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LIBRARY
Michigan State
University

 

 

 

This is to certify that the
dissertation entitled

MODEL BASED ARTERIAL PRESSURE WAVEFORM
ANALYSIS FOR MONITORING CARDIOVASCULAR
FUNCTION

presented by

GOKUL SWAMY

has been accepted towards fulﬁllment
of the requirements for the

 

 

 

 

Ph.D. degree in Electrical Engineering
’ Major Professor’s Signature
03/1 3/2009
Date

MSU is an Afﬁrmative Action/Equal Opportunity Employer

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5/08 K lProg/Acc8Pres/CIRC/Date0ue indd

 

MODEL BASED ARTERIAL PRESSURE WAVEFORM ANALYSIS
FOR MONITORING CARDIOVASCULAR FUNCTION

By

Gokul Swamy

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Electrical Engineering

2009

ABSTRACT

MODEL-BASED ARTERIAL PRESSURE WAVEFORM ANALYSIS FOR
MONITORING CARDIOVASCULAR FUNCTION

By

Gokul Swamy

Today, the workhorse in hemodynamic monitoring is the continuous (i.e.,
automated) measurement and display of blood pressure (BP) waveforms from peripheral
arteries. In particular, minimally invasive (or non—invasive) catheters are broadly utilized
in clinical practice to measure BP waveforms at these circulatory sites. However, it is
well known that the cardiac output (CO) left ventricular ejection fraction (EF), left atrial
pressure (LAP) and central aortic BP (ABP) are more useful in guiding therapy and more
predictive of patient outcome. The conventional methods for measuring each of these
critical central hemodynamic variables require an operator or an unacceptably high level
of invasiveness. We attempt to bridge this gap by presenting three novel techniques to
estimate the central ABP, CO and EF from mathematical analysis of routinely measured
blood pressure waveforms.

The ﬁrst technique is based on multi-channel blind system identiﬁcation in which two
or more measured outputs (peripheral artery pressure waveforms) of a single input, multi-
output system (arterial tree) are mathematically analyzed so as to reconstruct the common
unobserved input (central ABP waveform) to within an arbitrary scale factor. The
technique then invokes Poiseuille’s law to calibrate the reconstructed waveform to
absolute pressure. Proportional estimates of beat—to-beat CO can be estimated from the

reconstructed central ABP waveform by ﬁtting it to a Windkessel model. The second

 

technique aims to estimate EF and proportional left ventricular elastance from a BP
waveform by ﬁtting the waveform to a lumped parameter model of the arterial tree and
subsequently estimating the parameters of this model using an optimization scheme.
Finally, a procedure to quantify wave reﬂection in the arterial tree and its subsequent
application towards estimating the central ABP waveform from only a single peripheral
artery pressure waveform is discussed. These techniques are validated on data collected
from animal studies in which the hemodynamic parameters were varied over a wide

physiologic range.

-- g ._‘ a :«_i-.‘.-I., - ...

ACKNOWLEDGMENTS

First and foremost I would like to express my sincere gratitude and deepest
appreciation to my committee chair, Dr. Ramakrishna Mukkamala, whose technical
genius and passion for research have been instrumental in motivating me to attain greater
heights in my research career. This dissertation would not have been possible without his
constant professional and moral support.

I would also like to extend my sincere thanks to Dr. Satish Udpa whose invaluable
guidance and motivation during my formative years led me to choose a career in
research.

My heart felt gratitude is also due to Dr. Bari Olivier without whose support none of
the physiologic studies could have been completed. I will forever be indebted to him for
patiently explaining to me and teaching me the various surgical procedures used in
physiologic experimentation.

Finally, I would like to express my heartfelt gratitude to my father Mr. V
Doraiswamy Iyer and my mother Mrs. Sundari Swamy for their constant love and
encouragement during every stage of my career. Words cannot express how thankful I am
to my dearest wife Lakshmi for putting up with my late work hours and showering me
with her constant love and encouragement. It is to my family that I dedicate this

dissertation.

iv

TABLE OF CONTENTS

LIST OF TABLES ............................................................................................................ vii
LIST OF FIGURES ........................................................................................................... ix
NOMENCLATURE ......................................................................................................... xii
CHAPTER 1 ......................................................................................................... 1
INTRODUCTION ...................................................... .1
1.1 General ....................................................................................................................... 1
1.2 Current Monitoring Techniques ................................................................................. 5
1.3 Dissertation layout ................................................................................................... 12
CHAPTER 2 ....................................................................................................... 14

BLIND IDENTIFICATION OF THE CENTRAL AORTIC PRESSURE
WAVEFORM FROM MULTIPLE PERIPHERAL ARTERY PRESSURE

WAVEFORMS .................................................................................................................. 14
2.1 Introduction .............................................................................................................. 14
2.2 MBSI Technique ...................................................................................................... 16
2.3 Mathematical Derivation ......................................................................................... 20
2.4 Methods .................................................................................................................... 24
2.5 Results ...................................................................................................................... 25
2.5 Discussions .............................................................................................................. 31

CHAPTER 3 ....................................................................................................... 37

ESTIMATION OF THE AORTIC PRESSURE WAVEFORM AND BEAT-
TO-BEAT CARDIAC OUTPUT FROM MULTIPLE PERIPHERAL

ARTERY PRESSURE WAVEFORM .............................................................................. 37
3.1 Introduction .............................................................................................................. 37
3.2 Technique ................................................................................................................. 41
3.3 Methods .................................................................................................................... 51
3.4 Results ...................................................................................................................... 54
3.5 Discussions .............................................................................................................. 61

CHAPTER 4 ....................................................................................................... 69

CONTINUOUS LEFT VENTRICULAR EJECTION FRACTION

MONITORING BY AORTIC PRESSURE WAVEFORM ANALYSIS ......................... 69
4.1 Introduction .............................................................................................................. 69
4.2 Model Based Analysis Technique ........................................................................... 71
4.3 Materials and Methods ............................................................................................. 78
4.4 Results ...................................................................................................................... 81
4.5 Discussions .............................................................................................................. 90

CHAPTER 5 ....................................................................................................... 98

QUANTIFICATION OF FORWARD AND BACKWARD ARTERIAL
WAVES BY MODEL BASED ANALYSIS OF AORTIC AND F EMORAL

ARTERY PRESSURE WAVEFORM .............................................................................. 98
5.1 Introduction .............................................................................................................. 98
5.2 Technique ............................................................................................................... 100
5.3 Methods .................................................................................................................. 104
5.4 Results .................................................................................................................... 106
5.5 Discussions ............................................................................................................ 111

CHAPTER 6 ..................................................................................................... 1 13

ADAPTIVE TRANSFER FUNCTION FOR DERIVING THE AORTIC
PRESSURE WAVEFORM FROM A PERIPHERAL ARTERY

PRESSURE WAVEFORM ............................................................................................. 113
6.1 Introduction ............................................................................................................ 1 13
6.2 Methods .................................................................................................................. 114
6.3 Results .................................................................................................................... 122
6.4 Discussion .............................................................................................................. 128

CHAPTER 7 ..................................................................................................... 136

CONCLUSIONS .............................................................................................................. 136

REFERENCES ................................................................................................. 140

vi

LIST OF TABLES

Table 1.1 Alternative conventional methods for monitoring CO, LAP, and
EF. These methods suffer from substantial disadvantages that have generally
prevented them from supplanting the standard operator-dependent methods
employed in clinical practice, namely bolus thermodilution, pulmonary
capillary wedge pressure, and imaging. ............................................................................. 11

Table 2.1 Hemodynamic parameters of the swine evaluation datasets. The
multi-channel blind system identiﬁcation (MBSI) technique was
experimentally evaluated with respect to four swine in which femoral artery
pressure and radial artery pressure waveforms and a reference central aortic
pressure waveform were simultaneously measured over a wide
hemodynamic range. MAP is mean arterial pressure; SP, systolic pressure
from central aortic pressure; PP, pulse pressure from central aortic pressure;
EI, ejection interval from central aortic pressure; HR, heart rate; and CO,
cardiac output ..................................................................................................................... 27

Table 2.2 Summary of the experimental evaluation results of the MBSI
technique with respect to the four swine datasets. Overall, the MBSI
technique (see dark line and fonts in Figure (1.1)) as applied to the radial
and femoral artery pressure waveforms in the four swine datasets (see Table
(2.1)) reliably estimated the central aortic pressure waveform along with its
clinically signiﬁcant parameters and thereby signiﬁcantly reduced the wave
distortion in the measured peripheral artery pressure waveforms. RMSE is
root-mean-squared-error; TW, total waveform; SP, systolic pressure; PP,
pulse pressure; and EI, ejection interval. See example results of the MBSI
technique in Figure 2.2. ..................................................................................................... 28

Table 3.1 The new technique (see Figures (3.1) and (3.2)) as well as
alternative peripheral artery pressure (PAP) waveform analysis techniques
were evaluated with respect to four swine datasets consisting of
simultaneous measurements of femoral artery pressure (PAP) and radial
artery pressure (RAP) waveforms, a reference aortic pressure (AP)
waveform, and reference aortic ﬂow probe cardiac output (CO) during
diverse interventions. MAP is mean arterial pressure; SP, systolic pressure
from AP; PP, pulse pressure from AP; SEI, systolic ejection interval from
AP; and HR, heart rate. ..................................................................................................... 53

Table 3.2 Quantitative summary of the aortic pressure waveform estimation
results in terms of root-mean-squared-error. The new technique resulted in
more accurate AP waveform estimates in the four swine datasets (see Figure
(3.3)) than those obtained from alternative PAP waveform analysis
techniques including the conventional generalized transfer function (GTF).

vii

The root-mean-squared—errors are in units of mmHg for the total waveform
(TW), SP, and PP parameters and ms for the SE1 parameter. .......................................... 57

Table 3.3 Quantitative summary of the beat-to-beat proportional cardiac
output estimation results in terms of root-mean-squared-normalized-error.
The new technique (see Figure (3.2)) also resulted in more accurate beat-to-
beat proportional CO estimates in the four swine datasets (see Figure (3.4))
than those obtained from alternative PAP waveform analysis techniques.
The root-mean-squared-normalized-errors are in units of percent ................................... 60

Table 4.1 Summary of the hemodynamic range and results of the model-
based analysis technique for each dog. EF is left ventricular ejection
fraction; EDV, left ventricular end-diastolic volume; HR, heart rate; MAP,
mean aortic pressure; and RMSE, root-mean-squared-error. The BF and
EDV ranges were established with the reference echocardiographic
measurements .................................................................................................................... 85

Table 6.1 Aortic pressures (AP) and heart rate (HR) levels. Values are
expressed as meaniSD. MAP is mean arterial pressure; SP, systolic
pressure; and PP, pulse pressure ..................................................................................... 124

Table 6.2 Peripheral artery pressure (PAP) and derived AP waveform errors.
ATF is adaptive transfer ﬁrnction; GTFARX, autoregressive exogenous input-
based generalized transfer function; PATFTUBE, tube model-based partially
adaptive transfer function; TW, total waveform (i.e., sample-to-sample); and
RMSE, root-mean-squared-error. ................................................................................... 125

Table 6.3 Derived AP waveform errors using actual tube model parameter
values. TW RMSE values represent lower bound for the ATF technique ...................... 134

viii

LIST OF FIGURES

Figure 1.1 Annual percentage of the total population over the age of 60 in
the Americas (Adapted from the US Census Bureau). ....................................................... 2

Figure 1.2 Age distribution of critically ill patients from an intensive care
unit database [Saeed 2002]. ................................................................................................ 3

Figure 1.3 Annual national supply and demand projections for registered
nurses per 10,000 people (Adapted from the US Census Bureau and Bureau
of Health Professions). ....................................................................................................... 4

Figure 1.4 In the early stages of a bleed, arterial BP (ABP) is maintained
even while cardiac output (CO) is falling due to the cardiovascular control
system. Thus, ABP levels do not provide an early indicator of harmful

changes in circulatory status and, as a result, may not provide enough time
for successful therapy (adapted from [Barcroﬂ 1944]) ....................................................... 7

Figure 1.5 ABP levels are not sufﬁciently speciﬁc to permit diagnosis and
guide therapy. However, a differential diagnosis may be obtained by also
monitoring CO, left atrial pressure (LAP), and left ventricular ejection
fraction (EF). ....................................................................................................................... 8

Figure 2.1 The dark lines and fonts illustrate the single input, multi-output
model of the arterial tree upon which the multi-channel blind system
identiﬁcation (MBSI) technique introduced herein is based. The gray line
and fonts represent a contemporary model [Mc Combie 2005], where the
unobserved aortic ﬂow waveform is regarded as the input. ............................................. 18

Figure 2.2 (a, b) Example segments of the measured central aortic pressure
(solid), femoral artery pressure (dash), and radial artery pressure (dot-dash)
waveforms from the four analyzed swine datasets (see Table (2.1)). (c, (1)
Example segments of the central aortic pressure waveform measured (solid)
and estimated (dash) by applying the MBSI technique (see dark lines and
fonts in Figure (2.1)) to the two segments of peripheral artery pressure
waveforms in (a, b). .......................................................................................................... 30

Figure 3.1 Single input, multi-output model of the arterial tree providing the
basis for the technique introduced herein to estimate the clinically more
relevant aortic pressure (AP) waveform from multiple, less invasively
measured peripheral artery pressure (PAP) waveforms distorted by wave
reﬂections .......................................................................................................................... 39

ix

Figure 3.2 Lurnped parameter Windkessel model providing the basis for the
technique to estimate beat-to-beat relative changes in cardiac output (CO)

ﬁrom the estimated AP waveform (pa(t)) in which the wave distortion

should be greatly attenuated (see Figure (3.1)). (b) Illustration generally
indicating how the unknown timing parameters in Equation (3.12), namely

tbs (beginning time of the systolic ejection interval (SEI)), tes (ending time
of the SEI), T (cardiac cycle duration), and 1: = RaCa (time constant of the
Windkessel model), are determined from the estimated AP waveform. .......................... 50

Figure 3.3 (a) Example segments of the measured AP (solid), femoral artery
pressure (dash), and radial artery pressure (dot-dash) waveforms from one of
the four swine datasets (see Table (3.1)). (b) Example segments of the AP
waveform measured (solid) and estimated (dash) by applying the technique
to the two segments of the PAP waveforms (see Figure (3.1)). See Table
(3.2) for a complete quantitative summary of the AP waveform estimation
results. ............................................................................................................................... 58

Figure 3.4 Beat-to-beat CO estimated (and once calibrated) by applying the
technique to the estimated AP waveforms (see Figure (3.2)) plotted against
the gold standard reference aortic ﬂow probe CO measurements for each of
the four swine datasets (see Table (3.1)). The solid line in each plot is the
identity line. See Table (3.3) for a complete quantitative summary of the
beat-to-beat proportional CO estimation results. .............. _ ................................................ 59

Figure 3.5 Single input, multi-output model of the arterial tree providing the
basis for a previous MBSI technique to estimate the shape of the aortic ﬂow
waveform from multiple PAP waveforms. In contrast to the model of
Figure (3.1), the common input to be estimated is the aortic ﬂow waveform
(q(t)). However, the channels coupling the aortic ﬂow waveform input to
the PAP waveform outputs are not coprime, as they share the channel
relating the aortic ﬂow waveform to the AP waveform. Since common
channel dynamics cannot be estimated with MBSI methods [Xu 1995], the
previous technique required additional complicated signal processing. In
addition, this technique did not provide the scale factor of the aortic ﬂow
waveform and therefore cannot be utilized to monitor relative changes in

CO. .................................................................................................................................... 68

Figure 4.1 Model-based analysis technique for monitoring EF. (a) Lurnped
parameter model of the left ventricle and arteries upon which the technique
is based. (b) Elv (t) in the model is assumed to vary over time according to

a parametric raised cosine function (solid line in the lower panel). ................................. 76

Figure 4.2 Exemplary segments of the aortic pressure waveform measured
from the descending thoracic aorta in dog 2 during baseline, dobutarnine,
and volume infusion conditions. ....................................................................................... 84

Figure 4.3 Results for all six dogs in terms of Bland-Altrnan plots of the
average EF and average calibrated EDV errors as a function of the
corresponding reference echocardiographic values. The parameter it
indicates the mean of the errors; a, the standard deviation of the errors; and

RMSE the root-mean-square of the errors (i.e., p2 + 02 ). ........................................... 86

Figure 4.4 Results for all six dogs in terms of a plot of the average EF
estimates versus the corresponding reference echocardiographic EF values. .................. 87

Figure 4.5 Results for dog 2 in terms of the trends in the EF, calibrated
EDV, and CaEmax estimates (dark lines) as a function of time, along with

the corresponding intermittent reference echocardiographic values (gray
circles) and the employed interventions (underlines). ...................................................... 88

Figure 4.6 Results for dog 5 in terms of the trends in the EF, calibrated
EDV, and CaEmax estimates (dark lines) as a ﬁmction of time, along with

the corresponding intermittent reference echocardiographic values (gray
circles) and the employed interventions (underlines). ...................................................... 89

Figure 5.1 Arterial tube model upon which our technique is based. .............................. 101
Figure 5.2 Example segments of the (a) measured ascending aortic pressure

waveform, (b) measured femoral artery pressure waveform, and (c)
predicted abdominal aortic pressure waveform (dash), along with the

measured abdominal aortic pressure waveform (solid). ................................................. 107
Figure 5.3 Computed magnitude spectrum of the reﬂection coefﬁcient. ....................... 108
Figure 5.4 Predicted proportional femoral artery ﬂow. .................................................. 109

Figure 5.5 Measured ascending aortic and femoral artery pressure
waveforms and calculated zero-mean forward and backward waves at the
ascending aorta ................................................................................................................ 110

Figure 6.1 Adaptive transfer function (ATF) technique for deriving the
aortic pressure (AP) waveform from a peripheral artery pressure (PAP)
waveform. ....................................................................................................................... 116

Figure 6.2 Example segments of the reference AP (solid), measured PAP
(dot), and derived AP (dash) waveforms by the new ATF technique and
previous autoregressive exogenous input-based generalized transfer function
(GTFARX) and tube model-based partially adaptive transfer function
(PATFTUBE) techniques during the dobutamine and nitroglycerin conditions ................ 126

xi

Ca
CO
EI
Elv(t)
Emin
HR
LVEF

MBSI

nbs
nes
Pa(t)
PAP
pbwd
pfwd
PP

Ra
RMSE
RMSNE
SP

Ts

NOMENCLATURE

Windkessel time constant

Arterial compliance

Cardiac output

Ejection interval

Time varying ventricular elastance
Minimum left ventricular elastance

Heart Rate

Left ventricular ejection fraction

Mean arterial pressure

Multi-channel blind system identiﬁcation
Discrete time or sample index

Sample index denoting beginning of systolic ejection interval
Sample index denoting end of systolic ejection interval
Aortic pressure waveform

Peripheral artery pressure waveform
Reﬂected arterial pressure wave

Forward arterial pressure wave

Pulse pressure

Total peripheral resistance
Root-mean-squared error
Root-mean-squared normalized error
Systolic pressure

time

Sampling period

Time duration to reach Emax from Emin

Wave reﬂection coefﬁcient

xii

CHAPTER 1
INTRODUCTION

1.1 General

The proportion of the elderly population is projected to grow at the beginning of the
21m century (see Figure (1.1)). This projection can be partly attributed to advances in
biomedical technology, which have increased life expectancy. For example, new medical
devices such as implanted deﬁbrillators and stents have decreased the mortality rate
following a heart attack (but have increased the prevalence of heart failure). Moreover,
the elderly population contributes disproportionately to the overall prevalence of disease
(see, e.g., the age distribution of critically ill patients in Figure (1.2)). Because of the
evolving demographics as well as for other reasons, there is a simultaneous projected rise
in the deﬁcit of needed clinical staff (see, e. g., the nursing shortage projection in Figure
(1.3)). One important implication of these projections is the need for effective and easy-
to-use patient monitoring technologies for the new century.

This need is especially apparent in the context of hemodynamic monitoring of

cardiovascular disease.

 

 

Fraction of Total Population over Age 60
M)
N
O

15~
.l
1or—rrrijtrrrTfﬁrr1rrrr1
1950 1975 2000 2025 2050
Year

Figure 1.1 Annual percentage of the total population over the age of 60 in the Americas
(Adapted from the US Census Bureau).

300

 

 

 

.93 250—
C
.2
1'6 200
n.
H—
o
t. 150
a:
.o
E 100
z
50»
o
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Age

Figure 1.2 Age distribution of critically ill patients from an intensive care unit database
[Saeed 2002].

901

85*

80-“

75—

70-

65*

 

60—

Demand

Supply

 

Number of Registered Nurses I 10k People

2000

ﬁTI

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2010

I

2015

I 7 I

F i
2005 2020

Year

Figure 1.3 Armual national supply and demand projections for registered nurses per
10,000 people (Adapted from the US Census Bureau and Bureau of Health Professions).

