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CONTINUOUS LEFT VENTRICULAR EJECTION FRACTION
MONITORING BY CENTRAL AORTIC PRESSURE
WAVEFORM ANALYSIS

presented by

Jacob Andrew Gerrit Kuiper

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CONTINUOUS LEFT VENTRICULAR EJECTION FRACTION MONITORING BY
CENTRAL AORTIC PRESSURE WAVEFORM ANALYSIS

By

Jacob Andrew Gerrit Kuiper

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Electrical and Computer Engineering

2007

ABSTRACT

CONTINUOUS LEFT VENTRICULAR EJECTION FRACTION MONITORING BY
CENTRAL AORTIC PRESSURE WAVEFORM ANALYSIS

By
Jacob Andrew Gerrit Kuiper

Left ventricular ejection fraction is one of the most significant measures of
heart health used in medical practice today. Unfortunately, measuring left
ventricular ejection fraction through imaging (e.g. echocardiography), the method
used most commonly in clinical practice, requires heavy machinery and a skilled
operator. Thus, measurements can only be taken periodically. To combat these
disadvantages, we have developed an algorithm that uses central aortic blood
pressure waveform analysis to continuously (i.e. automatically) determine left
ventricular ejection fraction. To validate this technique, we have used
hemodynamic data collected from nine dogs placed under a variety of
hemodynamic stresses and instrumented to provide the data necessary for
analysis. The results of this data analysis show a strong agreement between the
measured left ventricular ejection fraction and that estimated by the algorithm.
With additional testing, this algorithm could be used to continuously and
automatically measure left ventricular ejection fraction in situations where an
aortic catheter is already used. Future efforts to adapt the algorithm to routinely

measured peripheral artery pressure waveforms are warranted.

ACKNOWLEDGMENTS

I would like to thank all of those who have aided me in my research that is
presented in this thesis.

First, I would like to thank my advisor Dr. Mukkamala who has inspired me
to do this work and supported it at every step. I would also like to thank him for
taking me onto his research team my freshman year when I had so little to offer.

I would also like to thank Javier A. Sale-Mercado, Robert L. Hammond,
Jong-Kyung Kim, Larry W. Stephenson, and Donal S. O’Leary from Wayne State
University and N. Bari Olivier from the Michigan State University Department of
Small Animal Clinical Sciences for their collection of experimental-data used in
this work.

Additionally, I would like to thank Zhenwei Lu, Ying Li, Ahmet Turkman,
Xiaoxiao Chen, Kabi Prakash Padhi, and Gokul Swamy for everything they have
done in helping me to develop as a researcher and for their aid with this research.

Lastly, I would thank my friends and family for all of their support

throughout my education and in this research.

iii

Table of Contents

List of Tables ........................................................................................ v

List of Figures ................................................................................... vi

. Introduction .................................................................................. 1
1.1. Significance1
1.2. Current Methods1
1.3. Limitations of Current Methods .............................................. 4
1.4. Proposed Solutron5
1.5. Similar Work ........................................................................... 6
. Method ................................................................................ 8
2.1. Introduction to the Method ..................................................... 8
2.2. The Lumped Parameter Model of The Circulatory System ............... 8
2.3. The Raised Cosine Model of Time Varying Elastance ............... 13
2.4. Derivation of the Mathematical Method ................................. 15
2.5. Faster Execution of the Method ............................................. 21
. Expirements ................................................................................ 26
3.1. Introduction to Experiments ....................................................... 26
3.2. Sonomicrometry Crystals and Chronic Pacing Experiments .......... 26
3.3. Dobutamine .................................................................... 30
3.4. Echocardiography and Pharmacological Interventions ........... 32
. Results .................................................................................................... 35
4.1. Verifying our Assumptions ............................................................. 35
4.2. Curve Fitting Examples ............................................................ 36
4.3. Results during Chronic Pacing Induced Heart Failure ................... 37
4.4. Results during Dobutamine Infusion ......................................... 41
4.5. Results during Drug Infusion Compared to Echocardiography ......... 43
4.6. Real Time Evaluation .................................................... 45
4.7. CaEmax Estimation ................................................ 46
. Discussion ...................................................................... 49
5.1. Parameters Being Estimated ................................................ 49
5.2. Algorithm Speed .......................................................... 56
5.3. Unstressed Volume ..................................................... 58
. Conclusion ......................................................................................... 60
6.1. Summary ............................................................ 60
6.2. Future Research ......................................................... 60
References .......................................................................................................... 62

iv

List of Tables

Table 1: A summary of the current methods for measuring left ventricular
ejection fraction and their advantages and disadvantages .............................. 4

Table 2: A summary of the parallels between hydraulic systems and electrical
systems ................................................................................ 9

Table 3: A summary of the mean values of certain important hemodynamic
parameters measured in the chronic pacing experiments during control
experiments. This table shows the wide range of hemodynamic parameters seen
between dogs in their control states ...................................... 29

Table 4: A summary of the mean values of certain important hemodynamic
parameters measured in the chronic pacing experiments during heart failure
experiments. This table shows the wide range of hemodynamic parameters seen
between dogs in their heart failure states as well as the large contrast between
the hemodynamic state of the dogs in their control and heart failure
states ....................................................................................... 29

Table 5: A summary of the mean values of certain important hemodynamic
parameters measured in the dobutmaine experiments during control experiments.
This table shows the wide range of hemodynamic parameters seen between
dogs in their control states ...................................................... 31

Table 6: A summary of the mean values of certain important hemodynamic
parameters measured in the dobutamine experiments during dobutamine
administration. This table shows the wide range of hemodynamic parameters
seen between dogs during dobutamine administration as well as the large
contrast between the hemodynamic state of the dogs during control and
dobutamine administration experiments ............................................. 32

Table 7: A summary of the mean values of certain important hemodynamic
parameters measured in the echocardiography experiment ............................ 34

Table 8: A summary of the results of two variations of our method in dog 1 of the
dogs used in the dobutamine experiments ................................... 56

Table 9: A summary of the results of two variations of our method in dog 2 of the
dogs used in the dobutamine experiments ................................. 56

Table 10: A summary of the results of two variations of our method in dog 3 of
the dogs used in the dobutamine experiments ................................ 56

List of Figures
Figure 1: Illustration of LVEF measurement through echocardiography .............. 3

Figure 2: A lumped parameter model of the circulatory system which uses a
current source with the value of cardiac output to represent the left ventricle ..... 10

Figure 3: A lumped parameter model of the circulatory system which uses a time
varying capacitor to represent the left ventricle ............................... 12

Figure 4: A summary of empirical data normalized to Emax, Em, and Ts ............ 13

Figure 5: A single heart beat taken from a CAP waveform with little to no effects
from wave reflections .................................................................... 19

Figure 6: A summary of the instrumentation of the subjects used in the first set of
experiments which concentrate on reduced cardiac function ............................ 28

Figure 7: A summary of the instrumentation of the subjects used in the second
set of experiments which concentrate on drug induced increased cardiac
funcfion ................................................................................................. 30

Figure 8: A summary of the instrumentation of the subject used in the last
experiment which analyzes the dog in states of both increased and decreased
heart function ............................................................................... 33

Figure 9: Examples of the effectiveness of the technique of Bourgeois et al. The
graph on the left is of a dog in control state while the graph on the left shows the
technique on the same dog in chronic pacing induced heart failure ............. 35

Figure 10: Examples of the results of the left and right sides of equation 8 for the
optimal elastance curve for two different beats .................................................. 37

Figure 11: Summary of the results of all five dogs used in the chronic pacing
induced heart failure experiments ............................................................ 38

Figure 12: Beat by beat scatter plot comparing sonomicrometry LVEF to LVEF
estimated by our method on a dog in chronic pacing induced heart failure ......... 40

Figure 13: Results of our method on dog one of the dobutamine experiments.
The middle of each bar is the mean of the results under the conditions listed on
the x axis and the bottom and top halves represent the standard deviation of the
measurements .......................................................................... 41

Figure 14: Results of our method on dog two of the dobutamine
experiments ............................................................................ 42

vi

Figure 15: Results of our method on dog three of the dobutamine
experiments .................................................................................. 42

Figure 16: Results of our method on our last experiment. The line corresponds
to the estimated LVEF based on our method while the circles correspond to
actual LVEF based on echocardiography measurements. ......................... 44
Figure 17: CaEmax results of our method on our last experiment ..................... 47
Figure 18: Graphs of the normalized elastance curve used to model the left
ventricle in our method and a central aortic blood pressure waveform with
important timing parameters note .................................................. 51

Figure 19: A standard ECG waveform with important characteristics
labeled ............................................................................. 54

vii

1 . Introduction

1.1 Significance

Left Ventricular Ejection Fraction (LVEF), the ratio of stroke volume (SV)
to end—diastolic volume (EDV), is one of the most powerful clinical measures of
cardiovascular function [Katz 1992]. For instance, if a patient being monitored
has been shown to have low cardiac output (CO), the rate of blood being
pumped out of the heart, then the ejection fraction can be used to help determine
what kind of problem the heart is having. If LVEF is low, then there is a problem
with the left ventricles ability to pump blood out (systolic failure); however, if the
LVEF is normal or high, then the problem is somewhere within the filling
mechanism of the ventricle (diastolic failure). In addition, epidemiological data
has shown a strong relationship between LVEF and outcome in outpatients with
heart failure [Curtis et al. 2003]. Moreover, LVEF is being considered as a
means to determine what type of treatment patients should receive. It has been
recommended that an implantable device, the internal defibrillator, which is very
expensive but potentially life-saving, should be used in cases where LVEF is less
than or equal to thirty percent [Moss et al. 2002]. These, along with other
applications, make the measurement of LVEF an important part of hemodynamic

monitoring.

