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Theses and Dissertations
2003
Hemodynamic modeling in human left ventricle
Kwamina Bedu-Amissah
Lehigh University
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Bedu-Amissah,
Kwamina
Hemodynamic
Modeling in
Human Left
Ventricle
May 2003
Hemodynamic Modeling in Human Left Ventricle
By
Kwamina Bedu-Amissah
A Thesis
Presented to the Graduate and Research Committee
Of Lehigh University
in Candidacy for the Degree of
Master of Science
III
Chemical Engineering
Lehigh University
4/25/03
Table of Contents
List of Tables
---------------------------------------
List of Figures
--------------------------------------
Abstract
--------------------------------------
Chapter 1 - Echocardiography
--------------------------------------- Pg. 2
Chapter 2 - Doppler Echocardiography
--------------------------------------- Pg. 3
Chapter 3 - Circulation in Heart
---------------------------------------- Pg. 5
Chapter 4 - Research
---------------------------------------
Chapter 5 - Results and Discussion
Patient I
--------------------------------------- Pg. 14
Patient 2
-------------------------------------- Pg. 18
Chapter 6 - Direction of Research
-------------------------------------- Pg. 22
111
IV
Pg. I
Pg. 8
Appendix
Appendix A - Patient 1
-------------------------------------- P g. 23
Appendix B - Patient 2
-------------------------------------- Pg. 26
Bibliography-
-------------------------------------- Pg. 30
Vita
---------------------------------~-----
11
Pg. 31
List of Tables
Table 1- Patient 1 ventricular volume from beginning to end diastole --- Pg 15
Table 2- Patient 2 ventricular volume from beginning to end diastole --- Pg 19
111
List of Figures
Figure 1: A Schematic and Real Doppler Echocardiography showing the fourchamber apical view -- Pg. 4
Figure 2: Schematic diagram showing pathways of blood flow in human heart-Pg.5
Figure 3: Pressure-Volume relation in left ventricle -- Pg. 3
Figure 4: Patient 1 A and E peak information showing variation of 2-D volumetric
flow rate across mitral valve vs % Total Diastolic Time -- Pg. 11
Figure 5: Patient 2 A and E peak information showing variation of2Dvolumetric
flow rate across mitral valve vs % total Diastolic Time -- Pg 11
Figure 6: Patient 1 A& E peak information -- Pg. 14
Figure 7: Patient 1 - Schematic diagram showing left atrial and ventricularmovement -- Pg. 15
Figure 8: Blood flow in left ventricle after time steps (a) 2 and (b) 50 -- Pg. 16
Figure 9: Blood flow in left ventricle after time steps (c) 150 and (d)350 -- Pg. 17
Figure 10: Patient 2 A& E peak information -- Pg. 18
Figure 11: Patient 2- Schematic diagram showing atrial and ventricular movement --
Pg. 19
Figure 12: Blood flow in left ventricle after time steps (a) 2 and (b)100 -- Pg. 20.
Figure 13: Blood flow in left ventricle after time steps (c) 200, (d) 300 and (e) 400-Pg.20
IV
Abstract
Heart failure is amongst the leading causes of death in all age groups around
the world. Modifications in the mechanical properties of cardiac tissue are due to the
effects of hormones, aging, unhealthy diets, smoking and others. These cause
alterations in the heart structure and hemodynamic properties, ultimately leading to
heart failure. In this study, the primary aim is to investigate the effect of left
ventricular shape on blood flow patterns. This will make it possible to study the
interaction between blood components and heart wall surface.
Depending on the flow patterns, blood components can be convected or
diffused into the heart wall tissue. The rate of transport depends on the heart shape
since diffusive or convective velocity is a function of ventricular shape. It is believed
that a hormone component ofthe blood, Angiotension II (ang II) is one cause of
hypertension. It docks on certain receptor sites on the wall and instigates
reprogramming of cardiac gene expression, resulting in cellular and molecular
changes. Sections of the heart wall with high normal velocity to it will tend to absorb
more hormones, thereby are more likely sites for the initiation of hypertension and
subsequent hypertrophy.