1.2 Current Monitoring Techniques

Today, the workhorse in hemodynamic monitoring is the continuous measurement
and display of BP waveforms. For example, invasive catheters are utilized in about 50-
80% of all critically ill patients to monitor BP waveforms from peripheral arteries,
especially the radial and femoral arteries. Moreover, over the past few decades, totally
non-invasive methods have been developed and reﬁned to measure peripheral ABP
waveforms via ﬁnger-cuff photoplethysmography and applanation tonometry. These
non-invasive methods are even available as commercial systems that are simple enough
to potentially be used by patients at home (see, for example, the Finometer and Portapres,
Finapres Medical Systems, The Netherlands and the T-Line Blood Pressure Monitoring
System, Tensys Medical Inc., San Diego, CA). Finally, new systems are continually in
development with much promise for future expansion of BP waveform monitoring. For
example, it may be possible one day to chronically monitor peripheral ABP waveforms
with simple wearable ring sensors. However, while these systems are continuous and
offer a level of invasiveness suitable for routine clinical use, they are limited in that the
measured and displayed BP levels are not very good indicators of circulatory status.

One reason is that BP levels in the peripheral arteries are not as clinically relevant as
its counterpart in the central aorta. (Note that catheterization of the central aorta is too
invasive and risky for routine clinical practice, as blood clot formation and embolization
here could lead to, for example, a stroke.) In particular, central ABP levels would be
preferred to peripheral ABP levels, as the latter are signiﬁcantly distorted by highly
complex wave reﬂections (see below). For example, both systolic pressure and pulse

pressure (PP, systolic minus diastolic pressure) are ampliﬁed in peripheral ABP

waveforms, with the extent of the ampliﬁcation dependent on the particular peripheral
site and circulatory state. Thus, it is the systolic and diastolic pressures measured
speciﬁcally in the aorta that truly reﬂect cardiac afterload and perfusion. Perhaps, as a
result, central measurements of systolic pressure and PP have been shown to be superior
in predicting patient outcome than corresponding measurements made in more peripheral
arteries.

Another reason is that BP levels do not provide an early indicator of changes in
circulatory status. For example, in the early stages of a bleed, the cardiovascular control
system maintains ABP at the expense of other hemodynamic variables (most notably CO)
in order to adequately perfuse all tissue beds of the body (see Figure (1.4)). While frank
hypotension may eventually occur (see Figure (1.4)), it is often too late to intervene at
this point (e.g., irreversible hemorrhagic shock). Thus, ABP levels do not provide as
early an indicator of circulatory changes as CO and, as a result, may not permit sufﬁcient

time for successful therapy.

 

 
  
    

 

 

 

 

 

 

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Figure 1.4 In the early stages of a bleed, arterial BP (ABP) is maintained even while
cardiac output (CO) is falling due to the cardiovascular control system. Thus, ABP levels
do not provide an early indicator of harmful changes in circulatory status and, as'a result,
may not provide enough time for successful therapy (adapted from [Barcroft 1944]).

(a)
ABP]

.// \\

Hypovolemia?

Diastolic Systolic
Dysfunction? Dysfunction?

(b)
ABpt
A// \\x
cot col

A// ;/’ \\\

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/ EFI Hypovolemia

Diastolic Systolic
Dysfunction . Dysfunction

Figure 1.5 ABP levels are not sufﬁciently speciﬁc to permit diagnosis and guide therapy.
However, a differential diagnosis may be obtained by also monitoring CO, left atrial
pressure (LAP), and left ventricular ejection fraction (EF).

A third reason is that BP levels are dependent on multiple physiologic factors and are
therefore not sufﬁciently speciﬁc to permit diagnosis and direct therapy. For example,
when the cardiovascular control system eventually fails so as to result in hypotension,
this could indicate sepsis (low vascular resistance due to bacterial blood poisoning),
diastolic dysfunction (reduced cardiac ﬁlling due to, e.g., tamponade), systolic
dysfunction (weakened cardiac contraction due to, e.g., a heart attack), or hypovolemia
(due to, e.g., an internal bleed). To distinguish amongst these possibilities so as to guide
therapy, it is well known that the CO, LAP, and EF must also be monitored (see Figure
(1.5)). Note that EF has also proven to be a powerful predictor of outcome in heart failure
patients.

The standard clinical method for monitoring CO involves the use of the balloon-
tipped, ﬂow-directed pulmonary artery catheter. CO is speciﬁcally estimated via the
bolus thermodilution method. This method involves injecting a bolus of cold saline in
the right atrium, measuring temperature downstream in the pulmonary artery, and
computing the average CO based on conservation laws. The standard clinical method for
monitoring EF (i.e., the ratio of the stroke volume (SV) to the left ventricular end-
diastolic volume (EDV)) is by imaging the left ventricular volume. Commonly employed
imaging methods include echocardiography, radionuclide techniques (ﬁrst pass or
equilibrium), contrast angiography, ultra-fast computed tomography (CT), and magnetic
resonance imaging. However, these clinical methods generally share the major limitation
of requiring a trained operator for their implementation. While alternative measurement
methods are available, these methods suffer from substantial limitations that have

generally prevented them from supplanting the operator-dependent methods in clinical

practice (see Table (1.1)). Thus, the monitoring of these three critical central

hemodynamic variables is limited today and likely to be even more so in the new century.

10

Table 1.1 Alternative conventional methods for monitoring CO, LAP, and EF. These
methods suffer from substantial disadvantages that have generally prevented them from
supplanting the standard operator-dependent methods employed in clinical practice,
namely bolus thermodilution, pulmonary capillary wedge pressure, and imaging. 1

Hemo-

 

 

. Me r men .
dynamic asu e t Advantages Disadvantages
. Method
varrable
aortic ﬂow probe continuous thorocotomy
accurate
continuous, with
continuous estimates of less accurate than
thermodilution right ventricular bolus thermodilution
ejection fraction
CO . . x r t r
doppler ultrasound non-invasrve e pert ope a o
expensrve
accurate two catheterizations
oxygen ﬁck . .
inexpensrve operator
. . . inaccurate,
thoracrc non-invasrve . .
. . . especrally 1n
brormpedance contrnuous . . . .
critically 111 patients
too invasive and
conductance . .
continuous rrsky
catheter . .
repeated calibrations
EF non-imaging non-invasive too difﬁcult to
nuclear monitor continuous position
thorocotomy, with
sonomicrometry continuous crystals sutured to

11

the ventricle

1.3 Dissertation layout

To address the above limitations we propose several techniques to mathematically
estimate the central hemodynamic parameters from routinely measured blood pressure
waveforms.

The rest of the dissertation is organized as follows. Chapter 2 describes a technique to
estimate the central ABP from two or more peripheral artery pressure waveforms. The
technique is based on multi-channel blind system identiﬁcation in which two or more
measured outputs (peripheral artery pressure waveforms) of a single input, multi-output
system (arterial tree) are mathematically analyzed so as to reconstruct the common
unobserved input (central aortic pressure waveform) to within an arbitrary scale factor.
The technique then invokes Poiseuille’s law to calibrate the reconstructed waveform to
absolute pressure. Chapter 3 proposes a reﬁnement to this technique wherein more
efﬁcient representations of arterial tree transfer ﬁmctions are utilized to reduce the
reconstruction error. This reduction in error permits beat-to-beat cardiac output to be
estimated from the reconstructed central ABP waveform with high accuracy. Chapter 4
lays the foundation for wave reﬂection analysis in which measured aortic and peripheral
artery pressure waveforms are represented in a transmission line model to estimate the
forward and backward pressure and ﬂow waves in the arterial tree. This model in
conjunction with the knowledge that aortic ﬂow is zero during diastole is utilized in
chapter 5 to estimate the central ABP waveform from a single peripheral artery pressure
waveform. Chapter 6 utilizes a Windkessel model for the arterial tree and a variable

capacitance model for the left ventricle to derive the ejection fraction and proportional

12

left ventricular elastance from measurement of a central ABP waveform. Finally, some

conclusions and directions for future research are outlined in chapter 7.

l3

CHAPTER 2

BLIND IDENTIFICATION OF THE CENTRAL AORTIC PRESSURE
WAVEFORM FROM MULTIPLE PERIPHERAL ARTERY PRESSURE

WAVEFORMS

2.1 Introduction

As the arterial pressure wave traverses from the central aorta to the peripheral
arteries, its contour becomes signiﬁcantly distorted due to complex wave reﬂections in
the distributed arterial tree [O’Rourke 1991]. For example, both systolic pressure and
pulse pressure usually become ampliﬁed with the extent of the ampliﬁcation dependent
on the particular peripheral site and state of the arterial tree [Soderstrom 2002]. Thus, it is
the systolic and diastolic pressures measured speciﬁcally in the central aorta that truly
reﬂect cardiac afterload and perfusion [Chen 1997]. Perhaps, as a result, central
measurements of systolic pressure and pulse pressure have been shown to be superior in
predicting patient outcome than corresponding measurements made in more peripheral
arteries [Safar 2002, Wadell 2001]. Moreover, since central aortic pressure is not
signiﬁcantly complicated by the wave reﬂections [Bourgeois 1976, Noordergraf 197 8 ],
the entire waveform clearly reveals the cardiac ejection interval through the dichrotic
notch [Fetics 1999].

The measurement of the central aortic pressure waveform usually involves
introducing a catheter into a peripheral artery and guiding the catheter against the ﬂowing

blood to the central aorta. However, placement of an aortic catheter is not commonly

l4

performed in clinical practice [Chen 1997] because of the risk of blood clot formation
and embolization. On the other hand, related, but distorted, peripheral artery pressure
waveforms may be measured less invasively and more safely via placement of a catheter
in a distal artery. Indeed, radial and femoral artery catheterizations are routinely
performed in clinical practice [Marino 1998]. Moreover, over the past few decades,
totally non-invasive methods have been developed and reﬁned to continuously measure
peripheral artery pressure based on ﬁnger-cuff photoplethysmography [Imholz 1998] and
applanation tonometry [Kenner 1988]. These non-invasive methods are even available as
commercial systems at present (see, for example, the Finometer and Portapres, Finapres
Medical Systems, The Netherlands and the T-Line Blood Pressure Monitoring System,
Tensys Medical Inc., San Diego, CA).

Several techniques have therefore been recently developed to mathematically derive
the clinically more relevant central aortic pressure waveform from less invasively
measured peripheral artery pressure waveforms. Most of these techniques have involved
1) initially obtaining simultaneous measurements of central aortic and peripheral artery
pressure waveforms in a group of subjects; 2) estimating a group-averaged transfer
function relating the measured peripheral artery pressure to the measured central aortic
pressure; and 3) subsequently applying this transfer function to peripheral artery pressure
measured from a new subject in order to predict the unobserved central aortic pressure
waveform [Soderstrom 2002, Chen 1997, Fetics 1999, Karamanoglu 1993]. The
principal assumption underlying these “generalized transfer function” techniques is that
arterial tree properties are constant over all time and between all individuals. Because of

known inter-subject and temporal variability of the arterial tree, a few techniques have

15

been more recently proposed towards partial individualization of the transfer function
relating peripheral artery pressure to central aortic pressure through modeling
[Karamanoglu 1997, Sugimachi 2001, Segers 2000].

It would be desirable to be able to estimate the central aortic pressure waveform from
peripheral artery pressure in an entirely patient and time speciﬁc manner. One possible
way to do so is with the multi-channel blind system identiﬁcation (MBSI) approach of
recent interest in signal processing [Abed—Meraim 1997, Xu G 1995]. In this approach,
two or more outputs of a single input, multi-output system are analyzed so as to
reconstruct the common input. To our knowledge, the very recent study by McCombie et
a1. represents the ﬁrst application of MBSI to the ﬁeld of hemodynamic monitoring [Mc
Combie 2005]. However, their study speciﬁcally aimed to estimate the shape of the
aortic ﬂow waveform from peripheral artery pressure measurements (see Discussion
section).

In this study, we introduce a new technique to reconstruct the central aortic pressure
waveform from multiple peripheral artery pressure waveform measurements without the
need for a generalized transfer function using the MBSI approach. We then demonstrate
the validity of the MBSI technique with respect to four swine in which femoral and radial
artery pressure waveforms and a reference central aortic pressure waveform were

simultaneously measured during diverse hemodynamic interventions.

2.2 MBSI Technique

Our technique, which was initially presented in abbreviated form in [Swamy 2006],
applies standard MBSI algorithms from the signal processing literature [Abed-Meraim

1997, Xu 1995] to two or more peripheral artery pressure waveforms in order to

16

reconstruct the central aortic pressure waveform to within an arbitrary scale factor and
then calibrates the reconstructed waveform to absolute pressure based on known
physiology. Below, we describe the technique at a conceptual level while stating its
underlying assumptions. See the next section for the mathematical derivation and the
Discussion section for a justiﬁcation of its assumptions.

Figure (2.1) (dark lines and fonts) illustrates the single input, multi-output model of
the pressure waveforms in the arterial tree upon which the technique is based. Here, the

m (> 1) measured and sampled peripheral artery pressure waveforms (ppi(t),l _<_ i S m)

are modeled as outputs of m unknown systems or channels driven by the common
unobserved and likewise sampled central aortic pressure waveform (pca(t)) input. Each
of the discrete-time channels coupling the common input to each of the distinct outputs
characterizes the dynamic properties of a different arterial tree path. These channels are
assumed to be linear and time-invariant (LTI) over each one-minute interval of analysis
(see Methods section). The LTI channels are further assumed to be well approximated by

impulse responses (i.e., time-domain version of transfer functions; (hi(t), 1 S i S m) that

are ﬁnite in duration and different from each other.

17

'11“) PM It)

 

ppZIt)

 

pimp)

Figure 2.1 The dark lines and fonts illustrate the single input, multi-output model of the
arterial tree upon which the multi-channel blind system identiﬁcation (MBSI) technique
introduced herein is based. The gray line and fonts represent a contemporary model [Mc
Combie 2005], where the unobserved aortic ﬂow waveform is regarded as the input.

18

Different here precisely means that the ﬁnite impulse responses (F IRs) are coprime with
each other (i.e., the Z-transforms of the impulse responses share no common zeros or
roots). In this way, all of the commonality in the measured outputs may be attributed to
the input, and the differences in the measured outputs (see Discussion section) may then
be deciphered so as to estimate the FIRs and ultimately reconstruct the common central
aortic pressure waveform input. Note that it is generally impossible to determine the
scale factor of the FIRs and therefore the common input, because any scaling of the
common input may be offset with a reciprocal scaling of the FIRs. Thus, physiologic
knowledge must also be employed to clarify the ambiguity.

More speciﬁcally, ﬁrst, the Fle are mathematically estimated based on the cross
relations between pairs of measured outputs. These cross relations may be derived from
the fundamental properties of the convolution operation governing LTI input-output

behavior as follows:
Ppi(t)® hj1tl= (Pcalt)® hiitll 8’ hj“) =
hi(t)®(pca(t)®h,-(t»= hirtieppjrt).

where i 1: j and ® denotes the convolution operation. The FIRs are speciﬁcally estimated
to within an arbitrary scale factor by solving the homogenous system of equations
resulting from the cross relations using the convenient eigenvector algorithm [Xu 1995].
The implicit assumption here is that the central aortic pressure waveform input is
persistently exciting of high enough order (i.e., containing at least as many frequency
components as the ntunber of estimated FIR samples) [Xu 1995].

Then, the central aortic pressure waveform input is reconstructed to within an

arbitrary scale factor by deconvolving the estimated Fle from the measured peripheral

19

artery pressure waveforms. In particular, a single reconstructed waveform is obtained by
employing a multi-channel least squares deconvolution algorithm [Abed-Meraim 1997].

Finally, the reconstructed waveform is calibrated to absolute pressure by sealing it to
have the same mean value as the measured peripheral artery pressure. This scaling step is
well justiﬁed, since the paths from the central aorta to peripheral arteries offer very little
resistance to blood ﬂow due to Poiseuille’s law [Noordergraf 1978].

It should be noted that the reconstructed absolute central aortic pressure waveform
will be slightly delayed with respect to the actual central aortic pressure waveform,
because the time delay shared by the FIRs cannot be identiﬁed with MBSI. However,

this delay, which is usually < 0.1 sec, is not important for most clinical applications.

2.3 Mathematical Derivation

We outline below the mathematical steps of the MBSI technique for the simplest case
in which two peripheral artery pressure waveforms are analyzed. See [Abed-Meraim
1997, Xu 1995] for a more general mathematical treatment of the employed MBSI
algorithms.

First, the Fle in Figure (2.1) (dark lines and fonts) are mathematically estimated to
within an arbitrary scale factor based on the following cross relation between the two

measured outputs:

L—1
2 h1(k)*pp2(t-k)—k 20h2(k)*pp1(t —k) = e(t), te [L —1,N —1]. (2.1)

Here, the convolution sum has been explicitly written (rather than using shorthand
notation as in the initial cross relation equation of the manuscript), and the term e(t) has

been included to account for any measurement noise and/or modeling error. The

20

variables L and N in Equation (2.1) respectively represent the number of samples of each
FIR (channel order) and the number of measured peripheral artery pressure waveform
samples. This equation can be expressed in matrix form by stacking each individual

equation corresponding to each time t, one on top of the other, as follows:

 

[p332 3-93,] [32]” (2.2)
h
where
' ppi(0) ppm) ppi(L—1)“
133: pp,(1) ppi:(2) ppifL) . ie[1,2],
_pp;(f‘l-L) ppi(N:-L+1) --: ppi(N—1)_

 

 

are [(N-L+l)xL] Hankel matrices comprising the respective measured output samples;
h-h-(L—1)h~L 2 h-O T ' 12
i_[| |(—)--- Kl]. |E[.].
are [Lxl] vectors specifying the samples or parameters of the two respective FIRs; and
T
e = [e(O) e(1) e(N—L)]

is an [(N-L+1)x1] vector of the noise samples. For a ﬁxed channel order L, the vector h
in Equation (2.2) is estimated to a certain non-trivial constraint by minimizing the energy

in the vector e. This optimization problem is speciﬁcally solved in closed-form by

selecting the eigenvector associated with the minimum eigenvalue of the matrix PTP as
a unit-energy estimate of the vector 11. The channel order L is determined by 1) forming

a P matrix of dimension [(N-Lmax +1)><2Lmax], where Lmax =15 is assumed to

encompass the true channel order; 2) computing the eigenvalues of the matrix P1P; and

21

3) establishing the Optimal value of L as half the number of eigenvalues (rounded up
when odd) that are at least 5% of the maximum eigenvalue.

Second, the common central aortic pressure waveform input in Figure (2.1) (dark
lines and fonts) is determined to within an arbitrary scale factor from the two determined
FIRs (i.e., the estimated vector h) and the two measured outputs through multi-channel
least squares deconvolution. That is, the two measured outputs may be expressed in

terms of their common input via the convolution sum as follows:
L — 1

Ppi“) = Z hi(k)pca(t — k) + ni(t). i e [1.2], t e [0,N — 1], (2.3)
k = 0

where ni(t)accounts for any noise. This equation may also be expressed in matrix form

by stacking each individual equation for each t and i, one on top of the other, as follows:
P H
p1 3 1 Pca + "1 (2.4)
ppz H2 n .
H
pp "
where
pp, = inspire) ppi<2) ppim -1)1T. ie [1,2].

are [N)( 1] vectors of the respective measured output samples;

hi(L—1) hi(0) 0
Hi: 2 s , ie[1,2],
o hi(L—1) hi(0)

are the [Nx(N+L-1)] Toeplitz matrices including the estimated samples of the respective

FIRs;

pca=[pca(_L-1) pca(_L"2) pca(0) pea“) pca(N'—1)]T

22

is a [(N+L-1)xl] vector of unmeasured common input samples; and
T .
ni = [ni(0) ni(1) ni(N—1)] , Ie [1,2],

are [le] vectors of the respective noise samples. The vector pca in Equation (2.4) is

then estimated to within an arbitrary scale factor by minimizing the energy in the vector
n. This optimization problem is speciﬁcally solved in closed-form using the following

linear least-squares solution:
pca : (HTH)_1HTppv (25)

where the inverse here is computed efﬁciently as described in [Jain, 1978]. Following
the deconvolution, the reconstructed input is lowpass ﬁltered with a cutoff frequency of

10 Hz.

Third, the reconstructed waveform (i.e., the determine pea vector) is calibrated to

absolute pressure by scaling it to have the same mean value as that of the measured
peripheral artery pressure as follows:

N—1

p80,“): pcam—Li‘,’ - (2'6)

2 pca(t)
t= 0

Here, p3ca is the absolute (scaled) estimated central aortic pressure waveform.

Finally, if the average systolic pressure of the reconstructed central aortic pressure
waveform is greater than that of the measured peripheral artery pressure, then the solution
is considered to be invalid and the above steps are repeated but with the channel order

reduced by one. We note that this technique always resulted in a valid estimate of the

central aortic pressure waveform for every interval of analysis in the present swine study.