1.2 Current Methods
LVEF is currently measured in clinical settings through a variety of

methods. The most common clinical techniques for measuring LVEF are

radionuclide angiography, echocardiography, and magnetic resonance imaging
(MRI). Radionuclide angiography, the most invasive of the three methods, is
done by injecting a small amount of radioactive material into the blood stream
and using cameras designed to detect the radioactively marked blood as it
travels through the body. A specialist will then review the recordings of this
camera and use them to determine what problems may exist with the heart and
circulatory system, as well as LVEF. Radionuclide angiography is considered the
most accurate of the commonly used clinical techniques for measuring LVEF.
Echocardiography, also known as trans-thoracic echocardiography, is the most
commonly used method for determining LVEF in the clinical setting [Rumberger
et al. 1997]. LVEF is measured through echocardiography by placing a person
under a machine which records images through the use of sound waves. These
images are then reviewed by an expert and LVEF is determined based on the
volume of the left ventricle at the end of diastole and at the end of ejection. An
illustration of this method can be seen in Figure 1. Echocardiography has been
shown to compare closely with radionuclide angiography but the measurements
do not match exactly [Habash-Bseiso et al. 2005]. MRI and other imaging
techniques are also sometimes used following a similar method to that of
echocardiography. These methods are less common than echocardiography
because they come at a higher cost.

There are also several methods for measuring LVEF that have been

suggested or proven in research settings but have not been put in to clinical

 

Figure 1. Illustration of LVEF measurement through echocardiography
practice due to the invasiveness or other difficulties of the procedures. The first
of these methods is the non-imaging nuclear monitoring method [Dellegrottaglie
et al. 2002]. This method relies on the same principles as radionuclide
angiography but uses an automated nuclear probe which must be placed on the
body in lieu of the imaging camera and professional analysis. Additionally, a
conductance catheter method has been developed to automatically and
continuously monitor LVEF [Burkhoff 1990]. This method requires the insertion
of a catheter into the body to measure left ventricular volume and from that
ejection fraction is calculated. Implanted sonomicrometry crystals can also be
used to measure left ventricular volume and LVEF [Rushmer et al. 1956]. This
method requires open heart surgery to implant sonomicrometry crystals at certain
locations on the heart. These crystals then continuously measure certain
dimensions of the heart which can be used to calculate the volume of the heart
based on the known geometry of the heart. A summary of the available methods
for measuring LVEF along with their advantages and disadvantages can be

found in Table 1.

 

METHOD IADVANTAG ES 1DISADVANTAGES

 

Conductance catheter continuous Jinvasive, inaccurate

 

continuous therrnodilution continuous right ventricular EF only

 

 

 

Imaging (e.g., echo) non-invasive lexpert operator, expensive

 

non-imaging nuclear monitorlnon-invasive, continuous 00 difficult to position

 

 

 

Ultrasonic crystals continuous llhoracotomy

 

 

Table 1. A summary of the current methods for measuring left ventricular
ejection fraction and their advantages and disadvantages
1.3 Limitations of Current Methods

All of the current methods have limitations which keep them from being an
optimal solution for patient care. The imaging methods (e.g. echocardiography)
generally share the downfall of being operator and equipment intensive. This
causes two major problems. First, there is a high cost to do imaging based
measurements of LVEF. This high cost results from the high cost of equipment
and the high cost of skilled operators for a long period of time per measurement.
This high cost may be one factor that prohibits there being enough
measurements of LVEF to properly track the progress of the patient. It is
uncommon in clinical settings to have LVEF measured more than one or two
times a day and that is only done when there is reason to believe that there has
been a significant change in hemodynamic state. The fact that LVEF
measurement through imaging techniques is so operator intensive also prevents

the measurements from being continuous. A continuous measurement of LVEF

would be advantageous because it would allow the physicians caring for the
patient to immediately see the effects of any treatments they may be
administering.

The continuous and automatic methods mentioned in the previous section
(e.g. sonomicrometry crystals) allow for continuoUs measurement of LVEF but
also have their own limitations. Although the specifics of these limitations vary,
the basic reasons these methods are not used clinically are generally the same.
First, these measurements require highly invasive procedures which add a large
amount of risk to the measurement of LVEF. Since this risk can be avoided by
using imaging methods, imaging methods are much more commonly used at the
cost of continuous measurement. These methods also do not alleviate the
limitations caused by high cost. Since these methods require highly invasive
procedures, they also require highly skilled medical professionals to administer
them. These limitations prevent the current continuous and automatic methods

from being used in practice.

1.4 Proposed Solution

In order to measure LVEF continuously, automatically, and non-invasively
(or minimally invasively), we propose, as our overall hypothesis, using
mathematical analysis of routinely measured blood pressure waveforms to
estimate LVEF. Specifically, the ultimate goal of this research is to use
peripheral blood pressure waveforms, along with a model of the hearts

functionality, to calculate LVEF continuously. In this thesis, I present a method

for calculating LVEF from the central aortic blood pressure (CAP) waveform.
Although this waveform is not commonly measured in clinical practice due to the
invasive nature of its measurement, analysis of these waveforms is simpler than
analysis of peripheral blood pressure waveforms as will be described later. Thus,
this thesis may be viewed as a first step towards proving our overall hypothesis
and monitoring LVEF from routinely measured peripheral blood pressure

waveforms.

1.5 Similar Work

There have been related attempts in the past to use a modeling based
technique estimate hemodynamic parameters from blood pressure. First, a
method presented by Guarini et al. uses a model based approach to measure
many different hemodynamic parameters including EDV (from which LVEF can
be calculated since CO is also measured in this case) from arterial blood
pressure waveforms and a cardiac output measurement [Guarini et al. 1996].
This methodology yielded good results using computer generated data but was
not verified against a gold standard in experimentally collected data. This lack of
accuracy with actual data suggests although some assumptions about some of
the parameters were made, the method was not able to predict this many
parameters. The other downfall of this method is that it requires the continuous
measurement of CD which requires an operator and invasive procedures to
measure. For this reason, the method has little advantage over currently

available methods for measuring LVEF.

In addition, a method developed by Xiao et al. attempts to use certain non-
invasive blood pressure measurements and a hemodynamic model to calculate
certain parameters of heart functionality. The method showed promise in
calculating total systemic resistance and some promise in calculating other
parameters such as EDV and maximum left ventricular elastance (Emax). This
method, however, did not show promise in accurately estimating LVEF or SV

(from which LVEF could be calculated given the estimation of EDV).

2. Method

2.1 Introduction to the Method

Our model based method for estimating LVEF from the CAP waveform is
based off of two previously developed models. The first model used in our
method, a lumped parameter model using the Windkessel model to represent the
arterial branches, is a model of the arteries. The second model used is a model
of left ventricular elastance (Elv) over time which reduces the elastance over a
heart beat to a function of several parameters. These two models will allow us to
do a minimum square error parameter fitting which will allow for estimation of

LVEF.

2.2 The Lumped Parameter Model of the Circulatory System

A lumped parameter model of the circulatory system allows for the
physiology of the system to be represented by only a few values which
summarize the effects of all of the blood vessels throughout the body and the
heart. The cardiovascular system can be viewed as a hydraulic system for the
purpose of these models. Hydraulic systems parallel electrical systems in many
ways. A summary of the analogous terms used in hydraulic systems versus
electrical systems can be seen in Table 2. All relationships found in electrical
systems between the values summarized are maintained in hydraulic systems.
This terminology will be used in order to treat the analysis of the cardiovascular
system as circuit analysis to derive the necessary equations to accomplish our

goal.

 

 

 

 

 

 

Hydraulic Variable Electrical Variable
Pressure Voltage
Flow Current
Volume Charge
Resistance Resistance
Capacitance Capacitance

 

 

 

 

Table 2. A summary of the parallels between hydraulic systems and
electrical systems

Both of the lumped parameter models I will discuss in this thesis involve
the Windkessel model. The Windkessel model is a model of the body’s blood
vessels which assumes that they can be modeled by a parallel combination of
resistance, which represents the smallest blood vessels, and capacitance, which
represents the large arteries that store blood. The heart can then be modeled in
one of two ways to complete the cardiovascular picture. First, the heart can be
modeled as a current source where the current represents cardiac output, the
blood flowing out of the heart. Figure 2 shows a lumped parameter model of the
circulatory system using a current source to represent the heart.

