CONCLUSIONS The overall goal of this project was to real istically model cardiac electrical anisotropy and run sensitivity analyzes for different input and boundary conditions. Such simulations were primarily done based on a steady-state case a nd preliminary studies were done on a timedependent model. This project also included simulations of infarcted myocardium based on reported characteristics of such tissue. As a resu lt, the general objectives of this project were achieved together with the possibility for expansion in many directions. Although an accurate heart geometry was used, there are certain limitations to the model that need to be addressed. Heart tissue consists of different kinds of cells, i.e. Purkinje fibers, SA node cells etc. These cells have different elect rical characteristics that were not taken into account.37 The tissue was assumed anisotropic throughout but with the same electrical excitation characteristics. This clearly affects the propagation patterns in heart tissue, but these issues will be addressed in future studies.

PAGE 52

52 APPENDIX A DATA PROCESSING AND TISSUE SEGMENTATION SUBROUTINE test3a_cuts_atria.m %Get eigenvalues and eigenvect ors from .flt file DTI data %Output the G tensor %and separate files containing th e anisotropic matrix values %Organizes these values in rows of 80 columns (or the size of the image) in %the x-direction %Cuts atria % DT=openFLT('dti.flt'); eigen_vec=fopen('eigen_vec.txt',' w'); %Eigenvector file eigen_val=fopen('eigen_val.tx t','w'); %Eigenvalues Gtensor=fopen('Gtensor.txt', 'w '); %Conductivity tensor e11=fopen('e11.txt','w'); e12=fopen('e12.txt','w'); e13=fopen('e13.txt','w'); e22=fopen('e22.txt','w'); e23=fopen('e23.txt','w'); e33=fopen('e33.txt','w'); sur=fopen('surface.txt','w'); %Electrical Conductivity gl=0.625; % (S/m) Parallel to myofibers gt=0.236; % (S/m) Transverse to myofibers but in the same plane gn=0.1087; % (S/m) Normal to the layer G=zeros(3,3); %Initialize matrices g=zeros(3,3); % D=zeros(80,80); E11=zeros(1040,80); E12=zeros(1040,80); E13=zeros(1040,80); E22=zeros(1040,80); E23=zeros(1040,80); E33=zeros(1040,80); for k=1:5 %Number of slices for j=14:68 %Size of the regi on of interest where the image is for i=25:70 %Size of the re gion of interest where the image is if ((i-40)^2+(j-40)^2<=33^2) [v,l]=eig(matr(DT,i,j,k)); %Function that gets eige nvalues and eigenvectors trace=(l(1,1)+l(2,2)+l(3,3))/3;

PAGE 53

53 FA=(sqrt(3*((l(1,1)-trace)^2+(l(2,2)-trace)^2+(l(3,3)trace)^2)))/(sqrt(2*(l(1,1) ^2+l(2,2)^2+l(3,3)^2))); if (FA<0.2) G(1,1)=0.0001; G(1,2)=0.0001; G(2,2)=0.0001; elseif (F A>=0.2)%Sorts and assigns values diag=[l(1,1),l(2,2),l(3,3)]; [lam,idxMax]=max(diag); [lam,idxMin]=min(diag); signEv=sign(diag); if idxMax==1 if idxMin==2 g(1,1)=gl; g(2,2)=gt; g(3,3)=gn; elseif idxMin==3 g(1,1)=gl; g(2,2)=gn; g(3,3)=gt; end elseif idxMax==2 if idxMin==1 g(1,1)=gt; g(2,2)=gl; g(3,3)=gn; elseif idxMin==3 g(1,1)=gt; g(2,2)=gl; g(3,3)=gn; end elseif idxMax==3 if idxMin==2 g(1,1)=gt; g(2,2)=gn; g(3,3)=gl; elseif idxMin==1 g(1,1)=gn; g(2,2)=gt; g(3,3)=gl; end

PAGE 54

54 end G=v*g*v'; end end if (k==6) if (FA>=0.2) aa=1; else aa=2; end D(i,j)=aa; %fprintf(s ur,'%d\t %d\t %+15.6e\n',i,j,aa); end E11(j+80*(k-1),i)=G(1,1); E12(j+80*(k-1),i)=G(1,2); E13(j+80*(k-1),i)=G(1,3); E22(j+80*(k-1),i)=G(2,2); E23(j+80*(k-1),i)=G(2,3); E33(j+80*(k-1),i)=G(3,3); fprintf(e11,'%+15.6e\t',G(1,1)); fprintf(e12,'%+15.6e\t',G(1,2)); fprintf(e13,'%+15.6e\t',G(1,3)); fprintf(e22,'%+15.6e\t',G(2,2)); fprintf(e23,'%+15.6e\t',G(2,3)); fprintf(e33,'%+15.6e\t',G(3,3)); end end end fclose('all');