23

2.4 Methods

We evaluated the MBSI technique with respect to previously collected hemodynamic
measurements from swine, which are described in detail elsewhere [Mukkamala 2006].
Below, we brieﬂy describe the experimental procedures employed for collecting these
hemodynamic data and then present the methods for data analysis utilized herein.
Hemodynamic Data

Six Yorkshire swine (30-34 kg) were studied under a protocol approved by the MIT
Committee on Animal Care. Following the induction of general anesthesia and
mechanical ventilation, physiologic transducers were placed in each animal as follows. A
micromanometer-tipped catheter was fed retrograde to the thoracic aorta via a femoral
artery for reference central aortic pressure. Fluid-ﬁlled catheters were then inserted in the
opposite femoral artery for femoral artery pressure and in an artery as distal as possible to
the brachial artery for “radial” artery pressure. Finally, an ultrasonic ﬂow probe was
placed around the aortic root following a midline stemotomy for cardiac output. In each
animal, a subset of the following interventions was then performed over the course of 75
to 150 minutes to vary arterial pressures as well as other hemodynamic parameters:
infusions of volume, phenylephrine, dobutamine, isoproterenol, esmolol, nitroglycerine,
and progressive hemorrhage. Several infusion rates were implemented followed by brief
recovery periods. The hemodynamic waveforms were continuously recorded throughout
the intervention period at a sampling rate of 250 Hz and 16-bit resolution.

Data Analysis
We discarded two of the six swine datasets from the study due to excessive damping

of the femoral artery pressure waveform in one dataset [Mukkamala 2006] and an

24

improperly calibrated reference central aortic pressure waveform in the other dataset. We
then applied the technique to all one-minute, non-overlapping intervals of the femoral
and radial artery pressure waveforms (resampled to 50 Hz) in the remaining four swine
datasets. We evaluated the resulting central aortic pressure waveform estimates with

respect to the measured reference waveforms (likewise resampled to 50 Hz) in terms of

the root-mean-squared-error (RMSE = ([112 +02 , where u is the bias and o is the

precision) of the following parameters: total waveform (i.e., sample-to-sample), beat-to-
beat systolic pressure, beat-to-beat pulse pressure, and beat-to-beat ejection interval. For
comparison, we likewise evaluated the peripheral artery pressure waveforms with respect
to the measured central aortic pressure waveforms in terms of the ﬁrst three parameters.
(Note that we did not attempt to determine the ejection intervals from the peripheral
artery pressure waveforms, because the dichrotic notch was generally obscured by wave
reﬂections.) Prior to conducting these evaluations, we advanced the central aortic
pressure waveform estimates so that they were temporally aligned with the measured
central aortic pressure waveforms. To make a fair comparison, we likewise time aligned

the peripheral artery pressure waveforms.

2.5 Results

Tables (2.1) and (2.2) respectively summarize the hemodynamic parameters of the
four analyzed swine datasets and the evaluation results of the MBSI technique with
respect to these datasets. These tables generally indicate that the technique was able to

reliably estimate the central aortic pressure waveform over a wide hemodynamic range

25

with a level of accuracy that was far better than no mathematical analysis of the
peripheral artery pressure waveforms.

More speciﬁcally, the overall total waveform RMSE of the estimated central aortic
pressure was 4.7 mmHg (after a modest time alignment as described above). For
comparison, the average overall total waveform RMSE between the measured peripheral
artery pressures and the measured central aortic pressure was 8.6 mmHg (after a more
signiﬁcant time alignment). Thus, the technique was able to effectively reduce the total
wave distortion in the measured peripheral artery pressure waveforms by 45%.
Furthermore, the overall beat-to—beat systolic pressure RMSE and the overall beat-to-beat
pulse pressure RMSE of the estimated central aortic pressure were 7.5 mmHg and 8.2

mmHg, respectively.

26

Table 2.1 Hemodynamic parameters of the swine evaluation datasets. The multi-channel
blind system identiﬁcation (MBSI) technique was experimentally evaluated with respect
to four swine in which femoral artery pressure and radial artery pressure waveforms and a
reference central aortic pressure waveform were simultaneously measured over a wide
hemodynamic range. MAP is mean arterial pressure; SP, systolic pressure from central
aortic pressure; PP, pulse pressure from central aortic pressure; EI, ejection interval from
central aortic pressure; HR, heart rate; and CO, cardiac output.

 

 

MAP SP PP EI HR CO

Animal Range Range Range Range Range Range
immHg] [mmH g] [mmH g] [ms] [beats/min] [L/min]

1 54— 136 70— 168 28—48 237—267 100—223 2.3 —4.1

2 58—117 73-143 19—49 235-337 92—190 1.7—6.0

3 45— 114 60- 140 22—45 204—296 91 -243 2.4—5.7

4 48—119 70—142 22-51 198-261 102—207 1.3—6.2
Total 45—136 60-168 19—51 198—337 91 —243 1.3—6.2

 

27

Table 2.2 Summary of the experimental evaluation results of the MBSI technique with
respect to the four swine datasets. Overall, the MBSI technique (see dark line and fonts in
Figure (1.1)) as applied to the radial and femoral artery pressure waveforms in the four
swine datasets (see Table (2.1)) reliably estimated the central aortic pressure waveform
along with its clinically signiﬁcant parameters and thereby signiﬁcantly reduced the wave
distortion in the measured peripheral artery pressure waveforms. RMSE is root-mean-
squared-error; TW, total waveform; SP, systolic pressure; PP, pulse pressure; and EI,
ejection interval. See example results of the MBSI technique in Figure 2.2.

 

Estimated Central Aortic Femoral Artery Radial Artery Pressure
Dog Pressure RMSE Pressure RMSE RMSE
TW SP PP E1 TW SP PP TW SP PP
[mmHg] [ms] [mmHg] [mmHg]
1 4_4 4.7 4.8 82 11.6 19.2 10.5 4.0 5.1 4.9

2 4,3 5,2 6,4 57 7.8 13.5 14.3 10.5 24.5 19.2
3 4_7 62 8.5 55 6.3 12.0 10.2 9.4 16.7 10.3
5.6 7,8 7,7 67 7.0 10.7 14.3 9.3 22.7 21.6

 

Total 4.7 7.5 8.2 59 8.1 13.7 12.6 9.1 19.7 15.9

 

28

These errors represent an average overall improvement of 57% and 41% with respect
to the corresponding parameters from the measured peripheral artery pressure
waveforms. In addition, the overall beat-to-beat ejection interval RMSE of the estimated
central aortic pressure was 59 msec. Finally, though not shown in the tables, the errors in
the four studied parameters of the estimated central aortic pressure were virtually
uncorrelated with the respective reference values of these parameters.

Figures (2.2a) and (2.2b) provide two visual examples illustrating the signiﬁcant
differences between the measured peripheral artery pressure waveforms (dash and dot-
dash) and the corresponding measured central aortic pressure waveforms (solid), while
Figures (2.2c) and (2.2d) show the resulting central aortic pressure waveforms estimated
from these peripheral artery pressure waveforms (dash) along with the reference central
aortic pressure waveforms (solid). As is evident in these examples at two different mean
pressure levels, the estimated and reference central aortic pressure waveforms agree very
closely, and much of the wave distortion in the measured peripheral artery pressure

waveforms has been eliminated.

29

140' 140'
120 120 .
100’

80

 

 

 

 

 

 

 

 

 

 

 

'5':
:1: 60-
E
E. 40 .
2 o 05 1 15 2
3
(0
'8
" (C) d
9.: 140' 140- ( )
.2
3
1: 120+ 120-
<
100’ 100.
so so,
601 so.
400 0:5 1 1:5 2 400 0:5 1 1:5 2

Time [see]

Figure 2.2 (a, b) Example segments of the measured central aortic pressure (solid),
femoral artery pressure (dash), and radial artery pressure (dot-dash) waveforms ﬁom the
four analyzed swine datasets (see Table (2.1)). (c, d) Example segments of the central
aortic pressure waveform measured (solid) and estimated (dash) by applying the MBSI
technique (see dark lines and fonts in Figure (2.1)) to the two segments of peripheral
artery pressure waveforms in (a, b).

30

2.5 Discussions

In summary, we have introduced a new technique to mathematically reconstruct the
clinically more relevant central aortic pressure waveform from multiple, less invasively
measured peripheral artery pressure waveforms distorted by wave reﬂections. Our
technique capitalizes on the powerful MBSI approach of recent interest in signal
processing in which the differences in the outputs of a single input, multi-output system
are assessed so as to reconstruct the common input to within an arbitrary scale factor.
Then technique then calibrates the reconstructed waveform to absolute pressure using
Poiseuille’s law. As a result, in contrast to previous, related efforts, our technique neither
employs a generalized transfer function nor requires any training data and is therefore
entirely patient and time speciﬁc. We have also presented an experimental evaluation of
the technique in four swine in which radial and femoral artery pressure waveforms and a
reference central aortic pressure waveform were simultaneously measured over a wide
hemodynamic range. Our results show that the technique was able to reliably estimate
the entire central aortic pressure waveform and thereby signiﬁcantly improve upon the
determinatidn of systolic pressure, pulse pressure, and the ejection interval as compared
to measuring these clinically signiﬁcant parameters directly from the peripheral artery
pressure waveforms.

Assumptions of the MBSI Technique

As stated above, our MBSI technique is based on a set of assumptions. We make
physiologic and empirical arguments to justify each of the underlying assumptions below.

Assumption 1: the channels relating the common input to each distinct output in

Figure 1.1 are LTI over each one-minute interval of analysis. Over such short time

31

intervals, the arterial tree is usually operating in near steady-state conditions in which the
statistical properties of the arterial pressure waveforms vary little over time. Such steady-
state conditions clearly justify the time-invariance approximation. Moreover, these
conditions also support the linearity approximation as argued in [McCombie 2005] and
references therein.

Assumption 2: the LT] channels are characterized with FIRs. Although not widely
appreciated, it is known that arterial pressure waveforms measured from distinct sites
only differ signiﬁcantly in terms of their high frequency detail while being quite similar
at lower frequencies [Noordergraf 1978, Mukkamala 2006]. Thus, the dynamics of each
of the channels in Figure (2.1) (dark lines and fonts) are fast (e.g., effectively vanishing
within ~0.5 sec [Zhang 2002]), thereby supporting the FIR approximation.

Assumption 3: the FIRs are coprime with each other. If the F IRS were not coprime
with each other, then the non-coprime or common FIR dynamics would be erroneously
attributed to the common input. As discussed above, peripheral artery pressure
waveforms from distinct sites in the arterial tree appear different. Thus, the dynamics of
each channel cannot be the same, and the coprime channel approximation is at least
somewhat tenable.

Assumption 4: the central aortic pressure waveform is persistently exciting of high
enough order. This assumption means that the central aortic pressure waveform contains
at least as many frequency components as the number of estimated FIR samples. As
described above, the channel dynamics are short duration. Thus, the number of FIR
samples to be estimated is small, thereby buttressing the persistence of excitation

approximation.

32

Any violation to the four above assumptions in the present swine study may indeed
represent a source of error of our MBSI technique. However, we note that each of the
assumptions must have been at least largely valid here given that the discrepancy
between the estimated and reference central aortic pressure waveforms was relatively
small (see Table (2.2)).

MBSI Technique in the Context of Previous Efforts

The MBSI technique that we have introduced herein was inspired by the seminal
contributions of several previous investigations described in the hemodynamic
monitoring literature. In particular, the idea of mathematical deriving central aortic
pressure from measured peripheral artery pressure stems from the body of generalized
transfer function literature [Soderstrom 2002, Chen 1997, Fetics 1999, Karamanoglu
1993, Karamanoglu 1997, Sugimachi 2001, Segers 2000], whereas the idea of employing
MBSI to do so in an entirely patient and time speciﬁc manner is based on the very recent
study by McCombie et al. [McCombie 2005].

McCombie et al. speciﬁcally proposed a technique using MBSI to reconstruct the
shape of the common aortic ﬂow waveform input from multiple peripheral artery
pressure waveform outputs and demonstrated its feasibility in a pilot swine experiment.
Figure (2.1) (all lines and fonts) illustrates the single input, multi-output model upon
which their technique was based. As can be seen from this model, the channels coupling
the aortic ﬂow waveform to each peripheral artery pressure waveform include common

dynamics, namely the channel relating the aortic ﬂow waveform to the central aortic

pressure waveform (h0(t)), and are therefore not coprime. As a result, these

investigators had to develop additional signal processing to estimate the common channel

33

dynamics, which resulted in a considerably more complicated algorithm than standard
MBSI. Moreover, their framework did not provide an obvious means to determine the
uncontrolled scale factor of the reconstructed input. Thus, the technique cannot be
utilized to monitor relative changes in cardiac output. In contrast, our MBSI technique
aimed to estimate the central aortic pressure waveform input in which the coprime
channel assumption is more tenable (thereby rendering a relatively straightforward
algorithm) and the arbitrary scale factor of the input is conveniently determined by
invoking Poiseuille’s law.

It is not strictly valid to compare the results of the MBSI technique reported here
with those of previous studies employing generalized or partially individualized transfer
functions due to variations in evaluation datasets (both subjects and experimental
conditions) and methods for evaluation. Nevertheless, we ﬁnd that the total central aortic
pressure waveform error of 4.7 mmHg obtained by our technique (see Table (2.2)) is
somewhat higher than the approximately 2 to 4 mmHg errors reported in four of the
previous studies [Chen 1997, Fetics 1999, Karamanoglu 1993, Karamanoglu 1997].
However, we note that the transfer functions that were utilized in each of these studies
were trained on the same subjects and/or the same experimental conditions that were
subsequently employed for testing. The impetus for the present research is that the MBSI
technique should demonstrate an improved performance when applied to the diverse
population of patients and patho-physiologic conditions seen in clinical practice. On the
other hand, we acknowledge that the cost of this potential improvement in accuracy is the
requirement of more than one peripheral artery pressure waveform for analysis.

However, several convenient methods are currently available for measuring peripheral

34

artery pressure waveforms (see above) and new systems are continually in development.
For example, it may be possible one day to chronically monitor peripheral artery pressure
waveforms with wearable ring sensors [Asada 2003].

Future Directions

The present study opens up the possibility of several different avenues of future
investigation. In terms of subsequent mathematical efforts, it would be worthwhile to
attempt to improve upon the accuracy of the MBSI technique by compactly representing
the FIR channels with more appropriate basis functions and thereby alleviate the
persistence of excitation demands. In addition, it would be extremely desirable, from a
clinical point of view, to be able to extend the MBSI technique to also estimate relative
changes in aortic ﬂow through, for example, physical modeling. In terms of subsequent
experimental efforts, it may prove useful to seek correlation between the estimated FIR
channels and phenomena that are local to the respective peripheral artery pressure
measurement sites (e. g., plaque development) so as to further extend the monitoring
capabilities of the technique. Furthermore, while the application of the technique to
femoral and radial artery pressure waveforms may be most suitable for clinical practice
(see above), it would be interesting, from a scientiﬁc point of view, to establish the
optimal sites and number of peripheral artery pressure measurements (e.g., the arterial
tree sites that result in the most coprime channels and the smallest number of
measurements that does not signiﬁcantly compromise estimation accuracy). Finally,
future evaluations of the MBSI technique in humans and with respect to non-invasive
peripheral artery pressure waveforms are certainly warranted.

Potential Applications of the MBSI Technique

35

Our MBSI technique mathematically derives the clinically more relevant central
aortic pressure waveform from multiple, less invasively measured, but distorted,
peripheral artery pressure waveforms without using any training data. The technique
may easily be implemented in near real time (with a one minute delay) using a standard
home personal computer. With further development and successful testing, the technique
may ultimately be utilized for more precise monitoring and titration of therapy [Chen
1997] in, for example, critically ill patients with invasive catheters installed and
hypertension patients instrumented with non-invasive arterial pressure transducers.
Advancements in arterial pressure monitoring technology hold further promise for the

application of the technique in the context of chronic ambulatory and home monitoring.

36

CHAPTER 3

ESTIMATION OF THE AORTIC PRESSURE WAVEFORM AND BEAT-TO-
BEAT CARDIAC OUTPUT FROM MULTIPLE PERIPHERAL ARTERY

PRESSURE WAVEFORM

3.1 Introduction

Blood ejected by the left ventricle initiates pressure and ﬂow waves that propagate
through the arterial tree. These waves are reﬂected at multiple sites of impedance
mismatch caused by arterial bifurcations, narrowing, and stiffening [Donald 1960],
[Noordergraf 1978]. For example, wave reﬂections are especially pronounced at the sites
of arterial terminations [Donald 1960, Westerhof 1972]. The pressure waveform
measured at a given site in the arterial tree therefore represents the sum of the forward
and backward traveling pressure waves at that particular site [Westerhof 1972, Berger
1993]. As a result, the arterial pressure waveform becomes progressively distorted as its
site of measurement becomes more distal to the aorta [Donald 1960]. Most notably,
systolic pressure (SP) and pulse pressure (PP) become increasingly ampliﬁed [Donald
1960, Soderstrom 2002] and therefore less indicative of cardiac performance [Chen
1997]. Indeed, central measurements of SP and PP have been shown to be superior
predictors of patient outcome than corresponding measurements made in more peripheral
arteries [Safar 2002, Wadell 2001]. In addition, aortic pressure (AP, especially from the
descending thoracic aorta) is less complicated by wave reﬂections than peripheral artery

pressure (PAP) due in part to attenuation and destructive interference of the reﬂected

37

waves that reach the aorta [Noordergraf 1978, Bourgeois 1974, Bourgeois 1976]. Thus,
the entire AP waveform usually reveals the ventricular systolic ejection interval (SE1)
through the dicrotic notch [Fetics 1999] and may be represented with a lumped parameter
Windkessel model in order to accurately estimate beat-to-beat relative changes in cardiac
output (CO), as convincingly demonstrated in [Bourgeois 1976]. On the other hand, PAP
may be measured more safely than AP through catheterization and even non-invasively
via ﬁnger-cuff photoplethysmography [Irnholz 1998] or applanation tonometry [Kenner
1998]. It is therefore PAP waveforms that are routinely monitored in humans [Marino
1998], even though the AP waveform is known to be of greater clinical value.

As a result, over the past 15 years, there has been considerable interest in estimating
the AP waveform from measured PAP waveforms using generalized transfer function
techniques [Soderstrom 2002, Chen 1997, Fetics 1999, Karamanoglu 1993, Karamanoglu
1996]. These techniques essentially involve: 1) initially obtaining simultaneous
measurements of AP and PAP waveforms in a group of subjects; 2) estimating a group-
averaged transfer function relating the measured PAP to the measured AP; and 3)
subsequently applying this transfer function to PAP measured from a new subject in
order to estimate the AP waveform. However, these techniques do not account for
known inter-patient and temporal variability of arterial tree properties (e.g., [Hallock
1937, Guyton 1996]) and may therefore be prone to signiﬁcant estimation error when

applied to the diverse patient population encountered in clinical practice.

38

Pp1It)

PpZIt)

palt)

 

Pmet)

Figure 3.1 Single input, multi-output model of the arterial tree providing the basis for the
technique introduced herein to estimate the clinically more relevant aortic pressure (AP)
waveform from multiple, less invasively measured peripheral artery pressure (PAP)
waveforms distorted by wave reﬂections.

39

We have recently developed a patient and time speciﬁc technique for estimating the
AP waveform from multiple PAP waveforms by capitalizing on the powerful multi-
channel blind system identiﬁcation (MBSI) approach [Swamy 2007]. The technique
speciﬁcally involves: 1) modeling two or more measured PAP waveforms as outputs of
distinct ﬁnite impulse response (FIR) channels driven by the common AP waveform
input (see Figure (3.1)); 2) estimating the FIRs to within an arbitrary scale factor by
applying the standard eigenvector method to the cross relations between pairs of
measured outputs (see Equations (3.1) and (3.2)); 3) reconstructing the AP waveform to
within an arbitrary scale factor by deconvolving the estimated Fle from the measured
waveforms; and 4) scaling the reconstructed waveform to absolute pressure by invoking
Poiseuille’s law. In this way, the technique is able to estimate the AP waveform from
PAP waveforms without the need for a generalized transfer function. We have tested this
technique with respect to four swine datasets consisting of simultaneous measurements of
two PAP waveforms from the femoral and radial arteries and a measured reference AP
waveform during diverse hemodynamic interventions [Swamy 2007]. Our results
showed that the technique provided more accurate AP waveform estimates than a
generalized transfer function developed from a subset of the same datasets. In this paper,
we introduce an improved technique for estimating the AP waveform as well as beat-to-
beat relative changes in CO from multiple PAP waveforms. The technique speciﬁcally
involves: l) estimating the AP waveform based on our new MBSI method in which the
Fle are represented with more efﬁcient basis functions than the impulse basis functions
assumed by the standard eigenvector method in order to reduce the number of parameters

to be estimated and therefore enhance the estimation accuracy; and 2) estimating beat-to-

40

 

 

 

beat relative changes in CO by ﬁtting a Windkessel model to the estimated AP waveform
in which wave distortion should be greatly attenuated. We show that this new technique
is able to estimate the AP waveforms in the four aforementioned swine datasets with
greater accuracy than our initial technique. We further show that this enhanced accuracy
permits reliable estimation of beat-to-beat relative changes in CO as compared to gold

standard reference aortic ﬂow probe measurements also available in the swine datasets.