This version of the lumped parameter model has been shown to work in a
method to estimate proportional SV from the CAP waveform [Bourgeois et al.
1976]. Bourgeois et al. used a CAP waveform measured through aortic
catheterization and this model to estimate proportional SV. The test animals
were also instrumented with a highly invasive aortic flow probe which gave the

experimenters the exact value of SV to compare to their results. The

experimental animals were also subjected to a number of pharmacological _
interventions which allowed the experimenters to verify the method over a wide
range of hemodynamic states. When the estimated SV was compared to the
actual value of SV there was a linear relationship with a slope equal to the
arterial compliance (Ca). This was the result that was expected based on there
equations derived from the lumped parameter model above and the CAP

waveform.

Plv (t) Pa (t) Ra
. ’VVV—

 

 

COCD 2:: Ca

 

 

 

Figure 2. A lumped parameter model of the circulatory system which uses
a current source with the value of cardiac output to represent the left
ventricle

The successful implementation of this method proves two key facts about
the lumped parameter model and the cardiovascular system. First, it proves that

the Windkessel model of the arterial system along with a current source

representing the heart is a reasonably accurate model of central aortic pressure.

10

Second, it proves that Q, does not vary significantly over a monitoring period
even under highly varying hemodynamic conditions. If Ca did vary, the results
would not have shown a linear relationship between estimated proportional SV
and measured SV, because estimated proportionalSV is multiplied by a factor of
Ca. Based on those results, we will later assume that Ca is constant over a
monitoring period to measure proportional SV or a single beat to measure LVEF.

The second version of the lumped parameter model using the Windkessel
model of the blood vessels is the version that is used in our algorithm. This
model replaces the current source as the model of the heart with a time variant
capacitor [Sagawa et al. 1977]. This model of the heartis based on the fact that
the heart acts like a capacitor for blood that varies in capacity as the heart
squeezes during systole. This model for the circulatory system can be seen in
Figure 3.

We choose to use the model of the circulatory system which models the
left ventricle as time varying capacitance as opposed to a current source with the
value of cardiac output for two reasons. First, we have a model for elastance,
which is equal to one divided by the capacitance, which reduces the problem of
determining elastance to the estimation of only a few parameters of the elastance
function. This model will be introduced in the next section. The second reason is
that this elastance model provides the framework necessary to estimate ejection
fraction from a central aortic blood pressure waveform. The model using a
current source to represent the left ventricle does not take advantage of enough

information to accurately estimate ejection fraction.

11

Plv (t) Pa (t) Ra

 

 

 

 

Ca

# El: (t) :

 

 

 

Figure 3. A lumped parameter model of the circulatory system which uses
a time varying capacitor to represent the left ventricle

Notice that in both models of the circulatory system shown here, there is
an ideal diode between the portion of model representing the left ventricle and
the portion representing the arterial system. This diode represents the heart
valve that allows blood to flow out of the heart into the aorta but does not allow
blood to flow back into the heart. We model this as an ideal diode with no
resistance. There is some resistance in the true cardiovascular system in series
with the valve but this resistance is assumed to be negligible which has been
shown to be valid when the heart valve does not show significant signs of
stenosis. The result of this is that during systole, when the valve is open, P.,,(t) is

equal to Pa(t).

12

2.3 The Raised Cosine Model of Time-Varying Elastance

Using only the lumped parameter model of the heart suggested in the last
section and making no assumptions about the elastance curve, we find that
knowledge of CAP yields less equations than the unknowns provided by the
system. In that case the system is underdeterrnined and cannot be used to find
any additional data about the hemodynamic state of the subject.

For this reason, we make an additional assumption about the form of the
elastance curve. Empirical data has shown that when the elastance curve is
normalized over maximum left ventricular elastance (Emax), minimum left
ventricular elastance (Emin), and duration of systole (Ts, the time over which
elastance is increasing) it will always have the same form even for a wide variety

of patients [Heldt et al. 2002]. A summary of this data is shown in Figure 4.

 

1 .4 .
T, } Td

 

l
I
II

1.2

 

 

 

Elv‘tyEmax
. . .° .
a:
fill] I'LLIIFIIIITTTIIIIIIWIIITTTT\TIW

 

1 2
Tlme (normalzed units)
Figure 4. A summary of empirical data normalized to Em”, Em... and T3

13

This empirical data was then analyzed and found to closely match the
form of a raised cosine that is a function of Emax, Emm, Ts, and time. This cosine
curve can also be seen in Figure 4. Note that the area of closest agreement
between the empirical data and the raised cosine is near the peak of the raised
cosine. This close approximation at the top of the elastance curve is what is
most important for our method as our method will concentrate on the period of
ejection which is found to be around the top of the elastance curve. The

equation of the raised cosine function used to parameterize elastance is:

Emin+Emax—Emin 1"‘COS E , OSt<71
E 2E T

 

 

max max S

—E"’(t) = < ——E'“‘" + Em” _ Em‘“ {1+ cosen—UTLZL)», T s t < 13:-T

 

E E 2E

max max m S

_Efli.n_’ 2]"; St,
L E 2

max

 

We have also chosen to further reduce the parameters being searched
over by assuming Em,n to be .05*Emax. This assumption is based off of empirical
data as well as the fact that the effect of Emir. on the fitting of the curve should be
minimal since the fitting will be done during the ejection interval which is near the
top of the elastance curve. With this equation and the lumped parameter model
described in section 2.2, we are now able to develop a system of equations
which will allow us to estimate some unknown hemodynamic parameters in order

to estimate LVEF. The derivation of this method will be shown in the next section.

14

2.4 Derivation of the Mathematical Method

Our model for the circulatory system and our model for the function of the
heart can be combined in order to estimate some of the hemodynamic
parameters of the system. The combination of the two models requires doing a
circuit analysis of the circuit using all that is known about the parameters of the
circuit. The derivation is started based on the fact that the blood flowing out of
the heart must equal the blood flowing into the peripheral blood vessels. The

equation for blood flowing out of the heart is:

z : d<C..(t> *Pa»
out dt ’ (1)

where iout is the blood flow leaving the left ventricle, P(t) is the left ventricular

 

pressure and the aortic pressure which are assumed to be equal, and:

1

sz (t) = m. (2)

The equation for the blood entering the blood vessels is:

. _ "Ca ”‘de _ P(t)
1"” _ dt R . <3)

(I

 

where in is the blood flow entering the aorta and Ca and R3 are the arterial
capacitance and resistance respectively. The fact that blood flowing into the

heart must equal blood flowing out of the heart is summarized as:

out = lin . (4)

15

By combing equations 1, 2, 3, and 4 we arrive at:

 

 

 

 

 

d P(t)
Elva) _ — Ca * dPU) P(t)
dt — dt - 72a— - (5)
Then the equation is divided by Ca on both sides and integrated to yield the
following:
.
IP(t)dt
P(t) =—P(t)—° +C
Ca * Elv (t) T ' (6)

where T is equal to the product of total peripheral resistance, Ra, and arterial

capacitance, Ca, and C is the constant of integration. Time 0 is defined to be at

the start of ejection. By observing the system at the start of ejection, C can be

 

 

 

defined as:
C 2 HO) + P(O)

Ca * Elv (0) . (7)
The combination of equations 6 and 7 is:

P(t) :
Ca * Elv (t)
JP(t)dt _ (8)

— P(t) — 0 10(0) + P(O)

———+
r C *E,v(0)

16

The left side of equation 7 can easily be recognized as the volume of the left
ventricle divided by 0,... The right side of the equation is the total blood flow
through the aorta plus the volume of the heart at time 0 divided by Ca.

From equation 7 it is easy to derive the equation for proportional SV, SV
divided by Ca, developed by Bourgeois et al. To do this, simply subtract
proportional left ventricular volume at the end of ejection from the proportional

volume at the beginning of ejection. This yields:

EE— 2 P(eej) — P(O) + 54
C r

a

. (9)

where SA is the integral of CAP over the period of ejection and P(eej) is CAP at
the end of ejection. This is simply a function of CAP which is assumed to be a
known waveform in this research; therefore, proportional stroke volume can
simply be calculated by CAP waveform analysis.

The most difficult parts of calculating proportional stroke volume from CAP

is properly identifying the beginning and end of ejection on the CAP waveform
and calculating 1:. The morphology of 3 CAP waveform containing little corruption

due to reflections can be seen in Figure 5. As shown on the figure, when
ejection starts there is a large spike in CAP as a result of the large increase in
volume which is being pumped from the heart. The end of ejection is also
recognizable on the waveform as a slight bump, called the dicrotic notch, in the
decreasing side of the blood pressure waveform which occurs as a result of the
valve closure and a small amount of blood flow back into the heart during the

valve closure. To detect these points on the blood pressure waveform we used a

17

previously developed method for detecting the beginning of ejection and the
maximum point of the blood pressure waveform for each beat. This algorithm
searches the waveform for a local maximum over a window of about half of the
size of the period of the waveform. The period of the waveform is assumed to be
one divided by the heart rate although there is some beat to beat variation in this.
Once the location of the maximum value of CAP for this beat is detected, the
algorithm steps backwards in time through the data looking for the point at which
the slope goes from positive to negative. This point is marked as the start of the
upslope of blood pressure, which is also the start of ejection for that beat. After
the operation of this algorithm, it is still necessary to detect the dicrotic notch
signifying the end of ejection. In order to do this, we use a similar method as that
used to detect the beginning of ejection. The algorithm steps forward in time
from the maximum looking for the point at which the slope goes from negative to
positive. This point is marked as the end of ejection.