In this work, ventricular wall velocities obtained from patient cine-angiograms
are used to simulate blood flow patterns in the heart of a male and female patient.
Since the heart wall absorbs the hormones during the filling of the heart, only the
period of time for isovolumic relaxation and filling of the left ventricle is simulated.
Evaluation of the blood flow patterns employed a finite difference method to solve
the Navier-Stokes equation. Patient ventricular cavity volumes are also determined.
1
Chapter 1
Echocardiography
Echocardiography (ultrasound imaging) is a non-invasive tool for imaging the
heart as well as its internal and surrounding structures. The basis of ultrasound
imaging is reflection oftransmitted waves from internal organs of the body.
Ultrasound transducers using piezoelectric crystals generate sounds waves that are
mechanically or electrically swept across a tomographic plane (Otto and Pearlman [1]
- Pg. 10). The reflected waves are received by the transducer's piezoelectric crystal.
The ultrasound waves used have frequencies of 1-20MHz and travel at about 1540rn/s
in soft internal organs. Since speed is a function of medium density, this value
increases considerable for higher dense organs such as in bones.
The usefulness of echocardiography is limited by the quality of the images
that are obtained from the patient. Air-free contact between the transducer and body
is necessary by using gels to form an airless contact. The transducer's position is
adjusted to find the best acoustical window allowing clear visualization of the heart.
Also, the depth of penetration of the wave into the body is directly related to
wavelength. The shorter the wavelength, the shorter the penetrating distance and vice
versa. Thus, depending on what internal organ is being examined, one can use
shorter wavelength (higher frequency, lower depth penetration but decent image
resolution) or longer wavelength (lower frequency, higher depth penetration but
mediocre resolution) (Otto and Pearlman - [1] Pg. 2). Optimal return of reflected
ultrasound occurs at a perpendicular angle (90 of incidence. Small structures result
0
)
in scattering of the ultrasound signal instead of reflection.
2
Chapter 2
Doppler Echocardiography
The basis of Doppler Echocardiograph is scattering of ultrasound from
moving blood cells. Even though red blood cells are individually much smaller than
the wavelength of the sound waves used (7-10Jlm radius), detectable scatter occurs
due to thousands of blood cells being present in the path of the beam (Otto and
Pearlman [1] Pg. 4). A stationary target of dimension smaller than the wavelength of
the beam will scatter ultrasound in all directions, with the frequency of the scattered
signal being the same as the transmitted when observed from any direction. On the
contrary, a moving target will backscatter ultrasound to the transducer so that the
frequency observed when the target is moving toward and away from the transducer
is higher and lower than the original transmitted frequency respectively. The
difference in frequency between the transmitted and backscattered signal received at
the transducer is the Doppler shift, shown in the Doppler equation below:
DopperShift =
c(F -F )
s
r
2Fr(cosE»
(1)
Where c is the speed of sound in blood, FT and Fs are the transmitted and
backscattered frequencies respectively, e is the intercept angle between the
ultrasound beam and direction of blood flow. The multiplying constant 2 is a
correction factor for the transit time both to and from the scattering source (Otto and
Pearlman [1] Pg. 14). When the backscattered signal is received at the transducer, the
difference between the transmitted and backscattered signal is determined by
comparing the two waveforms using Fast Fourier Transform.
3
Below in Figure 3 is a schematic and real four-chamber Doppler
echocardiograph image of a human heart. The first figure is labeled to show the left
atrium (LA), right atrium (RA), left ventricle (LV) and right ventricle (RV). The
closed mitral and tricuspid valves can also be seen in the left and right ventricles
respectably.
Figure 1: A Schematic and Real Doppler Echocardiography showing the fourchamber apical view.