3.2 Technique

Our technique applies a new MBSI method that we have developed to two or more
PAP waveforms to estimate the AP waveform and then ﬁts a Windkessel model to the
estimated waveform to estimate beat-to-beat relative changes in CO. We fully describe
the technique below for the simplest case in which two PAP waveforms are available for
analysis. Generalization of the technique to more than two measured waveforms readily

follows analogous to [Xu 1995].

First, two measured and sampled PAP waveforms (ppi(t), i e [1, 2]) are modeled as

individual outputs of two unknown channels driven by the common unknown and
likewise sampled AP waveform (pa(t)) input as shown in Figure 1 with m = 2. The two
discrete-time channels coupling the common input to the two distinct outputs represent
the dynamic properties of a different path in the arterial tree. A principal assumption
underlying the model is that the channels may be well characterized by coprime FIRs
(hi(t) , i e [1, 2]) over each one-minute interval of analysis (see below). Over such short

time intervals, the arterial tree is usually operating in near steady-state conditions, thereby

clearly supporting the implicit time invariance assumption as well as buttressing the

41

implicit linearity assumption as argued in [McCombie 2005] and references therein.
Moreover, since pressure waveforms measured from distinct sites in the arterial tree only
differ signiﬁcantly over short time scales (e. g., within a cardiac cycle) while being quite
similar over longer time scales (e. g., mean values as described below) [Noordergraf 1975,
Zhenwei 2006], the FIR assumption is also well justiﬁed. Finally, the coprime
assumption, which is needed to subsequently estimate the FIRs [Xu 1995], may be
largely valid due to the signiﬁcant differences in PAP waveforms measured from distinct
arterial sites.

Then, by applying the fundamental properties of convolution to the single input,
multi-output model of Figure (3.1), the two FIRs may be estimated based on the resulting

cross relation between the two measured outputs:
L —1 L —1

Z h1(t)*pp2(t—t)— z h2(t)*pp1(t— r) = e(t), te [L —1,N—1]. (3.1)
T = 0 I = 0

Here, e(t) accounts for any measurement and/or modeling error, and the variables L
and N respectively represent the maximum duration of the Fle (channel order) and the
number of measured PAP waveform samples in a one-minute interval of analysis [Xu
1995]

The standard method for estimating the FIRs in Equation (3.1) is to ﬁrst
determine the channel order through eigenvalue analysis and then to estimate the FIR
samples or parameters to within an arbitrary scale factor by least squares minimization of
e(t) via the eigenvector method [Xu 1995]. More speciﬁcally, Equation (3.1) may be
expressed in matrix form by stacking each individual equation, corresponding to each t,

one on top of the other as follows:

42

[9,2 41,1] ["‘]=e, (3.2)

 

. , n,
P T

where
' ppi(0) ppi(1) ppi(L—1)'

ppi— pp:(1) ppile ppif") ,ie[1,2], (3.3)
3ppitN-L) ppi(N;L+1) --: ppi(N—1)_

 

 

are [(N-L+l)><L] Hankel matrices comprising the respective measured output samples;

hi=[hi(L—1) hi(L—2) hi(0)]T, ie[1,2], are [Lxl] vectors specifying the

parameters of the two respective FIRs; and e = [8(0) 6(1) e(N-L)]T is an [(N-
L+l)><1] vector consisting of the error samples. The channel order L may then be

determined by 1) forming a matrix P of dimension [(N—Lmax +1)x2Lmax], where

Lmax is assumed to encompass the true channel order; 2) computing the eigenvalues of

the matrix PTP; 3) identifying the number of insigniﬁcant eigenvalues niei and 4)
selecting the optimal value of L as Lopt = Lopt —(nie —1). Then, the least squares, unit
two-norm estimate of the vector h may be conveniently obtained by selecting the

eigenvector associated with the minimum eigenvalue of the matrix PTP, where P is of

dimension [(N "Lopt +1)x 2Lopt]- The implicit assumption of this standard eigenvector

method, which was employed by our initial MBSI technique [Swamy 2007], is that the
AP waveform contains at least as many frequency components as the number of

eStimated FIR parameters 2Lopt [Xu 1995].

43

To reduce the number of parameters to be estimated so as to alleviate the frequency
content demands on the AP waveform and thereby improve the estimation accuracy, the
Fle are instead compactly represented with damped sinusoidal basis ﬁrnctions as
follows:

n t . .
hi(t) = '31). (aik cos(mikt) + bik s1n(mikt)), | e [1, 2], (3.4)

where {k'aik'bik'wikhs a set of unknown parameters and n is an unknown number of

basis functions. These basis functions were chosen empirically based on our swine
datasets (see section 3.5). Thus, only a small number of basis functions should be needed
to represent the F IRs, thereby resulting in a signiﬁcant reduction in the parameters to be
estimated. Then, for a ﬁxed number of basis functions 11, the set of parameters is
estimated based on least squares minimization of e(t) in the following cross relation

equation resulting from substitution of Equation (3.4) into Equation (3.1):

n Lopt'1 3 l-opt"' I .
2 31k 2 A. COS(0)1kT)pp2 (I - 1’) + b1k Z A SII'I((D1k‘IZ)pp2 (I - I)
k=1 1:0 1:0
L —1 L -1
n opt opt
_ 2 32k 2 At COS((1)2kT)pp1(I- I) + b2k 2 X1: Sin(w2kt)pp1(t— T) (3.5)
k =1 1: = O T = 0

= e(t), te [Lopt — 1,N — 1],
where Lopt is established through eigenvalue analysis as described above with Lmax=

15. To estimate the coefﬁcient parameter sets {aikwbik} uniquely (rather than to within

an arbitrary scale factor), one FIR is constrained to have unity gain as follows:

11 I-opt '1 n Lopt ‘1

2 31k 2 At COS(C01kt) + Z b1k 2 At SID((1)1kt) =1. (3.6)
k =1 1: O k =1 1: 0

44

The constraint here is well justiﬁed, as the paths from the aorta to the peripheral
arteries offer very little resistance to blood ﬂow due to Poiseuille’s law (i.e., the mean
values of pressure waveforms from different sites in the arterial tree are nearly the same)
[Noordergraf 1978]. To simplify this constrained optimization problem, the damping

parameter 7t is set to exp(-3 / Lopt ) so that the FIRs approximately decay to zero, while
the frequencies in the parameter set (”ik are allowed to take on only discrete values
according to the Fourier Series (i.e., 21t|l Lopt for l=0,1,..., C9"((Lopt —1)/ 2)), where
ceil(x) is the smallest integer 2 x). For each set of frequency parameters {wik}

considered (see below), the corresponding coefﬁcient parameter sets are estimated
through the linear least squares solution. More speciﬁcally, similar to Equation (3.1),

Equations (3.5) and (3.6) may be expressed in matrix form as follows:

 

 

 

 

 

 

 

a1 0
b M
[P21 I,22 _P11 4312] a1 = 0 +9, (3.7)
E ﬂ , 2
A -b2_ L1-
,1 b

P. kLopt’1f((Lopt—1)*mi1) ALOPt"1f((Lopt—1)*min)

<9 tonomn) Naomi")

 

 

45

Ppiis deﬁned in Equation (3) with L = Lopt§ (p is a [1x Lopt] vector whose elements are

all zeros for i = l and all ones for i = 2; and a'=[ai1 ain:|T and

b- = b- b- T, ie [1,2], are [nxl] vectors specifying the coefﬁcient parameters
I I1 In

of the two respective F IRs. Then, the least squares estimate of the vector x in Equation
(7) is obtained asx = (ATA)—1ATb. Amongst all of the linear least squares solutions
computed for each considered set of frequency parameters {wik }, the one that minimizes

the two-norrn of the vector e in Equation (7) is selected so as to provide the optimal
estimate of the parameter set {k,aik,bik,mik}. Finally, the number of basis functions n is
determined iteratively by starting with a single basis function representation and then

adding one basis function at a time until the two-norm of the vector e becomes < 10% of

the two-norm of the vector (pm + sz ) I 2 , where pm is deﬁned in Equation (3.10). For

further simplicity, in the kth iteration, the frequency parameters are only estimated for
the newly added basis function with the frequency parameters of the previous (k-l) basis

functions set to the estimates obtained from the (k -1 iteration. lln the kth iteration,

the number of sets of frequency parameters considered is
speciﬁcally(ceil((Lopt—1)/2))—k+2)2. Thus, the two Fle are assumed to be

represented by the same number of basis functions but of generally different frequencies.

Next, with the set of basis function parameters {x’aik’blk’mik} estimated and the two

Fle fully deﬁned through Equation (4), the common AP waveform input of the model of

Figure (3.1) is estimated through multi-channel least squares deconvolution [Abed-

46

Meriam 1997]. That is, the two measured outputs may be expressed in terms of their

common input via convolution as follows:

Lem—1
Ppi(t)= k2 hi(k)Pa(t—k)+ni(t), ie[1,2], te[O,N—1], (3.8)

where ni (t) accounts for any noise. The common input is then estimated by least squares
minimization of ni(t), ie[1,2] [Donald 1960, Noordergraf 1978]. More speciﬁcally,

Equation (3.8) may also be expressed in matrix form by stacking each individual equation

corresponding to each t and i, one on top of the other as follows:

Pp1 _ H1 "1 (3.9)
lwlllll

pp H n

where

ppi=lppi(0) ppm) ppi<N-1>1T.ie[1.2]. (3.10)

are [N>< 1] vectors of the respective measured output samples;

'hi(Lopt—1) hi(0) 0 ‘
H.= 3 = ,ie[1,2],

o hi(Lopt —1) him)‘

 

 

h

are the [Nx(N+Lopt-1)] Toeplitz matrices including the estimated parameters of the

respective FIRs;
pa=tpa(-Lopt—1) pa(—Lopt—2) paw) paw) pa(N—1)1T

is a [(N+ Lapt -1)><1] vector of the unknown common input samples; and

47

T .
ni =[ni(0) ni(1) ni(N—1)] , Ie[1,2],
are [le] vectors of the respective noise samples. The least squares estimate of the

vector p a in Equation (3.9) is then obtained through the following linear least squares

solution with Tikhonov regularization [Hansen 1987]:

pa = (HTH + uI)—1HTpp. (3.1 1)

The matrix (HTH+ pl) here is relatively large ((> 3000)><(> 3000)) for one-minute

analysis intervals and a sampling frequency of 50 Hz), and standard computation of its

inverse is therefore very expensive requiring 0(N3) operations. However, by exploiting
the Toeplitz structure of this matrix, its inverse is instead computed efﬁciently in
0(N log N) operations using the fast circular decomposition method described in [Jain
1978]. The estimated AP waveform is then lowpass ﬁltered with a cutoff frequency of 15
Hz in order to further attenuate any high frequency noise generated in the deconvolution
process. This cutoff frequency is well justiﬁed, as the relevant waveform features
generally fall within this frequency range [Chen 1997]. It should be noted that the
estimated AP waveform will be slightly delayed (< 0.1 s) with respect to the actual AP
waveform, because the time delay shared by the Fle cannot be identiﬁed with MBSI.
Finally, beat-to-beat relative changes in C0 are estimated from the determined AP
waveform by employing the Windkessel model of Figure (3.2a), which accounts for the

lumped arterial compliance (C a) of the large arteries and the total peripheral resistance
(R a) of the small arteries, as described in [Bourgeois 1976]. That is, since C a may be

nearly constant over a wide pressure range and on the time scale of months [Bourgeois

48

1974, Hallock 1937, Zhenwei 2006], proportional CO for each cardiac cycle is calculated

through the following governing model equation:

1 1tes
COoc? pa(tes)‘pa(tbs)+;tl pa(t)dt , (3.12)
bs

where pa(t) is again the estimated AP waveform, andtbs,tes,T, and T = RaCa are
timing parameters that are determined ﬁ'om this waveform as shown in Figure (3.2b).
More speciﬁcally, tbs , which denotes the beginning time of the SEI, is identiﬁed as the

time of the local minimum that immediately precedes the time of peak SP for the cardiac

cycle; tes’ which indicates the ending time of the SE1, is identiﬁed as the time of the

minimum of the derivative of the estimated waveform over the interval from the time of

peak SP to the tbs of the next cardiac cycle; T, which signiﬁes the cardiac cycle
duration, is determined as the difference between the tbs of successive cardiac cycles;

and 1:, which is the time constant of the Windkessel model, is estimated by least squares
ﬁtting of a single exponential decay to the reconstructed waveform over an interval

between tes and tbs of the next cardiac cycle, in accordance with the model prediction.
Since the time constants governing Ra changes are signiﬁcantly longer than the cardiac

cycle duration [Berger 1989], the t estimates are actually averaged over ﬁve cardiac

cycles to attenuate any noise.

49

 

 

C0 pa(t)

 

 

 

('3)

AA

nu

cc
1'!

tes

.x
.x
O

 

.L
O
O
4

AP [mmHg]
09
O

 

on
O
H-
5'
m

 

0 0.5 1 1.5
Time [sec]

N
D

Figure 3.2 Lurnped parameter Windkessel model providing the basis for the technique to
estimate beat-to-beat relative changes in cardiac output (CO) from the estimated AP

waveform (pa(t)) in which the wave distortion should be greatly attenuated (see Figure

(3.1)). (b) Illustration generally indicating how the unknown timing parameters in
Equation (3.12), namely tbs (beginning time of the systolic ejection interval (SEI)),

tes (ending time of the SEI), T (cardiac cycle duration), and r = RaCa (time constant of
the Windkessel model), are determined from the estimated AP waveform.

50

3.3 Methods

We evaluated the technique with respect to experimental datasets that were originally
collected to address related but different speciﬁc aims [Mukkamala 2006] and previously
utilized to investigate our initial MBSI technique for estimating the AP waveform using
the standard eigenvector method [Swamy 2007]. Brieﬂy, these datasets consist of
various hemodynamic recordings obtained from four swine (30-34 kg) under general
anesthesia and mechanical ventilation. The hemodynamic recordings include femoral
artery pressure (PAP) and radial artery pressure (RAP) waveforms measured with ﬂuid-
ﬁlled catheters, a reference AP waveform measured from the descending thoracic aorta
with a high frequency response micromanometer-tipped catheter, and gold standard
reference beat-to—beat CO measured with an aortic ﬂow probe. These hemodynamic
recordings are available at a sampling frequency of 250 Hz for a total of 253 minutes
during infusions of volume, phenylephrine, dobutamine, isoproterenol, esmolol,
nitroglycerine, and progressive hemorrhage. Table (3.1) shows that these interventions

imposed a wide hemodynamic parameter range for each of the four swine datasets.

We applied the technique to all one-minute, non-overlapping intervals of the PAP and
RAP waveforms resampled to 50 Hz. We evaluated the resulting AP waveform estimates
with respect to the measured reference waveforms (likewise resampled to 50 Hz) in terms
of the root-mean-squared-error (RMSE) of the following parameters: total waveform
(i.e., sample-to—sample), beat-to-beat SP, beat-to-beat PP, and beat-to-beat SEI. For
comparison, we likewise evaluated the PAP waveforms (with respect to the measured AP
waveforms) as well as the AP waveform estimates from our initial MBSI technique and

an autoregressive exogenous input (ARX)-based generalized transfer ﬁmction [Fetics,

51

1999] developed on a subset of the swine datasets as described in [Swamy 2007]. (Prior
to conducting these evaluations, we advanced the AP waveform estimates and the PAP
waveforms so that they were temporally aligned with the measured AP waveforms.)
Since it is customary to report CO errors in percent, we evaluated the resulting beat—to—
beat proportional CO estimates with respect to the reference aortic ﬂow probe
measurements in terms of the root-mean—squared-normalized-error (RMSNE). This
quantity was speciﬁcally computed by 1) scaling the proportional CO estimates to have
the same mean value as the reference CO in each animal; 2) normalizing each calibrated
CO error with the reference CO value; and 3) computing the RMS of the normalized,
calibrated CO errors. For comparison, we likewise evaluated the beat-to-beat
proportional CO estimates obtained by ﬁtting the Windkessel model directly to the PAP
and RAP waveforms as well as to the AP waveforms estimated by our initial technique

and the generalized transfer function.

52

Table 3.1 The new technique (see Figures (3.1) and (3.2)) as well as alternative
peripheral artery pressure (PAP) waveform analysis techniques were evaluated with
respect to four swine datasets consisting of simultaneous measurements of femoral artery
pressure (PAP) and radial artery pressure (RAP) waveforms, a reference aortic pressure
(AP) waveform, and reference aortic ﬂow probe cardiac output (CO) during diverse
interventions. MAP is mean arterial pressure; SP, systolic pressure from AP; PP, pulse
pressure from AP; SEI, systolic ejection interval from AP; and HR, heart rate.

 

Hemodynamic Animal
Parameter 1 2 3 4 Total
MAPRange 54—136 58—117 45—114 48—119 45—136
[mmHg]
SP Range
62—182 70—148 55—144 58—157 55—187
[mmHg]
PP Range
17—54 17—52 20—62 19—57 17—62
[mmHg]
SEixfge 160 — 400 160 — 340 120 - 240 140 — 300 120 — 400
HR Range 100—223 92 — 190 91 —243 102—207 91 —243
[beats/mm]
CO Range 2.3 — 4.1 1.7 — 6.0 2.4 — 5.7 1.3 — 6.2 1.3 — 6.2
[L/mm]

 

S3

3.4 Results

Table (3.2) includes the evaluation results for the AP waveforms estimated by the
technique introduced herein, which employs our new MBSI method, for each of the four
swine datasets as well as the corresponding results for the measured PAP waveforms.
The overall total waveform RMSE of the AP waveform estimates was 3.5 mmHg (after a
modest time alignment), whereas the average overall total waveform RMSE between the
PAP waveforms and the measured AP waveforms was 8.6 mmHg (after a more
signiﬁcant time alignment). Thus, the technique effectively reduced the wave distortion
in the measured PAP waveforms by 59%. As a result, the technique also reduced the
average overall SP and PP RMSEs by 71% and 61%, respectively. However, the
technique did not improve upon the surprisingly small, overall SEI RMSE obtained from
the FAP waveforms. Figure (3.3a) provides a visual example illustrating the signiﬁcant
differences between the PAP waveforms and the measuredAP waveform, while Figure
(3.3b) shows that the AP waveform estimated from these PAP waveforms closely agrees
with the directly measured reference waveform.

Table (3.2) also provides a comparison of the evaluation results for the AP
waveforms estimated by our new technique, our initial MBSI technique, which employs
the standard eigenvector method, and a previous ARX—based generalized transfer
function for each of the four swine datasets. The overall total waveform RMSE of the
AP waveform estimates from our initial technique was 4.6 mmHg, while the average
overall total waveform RMSE from the generalized transfer function was 5.4 mmHg.
Thus, both MBSI techniques provided more accurate AP waveform estimates than the

conventional generalized transfer function even though it was developed on a subset of

54

the swine datasets. However, the new technique enhanced the estimation accuracy of our
initial technique by 24%. As a result, the new technique provided similar improvements
in the estimation of SP and PP (but yielded similar results in the estimation of the SEI).
In addition, a more signiﬁcant advantage of the new technique over the initial technique
is indicated below.

Table (3.3) includes the evaluation results for the beat-to-beat proportional CO
estimated by the new technique via ﬁtting the Windkessel model of Figure (3.2a) to the
estimated AP waveform for each of the four swine datasets as well as the corresponding
results obtained by directly ﬁtting the model to each of the PAP waveforms. The overall
beat—to-beat CO RMSNE from the AP waveform estimates was 12.9%, whereas the
average overall beat-to-beat CO RMSNE from the PAP waveforms was 26%. Thus, the
technique enhanced the beat-to-beat CO estimation accuracy by 50%. Interestingly, the
overall beat-to-beat CO RMSNE from the PAP waveforms was 49% smaller than the
corresponding RMSNE from the RAP waveforms but still 33% larger than the analogous
RMSNE obtained with the technique. Figure (3.4) provides a visual illustration of the
close agreement between the estimated and once calibrated beat-to-beat CO from the
technique and the gold standard reference aortic ﬂow probe measurements for each of the
four swine datasets.

Table (3.3) also provides a comparison of the evaluation results for the beat-to-beat
proportional CO estimated by the new technique and via ﬁtting the Windkessel model to
the AP waveform estimates from our initial technique and the generalized transfer
function. The overall beat-to-beat CO RMSNE from the AP waveforms estimated by our

initial technique was 36.3%, while the average overall beat-to-beat CO RMSNE from the

55

 

1—

waveforms estimated by the generalized transfer function was 20.2%. Thus, the new
technique increased the beat-to-beat CO estimation accuracy of the former technique by
63% and the latter technique by 36%. Note that the beat-to-beat CO RMSNE obtained
with our initial technique was even larger than the corresponding RMSNEs from the PAP
waveforms. As discussed in Section 3.5, this result was mainly due to outliers in the AP

waveform estimation.