Calculating 1: from the CAP waveform requires some additional
understanding of the waveform. During the diastolic portion of the waveform,
when ejection is not occuning, the heart portion of the lumped parameter model
has no effect on the CAP waveform. For this reason, the pressure is simply
decaying across a parallel combination of a resistive load and a capacitive load.

From circuit theory, it is known that the pressure will decay exponentially with a
rate of speed based on the time constant, 1:, which is the product of the

capacitance and the resistance. Therefore, if the diastolic decay of the CAP

18

.83

Te]

h.
T

 

coco
601

 

tht) immhgi

3

 

 

 

 

 

 

0 . 1
To Time [s]
Figure 5. A single heart beat taken from a CAP waveform with little to no
effects from wave reflections

waveform is plotted on a semi-logarithmic scale it will become a line with slope T.
This methodology is used to calculate T in both the calculation of proportional

stroke volume and the estimation of LVEF.

LVEF is somewhat more difficult to estimate than proportional SV is to
calculate since it relies on more than just the difference between the pre-ejection
and post-ejection volumes. LVEF also relies on the pre-ejection volume itself
which does not allow it to be derived from equation 8 without the appearance of
Elv. For this reason the raised cosine model of E, is now used to make both
sides of equation 8 a function of four parameters. Each side is a function of
CaEmax, the maximum left ventricular elastance multiplied by the aortic
capacitance, T5, the duration of time the elastance of the left ventricle is

increasing, Tbegin, the time at which ejection begins relative to the beginning of

systole, and time. Note that both sides would also be a function of CaEmin if we
did not previously assume that CaEmin could be approximated as .05*Ca.Emax for
this application. Since digitized measurements of CAP have samples at many
different points in time during ejection, we are left with many equations and three
unknowns.

To estimate the values of our three remaining unknowns, least square
error fitting is implemented. Initially, we use a very basic algorithm to do least
square error fitting. First, an acceptable range for each of the three unknowns is
selected. The acceptable values for CaEmx are based on previously collected
hemodynamic data and are left at a very wide range as this is the most difficult
variable to parameterize. The values of CaEmax searched over in order to
produce the results shown later in this thesis are .5 to 20.5. Values for Ts and
Tbegm are based on the definitions of these values and the morphology of the
pressure waveform. The value of Ts is limited to being a fraction of the period of
the blood pressure waveform since the systolic interval cannot be greater than
the entire period of the heart. The range of Ts considered acceptable is also
based on previously analyzed data. Ts is limited to values between one fifth and
one half of the period. The range of acceptable values for Tbegin range from 0,
which means that ejection starts as soon as the heart starts to squeeze, to Ts,
which means that the heart begins to eject at the last moments of systole. We
then search over a set of equally spaced combinations of these values to find the

value with the minimum squared error. The values of the three unknown

20

parameters which result in the minimum value of square error are considered the
estimated hemodynamic state of the subject.

These values are then plugged in to the left hand side of equation 8 to
calculate proportional stroke volume and end diastolic volume. The last step in
calculating LVEF is to make an assumption about the unstressed volume of the
heart. The unstressed volume of the heart is the volume at which there will be a
blood pressure of zero. This volume cannot be determined from central aortic
blood pressure and therefore must be accounted for now. An assumption of this
value divided by Ca is made based on the size of the dog and previously
analyzed hemodynamic data, and this value is then added to the previously
estimated value of end diastolic volume. The final equation for left ventricular

ejection fraction is:

 

 

 

12(0) _ P.(es>
* at:
LVEF = C“ Elv(0) Ca Elv(eS)
130(0) + IflvO ’ (10)

 

 

Ca *Elv(0) C

a

where es is the time at which systole ends and V.,,o is the unstressed volume of
the left ventricle.
2.5 Faster Execution of the Method

As stated in the previous chapter, the purpose of this method is to obtain a
continuous and automatic measure of LVEF from CAP waveform analysis. In
order to have a truly continuous measure it is necessary that the algorithm be

able to operate on a real time basis. This means that the time the algorithm

21

requires to execute the least square error parameter fitting for a single beat must
be less than the period of the CAP waveform. The execution time of the
algorithm also includes all waveform morphology detections and all other
calculations needed to execute the method. A three parameter search method
as described in the previous section has an operational time proportional to the
product of the number of sample points used for each parameter. Therefore,
reducing the number of parameters which are estimated via a search method will
reduce the execution time of the method by a factor of the number of sample
points used for each parameter reduced. Since the three parameter search
method described in the previous section was found to need a longer execution
time per beat then the period of the waveform, we have developed an alteration
to the algorithm which reduces the number of parameters to be searched over.

Linear algebra theory states that the value of a parameter which results in
the least square error between two sides of a function can be calculated through
a series of matrix multiplications if the function can be expressed as a linear

function of the unknown parameter. This means that if a function is of the form:

A=XB, (11)

then if the vectors A and B are known, the unknown value, X, can be calculated
through a series of matrix functions rather than requiring that all possible values
of X be searched over as done in the previous section. Therefore, we have
modified the equations from the previous section to derive a linear function of

CaEmax so that CaEmax can be calculated through a series of matrix calculations,

22

and the algorithm can be executed in a much shorter time as it does not require
searching over a very wide range of possible values for CaEmax.
In order to make these equations more easily read, we will first define C(t),

the part of our model equation for Eiv(t) not dependant on Emax as:

 

 

”a:
C(t)=1—cos OSt<TS
_ 3
=1+cos 2”“ T‘) T, -<—t<—TS
, ' 2
= 0 OtheI'Wise. (12)

Using this equation, we represent left ventricular elastance as:

 

E —E .
E,,(z) = Emin + m 2 mm C(t)-

(13)

This equation can be further reduced because we have assumed that Emir, is

equal to .05*Emax. We therefore rewrite equation 13 as:

Eh, (t) = .05 * Em + .475 * Em * C(t) _ (,4,

23

Using this equation for E.,, we can rewrite equation 8 as:

P(t) _
.05 * CaEm + .475 * CaEmax * C(t)

 

IP(t)dt P(O)

 

 

— P(t) — ° + P(O) +
2' .05 * CaEm + .475 * CaEm * C(O) -

(15)

This equation can be adjusted as follows:
.05 * CaEm + .475 * CaEmax * C(t) =

P(t)

 

:[P(t)dt

— P(t) — O 2' + P(O) + P(O)

.05*C E +.475*C E *C(O)

a max a max

 

 

(16)

24

Multiplying both sides by one over CaEmax and rearranging terms results in:

f t \

JP(t)dt

-P(t) - ° + P(O) *CaEm... =
2'

 

 

 

P(t) _ P(O)
.05 + .475 * C(t) .05 + .475 * C(O) - (17)

 

 

This equation is now a linear equation of CaEmax with all of the other values being
known for a give pair of Ts and Tbegin. The method can now achieve the same
results as before by searching over only two parameters and calculating the

value CaEma,x which minimizes square error.

25

3. Experiments

3.1 Introduction to Experiments

In order to verify that the mathematically developed method presented in
chapter two can work on real data it is necessary to collect such real data in a
manner that allows for the verification of the method. This is best done when the
most information about the state of the subject is available. Collection of
information such as stroke volume, left ventricular volume, central aortic blood
pressure, and left ventricular ejection fraction is desired in such an experiment.
The measurement of peripheral blood pressure is also desired as it will allow us
to validate are overall hypothesis in the future. It is also important that the
method be verified over as many different hemodynamic states as possible. This
means that the hemodynamic data should be collected while the subject of the
experiment is put under a number of experiments which affect the waveforms
being measured. Since the collection of some of these waveforms requires very
invasive procedures and the hemodynamic state of the subject must be varied,
collection of this type of data in humans is too dangerous. For that reason,
varying experiments on nine dogs have been done which collect some data
which can be analyzed to verify this method. Specifics of the experiments and
the data gathered will be covered in the sections that follow.
3.2 Sonomicrometry Crystals and Chronic Pacing Experiments

The first set of experiments used in order to verify the functionality of the

method presented in chapter two were performed on five dogs. The purpose of

26

these experiments was to test the method under conditions of reduced cardiac
function in order to verify that the method can detect this state.

These experiments measured all of the important hemodynamic
waveforms to verify the method. First, they measured left ventricular volume
through a two-dimensional sonomicrometry crystal method. This method
requires the implantation of four sonomicrometry crystals in to the walls of the
heart such that two dimensions of the heart are accurately measured. These
dimensions of the heart are then used in conjunction with the known geometry of
the heart to calculate left ventricular volume. In addition, CO and SV were
measured using an aortic flow probe which is placed around the aorta and
measures the rate at which blood leaves the heart. LVEF is then calculated as
the ratio of stroke volume from the aortic flow probe to end diastolic volume
measured by the sonomichrometry crystals. The last needed parameter for the
verification of the technique is central aortic blood pressure. The dogs in these
experiments were also instrumented with an aortic catheter which was used to
measure central aortic blood pressure.