4
INTENTIONAL SECOND EXPOSURE
Below in Figure::; is a schematic and real four-chamber Doppler
echocardiograph image of a human heaJ1. The first figure is labeled to show the left
atrium (LA). right atrium (RA). left ventricle (LV) and right ventricle (RV). The
closed mitral and tricuspid valves can also be seen in the left and right ventricles
respectably.
Figure 1: A Schematic and Real Doppler Echocardiography showing the fourchamber apical view.
4
Chapter 3
Circulation In Heart
PUlMONARY
ARTeRY
lEfT
AmlUM
ONE·WAY
V~VES
(J:lE.WAY
V~VE
AIGIff
ATRIUM
ONE·WAY
VAlVE
LEFT
RIGIff
VENTRIClE
VENTRIClE
Figure 2: Schematic diagram showing pathways of blood flow in the human heart
Figure 2 above is a schematic diagram of the heart illustrating its chambers, blood
vessels and direction of flow of blood. It is a hollow organ with four chambers and
muscular walls (myocardium). The right atrium collects deoxygenated blood from
the body. When the atrial pressure exceeds that of the ventricle's, the tricuspid valve
is forced open and blood flows into the later. The increased ventricular pressure
pushes the pulmonic valve open and blood is pumped at low pressure into the lungs
by way of the pulmonary artery. Oxygenated blood returning from the lungs through
the pulmonary veins is collected in the left atrium before being delivered to the left
5
ventricle via the mitral valve (diastole). The left ventricle then pumps oxygenated
blood at high pressure to the rest of the body by way of the aorta (systole). During
diastole, the left ventricular wall-thickness is 20-30% ofthe external cardiac radius.
The inner wall of the ventricle is irregularly corrugated by longitudinal columns of
muscle of varying thickness. The papillary muscles project into the cavity to
complicate its shape (Caro et al. [2] Pg. 208-209).
An example of a pressure - volume relation in the left ventricle is
shown in Figure 3 below:
200
""'
a.
~
~
.....E
.. -- ..
...d>
100
~
~
Q)
(.Q
-
q;.
a..
(
/'
... EJ '
\
~
t
I
tiC
IRl
r
o
o.l--~~==::===--~~
o
100
~O
WO
I
VentrIcular volume (em')
Figure 3: Pressure-Volume relation in left ventricle (adopted from Milnor [8])
The notation is as follows:
The D-curve depicts diastolic filling. During this time, blood passively fills
the left ventricle from the atrium, followed by an additional (smaller) volume
due to atrial contraction. It is characterized by the mitral valve opening whilst
6
the aortic valve is closed. Pressure changes are low but the volume increases
immensely (almost doubles).
Curve IC portrays isovolumic contraction. Ventricular pressure highly
increases. The mitral and aortic valves are closed.
Curve EJ shows the period of ejection of blood from left ventricle. Higher
pressure in the ventricle than in aorta causes the aortic valve to open and
blood flows into arterial system. This is coupled with mitral valve being
closed, little pressure changes and high volume decrement.
Finally, the IR Curve illustrates isovolumic ventricular relaxation (systole).
Mitral and aortic valves are closed and the ventricular pressure decreases
greatly.
Doppler echocardiography measurements of the flow across the mitral
valve show two peaks: an early peak velocity (E wave) reflecting passive diastolic
filling and a late peak velocity due to atrial contraction (A wave). The normal E and
A velocity in healthy young individuals are about 1.0 m/s and 0.2 to O.4m/s
respectively (Otto and Pearlman [1] - Pg. 55). With aging, there is a gradual
reduction in E velocity and an increase in A velocity such that the ratio of E to A
velocity changes from> I in young individuals, to about 1 at ages 50 to 60, to < 1 in
older normal individuals.
7
Chapter 4
Research Work
Doppler echocardiograph images of two patients were analyzed. It has been found
that the most useful images were those showing the apical 2-chamber or 4-chamber
views because there are few intrusions (artifacts), such as from the papillary muscles.