56

Table 3.2 Quantitative summary of the aortic pressure waveform estimation results in
terms of root-mean-squared-error. The new technique resulted in more accurate AP
waveform estimates in the four swine datasets (see Figure (3.3)) than those obtained from
alternative PAP waveform analysis techniques including the conventional generalized
transfer function (GTF). The root-mean-squared-errors are in units of mmHg for the total
waveform (TW), SP, and PP parameters and ms for the SEI parameter.

 

 

 

 

 

 

Arterial Pressure IArterial Animal

Waveform P “55"“ 1 2 3 4 Total
arameter

Estimated AP TW 2.5 3.9 3.4 3.4 3.5
(New Techn. ) SP 1.9 5.1 5.0 5.6 4.8
““16 PP 2.8 5.4 6.4 6.1 5.6
SE1 22 14 21 22 19
TW 11.6 7.8 6.3 7.0 8.1
FAP SP 19.2 13.5 12.0 10.7 13.7
PP 10.5 14.3 10.2 14.3 12.6
SEI 22 19 19 20 19
TW 4.0 10.5 9.4 9.3 9.1
RAP SP 5.1 24.5 16.7 22.7 19.7
PP 4.9 19.2 10.3 21.6 15.9
SE1 20 38 18 48 33
. TW 4.4 4.4 3.8 5.7 4.6
ESt‘ﬁttﬁAP SP 4.1 5.0 5.0 9.4 6.1
Techni ue) PP 4.3 5.9 7.1 9.9 7.1
‘1 SE1 23 14 22 22 20
. TW 4.8 5.2 6.9 5.6 5.8
$5.??pr SP 7.4 7.7 12.4 9.6 9.7
( AP "’ PP 8.6 9.4 13.4 11.2 11.0
) SEI 24 28 46 42 35
Estimated AP TW 4.2 4.6 6.0 5.1 5.0
(GTF, RAP SP 5.3 6.6 8.7 6.5 7.1
AP " PP 4.8 6.3 8.5 6.8 7.0
) SEI 22 26 35 37 29

 

57

(a) (b)

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

90 - 9%
'— A .4 _r 5
a? so- :1 :4 4:1 :1 ao-
E 31 5 -'l 5 E _ ' i
E E g 'l . :1 1
hi I
2 " a 701' ‘
a
«n
D
g 60 .
E
L
g 50 ,
<
40 1 1 1 1 40 L 4 1 1
O 0.5 1 1.5 2 O 0.5 1 1.5 2

Time [sec]

Figure 3.3 (a) Example segments of the measured AP (solid), femoral artery pressure
(dash), and radial artery pressure (dot-dash) waveforms from one of the four swine
datasets (see Table (3.1)). (b) Example segments of the AP waveform measured (solid)
and estimated (dash) by applying the technique to the two segments of the PAP
waveforms (see Figure (3.1)). See Table (3.2) for a complete quantitative summary of
the AP waveform estimation results.

58

Animal 1 Animal 2

”horror:

 

 

 

00 2 4 s 8
Animal3

 

o0

 

NAG“
Nham

 

 

 

2 4 6 £3
Aortic Flow Probe Beat-to-Beat CO [lein]

 

Estimated & Calibrated Beat-to-Beat CO [Umin]

00

2468

0°

Figure 3.4 Beat-to-beat CO estimated (and once calibrated) by applying the technique to
the estimated AP waveforms (see Figure (3.2)) plotted against the gold standard reference
aortic ﬂow probe CO measurements for each of the four swine datasets (see Table (3.1)).
The solid line in each plot is the identity line. See Table (3.3) for a complete quantitative
summary of the beat-to-beat proportional CO estimation results.

59

Table 3.3 Quantitative summary of the beat-to-beat proportional cardiac output
estimation results in terms of root-mean-squared-normalized-error. The new technique
(see Figure (3.2)) also resulted in more accurate beat-to-beat proportional CO estimates
in the four swine datasets (see Figure (3.4)) than those obtained from alternative PAP
waveform analysis techniques. The root-mean-squared-normalized-errors are in units of
percent.

 

 

Arterial Pressure Animal
Waveform for Wrndkessel 1 2 3 4 Total
modelrng
Estimated AP
(New Technique) 11.8 12.3 15.5 10.5 12.9
FAP 16.6 14.8 17.7 19.6 17.2
RAP 39.0 25.6 22.6 48.6 33.9
Estimated AP
(Initial Technique) 44 7 37 8 28.0 37 l 36 3
Estimated AP

(GTF: FAP _) AP) 11.6 14.1 26.6 16.0 17.6

Estimated AP

(GTF: RAP—)AP) 20.0 25.0 19.9 26.0 22.8

 

6O

3.5 Discussions

The present study represents a follow-up to our initial efforts in developing a patient
and time speciﬁc technique for mathematically estimating the clinically more relevant AP
waveform from less invasively measured PAP waveforms distorted by wave reflections
[Swamy 2007]. In our initial study, we developed a technique to estimate the AP
waveform from two or more PAP waveforms by employing MBSI rather than using the
conventional generalized transfer function or any training data for that matter. In
particular, the technique models the measured waveforms as individual outputs of
coprime FIR channels driven by the common AP waveform input (see Figure (3.1)) and
then analyzes the differences in the measured outputs, while invoking Poiseuille’s law, so
as to estimate the FIRS through the standard eigenvector method and ultimately
reconstruct the common input via multi-channel least squares deconvolution. We applied
the technique to FAP and RAP waveforms measured from four swine during diverse
hemodynamic interventions, and our results Showed superior agreement to
simultaneously measured AP waveforms than a generalized transfer ﬁrnction developed
on a subset of the swine data. The main contributions of the present study are in: 1)
introducing a new MBSI method to more accurately estimate the FIRS and therefore the
AP waveform; 2) proposing to estimate beat-to-beat proportional CO by ﬁtting a lumped
parameter Windkessel model to the estimated AP waveform in which the wave distortion
should be greatly attenuated (see Figure (3.2)); and 3) testing the resulting new technique
for estimating both the AP waveform and beat-to-beat relative changes in CO, while

comparing it to several alternative techniques, based on our four previous swine datasets,

61

which also included gold standard reference aortic ﬂow probe CO measurements (see
Table (3.1)).

The basic idea of our new MBSI method is to represent the F IRS with more efﬁcient
basis functions than the standard impulse basis functions assumed by the eigenvector
method. In this way, the number of parameters to be estimated will be reduced thereby
potentially resulting in a marked decrease in the precision component of the FIR
estimation error. We speciﬁcally chose damped sinusoidal basis functions to compactly
represent the F IRS (see Equation (3.4)). This choice was made empirically by observing
that the estimated impulse responses relating the measured AP waveforms to each of the
PAP waveforms in one of the swine datasets generally appeared as damped sinusoids.
Thus, the bias component of the FIR estimation error may only increase modestly with
this choice of basis functions. For simplicity, we assumed that 1) the damping could be
represented with a single parameter whose values was set according to the estimated
channel order; 2) the frequencies of the sinusoids could take on only discrete values
according to the Fourier Series; and 3) each of the FIRS could be represented with the
same number of basis functions. Even with these simpliﬁcations, the complexity in the
least squares estimation of the parameters of the basis functions increases considerably
with the number of basis functions. Thus, to render a real-time technique (with a one
minute delay), the parameter estimation was designed to only be optimal for a single
basis function representation and sub-optimal for a multiple basis function representation.
We acknowledge that an orthogonal basis set could permit both optimal and practical
parameter estimation. However, it turned out that only one basis function was needed to

represent the FIRS in about three-quarters of the one-minute intervals in the swine

62

datasets. As a result, only six parameters were generally estimated by the new method,
whereas ten parameters were, on average, estimated by the standard eigenvector method.
In addition to reducing the number of parameters to be estimated, the new method is also
advantageous in terms of accommodating larger channel orders (by simply redeﬁning the
deﬁnition of an insigniﬁcant eigenvalue), which are more congruent with the estimated
impulse responses mentioned above. Such channel orders would result in large
estimation errors with the eigenvector method due to the limited frequency content of the
AP waveform.

Our new technique was able to reliably estimate the AP waveform from the F AP
and RAP waveforms in the four swine datasets with an overall total waveform RMSE of
3.5 mmHg (see Table (3.2) and Figure (3.3)). This error effectively represents a 59%
reduction in wave distortion in the measured PAP waveforms. As a result, the technique
was able to similarly reduce the RMSES of SP and PP, whiCh are perhaps the two most
clinically signiﬁcant parameters of the AP waveform. Signiﬁcantly, the aforementioned
overall total AP waveform RMSE represents a 24% improvement with respect to our
initial technique and a 35% improvement with respect to a generalized transfer function
developed on a subset of the swine datasets. The former improvement is likely due to a
reduction in the FIR estimation error variance, while the latter improvement is
presumably a result of accounting for the changes in arterial tree properties induced by
the interventions as well as any inter-subj ect variability.

As a result of greatly attenuating the wave distortion, our new technique was also able
to reliably estimate beat-to-beat relative changes in CO in the four swine datasets with an

overall RMSNE of 12.9% (see Table (3.3) and Figure (3.4)). This result is consistent

63

with a compelling, previous study in which beat-to-beat proportional stroke volume was
shown to be impressively estimated over a wide hemodynamic range by simply ﬁtting the
Windkessel model to the AP waveform (measured Speciﬁcally from the descending
thoracic aorta) [Bourgeois, 1976]. Signiﬁcantly, the aforementioned overall beat-to-beat
CO RMSNE of our new technique represents an improvement of Z 25% with respect to
ﬁtting the Windkessel model directly to each of the PAP waveforms and to the AP
waveforms estimated by our initial technique and the generalized transfer function.
Counter to intuition, the beat-to-beat proportional CO estimates obtained from the AP
waveforms estimated by our initial technique were the least accurate. As mentioned
above, this result is mainly due to outliers in the AP waveform estimation. These outliers
may have been caused by an insufﬁcient number of frequency components in the AP
waveform and have essentially been eliminated by the new technique, which required
fewer parameters for estimation.

As an interesting aside, the beat-to-beat CO RMSNE obtained by direct ﬁtting of the
Windkessel model to the FAP waveforms is nearly half that obtained by direct model
ﬁtting to the RAP waveforms (see Table (3.3)). This result may be due to the
surprisingly accurate SEI estimates obtained from the FAP waveforms (see Table (3.2))
and their relatively consistent morphology throughout the interventions. It is unclear
whether these features hold. in human FAP waveforms. On the other hand, the beat-to-
beat CO RMSNE obtained from the RAP waveforms was reduced by nearly a third aﬁer
the generalized transfer function was applied to these waveforms (see Table (3.3)). To
our knowledge, this result of the generalized transfer ﬁrnction has not been shown before.

However, application of the generalized transfer function to the FAP waveforms did not

64

further reduce the already relatively low beat-to-beat CO RMSNE (see Table (3.3))
perhaps because the AP waveform estimates here were not as accurate (see Table (3.2)).
Our new technique (as well as our initial technique) was inspired by the recent efforts
of McCombie et al. who we believe were the ﬁrst to apply MBSI in the ﬁeld of
hemodynamic monitoring [McCombie 2005]. These investigators speciﬁcally proposed a
technique to estimate the morphology of the common aortic ﬂow waveform input from
multiple PAP waveform outputs and Showed its feasibility in a single swine. Figure (3.5)
illustrates the Single input, multi-output model upon which their technique was based. As
can be seen from this model, the channels coupling the aortic ﬂow waveform to each
PAP waveform include common dynamics, namely the channel relating the aortic ﬂow
waveform to the AP waveform, and are therefore not coprime. As a result, these
investigators had to develop additional signal processing to estimate the common
channel, which resulted in a considerably more complicated technique than standard
MBSI. Moreover, since their technique does not provide the scale factor of the estimated
input, it cannot be utilized to monitor relative changes in CO. In contrast, our new
technique ﬁrst estimates the AP waveform input in which the coprime channel
assumption is more tenable and the arbitrary input scale factor is conveniently determined
through Poiseuille’s law and then estimates beat-to-beat relative changes in CO by ﬁtting
the Windkessel model to the estimated waveform in which the wave distortion is greatly
attenuated. We note that it is also possible to estimate the aortic ﬂow waveform by
applying the governing Windkessel model differential equation to the AP waveform

estimated by our technique. However, the aortic ﬂow waveform estimated in this manner

65

did not closely agree with the reference aortic ﬂow probe waveform in our swine datasets
perhaps due to noise arising from implementing the required derivative operation.

Our new technique is related to a previous technique that we have developed for
automated and less invasive monitoring of average relative changes in CO by long time
interval analysis of a single PAP waveform [Zhenwei 2006]. The basic idea of this
technique is to circumvent the highly complex wave reﬂections by effectively applying
the Windkessel model to the waveform variations occurring over time scales greater than
a cardiac cycle in which the distributed arterial tree appears to be lumped [Noordergraf
1978]. In contrast, the technique introduced herein aims to essentially remove the wave
reﬂections through MBSI and then apply the Windkessel model. The advantage of this
technique is in providing AP waveform estimates as well as beat-to-beat rather than
average proportional CO estimates. The obvious disadvantage is in requiring more than
one waveform for analysis. (Note that the CO RMSNE that we reported earlier for our
previous technique in [Zhenwei 2006] is not directly comparable to the results of this
study, as the swine evaluation data utilized herein represent only a subset of the data
employed in our earlier study.)

The new technique introduced herein permits both automated and less invasive
central hemodynamic monitoring through estimation of the AP waveform and beat-to-
beat relative changes in CO by mathematical analysis of multiple PAP waveforms. In
addition, the technique may possibly prove useful for local hemodynamic monitoring at
each PAP waveform measurement site through the corresponding estimated FIR. The
technique could potentially be applied to non-invasive PAP waveforms measured from

patients with various cardiovascular diseases (e. g., hypertension, heart failure, Shock) in a

66

number of different settings (e. g., emergency rooms, intensive care units, homes) as well
as to invasive PAP waveforms obtained from critically ill patients. While only one PAP
catheter is commonly used in the latter patients, a subset of these patients is also
instrumented with the more risky pulmonary artery catheter for operator-required
measurements of average CO via the standard thermodilution method [Marino 1998].
For this patient population in particular, the technique as applied to two invasive PAP
waveforms (e. g., measured from the routinely cannulated femoral and radial arteries
[Marino 1998]) may be preferred. Future investigations of the technique with respect to
invasive and non-invasive PAP waveforms measured from humans are needed to

eventually realize these potential applications.

67

F’p1(t)

F’p2(t)
pa(t) =

 

cllt)

p.11)

 

Figure 3.5 Single input, multi-output model of the arterial tree providing the basis for a
previous MBSI technique to estimate the shape of the aortic ﬂow waveform from
multiple PAP waveforms. In contrast to the model of Figure (3.1), the common input to
be estimated is the aortic ﬂow waveform (q(t)). However, the channels coupling the
aortic ﬂow waveform input to the PAP waveform outputs are not coprime, as they share
the channel relating the aortic ﬂow waveform to the AP waveform. Since common
charmel dynamics cannot be estimated with MBSI methods [Xu 1995], the previous
technique required additional complicated Signal processing. In addition, this technique
did not provide the scale factor of the aortic ﬂow waveform and therefore cannot be
utilized to monitor relative changes in CO.

68

CHAPTER 4

CONTINUOUS LEFT VENTRICULAR EJECTION FRACTION MONITORING

BY AORTIC PRESSURE WAVEFORM ANALYSIS

4.1 Introduction

Left ventricular ejection fraction (EF) — the ratio of the stroke volume (SV) to the
end-diastolic volume (EDV) of the left ventricle — is widely recognized as one of the
most clinically Signiﬁcant indices of cardiac function. This recognition is largely due to
its ability to predict mortality in patients with heart failure and coronary artery disease
(see, e.g., [Bosch 2005, Curtis 2003] and references therein) as well as to readily
distinguish between systolic and diastolic dysfunction [Katz 1992]. Serial changes in EF
at rest and transient rate of changes in EF druing exercise may offer additional prognostic
value [Cintron 1993, Sridhara 1993].

The standard clinical method for measuring EF is through imaging the left ventricular
volume via echocardiography, radionuclide techniques, contrast angiography, ultra-fast
computed tomography, or magnetic resonance imaging [Rumberger 1997]. Each of these
imaging methods offers certain advantages over the others in terms of, for example, level
of accuracy, invasiveness, and radiation exposure. However, they generally share the
major disadvantages of requiring a trained operator to make each individual measurement
and expensive capital equipment. While sonomicrometry, the conductance catheter, and
the non-imaging nuclear monitor are currently available for automated and continuous
monitoring of left ventricular volume [Burkhoff 1990, Dellegrottaglie 2002, Rushmer

1956] these alternative methods suffer from signiﬁcant practical disadvantages (e. g., high

69

level of invasiveness) that have prevented them from supplanting or even complementing
imaging methods in clinical practice.

Thus, there is a need for a practical method capable of automated and continuous EF
monitoring. Indeed, this need may be regarded as urgent due to the rapidly growing
population with chronic heart disease [Ansari 2001] together with the projected Shortage
of clinical staff.

Based on our previous work in the ﬁeld of hemodynamic monitoring [Lu 2006,
Mukkamala 2006, Swamy 2007, Swamy 2008] our hypothesis is that EF may be
accurately estimated by deciphering the information embedded in the temporal variations
of blood pressure waveforms. In this way, EF may be continuously monitored in various
inpatient settings with routinely employed invasive catheter systems [Marino 1998] as
well as automatically measured in outpatient clinics and at home with commercial non-
invasive transducers (see, e.g., the Finometer and Portapres, Finapres Medical Systems,
The Netherlands and the T-Line Blood Pressure Monitoring System, Tensys Medical,
San Diego, CA). As an initial step towards this ultimate end point, in this study, we
speciﬁcally developed a technique to continuously estimate beat-to-beat EF as well as
relative changes in beat-to-beat EDV and other important hemodynamic variables by
model-based analysis of the aortic pressure waveform. We then performed experiments
in six dogs in order to evaluate the technique with respect to intermittent reference EF
and EDV measurements via standard trans-thoracic two-dimensional echocardiography
during various pharmacological and volume interventions. Our results demonstrate,
perhaps for the ﬁrst time, the feasibility of estimating EF from only blood pressure.

Preliminary versions of this work have been reported [Mukkamala 2006, Swamy 2007].

70

 

 

 

4.2 Model Based Analysis Technique

Our model-based analysis technique is generally implemented in three steps. First,
the aortic pressure waveform is represented with a circulatory model. Second, the model
is ﬁtted to each beat of the waveform so as to estimate its parameters to within a constant
scale factor. Third, the proportional parameter estimates are utilized to compute beat-to-
beat absolute EF by cancellation of the scale factor as well as monitor other beat-to-beat
proportional hemodynamic variables. We describe the details of these steps below while
stating the underlying assumptions and justify the major assumptions in the Discussion
section.

First, the measured and sampled aortic pressure waveform is assumed to be well
represented with the lumped parameter model of the left ventricle (1v), aortic valve, and
arteries (a) shown in Fig. 1a in electrical analog form. Here, voltage is analogous to
pressure (P), charge, to volume (V), and current, to ﬂow. rate. In particular, the left
ventricle is represented with the variable capacitance or compliance (C) model whose
elastance (E = l/C) oscillates over time (t) so as to drive the ﬂow of blood [Sagawa 1977,
Suga 1974]. The aortic valve is modeled by an ideal diode (i.e., Plv (t) = Pa(t)) during the
systolic ejection interval wherein the valve is opened) thereby making aortic stenosis a
contraindication of the technique. The arteries are represented with a two-parameter
Windkessel model accounting for the compliance of the large arteries and the resistance
(R) of the small arteries [Noordergraaf 1978]. In addition to compliance, the left
ventricle (and large arteries) is parameterized with a zero-pressure (0) ﬁlling volume (i.e.,

unstressed volume). Finally, Ca is assumed to be constant over a monitoring period

71

 

(e. g., days to weeks), whereas the remaining parameters are assumed to be constant only
over each cardiac cycle.
Thus, the pressure-volume relationship of the left ventricle model and the differential

equation governing the entire model during the systolic ejection interval are given as

follows:
P (t) o
_ a__
\IIV (t)— El v(t) + VIV’ tbs < t S tes' (4.1)
CI a(t)_Pc1Pa(t)
t+ t t s t , 4.2
—d—tE—I—_v(t)= Cadt P+a() bs < es ( )

Ra

where the term Pa(t)/ Elv(t)is the stressed left ventricular volume during the systolic
ejection interval, and the subscripts bs and es respectively stand for the beginning and
end of the systolic ejection interval. Integrating Equation (4.2) from tbs to some time t
within the systolic ejection interval, while dividing by the constant scale factor Ca , yields

the following equation:

F’ (t ) F’ (t) 1 t
a bS _ a _
CaElv(tbs) CaEMt) P a“) Pa(tbs)+;1 P 8W“ tbs <t5tes' (4-3)

tbs

 

where t = Race is the Windkessel time constant. Note that the left-hand side of this

equation is proportional to the volume of blood that has been ejected by the left ventricle
by time t in the systolic ejection interval, while the right-hand side is proportional to the

volume of blood that has entered the arteries by this time. Thus, whent = tes , both sides
of the equation indicate proportional SV or SV / Ca . Further note that the term

Pa (t)/ CaE|v(t) here is proportional to the stressed volume that is remaining in the left

72

ventricle at time t in the systolic ejection interval. Thus, the terms Pa(tbs)/ CaEIv (tbs)

and Pa(tes)/CaElv(tes) respectively represent proportional stressed left ventricular

volume at the beginning time of the systolic ejection interval wherein ejection has yet to
commence (i.e., proportional stressed EDV or stressed EDV/Ca) and at the end time of

the systolic ejection interval wherein ejection has ceased (i.e., proportional stressed left

ventricular end-systolic volume (ESV) or stressed ESV / Ca ).