The experiments also used a technique known as chronic pacing to
reduce the functionality of the heart as measured by LVEF. In chronic pacing,
electrical pacing wires are attached to the heart to cause it to beat at a much
faster rate than the heart would normally pace itself. This pacing is continued
over a long period of time, in this case, several weeks. After the pacing wires are
removed and the heart is allowed to return to a self regulated state, the heart

stills shows semi-pennanent effects of the pacing. SV and LVEF are both greatly

27

reduced and end diastolic volume has been shown to increase. Since LVEF is
reduced by this process, these experiments mimic a heart which is performing at
reduced capacity due to chronic heart failure or other similar heart diseases. A
summary of the instrumentation of the dogs in these experiments can be found in

Figure 6.
3 Pacing Wires

Central Aortic Aortic Flow Probe
Pressure Catheter ..

    

Figure 6. A summary of the instrumentation of the subjects used in the
first set of experiments which concentrate on reduced cardiac function

The values of many key hemodynamic parameters vary greatly over the
many subjects in these experiments. LVEF, SV, mean CAP, and several other
parameters show a great deviation between the control measurements taken
before chronic pacing and the heart failure measurements take after chronic
pacing. A summary of the range of hemodynamic parameters found in the
control measurements of these experiments can be found in Table 3 and a
summary for the range of hemodynamic parameters found in heart failure can be
found in Table 4.

These experiments provide all of the necessary measurements needed to

verify the functionality of the method presented in this thesis. LVEF is measured

28

through the use of sonomicrometry crystals and an aortic flow probe, and CAP is
measured through central aortic catheterization. Chronic pacing allows the same
subject to have measurements taken for a heart functioning normally and a heart

functioning at a reduced level.

 

 

 

 

 

 

 

Minimum 1 Maximum Average
Mean Heart Rate 94.7 122.1 112.0 I
(beats/min)
Mean CAP 97.89 130.13 115.84
(mmHg) "
Mean LVEF (%) 50.45 69.74 61.56
Mean SV (ml) 32.75 49.12 39.01

 

 

 

 

 

Table 3. A summary of the mean values of certain important hemodynamic
parameters measured in the chronic pacing experiments during control
experiments. This table shows the wide range of hemodynamic parameters
seen between dogs in their control states.

 

 

 

 

 

 

Minimum Maximum Average
Mean Heart Rate 111.2 177.0 137.6
(beats/min)
Mean CAP 74.62 115.56 92.92
(mmHg)
Mean LVEF (%) 26.33 37.62 30.79
Mean SV (ml) 20.87 29.32 25.07

 

 

 

 

 

Table 4. A summary of the mean values of certain important hemodynamic
parameters measured in the chronic pacing experiments during heart
failure experiments. This table shows the wide range of hemodynamic
parameters seen between dogs in their heart failure states as well as the
large contrast between the hemodynamic state of the dogs in their control
and heart failure states.

29

3.3 Dobutamine

Another set of experiments used to verify this technique were done on
three dogs. These experiments were performed to test the effectiveness of the
method under drug induced improved hemodynamic state.

These experiments provide only limited knowledge of the hemodynamic
state. In these experiments there is no direct measure of LVEF, CO, or SV, so
the results of implementing the method can not be verified against exact
numbers. The results are instead verified only qualitatively against the known
increase in LVEF seen with the administration of a commonly experimentally
used drug, dobutamine. Like the dogs in the first set of experiments, the dogs in
these experiments were also instrumented with an aortic catheter which was
used to measure central aortic blood pressure. The instrumentation of the

subjects of these experiments can be seen in Figure 7.

Central Aortic
Pressure Catheter

  

Figure 7. A summary of the instrumentation of the subjects used in the
second set of experiments which concentrate on drug induced increased
cardiac function

30

In order to vary the hemodynamic state during these experiments
pharmacological intervention was used. Dobutamine, a drug used to temporarily
increase the functionality of the heart was administered to each of the dogs in
varying amounts. Measurements of central aortic blood pressure were made at a
control state when no drugs were being administered and at states where varying
levels of dobutamine were administered. It is known that LVEF increases with
increasing dosages of dobutamine so the results of our method can be verified .I‘

qualitatively without a direct measure of LVEF. Table 5 provides a summary of

 

 

the variations in hemodynamic data in the dogs during control and Table 6
provides the same data for the dogs under the effects of dobutamine.

These experiments provide a measure of CAP and a qualitatively known
variation in LVEF through which the method can be verified. This data allows for
the verification of the method in control subjects as well as in those same
subjects with increased heart functionality due to the administration of drugs.
This is a good compliment to the first data set which verifies the method in

subjects with reduced heart functionality.

 

 

 

 

 

 

 

 

 

Minimum Maximum Average
Mean Heart Rate 104.0 111.8 108.9
(beats/min)
Mean CAP 70.63 87.66 76.70
(mmliq)

Table 5. A summary of the mean values of certain important hemodynamic
parameters measured in the dobutmaine experiments during control
experiments. This table shows the wide range of hemodynamic parameters
seen between dogs in their control states.

31

 

 

 

 

 

 

 

 

Minimum Maximum Average
Mean Heart Rate 134.0 198.0 156.1
Qeats/min)
Mean CAP 71.50 92.73 80.90
(mmHg)

 

 

Table 6. A summary of the mean values of certain important hemodynamic
parameters measured in the dobutamine experiments during dobutamine
administration. This table shows the wide range of hemodynamic
parameters seen between dogs during dobutamine administration as well
as the large contrast between the hemodynamic state of the dogs during
control and dobutamine administration experiments.

 

3.4 Echocardiography and Pharmacological Interventions

The last experiment used in this study included only one dog. The
purpose of this experiment was to test the method under conditions of both
reduced cardiac function and increased cardiac function based on the effects of
different pharmacological interventions.

Like the first set of experiments, this experiment measured all of the
important hemodynamic parameters to verify the method. In this experiment,
LVEF was measured periodically through the most common clinical
measurement technique, echocardiography. To do this, echocardiograms were
taken and analyzed by specialists to determine the volume of the heart at the
beginning and end of ejection. Also like the first set of experiments, CO and SV
were measured using an aortic flow probe. LVEF is then calculated as the ratio
of stroke volume from the aortic flow probe or echocardiography (both values are
the same) to end diastolic volume measured by echocardiography. An aortic

catheter was used to measure central aortic blood pressure as in the first two

32

 

sets of experiments. A summary of the instrumentation of the dogs in these

experiments can be found in Figure 8.

Central Aortic
Pressure Catheter

fiAortic Flow Probe

    

. "311,.”

 

Figure 8. A summary of the instrumentation of the subject used in the last
experiment which analyzes the dog in states of both increased and
decreased heart function

In order to vary the functionality of the heart to provide a range of
hemodynamic states to test the method, two different pharmacological
interventions, dobutamine and phenylepherine, were used at different dosages.
The result of these pharmacological interventions is that the data set includes a
wide range of values for both SV and LVEF. This data simulates a subject with
normal, decreased, and increased heart function, and will allow us to prove that
the method described in chapter two works in all cases. A summary of the range
of hemodynamic values found in this data set can be found in Table 7.

All of the necessary measurements to verify the functionality of our
method are provided by this experiment. LVEF is measured through

echocardiography, the most common clinical way to measure LVEF, and CAP is

33

 

measured through central aortic catheterization. This experiment also is done
continuously so that the change in LVEF during the times where
echocardiography measurements are not being taken can still be estimated by
our method. The pharmacological interventions done during this experiment
allow the method to be tested on the same subject in a control state, and in
states of increased and reduced heart function. This experiment is excellent to

show the effectiveness of this technique while highlighting its benefits.

 

 

 

 

 

 

 

Minimum Maximum Average
Heart Rate 93.0 131.0 114.7
(beats/min)
Mean CAP 59.47 160.13 105.91
(mmHe)
LVEF (%) 48.0 90.0 71.9
SV (ml) 18.85 32.75 25.03

 

 

 

 

 

Table 7. A summary of the mean values of certain important hemodynamic
parameters measured in the echocardiography experiment.

34

4. Results

4.1 Verifying our Assumptions

The first step in verifying our new technique is to verify that the algorithms
used to detect waveform morphology are working acceptably well, and to verify
that our assumptions are founded. In order to do this, we used data from the
chronic pacing experiments to verify that the method for estimating proportional

stroke volume from central aortic blood pressure presented by Bourgeois et al.

 

By verifying his results using this data, we prove that most of our assumptions
are accurate and that are detection schemes are working. We also can verify
that the effects of chronic pacing do not invalidate any of the assumptions proven
by the technique of Bourgeois et al, while the data collected in their original paper
shows that drug intervention does not invalidate these assumptions. Figure 9
shows examples of the use of this technique on a dog in control state as well as

dog in chronic pacing induced heart failure.