The computation follows that outlined in Macpherson and Neti [7]. It is done on
three levels namely:
I) Continuum Scale: This involves macroscopic flows in the left ventricle. It
uses a two-step finite difference method to solve the Navier Stokes. The
continuity equation is used to develop the configuration of the source. The
source is placed in the middle of the atrium. Sinks are also place around the
ventricle to ensure conservation of mass.
2) Blood Cell Scale: This level analyzes the interaction at blood cell size level.
It uses a Monte Carlo process.
3) Receptor Interaction Level: This level studies the interaction of blood
components (specifically Angiotension II) with receptors (known as ATR1) on
the heart wall. It uses a direct simulation method known as molecular
dynamics.
The Continuum Level - Sources and Sinks
The simulation focuses on modeling flows in the left ventricle. Assuming the
pulmonary and aortic valves are closed, inflow to the atrium is simulated by
8
distributed sources within the atrium. According to Peskin [4], the continuity equation
can be written as:
V11 == cp(r,t) == Q(t)cpo (r) == 0
---- (1)
This is in order to have the source strength determined at any instant by the
volumetric rate of flow, Q (t).
From (1)
dV
dr
-
= Q(t)cp
0
(r)
---- (2)
Let:
1
v = Q(t) => dV =- Q(t)2 => CPo (r) =- 2rcr
2
2Jr.r
dr. 2rcr
---- (3)
Hence, the configuration ofthe sources can be expressed as:
qJ
o
1
=- - - - - ----
---- (4)
2Jr.((x-x a)2 +(y_ Ya)2)
where (xa,ya) is a point in the middle of the atrium.
Also, since the domain is periodic,
JV .v J
rp
0
(r) dv
o
---- (5)
Thus, a sink is needed corresponding to the atrial source (conservation of mass).
Using two non-negative weight functions with integral 1:
---- (6)
Wa and We represent the source and sink functions respectively.
9
It should be noted that since all modeling is in 2-D, we have to convert areas
to volumes of the cardiac chamber. Peskin and Printz suggest multiplying the area
obtained with a characteristic depth of 1.84cm (Peskin and Printz [5] -pg 40). Q (t)
used in the above equations has dimensions (cm2/s).
From (3):
Vet) = Q(t) => Q(t) =2m-V(t)
21r.r
---- (7)
Figures (4) and (5) below are the 2D volumetric flow rates across the mitral
valve for a female (Patient 1) and male patient (Patient 2) aged 39 and 72
respectively. These were obtained from the echocardiograph showing the A and E
peak (See Figures 1 in Appendices A and B). In Figure 4, the E peak is higher than
the A peak. This is normal and signifies a healthy functioning heart. The first peak
shows passive filling of the ventricle and it takes up about 60 % of the total diastolic
time. The small atrial contraction (Peak E) is necessary to totally empty the atrium.
This takes up the remaining 40% ofthe total diastolic time.
10
Patient 1: 2- D Volumetric Flow Across Mitral Valve vs %
Total Diastolic Time
1400
1200
1000
800
600
400
~
.
•• •
• •
• •
••
•
•
A •
•
•••
200 ~ •
•• .t.,
•
•
•
•
O.------r------,----r------.-----....
~
~
~
a
•
E
•
20
40
60
100
80
% Total Diastolic Time
Figure 4: Patient 1 A and E peak information showing variation of 2-D volumetric
flow rate across mitral valve with % Total Diastolic Time
Patient 2: 2 - D Volumetric Flow Across Mitral Valve vs %
Total Diastolic Time
E
en
e
() Q)
~ 1000
E
~ ~ ~
600
Q)
400
E
:::l
o
>
••
800
2>:0
()
:::l
_()
.-L..
"l
200
• • •• ••
..• E··
•• ••
••
••
A
•••
•
•
••
••
• ••
O-.--------r------,------.------,----~
o
20
40
60
80
% Total Diastolic Time
Figure 5: Patient 2 A and E peak information showing variation of2Dvolumetric
flow rate across mitral valve with % total Diastolic time
11
100
Figure (5) has an opposite peak height configuration. The A (second) peak is higher
than the first, and takes up about 60% of the total diastolic time. This suggests that
the ventricle is less able to relax. The cardiac muscles have enlarged through
thickening and shortening of its fibers due to increased resistance in circulation. This
is a hypertrophied heart. The large A peak signifies strong and prolonged atrial
contraction as it tries to empty its content into the stiff ventricle.