Second, to ﬁt the model to the samples of the aortic pressure waveform so as to
estimate its parameters, Equation (4.3) is discretized by replacing t with nT, where T is
the sampling period and n denotes discrete—time, and approximating the integral via the

trapezoidal formula as follows:

Pa(nbsT) _ Pa(nT)
CaElv(anT) CaElv(“T)

n
Pa(nT) — Pa(anT) + L 2 (Pa(kT) + Pa ((k —1)T)), nbs < n S n .

es
211<.—.an+1

 

(4.4)

Here, T and Pa(nT)for nbs SnSneS (i.e., aortic pressure samples within the
systolic ejection interval) are known, while 1: (i.e., proportional total peripheral
resistance) and CaElv(nT) for ”be S n S nes (i.e., proportional left ventricular elastance

samples within the systolic ejection interval) are unknown. It is evident that Equation
(4.4) does not provide a basis for uniquely determining these unknown proportional
model parameters and thus the proportional stressed left ventricular volume terms, as it

represents an underdetermined set of equations with heS ’nbs equations and

nes - nbs + 2unknowns (where nes — nbs z 80 for T = 4 msec).

73

To arrive at a solvable (i.e., overdeterrnined) set of equations, a parametric function is
assumed to succinctly characterize the temporal evolution of Elv(t)over each cardiac

cycle. In particular, the following, previously proposed parametric raised cosine function

is employed:

r

 

Emin + Emax ; Emin {1- COS (2%)}, tbl S t < tbi + T5 (4.5)
S

27tt(t - (tbi + T8 ))
TS

 

 

EN“) :1 Emin + EmaXZ—Emin {1+ COS{ J}, tbl '1" TS S t < tbl +1.5Ts

Emin! tbl +1 '5TS St

 

l

where Emin and Emax respectively represent the minimum and maximum ventricular
elastances over a cardiac cycle; TS indicates the time duration to reach Emax from

Emax ; and the subscript bi stands for the beginning of the isovolumic contraction phase

(see solid line in lower panel of Figure (4.1b)) [Heldt, 2002]. Substitution of Equation
(4.5) into Equation (4.4) reduces the number of unknowns to ﬁve, namely ‘t, CaEmax ,
CaEmins TS , and CaElv (nbsT) (i.e., proportional left ventricular elastance at the
beginning time of the systolic ejection interval). Further, CaEmin is assumed to be equal
to 0.05- CaEmax so as to reduce the number of unknowns to four (see gray font in Figure

(4.1b)). The four unknowns are then estimated for each beat in two steps.

In the initial step, 1' is estimated from the diastolic interval of the aortic pressure
waveform (i.e., from the time of the minimum of the ﬁrst difference of the waveform
between the time of the peak systolic pressure and the time of the local minimum

immediately preceding the subsequent peak systolic pressure (deﬁned as nesT of the

74

beat) to this latter time (deﬁned as nbsT of the next beat)). In particular, since the

lumped parameter model predicts that aortic pressure should decay like a pure
exponential during the diastolic interval, 1 is estimated by least squares ﬁtting of an

exponential to this interval (see Figure (4.1b)). Optimal ﬁtting is achieved in closed-form

after log transformation of aortic pressure. Since the time constants governing

Ra changes are signiﬁcantly longer than the cardiac cycle duration [Berger 1989], the r

estimates are then averaged over ten successive cardiac cycles to attenuate any noise.

75

 

 

*Nv

I."
14?
II
II

 

 

 

3
m
X

Pan“) [mmHg]

 

 

 

E.v(t) [mmHg/ml]

O

 

 

 

i I 0

Time
Figure 4.1 Model-based analysis technique for monitoring EF. (a) Lurnped parameter
model of the left ventricle and arteries upon which the technique is based. (b) Elv (t) 'in

the model is assumed to vary over time according to a parametric raised cosine function
(solid line in the lower panel).

76

In the subsequent step, the estimated 1: is substituted into Equation (4.4), and the
remaining unknowns in this equation, CaEmax ,Ts , and CaElv(nbsT), are estimated by
least squares ﬁtting of the equation to the systolic ejection interval of the aortic pressure

waveform (i.e., the remainder of the beat). Optimal ﬁtting is achieved by numerical

search over the following physiologic range of the three unknowns: l) 0 (physical

minimum value) < CaEmax S 15 (3 times the nominal value reported in the literature
[20], [24], [25], [26]; 2) % QT s T8 3 % QT, where the QT interval is obtained from a

simultaneous ECG measurement (with the underlying assumption that the electrical QT

interval is a rough approximation of the mechanical time interval between the start of the

upstroke to the end of the downstroke of Elv(t); and 3) CaElv(tbi +TS—Teje) S
CaElv(nbsT) S CaElv(tbi +Ts), where Teje is the duration of the systolic ejection

interval (to ensure that the end time of the systolic ejection interval does not occur prior

to the time of Emax ). With the estimated CaEmax , Ts , and CaElv (nbsT), CaE.v(nT) is
computed for nbs S n S nes through Equation (4.5).
Third, beat-to-beat absolute EF is computed from the resulting CaElv(nT) for

nbs S n S nes and the measured P(t) by cancellation of the Ca scale factor as follows:

SV Pa(nbsT) _ Pa(nesT)

Ca = CaElv (nbsT) CjEIV (nesT) (46)

EDV Pa(nbsT) +Y§
Ca CaEIVInbsT) Ca

 

 

EF

 

 

 

where V19 /Ca (i.e., proportional unstressed left ventricular volume), in contrast to the

other terms in the right-hand side of this equation, is neither estimated nor measured but

77

rather assumed to take on a nominal value (see below). Note that the term

Pa("esT)/CaElv(nesT) (i.e., proportional stressed ESV) should correspond to the
minimum value of Pa(nT) / CaEIV(nT)over "05 S n S nes (i.e., proportional stressed left

ventricular volume samples within the systolic ejection interval). However, such
correspondence may not always hold due to imperfect identiﬁcation of the end time of
the systolic ejection interval (or, equivalently, the beginning time of the diastolic

interval). In these instances, Pa (nesT)/ CaElv (nesT) in Equation (4.6) is replaced with
the minimum value of Pa(nT) / CaElv (nT) over nbs S n S nes-

Note that by-products of the above three steps are beat-to-beat proportional estimates

of EDV, SV, cardiac output (CO = HR-SV, where HR is heart rate), Ra, and Emax

(which is known to be a relatively speciﬁc index of ventricular contractility [Sagawa
1977, Suga 1974]. Thus, relative changes in these important hemodynamic variables

may be monitored as well.

4.3 Materials and Methods

Experimental Procedures

To evaluate the model-based analysis technique, experiments were performed in
six normal adult beagles (10-15 kg). All experimental procedures were reviewed and
approved by the MSU All-University Committee on Animal Use and Care.

In one dog, a sterile surgical procedure was employed for implanting chronic
recording transducers as follows. General anesthesia was induced with an intravenous
injection of propofol (2.2 — 6.6 mg/kg) and maintained with inhaled isoﬂorane (1.5 —

2.5%), and mechanical ventilation was instituted. A left lateral thorocotomy was

78

 

 

performed. An ultrasonic ﬂow probe was placed around the ascending aorta for gold
standard reference SV and CO (Transonic Systems, Ithaca, NY), while a tygon catheter
was placed in the left atrial appendage for unrelated purposes. The chest was evacuated
and closed in layers, with the cable and catheter tunneled subcutaneously and exteriorized
between the scapulae. The dog was then allowed ten days for recovery.

The chronically instrumented dog and the remaining ﬁve dogs were then studied
as follows. General anesthesia was induced and maintained as described above but
mechanical ventilation was not employed. A micromanometer-tipped catheter was
inserted into a femoral artery and positioned under ﬂuoroscopic guidance in the
descending thoracic aorta (see below) for the aortic pressure waveform (Millar
Instruments, Houston, TX). A similar catheter was also placed in the opposite femoral
artery of half the dogs for future studies. A catheter was inserted into a cephalic vein for
drug and isotonic ﬂuid administration, and surface electrodes were placed for two frontal
ECG leads. All of the analog transducer outputs were interfaced to a personal computer
through an A/D conversion system (DataQ Instruments, Akron, OH). The cardiovascular
measurements were then recorded in each dog at a sampling rate of 400-1000 Hz over the
course of 50—170 minutes during a subset of the following interventions to alter EF and
other hemodynamic variables: infusions of dobutamine, esmolol, verapamil,
phenylephrine, nitroprusside, and volume as well as progressive hemorrhage. Various
infusion rates were employed followed by brief recovery periods. During the recording
session, trans-thoracic two-dimensional echocardiography (GE Vivid 7, Horton, Norway)
was intermittently used for four-chamber left apical imaging in order to calculate

reference EF and EDV. (This single plane method was Shown to be nearly equivalent to

79

a biplane method in terms of measuring absolute EF and relative changes in EDV and
ESV [Sutton, 1998].) A 5-10 beat cine echocardiographic recording at > 90 ﬂames/sec
was obtained at a time for the chronically instrumented dog, whereas 5-10 beat cine
echocardiographic recordings at the same frame rate were obtained in triplicate for the
ﬁve acutely studied dogs.
Data Analysis

The model-based analysis technique was applied off-line to the aortic pressure
waveforms resampled to 250 Hz with the QT interval automatically detected from the

surface ECGS using a previously introduced. wavelet-based method [Maetinez 2004] and
v.8 / Ca set to the nominal canine value of 15 mmHg as prescribed in [Suga 1974,

Bourgeois 1976]. The resulting beat-to-beat EF and proportional EDV and Emax

estimates were then averaged over multiple beats for evaluation against the reference
measurements and known drug effects (see below).

Reference EDV and ESV, and thus EF, were established by manually tracing the
endocardial border of the single plane images, excluding papillary muscles, at end-
diastole and end-systole and then applying Simpson’s rule [Schiller 1991]. This method
was performed for two beats of each cine echocardiographic recording, and the resulting
values were then averaged over the two beats. For the ﬁve acutely studied dogs, the
values were further averaged over each set of triplicate cine echocardiographic
recordings. Thus, in these dogs, the reference EF and EDV represent six beat averages.
For the chronically instrumented dog, the reference echocardiographic values were
validated in part by noting a tight correspondence between the echocardiographic SV and

the gold standard aortic ﬂow probe SV (p = 0.92).

80

 

To compare the average proportional EDV estimates with the corresponding absolute
reference values, the estimates were ﬁrst scaled to have the same mean as the reference
values in each dog. The average EF and average calibrated EDV estimates were then
evaluated against their corresponding reference values through 1) standard Bland-Altrnan
analysis for a comprehensive illustration of the estimation errors as a function of the
reference values (rather than the average of the estimated and reference values) and an

indication of the bias [1 and precision o of the estimation errors [Bland 1986] and 2) the

root-mean-square of the estimation errors (RMSE=\/u2+02) for a simple scalar

metric indicating the overall error size. For the chronically instrumented dog, the
resulting beat—to-beat proportional CO estimates were likewise calibrated and then
compared to the corresponding absolute reference aortic ﬂow probe values through the
RMSNE (i.e., RMS of the estimation errors normalized (N) by the reference values), as
CO errors are customarily reported in percent [Crtichley 1999]. Finally, the average
proportional Emax estimates were qualitatively evaluated in terms of whether they
changed in the physiologically expected manner in response to the positive inotrope

dobutamine and the negative inotrope esmolol. (Note that veraparnil acted more like a

vasodilator in our study.)

4.4 Results

Figure (4.2) illustrates exemplary segments of the aortic pressure waveform measured
from one of the dogs during baseline, dobutamine, and volume infusion conditions. Note
that the aortic pressure waveform consistently exhibited a smooth upstroke during the

systolic ejection interval and an exponential decay during the diastolic interval despite

81

 

large variations in EF and mean aortic pressure (MAP). This result generally held for the
entirety of the waveform as well as for the waveforms of the remaining ﬁve dogs. Thus,
the lumped parameter model of Figure (4.1a), which accounts for smooth dynamics, was
representative of the measured aortic pressure waveforms.

The Table and Figure (4.3) summarize the results of applying the model-based
analysis technique to the measured aortic pressure waveforms. In particular, the Table
includes the hemodynamic range and average EF and calibrated EDV RMSES for each
dog, while Figure (4.3) illustrates Bland-Altman plots of all of the average EF and
calibrated EDV errors for the six dogs. As can be seen, the employed interventions
imposed a wide spectrum of hemodynamic conditions but did not result in a state of
severe systolic dysﬁmction. The average EF RMSE was 5.6% over all the dogs and was
generally consistent in each dog, deviating most in dog 1 with a value of 8.5%. Further,
the average EF RMSE showed only a small bias component of ~1%. While the average
EF errors did show a negative correlation with the corresponding reference
echocardiographic EF values (p = -0.61), removal of the few large errors in the very high
reference EF range (> ~80%; see below) substantially reduced this correlation. The
average calibrated EDV RMSE was 4.1 ml over all the dogs and was likewise generally
consistent in each dog, except for dog 1 with a value of 8.6 ml. The average calibrated
EDV RMSE has no bias component due merely to the calibration step. Further, the
average calibrated EDV errors were uncorrelated with the corresponding reference
echocardiographic EDV values (p = 0.06).

Figure (4.4) illustrates the results from all six dogs in terms of a plot of the average

EF estimates versus the corresponding reference echocardiographic values. (Note that an

82

analogous plot of the calibrated EDV results may misleadingly reveal strong correlation
simply due the calibration step.) Figures (4.5) and (4.6) illustrate the results from dogs 2
and 5 in terms of trends in the EF, calibrated EDV, and CaEmax estimates (dark lines) as
a function of time, along with the corresponding intermittent reference echocardiographic
values (gray circles) and the employed interventions (underlines). These ﬁgures not only
further reveal the correspondence between the model-based analysis technique and
echocardiography but also demonstrate the advantage of the new technique in providing

continuous hemodynamic monitoring. In addition, consistent with expectation, CaEmax

increased during dobutamine and decreased during esmolol.

83

120

Baseline (EF = 50%, EDV = 22 ml)

 

 

 

 

 

 

100 -
60 -
a 239.2 299.4 299.6 299.8 300 300.2 300.4 300.6 300.8 301
:E 120 - Dobutamine (EF = 82%, EDV = 24 ml)
E. 100 -
Q,
*5 so -
3 so
E
0 699.2 899.4 899.6 899.8 900 900.2 900.4 900.6
t
2 120 - Volume Infusion (EF = 33%, EDV = 29 ml)
80 -
60 -
40 l l I l g
2699 2699.4 2699.8 2700.2 2700.6 2700.9
Time [sec] *
Figure 4.2 Exemplary segments of the aortic pressure waveform measured from the

descending thoracic aorta in dog 2 during baseline, dobutamine, and volume infusion
conditions.

84

Table 4.1 Summary of the hemodynamic range and results of the model-based analysis
technique for each dog. EF is left ventricular ejection fraction; EDV, left ventricular end-
diastolic volume; HR, heart rate; MAP, mean aortic pressure; and RMSE, root-mean-
squared-error. The BF and EDV ranges were established with the reference
echocardiographic measurements.

 

 

Dog Number of EF EDV MAP EF EDV
Comparis Range Range Range RMSE RMSE

ons [%] [ml] [mmHg] [%] [ml]

1 7 53 - 90 28 - 46 67 - 156 8.5 8.6

2 9 33-84 17-29 60-94 6.1 1.9
3 10 56-74 21 -25 58—118 4.1 4.5
4 1O 52 - 69 18 - 23 42 - 82 3.4 4.3

5 17 37-74 10-15 36-116 5.5 1.4
6 9 43-74 13-18 53-128 5.8 3.3
Total 62 33 - 90 10 - 46 36 - 156 5.6 4.1

 

85

N
O

RMSE: 5.6%

a
0'!

d

O
o

°o

EF Error [%]
0 0|

0 l

of

Q) ° :
853;
950 i

i:

'C 1+

la

 

 

('n
[a
i
i”
:0
it

 

 

 

 

 

 

~10 ' o O
O
-15 1 O O
-20 - - - - -
20 40 60 80 100
Echocardiographic EF [%]
20»
RMSE: 4.1 ml 0
15’
= 10’ oo o
E. 5 W. 3.9. ..... U ......... 1.0
3 85
L-
m
>
o
u:
-20 ‘

0 1o 20 30 4o 50
Echocardiographic EDV [ml]

Figure 4.3 Results for all six dogs in terms of Bland-Altman plots of the average EF and
average calibrated EDV errors as a ﬁmction of the corresponding reference
echocardiographic values. The parameter [1 indicates the mean of the errors; 0, the
standard deviation of the errors; and RMSE the root-mean-square of the errors (i.e.,

[.12 +02 ).

86

100

 

 

 

80r
E":
u. 60-
Ill
'5
3
g 40-
'55
Ill
20-
o I n 1 1 j
0 20 40 50 80 100

Echocardiographic EF [%]

Figure 4.4 Results for all six dogs in terms of a plot of the average EF estimates versus
the corresponding reference echocardiographic EF values. .

87

Volume

 

 

 

 

 

 

Dobutamine Hemorrhage Infusion
90
.—. o 0 °
.5 70 ' ~“
u. so W
l.l.l
30
10 l l I I l l I l l I
0 5 10 15 20 25 30 35 40 45 50
=- 40 r
.§. 30~ e
3 20-
‘” 1o-
0 I I 1 l l l l l I I
0 5 10 15 20 25 30 35 40 45 50
_. 20
x m
g3 15
Ill 3
6'- 10
° 5..

 

v

0 5 10 15 20 25 30 35 40 45 50
Time [min]

GUI

Figure 4.5 Results for dog 2 in terms of the trends in the EF, calibrated EDV, and
CaEmax estimates (dark lines) as a function of time, along with the corresponding

intermittent reference echocardiographic values (gray circles) and the employed
interventions (underlines).

88

Esmolol Verapamil

 

 

 

 

 

 

 

 

 

 

 

E
II-
I“
25 1 1 1 1 1 1 1 1
0 20 40 50 80 100 120 140 160
='
.5.
>
D
I.“
0 J m 1 1 1 1 1 4
0 20 40 60 80 100 120 140 160
:— 20 a
" 3
“E 2 15 -
w
L; .E 10
E. 5 _
0 1 1 f 1 1 1 1 u
0 20 40 50 80 100 120 140 160

Time [min]

Figure 4.6 Results for dog 5 in terms of the trends in the EF, calibrated EDV, and
CaEmax estimates (dark lines) as a function of time, along with the corresponding

intermittent reference echocardiographic values (gray circles) and the employed
interventions (underlines).

89

Finally, the beat-to-beat calibrated CO RMSNE was 10.0% for dog 1. This RMSNE

had no bias component again because of the calibration step.

4.5 Discussions
In summary, we have developed a novel technique to estimate EF by model-based
analysis of the aortic pressure waveform (see Figure (4.1)). First, the aortic pressure

waveform is represented with a lumped parameter model comprising a three-parameter

variable elastance model of the left ventricle connected via a one-way, non-viscous valve

to a Windkessel model of the arteries accounting for Ra and the nearly constant Ca.

Then, this circulatory model is ﬁtted to each beat of the waveform so as to estimate its

lumped parameters to within a scale factor equal to Ca- Finally, these proportional

parameter estimates along with a nominal value for v.8 / C8 are utilized to compute beat-
to-beat absolute EF by cancellation of the Ca scale factor. In this way, in contrast to
conventional imaging methods, EF may be continuously monitored without making any

assumptions about the left ventricular geometry. In addition, with the proportional

parameter estimates, relative changes in EDV, SV, CO, Ra, and Emax may also be

monitored on a beat-to-beat basis. Note that the traditional method for determining

Emax , in particular, is much more cumbersome, requiring Simultaneous measurements of

left ventricular pressure and volume during alterations to the preload and/or afterload
[Sagawa 1977, Suga 1974]. To evaluate the technique, we measured aortic pressure
waveforms, intermittent average reference EF and EDV via standard echocardiography,
and other cardiovascular variables from six dogs during various pharmacological and

volume interventions (see Figure (4.2)). Our results showed overall RMSES in average

90

EF and average calibrated EDV of 5.6% and 4.1 ml, respectively, as well as reliable

estimation of relative changes in average Emax and beat-to-beat CO (see Table (4.1) and

Figures (4.3)—(4.6)). This study is notable in that it may be the ﬁrst to demonstrate the
feasibility of estimating EF from only blood pressure.
Assumptions of the Model-Based Analysis Technique

Our model-based analysis technique is based on four major assumptions. We
defend each of these assumptions below using known physiology and previous
experimental ﬁndings.