PAGE 55

55 APPENDIX B TISSUE ABOVE THRESHOLD SUBROUTINE tiss_nontiss.m %Program that evaluates nodal points clear all clc format long; fidEvec=fopen('infaaniso-point5v.txt'); Det=fopen('conductivitypoint-infa.txt'); tissue=0; tissue1=0; i=1; for r=1:70920 [Evec, cnt] =fscanf(fidEvec, '%25e %25e %25e %25e\n',[1,4]); E1=i; E4=Evec(4); vol{E1}=E4; [Idx2, cnt]= fscanf(Det,'%25e',[1,3]); [Evec2, cnt] =fscanf(Det,'%25e\n',[1,1]); E11=i; E41=Evec2(1); con{E11}=E41; i=i+1; end for q=1:70920

PAGE 56

56 if ((0.65>con{q}) & (con{q}>0.18)) tissue=tissue+1; if(vol{q}>=0.005) tissue1=tissue1+1; end end end fclose('all')

PAGE 57

57 LIST OF REFERENCES 1. Streeter DD, Ross J, Patel DJ, Spotnitz H M, Sonneblick EH. Fiber orientation in canine left ventricle during diastole and systole. Circulation Research, 1969. 24(3): p. 339-347 2. Tusscher K, Hren R, Panfilov A. Organiza tion of ventricular fibr illation in the human heart. Circulation Research, 2007. 100(12): p. E87-E101. 3. Salama G, Choi R. Imag ing ventricular fibrillation. Journal of Electrocardiology, 2007. 40(6): p. S56-S61. 4. Hsu EW, Muzikant AL, Matulevicius, Pe nland RC. Magnetic resonance myocardial fiber-orientation mapping with dir ect histological correlation. American Journal of Physiology, 1998. 274(5 PART 2): p. H1627-H1634. 5. Hsu EW, Xue R, Holmes A, Forder JR. De layed reduction of tissue water diffusion after myocardial ischemia. American Journal of Physio logy-Heart and Circulatory Physiology, 1998. 275(2): p. H697-H702. 6. Nielsen P, LeGrice I, Smaill BH, Hunter PJ. Mathematical-model of geometry and fibrous structure of the heart. American Journal of Physiology, 1991. 260(4): p. H1365H1378. 7. Lorange M, Gulrajani M. A computer hear t model incorporating anisotropic propagation Model construction and simulation of normal activation. Journal of Electrocardiology, 1993. 26(4): p. 245-261. 8. Vigmond EJ, Leon L. Effect of fiber rotation on the initiation of re-e ntry in cardiac tissue. Medical & Biological E ngineering & Computing, 2001. 39(4): p. 455-464. 9. Scollan D, Holmes A, Zhang J, Wnslow RL. Reconstruction of cardiac ventricular geometry and fiber orientation using magnetic resonance imaging. Annals of Biomedical Engineering, 2000. 28(8): p. 934-944. 10. Winslow R, Scollan DF, Greenstein JL, Y ung CK, Baumgartner W, Bhanot G, Gresh L, Rogowitz E. Mapping, modeling, and visu al exploration of structure-function relationships in the heart. Ibm Systems Journal, 2001. 40(2): p. 342-359. 11. Basser P, Mattiello J, Lebihan D, Estimation of the effective self diffusion tensor from the NMR-spin echo. Journal of Magnetic Resonance Series B, 1994. 103(3): p. 247-254. 12. Steendijk P, Mur G, Van Der Velde E, Baan J. The 4-electrode re sistivity technique in anisotropic media theoretical-analysis and applications on myocardial tissue in-vivo. IEEE Transactions on Biomedical Engineering, 1993. 40(11): p. 1138-1148.