The Continuum Level - Modeling the Flow
The Navier Stokes equations defined on an x-y Cartesian co-ordinate system
are:
J
p ( au
at + U"• V U" + vP = It V 2u" ~ ft
(8)
where it is the velocity vector, p is the density, t is the time, p is the pressure and
viscosity is f.l. ft is the boundary force arising from the heart muscles. It is defined
from Peskin [5] as:
ft(x,t) =J](s,t }5(x - X(s,t )~s
o
---- (9)
J is the boundary element force at the point s defined on a Lagrangian system and x
is defined on the Cartesian system.
X" is the nth point on the Lagrangian system.
The delta functions are defined as:
---- (10)
12
~
_I (1 +cos 1lX )Ixl 2h}
0" ()
x = 4h
2h
{
o
Ixl ~ 2h
---- (11)
These can be rewritten as:
2
l\(r)
3-21 rl-h!1+41 rl-41 r
8
1
,
1
for Ir I~ 2
rl~~
X
r=-
h
---- (12)
---- (14)
Implementation of the above equations employs a finite difference method using
three correction steps for the time advance of velocity. The boundary points X are
advanced according to the relation [7]:
i n+1(s)
= in (s) + MLU n+1(x)o" (x - in (s))h 2
x
---- (14)
The numerical method is fully detailed in Peskin and Printz [5], Peskin [4],
Macpherson and Neti [3], [7].
With the aid of software, the co-ordinates for the shape of the left ventricle
and atrium at various times in diastole are obtained. This data is differentiated with
respect to time to obtain the velocities at each point as a function of time, ensuring
that the correct ventricular shape variation is obtained. The blood flow velocity
closest to the membrane is used as the bulk flow input to the blood flow scale
calculation. Note that the mitral valve is excluded because of disturbances it causes
in the flow profile of the present model.
13
Chapter 5
Results and Discussion
Patient 1 Analysis
Patient 1 is a 39yr old female. As shown in figure 6, the E to A height ratio is about
1.72. Passive filling of the atrium takes about 60% of the total diastolic time and
involves high velocities. Correspondingly, ventricular volume increases rapidly
initially to accommodate this as shown in Figure 7. Atrial contraction time is 40%
of the diastolic time and is slow. The ventricular outward movement is slow towards
the end.
Patient 1: Flow Velocity vs %Time
100"T
~ .----------------------,
90
E
80
~.
70
~
••
•
• •
rJ)..--
60
co ~
40
'g
30
~
20
~
10 ~
\
0 .--------,------,,------,--------,----~
~ ~ 0
t3 E 5 ~~
~
IT:
•
•
•
•
o
20
•
•
•
• •
•
• ••
•••
••
•• • •
60
40
% Time
Figure 6: Patient 1 A& E peak information
14
80
100
Figure 7: Patient 1 - Schematic diagram showing left atrial and ventricular
movement
Table 1: Patient 1 ventricular volume for beginning and end diastole
Patient 1
Area (em.!)
First Frame
16.14
29.70
Last Frame
39.93
71.93
15
Volume (cm
j
)
Table 1 shows blood ventricular volume from beginning to end diastole. This is
obtained by multiplying the ventricular area by a depth of 1.84 cm as suggested in
Peskin and Printz[7]. The volume increases by a factor of about 2.