 

 

 

 

 

90
O
E sot E 0 ° 0
2 3 55 " o
s 3 8% °
.3 70* z 008’ o o
x
2 o 2 50 o 0%?g‘ioOOCbo 0
g 60 iii 0 6 o 06’
c o E 45 900239 6% °o
2 so 0 .9 @
t t O o o
g. a o oo
9, 40 o e ”I 0
1L 1:
O
30 - 1 ‘ ' 4 ‘ * 35 1 1 . J ' - . . .
15 20 25 30 35 40 45 50 18 19 20 21 22 23 24 25 26 27
Stroke Volume (ml) Stroke Volume (ml)

Figure 9. Examples of the effectiveness of the technique of Bourgeois et al.
The graph on the left is of a dog in control state while the graph on the left
shows the technique on the same dog in chronic pacing induced heart
failure.

35

Much like the results shown by earlier research, this figure shows a strong
linear correlation between proportional stroke volume as estimated by the
previously developed technique and stroke volume as measured by an aortic
flow probe. Since it is shown by the derivation of the equations used in this
technique that the scale factor between the proportional stroke volume shown
here and actual stroke volume is one divided by the arterial compliance, the
strong linear correlation between these values proves that arterial compliance is
constant over minutes to hours. Our assumption that arterial compliance is

constant throughout one heart beat is a simple consequence of that fact. The

 

successful implementation of the method of Bourgeois et al. verifies that our
waveform feature detection schemes are working and that most of the
assumptions used in deriving our algorithm are accurate.
4.2 Curve Fitting Examples

The second step in verifying that the methodology was working correctly
was to verify that for an appropriate elastance curve, the result of both sides of
equation 8 is a proportional volume curve. In order to do this, we found the
elastance curve that yields the best fit between the two sides of the equation.
After calculating each side of the equation based on the optimal elastance curve
and the portion of the blood pressure waveform corresponding to ejection, we
plotted these curves and verified that there morphology matched that of the
ejection portion of the left ventricular volume waveform. Figure 10 shows
examples of the plot of the left and right side of the equation from the heart

failure experiments.

36

u].
d

82’
/
a

a 3‘
E \ '2 so
3 ‘ 3 8°
970 9
IL E 70

 

 

50

 

 

a 5 1‘0 1'5 20 23 so 345 in 43 500 5 10 1s 20 25 30 as in is
Time (samples) Time (samples)
I I I I F”
Figure 10. Examples of the results of the left and right sudes of equation 8

for the optimal elastance curve for two different beats.

 

These proportional left ventricular volume curves show very close
correspondence to each other as is expected since the elastance curve was
parameterized to minimize the difference between the two curves. It is also quite
clear that both sides of the curve match closely to the morphology of a left
ventricular volume waveform during ejection. The blood volume drops rapidly at
the beginning of ejection while the heart continues to squeeze and then the
amount of blood leaving the heart tails off as the systolic interval ends. The close
correspondence of the estimated proportional volume curves shown in Figure 10
and real left ventricular volume waveforms helps to verify that our method is
producing the predicted results.

4.3 Results during Chronic Pacing Induced Heart Failure

The last step in verifying our method for estimating left ventricular ejection
fraction is to verify that the estimated ejection fraction yielded by the method
matches the expected values. As discussed in the previous chapter, the first

data set used to verify our method used chronic pacing to induce heart failure

37

which allows us to verify that the method works on subjects in both normal
hemodynamic conditions and in those with hearts functioning at a lower level.
This data set supplied beat by beat measurement of left ventricular ejection
fraction through the measurement of left ventricular volume by a two dimensional
sonomicrometry crystal method. The sonomicrometry crystal data verified that
after chronic pacing, both stroke volume and ejection fraction are greatly reduced. 1
Figure 11 shows a comparison of the estimated ejection fraction and the I‘

measured ejection fraction in all five of the dogs used in this experiment.

 

 

 

 

100~
l——l 80_
3‘3
U.
L]_] 60-
'O
(D
4.;
(U 40-
.E
.4-0
(0
LU 20-
O 1 l 1 I 1
0 20 40 6 80 100

0
Reference EF [%]

Figure 11. Summary of the results of all five dogs used in the chronic
pacing induced heart failure experiments

This figure shows that there is a strong correlation between the estimated

ejection fraction and the reference ejection fraction taken from the

38

sonomicrometry crystal data. The five data points with both reference and
estimated ejection fraction above fifty percent correspond to each of the five dogs
in its control state. The remaining five data points, which have reference and
estimated ejection fraction of below fifty percent, correspond to each of the five
dogs after chronic pacing induced heart failure. The stark contrast in the
estimated ejection fraction of a healthy dog and that of one in heart failure shows
the strong ability of the method to detect a subject with reduced heart
functionality. This ability of the method is the most important thing that the
method can do as the primary clinical use of the method would be to monitor a
patient with possible reduced heart functionality and to detect changes in that
patients clinical state.

The only notable negative feature shown in Figure 11 is that the method
appears to be very accurate in control while tending to always overestimate
ejection fraction in heart failure. We believe this to be the result of a change in
unstressed left ventricular volume resulting from chronic pacing. Proportional
unstressed volume is assumed to be a value (10 mmHg) based on normal
experimental values seen in other data sets. Since unstressed volume is
assumed to be the same for all of these experiments, the change in unstressed
volume results in the incorrect estimation of ejection fraction in the heart failure
state. This problem would be alleviated in clinical settings by using an initial
measurement of left ventricular ejection fraction taken via a current non-invasive
method to calibrate our method. If necessary, the method could then be

recalibrated any time there is a noticeable change in ejection fraction.

39

Using sonomicrometry crystals for continuous measurement of left
ventricular volume also allows for the validation of the method on a beat by beat
basis which will not be possible in the other experiments where we do not have a
gold standard measurement of ejection fraction or only have instantaneous
measurements of ejection fraction for a few beats.

50‘

h
01

h
D

00
01

 

Estimated LVEF [%]
W
o

N
01

 

2o - - . .
20 25 30 35 4o 45 50

Sonomicrometry LVEF [%]
Figure 12. Beat by beat scatter plot comparing sonomicrometry LVEF to

LVEF estimated by our method on a dog in chronic pacing induced heart
failure

Figure 12 shows a beat by beat comparison of LVEF estimated by our
method to the reference LVEF obtained from the volume waveform measured via
sonomicrometry crystals. The linear nature of this plot indicates that method is
working correctly since beats with higher reference ejection fraction correspond

to the beats with higher estimated ejection fraction.

40

4.4 Results during Dobutamine Infusion

As stated in the previous chapter, our second data set used dobutamine
infusion both to vary the hemodynamic state of the subject and to estimate the
expected result of the experiments. The three dogs used in this second set of
experiments were given varying amounts of dobutamine to raise stroke volume
and ejection fraction while instrumented with central aortic catheters to measure
central aortic blood pressure. The results of using our method on these central

aortic blood pressure waveforms are shown in Figures 13 through 15.

80)
75 -

70-

 

65-

Estimated EF [%]

60—

 

 

55

 

0 Control Low Medium High
Dobutamine Dosage
Figure 13. Results of our method on dog one of the dobutamine
experiments. The middle of each bar is the mean of the results under the

conditions listed on the x axis and the bottom and top halves represent the
standard deviation of the measurements.

41

«.1 \l 00
c: 01 1::

Estimated EF [%]
m
01

0')
O

 

55

 

0 Control Low High
Dobutamine Dosage

Figure 14. Results of our method on dog two of the dobutamine
experiments.

75-

'5‘
D

07
U1

0')
O

 

Estimated E F [%]

U1
U1

 

 

50

 

0 Control Low High
Dobutamine Dosage

Figure 15. Results of our method on dog three of the dobutamine
experiments.

42

All three dogs show that there is a significant increase in left ventricular
ejection fraction from the control state to the dobutamine states as expected
based on previous experiments. In all of the experiments there is a less notable
but still significant increase in ejection fraction with increase in dobutamine
dosage. The reduced effect of increasing the dosage of dobutamine is to be
expected since there is a limit to the amount of increase in ejection fraction that
can be caused by dobutamine infusion. These experiments show the ability of
our method to detect increase in cardiac function to drug infusion. This is
another important test for the method as drug infusion may be used in clinical
settings to increase ejection fraction in unhealthy patients.

4.5 Results during Drug Infusion Compared to Echocardiography

The last experiment used for testing our method shows all of the
methods benefits over echocardiography in a range of hemodynamic states. In
this data set, continuous central aortic blood pressure waveforms were taken
while periodic measurements of ejection fraction were taken via
echocardiography. The periodic echocardiography measurements mimic
standard practice for a patient being monitored for heart problems. Since central
aortic blood pressure is measured continuously, our method provides a
continuous measurement of left ventricular ejection fraction. This is a huge
advantage over echocardiography because changes in heart function can be
noticed immediately. Because of this, medical personnel could respond to a
failing heart immediately where in the past they would not be aware of the

condition.