Assumption 1: lumped parameter model representation of the aortic pressure
waveform. It is well known that the arteries are not simply a lumped system but rather a
complicated distributed system with impedance mismatches throughout due to vessel
tapering, bifurcations, and caliber changes. As a result, peripheral artery pressure
waveforms are heavily corrupted by wave reﬂections and generally require a distributed
model for their representation. However, aortic pressure waveforms are less complicated
by wave reﬂections due to destructive interference and attenuation of the reﬂected waves
reaching the aorta [Noordergraaf 1978] as well as signiﬁcant time delays between
forward and backward waves in the aorta [Albaladejo 2001]. Thus, aortic pressure
waveforms may be more amenable to a lumped parameter model representation. Indeed,
consistent with the prediction of the lumped parameter model utilized herein (see Figure
(4.1a)), Bourgeois et al. showed that aortic pressure waveforms measured Speciﬁcally
from the descending thoracic aorta exhibit pure exponential diastolic decays over a wide

hemodynamic range [Bourgeois 1974]. We therefore placed the aortic catheter in the

91

descending thoracic aorta in this study, and the resulting aortic pressure waveforms
generally conﬁrmed their ﬁnding (see Figure (4.2)).

Assumption 2: constant arterial compliance. To estimate EF and relative changes in
other hemodynamic variables including EDV and CO, the assumption is precisely that

C8 is constant over a monitoring period (e.g., days to weeks) in a given subject. Note
that this assumption is relaxed in terms of estimating just EF, as Ca must be invariant

only within each cardiac cycle (see Equation (4.6)). Indeed, it is well known that
Ca decreases gradually over the course of years in a subject (e.g., [Hallock 1937]).
However, previous studies have indicated that Ca tends to decrease with increasing
arterial pressure [Hallock 1937, Li 1994]. If this inverse relationship were strongly
present in the dogs studied herein, EF would be signiﬁcantly and consistently

underestimated, while EDV would be grossly overestimated at high arterial pressures and

underestimated at low arterial pressures. However, the EF error bias was small (see
Figure (4.3)), and the EDV error was only mildly correlated with MAP (p = 0.28). Thus,
Ca may have been sufﬁciently constant in our study even though MAP was varied over
a wide range (see Table (4.1)), which is consistent with previous studies including those
by Bourgeois et a1. [Bourgeois 1974, Bourgeois 1976] (see below) and us [Lu 2006,
R.Mukkamala 2006, Swamy 2008].

Assumption 3: three parameter raised cosine function representation of left
‘ ventricular elastance during the systolic ejection interval. In a surprising but compelling
study, Senzaki et al. showed that experimentally measured Elv (t), normalized both in

amplitude and time, were remarkably consistent in 87 patients despite extremely wide

variations in their ventricular states (see discrete values in the lower panel of Figure

92

(4.1b)) [Senzaki 1996]. Heldt et al. then showed that these normalized data could be well
ﬁtted, especially during the systolic ejection interval (i.e., the higher elastance range), by

a raised cosine function with three unknown parameters, Emin: Emax, and TS (see
Figure (4.1b) and Equation (4.5)) [Heldt 2002]. Further, Emin may be simply
approximated as 0.05-Emax as suggested by the data of Senzaki et a1 (see Figure (4.3) in
[Senzaki 1996]). Thus, Elv (1) during the systolic ejection interval may indeed be well
represented using a raised cosine function with three unknown parameters, Emax: Ts,

and Elv(nbsT)- Note that independent estimation of (proportional) Emin here would
likely be unreliable anyhow, as it would essentially amount to an extrapolation. That is,
Emin is not directly “seen” by aortic pressure, as the aortic valve is closed during
diastole. Further note that attempts to represent Elv(t) with a Gaussian, trapezoidal,

exponential or sinusoidal function (see [Guarini 1998]) and/or allow the ratio of the time

period of the downstroke of Elv(t) to T$ to vary (rather than be ﬁxed to 0.5 as assumed in

Equation (4.5)) did not improve our EF estimation results. Finally, it should be
emphasized that, while the assumption here constrains the shape of the left ventricular
elastance function, its amplitude and width are allowed to vary in order to account for any
changes that may occur in cardiac functioning.

Assumption 4: nominal value for proportional unstressed left ventricular volume. It

is generally appreciated that v.8 is usually much smaller than EDV. Thus, in many

subjects including the healthy dogs studied herein, v.8 / Ca may indeed be set to a

nominal value or even neglected (i.e., no calibration) without substantially sacriﬁcing the

93

EF estimation accuracy. In fact, varying VIE), / Ca from 10 to 20 mmHg in this study only
altered the EF RMSE by 2%. Alternatively, V's / Ca may be determined by obtaining an

independent measurement of EF via imaging and then ﬁnding the value of v.8 / Ca that

equalizes the EF estimated by aortic pressure waveform analysis to the independent EF
measurement (i.e., single calibration). This alternative approach would be preferred in

subjects with cardiac dilatation. Further, for these subjects, large changes in the EF

estimated by aortic pressure waveform analysis with the determined V18 / Ca may, in

turn, be utilized as a cue to re-image the heart (i.e., multiple calibrations). Finally, note

that it is possible that the ratio of SV to stressed EDV (i.e., V18 / C8 = 0) could prove to

be a valuable indicator of left ventricular function.
Potential Sources of Error

Any violation to the four above assumptions certainly represents a source of the
errors reported herein. Another source of the errors is likely imperfect reference
echocardiographic measurements due to, for example, individual variations in the trans-
thoracic imaging window [Rumberger 1997], inaccuracies in endocardial contour
deﬁnition [Hof 1998], and violation to the underlying ventricular geometry assumptions
[Nosir 1996]. Indeed, we suspect that the relatively large EF errors at reference EF
values > ~80% (see Fig. 3) were a result of the ventricular geometry assumptions of
echocardiography breaking down at the very small ESVs. In addition, we surmise that
the EF and EDV RMSES were largest for dog 1 (see Table (4.1)), because the reference
values were established by averaging over fewer beats than the other dogs (see above).

On the other hand, we do not believe that measurement inaccuracies in the aortic pressure

94

waveforms were a major source of the errors, as high ﬁdelity, micromanometer-tipped
catheters were employed.

Computational Speed

Our implementation of the model-based analysis technique included a brute force

search over a physiologic range to estimate the three parameters of the raised cosine left
ventricular elastance function during the systolic ejection interval (see above). However,
the technique may be implemented much more efﬁciently by exploiting the fact that the
parameters usually do not change much from one beat to the next. For example, we
estimated the parameters for one beat using the brute force search and then estimated the
parameters for subsequent beats using a local simplex method with the parameter
estimates of the previous beat speciﬁed as its initial seed. We actually implemented the
brute force search for every tenth beat rather than for just the initial beat in order to
attenuate any propagation of error due to, for example, ventricular ectopy. This
implementation produced almost the same average EF estimates as the original brute
force search and was ten times faster than real time when executed on MATLAB with a
standard PC.

Previous Related Techniques

We ﬁnd no techniques in the literature aiming to likewise monitor EF from only

blood pressure waveforms. However, we are aware of two closely related, previous
techniques.

Bourgeois et al. developed a technique to monitor relative changes in beat-to-beat SV
by analysis of the aortic pressure waveform using a two-parameter Windkessel model of

the arteries (see Figure (4.1a)) [Bourgeois 1976]. Their technique speciﬁcally involved

95

estimating SV to within a constant l/Ca scale factor for each beat using the right-hand
side of Eq. (3) with t = tes and ‘t determined by ﬁtting an exponential to each diastolic

interval of the waveform. These investigators applied their technique to canine aortic
pressure waveforms measured from the descending thoracic aorta wherein pure
exponential diastolic decays are visually apparent (see above). AS a result, they were
able to Show excellent agreement between their technique and the gold standard aortic
ﬂow probe over a wide hemodynamic range. Our study builds upon the seminal work of
Bourgeois et al. by modeling the left ventricle in addition to the arteries so as to also

permit the monitoring of beat-to-beat absolute EF and proportional EDV and Emax-

Guarini, Urzua, and co-workers developed a technique to monitor beat-to-beat
ventricular and arterial parameters based on analysis of the radial artery pressure
waveform [Guarini 1998, Urzua 1998]. Their technique speciﬁcally involved
representing the waveform with a parametric variable elastance model of the left ventricle
in series with a third-order lumped parameter model of the arteries and then estimating
the model parameters for each beat using a two-step procedure conceptually similar to
one utilized herein. However, since these investigators desired the absolute values of the
model parameters rather than EF, their technique also necessitated an operator-dependent
thermodilution CO measurement. Thus, the technique is not automated and continuous.
Moreover, unlike the aortic pressure waveform, the radial artery pressure waveform is
heavily corrupted by wave reﬂections and is therefore generally unsuitable for
representation with lumped parameter models, particularly those of low order (see
above). Indeed, these investigators did not experimentally verify their technique against

reference methods.

96

Potential Applications and Future Directions

Our model-based analysis technique may potentially be utilized in lieu of, or as a
complement to, imaging methods so as to permit continuous monitoring of EF as well as
other important proportional hemodynamic variables in subjects without aortic stenosis
and instrumented with an aortic catheter. Examples of such subjects include animals in
the context of research and patients undergoing cardiac catheterization or with intra-
aortic balloon pumps. To eventually realize these applications, we plan to conduct
ﬁrrther testing of the technique in both animals and humans, especially with severe
systolic dysfunction.

In addition, we intend to adapt the technique to peripheral artery pressure waveforms,
which, in contrast to aortic pressure waveforms, are routinely available in clinical
practice via invasive catheters and can be measured non-invasively (see above).
However, as we have discussed, analysis of peripheral artery pressure waveforms is
technically more difﬁcult due to signiﬁcant distortion by wave reﬂections. To overcome
this difﬁculty, we propose to ﬁrst mathematically reconstruct the aortic pressure
waveform from measured peripheral artery pressure waveforms using our multi-channel
blind system identiﬁcation technique [Swamy 2007, Swamy 2008] or possibly even the
conventional generalized transfer function (e. g., [Chen 1997, Soderstrom 2002]) and then
apply the model-based analysis technique to the reconstructed waveform.

Our hope is that these future directions ultimately lead to widespread automated and
continuous EF monitoring by arterial pressure waveform analysis in various inpatients
and outpatients as well as heart failure patients at home so as to help meet the increasing

patient monitoring demands.

97

CHAPTER 5

QUANTIFICATION OF FORWARD AND BACKWARD ARTERIAL WAVES BY
MODEL BASED ANALYSIS OF AORTIC AND FEMORAL ARTERY

PRESSURE WAVEFORM

5.1 Introduction

It is well appreciated that arterial pressure and ﬂow waveforms represent the sum and
difference of forward and backward traveling waves in the arterial tree. The relative
magnitude and timing of the backward or reﬂected wave likely play an important role in
health and disease. For example, a wave reﬂected at the periphery that reaches the
ascending aorta during systole would increase the cardiac afterload and thereby reduce
the stroke volume. On the other hand, a reﬂected wave that [arrives at the ascending aorta
during diastole would enhance myocardial perfusion. It is therefore important to be able
to quantify the wave reﬂections in order to obtain an improved understanding of
circulatory pathophysiology and potentially develop more effective indices for
monitoring cardiovascular health.

To this end, several techniques have been proposed to separate measured arterial
waveforms into their forward and backward wave components. Westerhof et al.
[Westerhof, 1972] modeled the arterial tree as a uniform frictionless tube in which the
forward and backward pressure waves are equal to the product of the corresponding ﬂow
waves and the characteristic impedance of the tube. These investigators obtained
simultaneous measurements of aortic pressure and ﬂow waveforms in order to ﬁrst

determine the tube characteristic impedance from the high frequency regime of the

98

arterial input impedance (i.e., the ratio of the Fourier Transforms of the aortic pressure
and ﬂow waveforms) and then calculate the forward and backward waves from the
measured waveforms and characteristic impedance by adding pressure waves and
subtracting ﬂow waves. Lee et al. [Lee, 1993] developed a more accurate technique to
determine the tube characteristic impedance by measuring the arterial pressure and ﬂow
response to a Single cardiac contraction using vagal stimulation. They were then able to
likewise calculate the forward and backward waves. Burattini et al. [Burratini, 1989]
employed a similar tube model but also represented its terminal load with a three-
parameter Windkessel model. These researchers then estimated the model parameters by
nonlinear least squares ﬁtting between the aortic pressure waveform predicted from the
model in response to a measured aortic ﬂow waveform input and a simultaneously
measured aortic pressure waveform. They were then able to calculate the forward and
backward waves ﬁ'om the estimated model parameters.

As described above, the initial techniques all involved the measurement of an arterial
ﬂow waveform, which either requires surgery to place a ﬂow probe around a blood vessel
or an expert operator to stabilize an ultrasound transducer. Thus, the applicability of
these techniques is limited. Recently, Westerhof et al. [Westerhof, 2006] proposed a
technique to compute the forward and backward waves from aortic pressure alone by
approximating the aortic ﬂow waveform with a triangular pulse. However, this
approximation could introduce signiﬁcant error in the calculated waves for at least some
pathophysiologic conditions.

In this study, we developed a technique to quantify forward and backward arterial

waves by model-based analysis of only aortic and femoral artery pressure waveforms,

99

without making any assumptions about ﬂow morphology. Furthermore, in contrast to
most previous related efforts, we validated the forward and backward waves through a set
of canine experiments by speciﬁcally Showing that the waves accurately predicted a third
arterial pressure waveform measurement and changed in the physiologically expected
manner to interventions of known effect. Finally, we calculated the forward and

backward waves during nine different hemodynamic conditions.

5.2 Technique

Our model-based analysis technique for quantifying forward and backward arterial
waves is conceptually similar to that of Burattini et al. but with three major differences.
Firstly, the model input and output are measured ascending aortic and femoral artery
pressure waveforms instead of ascending aortic ﬂow and pressure waveforms. Secondly, .
the tube load is effectively represented with a pole-zero model whose order is data-
deterrnined rather than being set to ﬁrst-order a priori. Thirdly, the model parameters are
determined using linear least squares estimation, which affords a convenient and reliable
closed-form solution, in contrast to nonlinear least squares estimation, which requires a

difﬁcult and often imperfect numerical search.

100

pasc“) pfem“)
I < T, > I, 4—T2—>

 

 

Figure 5.1 Arterial tube model upon which our technique is based.

101

More speciﬁcally, the technique is based on the arterial model of Figure (5.1) in
which a distributed uniform tube is terminated by a lumped load. The tube represents the
arterial conduit starting from the ascending aorta and passing through a femoral artery.

Similar to previous studies, this tube is frictionless and therefore has constant

characteristic impedance (ZC ). As a result, mean pressure is the same along the tube.

The terminal load represents the arterial bed distal to the femoral artery. This load has a
frequency-dependent impedance (Z(F), where F is frequency).
According to this model, the forward and backward waves at the terminal load

(pf (t) and pb(t)) are related as follows:

915(1) = 110839141). (5-1)
where ® is the convolution operation, and P(t) is the inverse Fourier Transform of
P(F) = (Z(F) — Zc ) / (Z(F) + Zc ) , which is the frequency-dependent reﬂection coefﬁcient
at the terminal load. The ascending aortic and femoral artery pressure waveforms
(pasc(t)and pfem(t)) may then be expressed as the sum of appropriately shifted forward
and backward waves as follows:
pasc(t) = pf(t + T1+ T2)+ F(t)® pf(t-T1—T2) (5.2)
pfemm = pf(t+T2)+F(t)®pf(t—T2), (5.3)
where T, is the measured time for the arterial wave to travel between the ascending aorta
and the femoral artery, while T2 is the unknown time for the arterial wave to travel

between the femoral artery and the terminal load. Substitution of Equation (5.3) into

Equation (5.2) gives the following equation:

102

E’fem(t + T1)‘ pasc (t? =
y(t)
1“(t — 2T2 ) ® Spasc (t) — pfem (t — T1ll+ e(t):
ﬁt)

 

(5.4)

 

where the term e(t) has been included to account for measurement and/or modeling error.

The unknown P(t) and T2 are calculated in Equation (5.4) from the measured “input”
x(t) and “output” y(t) by least squares minimization of e(t). This optimization problem is
speciﬁcally solved using standard autoregressive exogenous input (ARX) identiﬁcation

and the MDL criterion 0. In this way, F(t—2 T2) is efﬁciently represented with a pole-zero

model whose unknown parameters are estimated with the linear least squares solution and

whose unknown order and delay are determined from the measured waveforms. Then,
the computed P(t) and T2 are substituted into Equations (5.2) and (5.3), and pf (t) is
calculated ﬁ'om the measured pasc(t),pfem(t) and T1 by way of deconvolution. Rather

than applying conventional deconvolution to Equations (5.2) and (5.3) to arrive at two

versions of pf(t), a single, optimal pf(t) is speciﬁcally obtained from these equations

using multi-channel linear least squares deconvolution as described in [Abed-Meriam,
1997]

Note that, with the calculated pf (t) and pb(t) along with T2, the arterial pressure
waveform at any point along the tube (pa(t,T)) of the model of Figure (5.1) may be
computed using the following equation:
pa(t,T) = pf(t+T+T2)+pb(t —T—T2), (5.5)
where T is the measured time for the arterial wave to travel between the point of interest

and the femoral artery. Similarly, the arterial ﬂow waveform at any point along the tube

103

(qa(t,T)) may be computed to within a proportionality constant equal to1/Zc through

the following equation:
1

However, because of branching between the ascending aorta and the femoral artery,
the proportional qa(t,T) calculated at a given point in the tube does not have any
correspondence to anatomy, except at the point of the femoral artery (i.e., T = 0). Here,
the waveform speciﬁcally represents relative changes in femoral artery ﬂow. Finally,

note that, since F (t) and T2 are computed from zero-mean x(t) and y(t) in Equation (5 .4),
the mean value of proportional qa(t,T) may not be reliably determined. (However, the

mean value of pa(t,T) Should always be accurately determined, as it is effectively set to

the mean value of the measured arterial pressure.)

5.3 Methods

A. Experimental Procedures

Experiments were performed in three normal adult beagles (10-12 kg). Each dog was
studied under a protocol approved by the MSU All-University Committee on Animal Use
and Care as follows. General anesthesia was induced with an intravenous injection of
propofol (2.2-6.6 mg/kg) and maintained with inhaled isoﬂorane (1.5-2.5%). A
micromanometer-tipped catheter (Millar Instruments, Houston, TX) was inserted into a
either femoral or carotid artery and positioned via ﬂuoroscopic guidance for
measurement of the ascending aortic pressure waveform. A similar catheter was placed

for measurement of the femoral artery pressure waveform. A catheter was introduced

104

into a cephalic vein for drug and isotonic ﬂuid administration, and electrodes were placed
for measurement of surface ECGS. In the third dog, a micromanometer—tipped catheter
was also inserted into a femoral artery and placed for measurement of the abdominal

aortic pressure waveform (pabd(t)) as conﬁrmed with ﬂuoroscopy. In addition, a bipolar

electrode catheter was advanced through the jugular vein of this dog into the right atrium
via ﬂuoroscopic guidance for atrial pacing using an external pulse generator (Medtronic,
Minneapolis, MN). The analog transducer outputs were interfaced to a personal
computer through an A/D conversion system (DataQ Instruments, Akron, OH). The
cardiovascular measurements were then recorded at a sampling frequency of 1000 Hz
during a baseline condition and following infusions of dobutamine and esmolol in the
ﬁrst dog; infusions of volume and hemorrhage in the second dog; and infusions of
phenylephrine, nitroglycerin, and verapamil as well as atrial pacing in the third dog.

B. Data Analysis

The technique was applied to all 15-Second segments of the ascending aortic and

femoral artery pressure waveforms resampled to 250 Hz, with T1 measured for each

segment as the time delay between the onsets of upstroke of the two waveforms. Then,
similar to the study of Burattini et al.[Burratini, 1989], the abdominal aortic pressure

waveform was predicted (pa(t,T)) for the 15-second segments from the calculated

quantities, with T measured for each segment as the time delay between the onsets of
upstroke of the measured abdominal aortic and femoral artery pressure waveforms. The
predicted waveform was quantitatively evaluated with respect to the measured reference
waveform through the root-mean-squared-error (RMSE). For comparison, the ascending

aortic and femoral artery pressure waveforms were likewise evaluated with respect to the

105

measured abdominal aortic pressure waveform. In addition, the computed magnitude
spectrum of the reﬂection coefﬁcient (|F(F)|) and proportional femoral artery peak-to-
peak ﬂow were assessed during the phenylephrine and nitroglycerin conditions and
baseline, dobutamine, esmolol, vaSOpreSSin and HIS-pacing conditions respectively.
Finally, the computed forward and backward waves at the ascending aorta

(pa (1 + T1 + T2 ) and pa (t - T1 — T2 )) were examined during the baseline, phenylephrine

and nitroglycerin conditions.