Patient 1 Simulation Output at Different Time Steps
(a)
(b)
Figure 8: Blood flow in left ventricle after time steps (a) 2 and (b) 50
16
(d)
(c)
Figure 9: Blood flow in left ventricle after time steps (c)150 and (d)350
Figures 8 and 9 show advancing flows in the ventricle. The frames show the flow
velocity increasing with time in the ventricle (from (a) to (d». Assuming that
diastolic filling takes about 0.5s, we used a time step of 5 E- 4 in our calculation (thus
1000 times steps are needed to complete the simulation). Obtaining the correct and
complete simulation entails trail and error and experience. Currently, reasonably
flows have being obtained only to about the 400 th time step, after which the flows
become turbulent. This is because once the pressure difference between the atrium
and ventricle is not high enough, there is insufficient driving force and the velocity
vectors start changing direction rapidly and the flows become turbulent.
17
Figure 9-(d) above shows the flows arising at 350th time step (0.175s). The blood
velocity across the mitral valve is about 26cm/s. The highest normal velocities are
those vectors directed towards the bottom of the ventricle as shown by the arrow.
These are flowing at about 3.lcm/s.
Patient 2 Analysis
Patient 2 is a 72 yr old man. The E to A peak ratio (0.44) is very low. The
percentage of the total diastolic time occupied by the first peak is also low (about
40% instead ofthe healthy 66% time). The remaining time is occupied by the A-peak
since strong prolonged atrial contraction is needed to totally eject blood from the
atrium. The ventricle initially gradually moves outward. During atrial contraction,
the outward movement is more sudden
Patient 2: Flow Velocity vs % Time
100 + - - - - - - - - - - - - - - - - - - - - - - - - - ,
90 ~~
en_
en
en
0 __
5(IJ UE
~'Q;'
.>
u_
o (IJ
>
>
Q)
~
u::
80
•
70 ~~
60 ~~
50
•• •
••
~~
40
30
201
10
•••
• •
•
•
•
•••
••
• •
•
• •
•
•
••
••
....
O..--------r----,------,------r----.,..
o
20
40
60
80
100
% Time
Figure 10: Patient 2 A& E peak information
18
Figure 11: Patient 2- Schematic diagram showing atrial and ventricular movement
Table 2 shows blood ventricular volume from beginning to end diastole.
Table 2: Patient 2 ventricular volume for beginning and end diastole
Patient 2
Area (cm l )
Volume (em:';)
First Frame
19.52
35.92
Last Frame
34.29
63.09
19
Patient 2 Simulation Output at Different Time Steps
(~
(b)
Figure 12: Blood flow in left ventricle...after time steps (a) 2 and (b)100
..,.
'.
,;
(d)
(e)
Figure 13: Blood flow in left ventricle after time steps (c) 200, (d) 300 and (e) 400
(c)
20
Figures 12 and 13 above are the flow patterns for Patient 2 for 400 time steps. At this
time, the velocity across the mitral valve is about 26.2 cm/s. Again, the arrow shows
the direction of the highest normal velocities to the ventricular membrane. It is about
1.72cm/s for this patient.
Although the full simulation for both patients is yet to be done, both profiles
suggest that the lower part of the ventricle away from the septum could be the site
with absorbs most Angiotensin II and could be the potential site for the initiation of
hypertrophy.
21
Chapter 6
Direction of Research
Weare working to improve the simulation and hope to publish extensive
results soon. This means developing more code to mechanize the process as much as
possible. This will be a powerful tool for clinical and health industry because it will
enable measurements of the effects of specific loads (such as different drug dosages)
on hemodynamic properties in the heart. This will greatly improve drug
development. Also, we hope to be able to extend this research to flows in other
organs of the body.
22
Appendix A - Patient 1
0:06:57
li3
H -71b
10
m/s
Figure 1: A and E Peak Echocardiograph
Patient 1 Diastole Cycle
Figure 2: Frame 1
23
Figure 3: Frame 2
Figure 4: Frame 3
24
Figure 5 : Frame 4
Figure 6 : Frame 5
25
Appendix B- Patient 2
~
~~
J7l4-
~
""
w
,
...
1B'M1U:
:tm:~
'
DM.i~
:~;t=. ~ .
,
~-'l""3I~..
w
."
;~.