43

Dobutamine Phenylepherlne

 

 

 

 

 

100 —
o 0

8° W
.35
11. 60 —
111
3 e

40 —

200 20 40 60 80

Time [min]

Figure 16. Results of our method on our last experiment. The line
corresponds to the estimated LVEF based on our method while the circles
correspond to actual LVEF based on echocardiography measurements.
Figure 16 shows the results from this last experiment. The effects of
dobutamine and phenylepherine on LVEF in this experiment are clear.
Dobutamine infusion raises LVEF by slightly over ten percent while
‘ phenylepherine lowers LVEF by about twenty percent from its baseline value.
Due to these two effects, there is a range of LVEF in this data set of over forty
percent. Despite this wide range of LVEF values, it is clear that our method
corresponds very closely with the values measured by echocardiography. It is
also quite clear that the method shows a change in LVEF as a direct response to
drug infusion. Estimated LVEF rises upon dobutamine infusion and again upon
increased dosage of dobutamine (not labeled), while it decreases upon cessation
of dobutamine infusion and again upon administration of phenylepherine. To
additionally highlight the advantage of our method, it can be seen that our

method detects the effect of dobutamine at least ten minutes earlier than does

44

the echocardiography measurement. In an intensive care unit where
measurements of LVEF are only taken every several hours or days, the
advantage of our method would be even more distinct.
4.6 Real Time Evaluation

In order for our method to fully provide the advantages that have been
described, it would be usefulfor the algorithm to function in real time. In this
situation, real time functionality means that the data is processed at least as
quickly as it is generated. This is useful because one of the biggest advantages
of the algorithm is that it provides knowledge of changing heart condition as it
happens. This advantage does not exist if the method takes several minutes or
hours to find a solution for a small amount of data as is the case with
echocardiography which requires a technician to analyze images from a single
beat for several minutes to determine the ejection fraction. To verify that our
method can indeed run in real time we simply ran the algorithm using a standard
home personal computer processor and measured the run time of the process.
We then compared this time to the length of the data set that it was being tested
with and made sure that the run time was less than the duration of the data set.
This verifies that the method can operate in real time, possibly requiring a short
delay before the first beats are analyzed. For this experiment we used our last
data set which had a duration of about one hour and thirty minutes. With our
original version of the method, which did not take advantage of the linear
equation of C,.,.Emax to reduce run time, the method required slightly less than an

hour and a half to execute on a personal computer processor. Although this

45

would qualify for real time operation, because run time is slightly less than the
duration of the data set, it may not function well in an actual integrated systems
environment where there may be the desire to run more than one process on the
system’s microprocessor or to use a slower microprocessor to save on cost and
energy. Our linear solution of CaEmax provides a very good solution to these
problems. When the linear version of our technique is used, the entire run time
of the method, including the time taken for central aortic blood pressure
waveform morphology detection, is slightly less than three minutes running in
matlab on a personal computer. This duration suggests that doing real time
analysis on a similar system would require only one thirtieth of the systems
resources so that many other system functions could be executed. Alternatively,
the method could be run on a system with reduced processing power to reduce
cost and power consumption. In either case, the method has been shown to
maintain its stated advantage of real time constant monitoring due to the fact that
it can easily run in real time.
4.7 CaEmax Estimation

In addition to left ventricular ejection fraction, Em...x is considered a very
powerful measurement of heart functionality [Sagawa et al. 1977]. The elastance
curve is an exact measure of how the heart is functioning during systole which is
largely unaffected by the amount of blood filling the heart or being pumped out.
Unfortunately, Emax is almost never measured clinically due to the highly invasive
procedures necessary for measurement. In order to measure elastance directly,

it is necessary to have both a continuous measure of left ventricular volume and

46

a continuous measure of left ventricular pressure. Doing either of these
measurements requires left ventricular catheterization or another highly invasive
procedure which is both dangerous and expensive due to the highly skilled
operators required. In addition, to accurately measure Em,Ix it is necessary to
change the loading conditions on the heart which is also difficult and dangerous.
This is one of the reasons that left ventricular ejection fraction is much more
common clinically. Our method, however, estimates proportional elastance as a
necessary step in estimating ejection fraction. CaEmax as estimated by our
method could be used clinically to provide more specific information as to the
systolic functionality of the heart. Figure 17 shows the changes in CaEmax due to

the drug infusions in our last experiment.

 

 

 

 

Dobutamine Phenylepherlne
16
12
a
E
a .-
4 _
0 l 4 I 1
0 20 40 60 80
Time [min]

Figure 17. CaEmax results of our method on our last experiment.

It is quite clear in the figure that C...Emax rises with dobutamine infusion and
decreases with phenylepherine similarly to left ventricular ejection fraction. The

estimation of CaEmax could be used clinically to provide more specific information

47

than is currently available. Our method provides afar less invasive and more
cost effective way to measure both left ventricular ejection fraction and

proportional maximum elastance than current techniques.

48

5. Discussion

5.1 Parameters Being Estimated

In chapter two it was shown how the method was formulated using
assumptions and facts found in previous research. The lumped parameter model
used to represent the cardiovascular system was presented and the research of
Bourgeois et al. was used to support its validity. In addition, we presented a
model for the elastance curve which would be used to define the functionality of
the heart in only a few parameters. This model, which uses a raised cosine
elastance curve to model the heart as a time varying capacitor, is verified using
empirical data. The model, as it has been used, makes several assumptions
about the form of the elastance curve which are vital to our method as they
reduce the number of parameters to be estimated by the method to a number
which can be solved for. In this section, we will discuss these assumptions and
the possible effects of relaxing these assumptions or adding additional
assumptions.

CaEmax and C,.,.Emin are the two parameters of the elastance model which
determine its magnitude. In the model as it is presented in previous research,
there is no assumption regarding the relationship between CaEmax and C,..Emin
other than that CaEmin must be less than CaEmax. For this reason, the initial form
of our algorithm included CaEm,n as a parameter to be estimated. There were
three reasons that were factors in deciding to make an additional assumption
regarding CaEmin. The first reason was that any estimation of CaEmin would be

inaccurate given the data provided. The main source of data for this method, the

49

central aortic blood pressure waveform, contains no information about CaEmm
because the valve between the left ventricle and the aorta is closed when the left
ventricle is fully relaxed. This makes any estimation of C,.,.E.m,n by our algorithm

an extrapolation which cannot be expected to be accurate. The remaining
reasons for making an additional assumption about C;,.Emin result from the
algorithmic downsides of estimating the parameter. Both downsides of

estimating CaEm,n result from the fact that it is an additional parameter to estimate.
First, since the measured data contains some error and the general assumptions
about the hemodynamic system are known to be imperfect, less parameters can
be estimated accurately than the number of equations provided by the data.
Therefore, reducing parameters through valid assumptions should provide more
accurate estimations of the remaining parameters and a better estimation of left
ventricular ejection fraction. The second downside of estimating additional
parameters is that it significantly adds to the time required for the method to work.
The run time of the method is proportional to the product of the range of the
parameter multiplied by the accuracy to which the parameter is estimated of all of
the parameters estimated through non-linear means. Since the accuracy of the
method and its execution speed could be greatly improved by doing so, we made
the assumption discussed earlier that CaEmin was equal to .05*CaEmax. This
assumption was assumed to be valid based on empirical data that suggested this
to be relatively accurate and the fact that small differences between assumed
CaEmin and the actual value of C,.,.Em,n will not result in large differences in the

estimated value of left ventricular ejection fraction. For this reason, the amplitude

50

 

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_0 2 1 1 1 1 1 1 1 I 1 1 1 1
' 0 . ~11
Tbegin Ts 2*T3 3 T8
Time (normafized units)
Figure 18. Graphs of the normalized elastance curve used to model the left

ventricle in our method and a central aortic blood pressure waveform with
important timing parameters noted

 

51

of our estimated elastance curve is estimated completely through the estimation
of CaEmax; however, the method could be adapted to estimate both C,,,Emax and
CaEmin with the risk of losing accuracy and a guaranteed increase in execution
time.

Another parameter that is being estimated by our current method is Tbegin.
Tbegin is defined as the time between when the heart begins to squeeze and the
beginning of ejection. The beginning of ejection is clearly visible in the central
aortic blood pressure waveform as a sharp spike in blood pressure directly after
the valve between the left ventricle and the aorta opens. Figure 18 depicts both
a central aortic blood pressure waveform and the normalized elastance curve
and denotes Tbegin as well as some of the other timing parameters which will be
discussed during this section. Tbegin is a parameter that must be estimated by our
method and cannot be assumed based on previously obtained data. In order to
estimate this parameter it must be determined what range of values are
physically realistic for it. It is clear that Tbegin must be greater than zero because
ejection will not begin until the heart begins to squeeze. It is also clear that Tbegin
must be less than Ts because the ejection will begin before the left ventricle
reaches its maximum squeezing point or it will not be reached at all. Using this
as the acceptable range of values for Tbegin it is one of the parameters estimated
with our method.

During the previous research that resulted in the elastance curve used in
modeling the left ventricle in our method, it was found that empirical data

supported the assumption that Ts is about twice the duration of the time that it

52

takes for the heart to relax to the point where the elastance is once again equal
to CaEmin. This assumption, however, could be relaxed and the ratio of the
duration of these parameters could be estimated as part of the method. The
result of relaxing this assumption would be a modification of equation twelve,
which defined the raised cosine curve used in our model of the elastance curve
without including the scaling of CaEmax. If the assumption that Ts is twice the
duration of the relaxing of the left ventricle is removed, the equation for this part

of the equation would be:

 

 

:1:
C(t)=1—cos”t Ogt<TS
=1+cos ”ma—Ts) Ts._<_t<TS+—1—TS
s ’ a
= 0 otherwise, (13)

where a is the scale factor by which Ts is greater than the duration of the

relaxation time of the left ventricle. All other parts of our method would remain

the same so CaEmax could still be estimated through a linear method and all other

parameters, including a would be estimated through a non linear method of least

square error fitting. Including this additional parameter for estimation would have

the same downsides as were explained in the discussion of CaEmin but would be

necessary if it was found that the assumption that or is equal to two is not a good

one. Through our testing it has been found that assuming our a parameter is

53

equal to two appears to be valid and does not result in significantly different
results for left ventricular ejection fraction then does allowing it to vary. For this

reason, and the fact that empirical data also supports this assumption, we use

this assumption and do not include 01 as an additional estimated parameter of our

method.