PAGE 58

58 13. Hooks D, Trew M, Caldweel B, Sands G, LeGrice I, Smaill B. Laminar arrangement of ventricular myocytes influences el ectrical behavior of the heart. Circulation Research, 2007. 101: p. E103-E112. 14. .Heart Attack and Angina Statistics American Heart A ssosiation (2003). 15. Karlon W, Eisenberg S, Lehr J. Effect s of paddle placement and size on defibrillation current distribution: a three dime nsional finite element model. IEEE Transactions on Biomedical Engineering, 1993. 40(3): p. 246-255. 16. Sobel, Rachel K. A Shocki ng Story: Handy Defibrillators. US News & World Report. 28 September 1998 17. Chen JJ, Song SK, Liu W, McLean M, Alle n JS, Tan J. Remodeli ng of cardiac fiber structure after infarction in rats quantified with diffusion tensor MRI. American Journal of Physiology-Heart and Circulatory Physiology, 2003. 285(3): p. H946-H954. 18. Le Bihan D, Denis MD, Mangin JF, Poupon C, Clark CA. Diffusion tensor imaging: concepts and applications. Journal of Magnetic Resonance Imaging, 2001. 13: 543-546. 19. Tuch DS, Wedeen VJ, Dale AM, George JS. Conductivity tensor mapping of the human brain using diffusion tensor MRI. Proceedings of the National Academy of Sciences of the USA, 2001. 98(20): p. 11697-11701. 20. Plonsey R, Barr R. The four-electrode resi stivity technique as a pplied to cardiac muscle. IEEE Transactions on Biomedical Engineering 1982. 29(7):p. 541-546 21. Fallert M, Mirotznik M, Downing, S. Myocardial electrical impedance mapping of ischemic sheep hearts and healing aneurysms. Circulation, 1993. 87: 199-207. 22. .Mathews G, Cellular Physiology of Nerve and Muscle Fourth Edition, 2003. 23. FitzHugh R. Impulses and physiological states in theoretical models of nerve membrane. Biophysical Journal, 1961. vol 1: 445-466. 24. Filippi S, Cherubini C. Multiphysics models of biological systems. Exerpt from the Proceedings of COMSOL Users Conference 2006. 25. Pham D, Xu C, Prince J. Current methods in medical image segmentation. Annual review Of Biomedical Engineering 2000. Vol. 2: 315-337. 26. Eason J, Schmidth J, Dabasinskas A, Siek as G, Aguel F, Trayanova N. Influence of anisotropy on local and global measures of pot ential gradient in computer models of defibrillation. Annals of Biomedical Engineering, 1998. 26(5): p. 840-849.

PAGE 59

59 27. Liu F, Xia L, Zhang X. Analysis of the in fluence of the electri cal asynchrony on regional mechanics of the infarcted left ventricl e using electromechanical heart models. JSME International Journal 2003. Vol (46): 1. 28. Usyk T, LeGrice I, McCulloch A. Comput ational model of three-dimensional cardiac electromechanics. Computing and Visualization in Science 2002. 4 (4): p.249-257. 29. Filippi S, Cherubini C. Multiphysics models of biological systems. Excerpt from the Proceedings of COMSOL Users Conference 2006 30. Knisley SB, Trayanova N, Aguel F. Roles of electric field and fibe r structure in cardiac electric stimulation. Biophysical Journal, 1999. 77(3): p. 1404-1417. 31. Wei DM, Okazaki O, Harumi K, Harasawa E, Hosaka H. Comparative simulation of excitation and body surface electro cardiogram with isotropic and anisotropic computer heart models. IEEE Transactions on Biomedical Engineering, 1995. 42(4): p. 343-357. 32. Russo A, Sauer W, Gerstenfeld EP, Hsia HH, Lin D. Defibrillation threshold testing: is it really necessary at the time of implantable cardioverte r-defibrillator insertion?. Heart Rhythm 2005. 2(5) : p. 456-461. 33. Mower M, Mirowski M, Spear JF, Moore EN. Patterns of ventricular activity during catheter defibrillation. Circulation 1974. 44:858-861. 34. Blanchanard S, Ideker R. The process of defibrillation in implantable cardioverterdefibrillators, edited by N.A.M.I. Estes, A. Manoli and P. Wang. New York: Marcel Dekker 1994, p. 1-27. 35. Scollan D, Holmes A, Winslow R, Forder J. Histological valid ation of myocardial microstructure obtained from diffusi on tensor magnetic resonance imaging. Ame Physiological Society 1998., H2308-H2317. 36. Norbury J. Classical electrodynamics for undergraduates. University of Wisconsin 1997. p.63-89. 37. Tsalinkakis D, Zhang H, and Fotiadis, D. Phase response characteristics of sinoatrial node cells. Computers in Biology and Medicine, 2007. Vol 27 (1): 8-10.

PAGE 60

BIOGRAPHICAL SKETCH Ana Maria Saaibi was born in 1983 in Bucaramanga, Colombia, and in the fall of 2001 she received her high-school diploma from Colegio Panamericano in her home town. In the spring 2002, she began her engineering and her collegiate tennis career at Tulane University in New Orleans, Louisiana where she double majored in mechanical engineering and mathematics. She received her Bachelor of Scien ce in Engineering degree in the fall of 2005. In 2006 after starting graduate school in the Biomedi cal Engineering department at Tu lane University, she transferred to the Mechanical and Aerospace Engineering de partment at University of Florida. She will receive her Master of Science degree in mechan ical engineering with a minor in biomedical engineering from the University of Florida in August 2008.