&1
oiRl.-..:"
.3
-~.......- - - - "
n:ij'i~1
1=
r ~.
.........
1·:
.. ~
;\~;':~
.....
;,
..
~
.. "
~
--m
Figure 1: Patient 2 A& E peak echocardiograph
;!!!~.~v~~
~2:
~:s:
~~
::!D'~
na!l'e;iiGb
-m
26
INTENTIONAL SI;COND EXPOSURE
Appendix B- Patient 2
~~~',:':~:
~-~J.i
it.:.',,' _~ti:
7=-,,11""",: .... ~j,~~.;::';':::.
-;-s;;~
'i:':S_:,"
;:'=,?"
~=;~1
'-~S:.::""fIt ::~"'::
,_
W".::::. ,. .
-~
~
-r--.
'::fl"~._ ..:.:~.Z;
=""iI:;;'~;:
==_~
:::'-.=... JI."::
.~~,
~~'I=
:::.:r
;:;".,.~
r'~'F';':~=
-all;
:?=;;-
=~'~1
:f;:'~
-~~~
_
;P:~':~-:'::
4~---".---
Figure 1: Patient 2 A& E peak echocardiograph
.
~_-:~
"",,-::'.'
~~-2 ,JL_/::':!f-~~:.:I.
-~~-~
-;
~:=~;.::
I
.:
='~.~ L.::~
~;-:"::-~-:-,>.?:,.e~
=~ ~e:~-~~
'~
....
'--.'
'
.--~-,-;:
-~----;-
M§§§-- ~"'--''-","-, ..""",''--~~~~''--------='-~~"'~-'''':''-::....-.'-'
.~
.'- ..
=-F-~
26
Figure 2: M-mode tracing at left ventricular papillary muscle level
Figure 18 above is patient 2's m-mode tracing at the left ventricular papillary muscle
level. It shows the change in length from the shortest length (cm) with time (sec)
based on the echo data ofm-modes, mitral size and ventricle size. The change in
width of the ventricle at the papillary muscle is 4.5 cm to 6.5cm. The septum moves
quite a bit too.
Diastole Cycle
,
--"lIi..'--Figure 3: Frame 1
27
Figure 4: Frame 2
Figure 5: Frame 3
28
Figure 6 : Frame 4
Figure 7 : Frame 5
29
Bibliography
[1] Otto Catherine M. and Pearlman Alan S. Textbook of Clinical Echocardiography
[2] Caro CG, Pedley TJ, Schroter RC and Seed WA The mechanics ofcirculation
(Oxford University Press),pp.209 1978.
[3] Macpherson AK and Neti S 2001 A rapid procedure for initial drug evaluation,
Phys. In Med. And BioI. 46,pp 6-, 2001
[4] Peskin CS,1977 Numerical Analysis of Blood Flow in the Heart,J. Compo Phys
25,pp.220-252, 1977
[5] Peskin CS and Printz BF, Improved Volume Conservation in the Computation of
Flows with Immersed Elastic Boundaries J.Comp. Phys 105, pp.33-46, 1993.
[6] Hoppensteadt, Frank, C. and Peskin, Charles S. Modeling and Simulation in
Medicine and the Life Sciences (2nd Edition).
[7] Macpherson AK and Neti S 2001 Effect of Angiotensin II on heart blood flow and
hypertension, 2002.
[8] Milnor, William R. Cardiovascular Physiology 1990
30
Vita
I am the first of five children of Joseph and Kwansima Bedu-Amissah born on the
30 th ofDecember, 1978 in Takoradi, Ghana (West Africa). Six years ago, I opted to
attend college in the States over my native Ghana. I enrolled at Knox College
(Galesburg - Illinois) for a 3 semesters before transferring to Iowa State University
(Ames, Iowa) to finish my bachelor's degree. In May 2001, I received a B.Sc in
Chemical Engineering. In the fall ofthe same year, I enrolled in Lehigh University's
graduate program in Chemical Engineering.
31
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