The last parameter to be considered in this section is Ts. Much like Tbegin
and CaEmax, Ts must be determined in order to estimate left ventricular ejection
fraction. Ts can clearly be constrained to being a fraction of the duration of the
cardiac cycle, where the duration of the cardiac cycle is the time between two
consecutive heart beats. Using that as a constraint, Ts can be an estimated
parameter of the technique as described in chapter two. An alternative to
estimating Ts is to measure it based on the electrocardiogram and some basic

assumptions.

 

1' '1
Q S _ '1
QT Interval

Figure 19. A standard ECG waveform with important characteristics
labeled

54

Electrocardiograms are one of the most common measurements taken in
a clinical setting because they are completely noninvasive and provide important
information about electrical activity of the heart. Electrocardiograms are taken by
attaching electrodes to certain places on the body which then measure the
potential difference between these places. When systole begins, the SA node of
the heart starts an electrical signal which travels around the heart causing it to
squeeze. The Q wave denotes the beginning of the left ventricle receiving the
electrical signal, while the T wave denotes the electrical relaxation of the left
ventricle. Therefore, assuming cancellations of electromechanical delays, the QT

interval would be the duration of both the rise in elastance and the subsequent
fall in elastance. Ts is therefore simply a function of the QT interval and 11. Using

this knowledge we can avoid estimating a non-linear parameter, Ts, and measure
it instead. The added difficulty of doing this is that it is necessary to automatically
detect the Q and T waves to measure the QT interval. We have developed an
algorithm to do so based on the same methodology used to detect features of the
blood pressure waveform. There are also several other methods becoming
available for such detection that could also be used in place of this method which

is currently unproved in all but a small amount of data. A version of our method
which estimates a rather than assuming it to be two and uses the
electrocardiogram waveform to measure T8 was applied to the second set of
experiments where dobutamine was used to alter the hemodynamic state of the

subjects. The results of this alteration of the method compared to the version of

the method used in previous chapters can be found in Tables 8 through 10.

55

 

Mean estimated ejection
fraction using CaEmax, Ts,
Tbegin estimation method

Mean estimated ejection
fraction using CaEmax, a,
Tbggin estimation method

 

 

 

 

 

Control .6233 .6492

Low Dose dobutamine .7051 .7042

Medium Dose .7476 .7527
dobutamine

High Dose dobutamine .8251 .8281

 

 

 

Table 8. A summary of the results of two variations of our method in dog 1
of the dggs used in the dobutamine experiments

 

Mean estimated ejection
fraction using CaEmax, Ts,
Tbegin estimation method

Mean estimated ejection
fraction using CaEmax, a,
Tbmestimation method

 

 

 

 

Control .6123 .6545
Low Dose dobutamine .782 .7285
High Dose dobutamine .8248 .8271

 

 

 

Table 9. A summary of the results of two variations of our method in dog 2
of the dogs used in the dobutamine experiments

 

Mean estimated ejection
fraction using CaEmax, Ts,
Tbegin estimation method

Mean estimated ejection
fraction using CaEmax, 01,
Tbegin estimation method

 

 

 

 

Control .5507 .6714
Low Dose dobutamine .6584 .717
High Dose dobutamine .6648 .8247

 

 

 

Table 10. A summary of the results of two variations of our method in dog

3 of the dogs used in the dobutamine experiments

It can be seen from Tables 8 through 10 that ejection fraction is estimated

to be similar using both variations of our method. Since the time required to use

the linear method estimating Ts is less than the time required to do QT interval

56

 

 

 

detection and use the method with a measured value of Ts, the method currently
estimates CaEmax, Ts, and Tbegin while using assumptions and measurements to

complete the parameterization of each cardiac cycle.

5.2 Algorithm Speed

In chapter four the usefulness of having a real time algorithm was
discussed. Using a linear estimation of C,,.Emax and searching for T9, and Tbegin
over an acceptable range yields well better than real time run speed on a
standard home PC processor. However, making an algorithm more optimized for
speed would only improve the situation by allowing the designers of systems
which would run this algorithm more flexibility in hardware choices. The key
factor in optimizing our method for speed is to reduce the number of parameters
requiring a non-linear minimization as it was discussed eariier that each
parameter increases run time by a large factor. In the previous section it was
shown that the algorithm could be optimally done by estimating three
parameters; two of these parameters will require non-linear minimization while
CaEmax can be solved for using very quick linear minimization techniques for any
given pair of Tbegin and T5. The method as described in previous chapters simply
uses a grid over all the physiological possible values of the remaining parameters
and finds the combination of the three estimated parameters which minimizes the
difference between the sides of equation 8. To further increase the speed of the
method it is possible to use improved non-linear minimization techniques. Since

the functioning of the heart does not vary much from one beat to the next, it

57

would be possible to use the values of the parameters estimated for the last beat
as an initial guess for the values of the current beat. This initial guesscould then
be used to form a much smaller grid than the one containing all of the
physiologically possible values which would allow for finding the minimum point
much more quickly. There are also several other techniques available for non—
linear minimization which use an initial guess for a value to quickly arrive at the
minimum point. The two dimensional golden search method or the simplex
method are two examples of such methods which could be used to optimize the
speed of our algorithm. Since the algorithm runs well within real time constraints
in its current, robust form it could be used in this way or with more consideration
to optimizing the speed at which it runs.
5.3 Unstressed Volume

In previous chapters, we have discussed the necessity of making a guess
as to the value of unstressed volume. The unstressed volume, or dead volume,
is the volume of blood the heart contains with zero blood pressure. This value
cannot be estimated from aortic blood pressure. In addition, our method does
not use just the unstressed volume of the heart, but actually needs this value
divided by the arterial compliance. Since both of these values are unknown, the
only way to use the method with no additional testing is to use normal values for
each of these parameters to estimate the value in our method. Doing this would
result in some inaccuracy but would not decrease the ability of the algorithm to
detect a change in ejection fraction. As mentioned in chapter four, an alternative

to simply using a normal value for this parameter is to take a single measurement

58

of left ventricular ejection fraction using another method, such as
echocardiography, and calculating unstressed volume divided by arterial
compliance based on that measurement. The method could then be recalibrated
whenever there is a notable change in ejection fraction or any other reason to
believe that unstressed volume may have changed. While this is a shortcoming
of our method, the advantages of having a continuous measurement of left
ventricular ejection fraction more than make up for the small inaccuracies that
could be caused by misestimating unstressed volume or the need for periodic
echocardiography measurements since echocardiography measurements are

currently needed more often than would be the case with our method.

59

6. Conclusion

6.1 Summary

This thesis presents a novel, model based technique for measuring left
ventricular ejection fraction, an important hemodynamic indicator, through central
aortic blood pressure waveform analysis. This technique was tested using data
collected from nine dogs and showed ejection fraction estimated using this
technique showed a strong correlation with the expected value of ejection
fraction and the ejection fraction monitored using common and accepted
techniques.
6.2 Future Research

Chapter two describes the constraints on the cardiovascular system in
terms of two models. The first model is a lumped parameter model of the
cardiovascular system which is widely accepted and shown to be approximately
accurate by many past researchers. The second model is the model the left
ventricle as a time varying capacitor with elastance defined by a raised cosine
which is a function of only a small number of parameters. The assumption made
in forming our method is that since there are many equations corresponding to
each time at which central aortic blood pressure is sampled and only three
unknowns, there is a unique solution to these three variables. We have shown
that by following this assumption and implementing the method as described that
our results correlate very well with measured and expected results. In the future,
we would like to prove mathematically that the solution to these three parameters

is unique based on the factual constraints of the situation. This would be the

60

same as proving that our large number of equations are each unique and do not
all contain some or all of the same data. This proof would give theoretical validity
to our method.

Another area that will be addressed by future research is adapting our
method to minimally invasive or non-invasive measurements. Our current
method relies on a continuous, accurate measurement of central aortic blood
pressure. Central aortic blood pressure is not commonly measured in clinical
settings due to the highly invasive nature of inserting a catheter in to the aorta.
For this reason our current method would not be usable in a standard clinical
setting. In the future, we would like to adapt the method to work using blood
pressure measurements from one or more locations where blood pressure can
be measured minimally invasively or non-invasively. To do this we will
investigate a technique which will reconstruct the central aortic blood pressure
waveform via analysis of more readily available peripheral blood pressure
waveforms. The algorithm presented in this thesis will then be used to estimate

ejection fraction from this reconstructed central aortic blood pressure waveform.

61

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63

 

 

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