5.4 Results

The overall RMSE of the abdominal aortic pressure waveform predicted by the
forward and backward waves, which were calculated from the ascending aortic and
femoral artery pressure waveforms, was 2.4 mmHg. For comparison, the overall RMSE
of the ascending aortic and femoral artery pressure waveforms with respect to the
measured abdominal aortic pressure waveform was 5.9 and 4.5 mmHg, respectively.
Figure (5.2) illustrates an example of the measured ascending aortic and femoral artery
pressure waveform segments and the predicted abdominal aortic pressure waveform
segment, along with the measured reference abdominal aortic pressure waveform
segment.

Figure (5.3) illustrates the magnitude Spectrum of the reﬂection coefﬁcient calculated
during the baseline, phenylephrine, and nitroglycerin conditions. Consistent with known
physiology, this spectrum was ampliﬁed under the intense vasoconstrictor
(phenylephrine) and blunted under the strong vasodilator (nitroglycerin). Figure (5.4)
Shows the reference (solid line) and predicted (dashed line) peak-peak femoral artery

ﬂow (solid line) for one of the dogs.

106

  
 

 

 

 

 

 

 

 

 

 

 

 

 

 

_1101 ,_,11oi 110

a: m
I I .—

E E m
g oo- , g so I so»
A . A E
t: ,3 3:: E

.3; 1' 3:. A

0. 7o- 1 a. 70» 32:70.

a I a
1.7 1 i 1.7 "
7 ' ‘g ‘1 . .

I to-

n 5 - . D- 5 - - so - -

8 3 0.85 1.4 3 3 0.85 1.4 0.3 0.85 1.4

Time [see]

Figure 5.2 Example segments of the (a) measured ascending aortic pressure waveform,
(b) measured femoral artery pressure waveform, and (c) predicted abdominal aortic

pressure waveform (dash), along with the measured abdominal aortic pressure waveform
(solid).

107

(L8

    

 

 

 

7" Phenyleprhine

33 045-

:22

C

.2. 0.4-

E

-'= 0.2-

wtrogwcerine
0 . . . - :===
0 5 10 15 20 25
Frequency [Hz]

Figure 5.3 Computed magnitude spectrum of the reﬂection coefﬁcient.

108

Baseline Dobutamine Esmolol

 

 

 

 

Vasopressln HIS Pacing
<——> ‘L > <————» ‘ﬁ : <———>
800 .
g- l“.
3; '5' P" i at]; '
a? 2 400 " I
E l: 1" r“ ’h
a g - , ' I .’
a 200+ ‘ ' . “‘ '
ﬂ ..
O I I I I I I I I
20 40 60 80 100 1 20 1 40 1 60
Time [min]

Figure 5.4 Predicted proportional femoral artery ﬂow.

109

Baseline Phenyleprhlne

    

 

 

 

 

70.1 70.8 71.6 856.8 857.5 858.3
Time [sec] Time [sec]

030)
O

[mmHg]
OI

 

Nitroglycerln

 

516.3 517.0 517.8

Time [sec]

Figure 5.5 Measured ascending aortic and femoral artery pressure waveforms and
calculated zero-mean forward and backward waves at the ascending aorta.

110

Figure (5.5) illustrates the measured ascending aortic and femoral artery pressure
waveforms and the calculated zero-mean forward and backward waves at the ascending
aorta during each of the nine experimental conditions. The magnitude of the backward
wave relative to the forward wave appeared smallest during nitroglycerin and dobutamine
and largest during phenylephrine and hemorrhage, while the time delay between the two
waves was smallest during atrial pacing and about the same during the remaining eight
conditions. Overall, the backward waves generally appeared like a time Shifted and
scaled version of the forward waves. This result is congruent with those of Lee et al.
who, as described above, computed the waves using aortic pressure and ﬂow

measurements during vagal stimulation [Lee, 1993].

5.5 Discussions

In a companion study [Swamy, 2008], we modeled an arterial conduit with a tube
model terminated by a ﬁrst-order load and then utilized a numerical search to estimate the
model parameters from only a femoral artery pressure waveform by exploiting the fact
that aortic ﬂow is zero during diastole. While the forward and backward arterial waves
could be calculated from the estimated parameters, the main aim of that study was to
estimate the aortic pressure waveform from a peripheral artery pressure waveform via an
adaptive transfer function. In contrast, in this study, we aimed to accurately calculate the
forward and backward waves by analysis of aortic and femoral artery pressure
waveforms. By including both waveforms in the analysis, a higher order terminal load
could be conveniently and accurately calculated in closed-form. We veriﬁed the forward
and backward waves computed here by showing that these waves accurately predicted a

third arterial pressure waveform measurement and changed in the physiologically

lll

expected manner to interventions of known effect. We then examined the forward and
backward waves during various hemodynamic conditions. In the future, we plan to
continue the evaluation of the technique by quantitatively comparing the predicted
proportional femoral artery ﬂow with a corresponding independent reference
measurement. With such future successful testing, the technique may ultimately be used

to enhance the understanding of arterial wave phenomena in health and disease.

112

 

CHAPTER 6

ADAPTIVE TRANSFER FUNCTION FOR DERIVING THE AORTIC PRESSURE

WAVEFORM FROM A PERIPHERAL ARTERY PRESSURE WAVEFORM

6.1 Introduction

Since its introduction by O’Rourke and co-workers in 1993 [Karamanoglu, 1993], the
generalized transfer function has received attention for providing a convenient and safe
means for monitoring central aortic pressure (AP) by mathematical transformation of a
peripheral artery pressure (PAP) waveform. The basic premise of the transformation is
that a single, universal transfer function exists that can faithfully relate the PAP
waveform to the AP waveform of all individuals for all time. However, the transfer
function linking PAP to AP would ideally be able to adapt to the inter-subject and
temporal variability of the arterial tree due to, for example, age-related arterial
compliance differences, baro- and thermo-regulatory modulation of peripheral resistance
in response to physiologic perturbations, progression of peripheral vascular disease, and
therapeutic administration of vasoactive agents. To this end, Sugimachi et al. [Sugimachi
2001] and Westerhof et al. [Westerhof 2007] have previously proposed a technique to
partially adapt the transfer function by deﬁning it through an arterial tube model with a
personalized value for a model parameter reﬂecting the wave propagation delay time and
population averages for the remaining parameters. We recently introduced perhaps the

ﬁrst entirely adaptive technique for mathematically deriving the AP waveform by

113

exploiting the commonality in multiple PAP waveforms through the powerful multi-
channel blind system identiﬁcation approach [Swamy 2007, Swamy 2008]. However, the
requirement of more than one PAP waveform is a practical disadvantage of this
technique.

In this study, we conceived a fully adaptive technique for deriving the AP waveform
from only one PAP waveform. The new technique similarly deﬁnes the transfer function
relating PAP to AP through a parallel tube model of pressure and ﬂow in the arterial tree
but then estimates all of its parameters by capitalizing on pre-knowledge of aortic ﬂow
with the model. The parameters are periodically re-estimated for each subject so as to
derive an adaptive transfer function (ATF). We performed canine experiments in order to
evaluate the ATF technique as well as compare it to previous transfer function techniques
over a broad array of controlled and signiﬁcant hemodynamic perturbations. A

preliminary version of this study has been reported in abbreviated form [Swamy 2008].

6.2 Methods

Adaptive Transfer Function (A T F) Technique
Figure (6.1) illustrates the ATF technique. As shown in the top panel of this ﬁgure,

the arterial tree is modeled as a parallel arrangement of m uniform tubes in series with

terminal loads. The ith tube represents the path between the aorta and the 1th peripheral

artery. Each tube is frictionless and therefore has constant characteristic impedance
(Zci = ,/(Ii ’91) , where Ii and Ci are the tube’s total inertance and compliance) and
allows waves to propagate with constant velocity and delay time from one end of the tube

to the other (Tdi = ,Klici) ). Thus, consistent with Poiseulle’s law, mean pressure is

114

 

identical throughout the tubes. The ith terminal load signiﬁes the arterial bed distal to the

ith peripheral artery. Like the studies of Sugimachi et al [Sugimachi 2001] and

Westerhof et al. [Westerhof 2007], each terminal load has a frequency-dependent

impedance (Zi(co), where (0 is frequency) characterized by two parameters that are
dependent on the peripheral resistance and compliance (A1 and Bi , where 0 < Ai < Bi)

as well as the characteristic impedance of the corresponding tube (i.e., Zci)- Thus, the

wave reﬂection coefﬁcient at each terminal load is also frequency-dependent

(17(0)) = (Zi((D) - ZOO/(21(0)) + ZCI) ).

115

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

f\_ O 26) 91(0))
5 [z (8-2 We)
Aortic Flow 5 1 ’ jw+A1
{\— Qai(w) 4 fdl b l
n— O 2‘9 rpm)
Pa(w) : z _z (jw+Bi)
i i (w jw+Ai
Qam(w)4 Tdm D 7 ¢
“— z N .miw)
my I I
, _Z (]w+B )
z (wl- °Tw+Am m 1
Jr

 

 

 

0 diastole

Figure 6.1 Adaptive transfer function (ATF) technique for deriving the aortic pressure
(AP) waveform from a peripheral artery pressure (PAP) waveform.

116

Generally speaking, pressure and ﬂow waves propagate along each tube without
distortion and are proportional to each other through the tube characteristic impedance.
Waves traveling in the forward direction (left to right) are reﬂected in the opposite or
backward direction at the terminal load with relative magnitude and phase based on the
frequency according to the wave reﬂection coefficient. The actual pressure (or ﬂow)
waveform at any point along a tube may therefore be expressed as the sum (or difference)
of the forward and backward traveling pressure (or ﬂow) waves appropriately shifted in
_ time based on the distance between the point and the tube end. In this way, the model is
able to mimic the well-known progressive distortion that experimental arterial pressure
and ﬂow waveforms undergo with increasing distance from the aorta. Further, from
these expressions, a transfer function relating the arterial pressure or ﬂow waveform at
any point along a tube to the arterial pressure or ﬂow waveform at any other point on the
tube may be established in terms of the model parameters. '

More speciﬁcally, according to the arterial tree model, a PAP waveform (ppi(t)) is

related to the AP waveform (pa(t)) through the transfer function shown in the middle

panel of Figure (6.1) (“pressure—>pressure transfer function”) with unknown model
parameters (see detailed derivation in [Stergiopulos 1998]). Thus, this transfer function

may be applied to a measured PAP waveform so as to derive the AP waveform, if its

parameters, namely Tdi , Ai and Bi , could be determined.
To this end, Tdi the wave propagation delay time between the aorta and peripheral

artery measurement site, is ﬁrst measured non-invasively (see Discussion section). Only

one Tdi measurement is made for a subject during a monitoring period (e. g., on order of

days), as arterial inertance and compliance may not greatly vary over this time period.

117

Thereafter, the three parameters are determined from each 15 sec segment of the
measured PAP waveform and initial Tdi value by exploiting the fact that aortic ﬂow is
negligible during each diastolic interval due to aortic valve closure (in absence of aortic
regurgitation). Thus, as indicated in the t0p panel of Figure (6.1), the arterial flow at each
tube entrance in the arterial tree model (“arterial entry ﬂow”) may likewise be small
during these time intervals. In particular, according to this model, the PAP waveform is

related to the arterial entry ﬂow waveform to the corresponding peripheral artery (qai(t))

through the transfer function shown in the bottom panel of Figure (6.1) (“pressure—>ﬂow
transfer function”) with the same unknown model parameters as the pressure—>pressure
transfer function. The common parameters are then estimated by ﬁnding the
pressure—mow transfer function, which when applied to the PAP waveform segment,
minimizes the energy (sum-of-squares) of the arterial entry ﬂow waveform (scaled by
qai(t)) output over its diastolic intervals. In other words, as indicated in the bottom
panel of Figure (6.1), the parameters are selected so as to map the PAP waveform to an

arterial entry ﬂow of zero during diastole. The detailed steps of the technique are given

as follows.

First, since the mean or DC value of pa(t) is already known (i.e., approximated as
the corresponding value of ppi(t)due to Poiseuille’s law), the DC value of ppi(t) is

removed in order to focus the mapping on the unknown zero—mean or AC components.

Then, AC qai(t) is calculated to within a 1/Zci scale factor by applying the

pressure—mow transfer function to AC ppi(t) for a set of Ai and Bi values over a

physiologic range with the Tdi value as measured. Next, the end of each diastolic

118

interval in each candidate AC proportional qai(t) is determined by identifying the

minimum preceding the peak amplitude of a cardiac cycle, and the start of each

corresponding diastolic interval is approximated based on Malik’s formula [Malik 1996]

relating the cardiac cycle length to the systolic interval length. Then, the values of Al and
Bi are selected that provide the minimum variance of AC proportional qai(t) over its

diastolic intervals and yield physiologically reasonable pressure and ﬂow waveforms

(i.e., AC proportional qai(t) exhibits an undershoot during diastole as typically seen in
experimental waveforms [McDonald 1974] and pa(t), computed as described below,
does not reveal double peaks). In the event that none of the Ai and Bi values in the set
result in physiologically reasonable waveforms, Tdi is successively incremented when
mean pressure decreases relative to the Tdi measurement period or decremented when
mean pressure increases relative to this period (as Tdi tends to have an inverse

relationship with mean pressure for signiﬁcant changes [Gribbin 1976]) until the criterion
is satisﬁed. Finally, the pressure—>pressure transfer function, with the selected values

for Ai, Bi, and Tdi , is applied to ppi(t) (including its DC value) so as to derive pa (t).

(Note that the DC value of proportional qai(t) may be determined by shifting the

diastolic intervals of the AC waveform to zero amplitude.)

Data Collection

Data were collected from six healthy adult beagles (10-12 kg) under an experimental
protocol approved by the MSU All-University Committee on Animal Use and Care. For
each dog, general anesthesia was induced by an intravenous injection of propofol (2.2-6.6

mg/kg) and maintained with an inhaled mixture of oxygen and isoﬂorane (1.5-2.5%). A

119

micromanometer-tipped catheter (Millar Instruments, Houston, TX) was placed in a
femoral artery for the PAP waveform for analysis. A similar catheter was inserted in the
opposite femoral artery or a carotid artery and positioned in the ascending aorta for the
reference AP waveform. A catheter was also placed in a cephalic vein for drug and
isotonic ﬂuid administration, and electrodes were positioned for standard ECG
measurements. In the ﬁfth dog, a bipolar electrode catheter (EP Technologies, Boston
Scientiﬁc, Sunnyvale, CA) was inserted into a jugular vein and advanced to the right
atrium for high rate pacing with an external pulse generator (Medtronic, Minneapolis,
MN). In the sixth dog, a quadrapolar ablation catheter (EP Technologies) was inserted
into a femoral vein and positioned to ablate the AV node and then apply bipolar electrical
stimulation to the His bundle as previously described [Sanders 2004] for low rate pacing
with the external pulse generator. (Additional instrumentation was also installed in the
ﬁfth and sixth dogs to address different speciﬁc aims.) Placement of all central catheters
was accomplished using single-plane lateral projection ﬂuoroscopic guidance. The
analog transducer outputs were interfaced to a personal computer (DataQ Instruments,
Akron, OH). The arterial pressure waveforms and ECG measurements were recorded at
a sampling rate of 1000 Hz during a baseline period and following infusions of
phenylephrine and nitroglycerin in the ﬁrst dog; dobutamine and esmolol in the second
dog; norepinephrine and xylazine in the third dog; saline and progressive hemorrhage in
the fourth dog; verapamil and high rate pacing in the ﬁfth dog; and vasopressin (prior to
AV node ablation) and low rate pacing in the sixth dog. Several infusion and pacing
rates were employed followed by recovery periods.

Data Analysis

120

 

The ATF technique was applied off-line to approximately 300 min of recorded PAP

waveforms re-sampled to 250 Hz, with Tdi measured for each dog as the time interval

between the onsets of upstroke of the AP and PAP waveforms during the initial beats of
the baseline period. The resulting derived AP waveforms were lowpass ﬁltered with a
cutoff frequency of 15 Hz as previously justiﬁed [Fetics 1999] and then quantitatively
evaluated against the (unﬁltered) reference AP waveforms in terms of the sample-to-
sample (total waveform, TW), beat-to-beat systolic pressure (SP), and beat-to-beat pulse
pressure (PP) root-mean-squared-error (RMSE) values. The unprocessed PAP
waveforms were likewise assessed with respect to the reference AP waveforms after
time-aligning the two waveforms to eliminate error due merely to the wave propagation
delay. For further comparison, AP waveforms were also derived by an autoregressive
exogenous input-based generalized transfer function (GTFARX) technique [Fetics 1999]
(which was shown to be the most accurate amongst three generalized transfer function
techniques) and the tube model-based partially adaptive transfer function (PATFTUBE)
technique [Sugimachi 2001, Westerhof 2007] (i.e., the transfer function shown in the top

panel of Figure (6.1) with Tdi measured for each dog as described above and the same
values for the Ai and Bi parameters for all dogs). More speciﬁcally, the two previous

transfer functions were established by averaging over set of transfer functions computed
from each 15 sec segment of the PAP and AP waveforms of one dog and then applied to
the PAP waveforms of the remaining dogs (see Discussion section). The resulting
derived AP waveforms were then similarly lowpass ﬁltered and evaluated. This
procedure was repeated for each dog in order to avoid any bias, and the results were

averaged.

121

 

6.3 Results

Table (6.1) illustrates the AP and heart rate (HR) levels during each of the
experimental conditions. Table (6.2) shows the TW, SP, and PP RMSE values of the PAP
waveforms after time-alignment and the AP waveforms derived by the new ATF
technique as well as the previous GTFARX and PATFTUBE techniques for each condition
and overall. Figure (6.2) provides visual examples of the measured AP and PAP
waveform segments and the corresponding derived AP waveform segments during the
dobutamine and nitroglycerin conditions.

The AP and HR levels (meaniSD) varied widely over the different experimental
conditions. Mean arterial pressure (MAP) ranged from 62i3 to 134i15 mmHg, SP, from
7818 to 162i22 mmHg, PP, from 183:1 to 552tll mmHg, and HR, from 73120 to 197i16
bpm.

In general, the PAP waveforms were signiﬁcantly different from the reference AP
waveforms, especially in terms of SP and PP. The overall TW, SP, and PP RMSE values
of the time-aligned waveforms were 8.6, 16.9, and 19.9 mmHg, respectively. The level
of discrepancy between the unprocessed PAP and reference AP waveforms likewise
varied over the different experimental conditions. The RMSE values were by far the
smallest during the nitroglycerin condition (average of 2.7 mmHg) and largest during the
norepinephrine condition (average of 26.4 mmHg).

Table (6.1) illustrates the AP and heart rate (HR) levels during each of the
experimental conditions. Table (6.2) shows the TW, SP, and PP RMSE values of the
PAP waveforms after time-alignment and the AP waveforms derived by the new ATF

technique as well as the previous GTFARx and PATFTUBE techniques for each condition

122

 

and overall. Figure (6.2) provides visual examples of the measured AP and PAP
waveform segments and the corresponding derived AP waveform segments during the

dobutamine and nitroglycerin conditions.

123

 

Table 6.1 Aortic pressures (AP) and heart rate (HR) levels. Values are expressed as
mean-LSD. MAP is mean arterial pressure; SP, systolic pressure; and PP, pulse pressure.

 

 

Condition [$an @3191 Mme] [£151.]
Baseline 88:20 104124 3419 12717
Phenylephrine 132:14 158:19 49:9 98:9
Nitroglycerin 62:3 79:3 27:1 96:1
Dobutamine 89:2 1 16:3 51 :2 172:8
Esmolol 71 :3 85:3 24:0 1 13:1
Norepinephrine 134:15 162:22 55:1 1 117:13
Xylazine 65:8 78:8 25:1 1 00:10
Saline 98:1 119:1 45:1 137:1
Hemorrhage 66:5 80:5 25:1 105:4
Verapamil 74:10 90:1 1 33:3 115:10
High Rate Pacing 75:3 85:2 18:1 197:16
Vasopressin 94:1 5 108:16 28:3 103:4
Low Rate Pacing 73:7 95:5 42:6 73:20

 

124

 

Table 6.2 Peripheral artery pressure (PAP) and derived AP waveform errors. ATF is
adaptive transfer ﬁmction; GTFARX, autoregressive exogenous input-based generalized
transfer function; PATFTUBE, tube model-based partially adaptive transfer function; TW,
total waveform (i.e., sample-to-sample); and RMSE, root-mean-squared-error.

 

Time Aligned PAP ATF GTFARX PATFTUBE
Condition TW SP PP TW SP PP TW SP PP TW SP PP

RMSE[mmHg] RMSE [mmHg] RMSE[mmHg] RMSE[mmHg]

 

Baseline 10.3 20.0 21.8 6.3 10.2 8.8 5.1 5.9 4.9 5.4 8.5 7.1

Phenrﬁeph" 10.6 19.8 22.6 3.3 4.8 4.4 5.4 6.5 6.1 7.9 4.7 4.2
Ni"°%'yce" 3.1 1.9 2.0 3.8 6.9 6.6 5.3 9.4 8.2 4.4 8.0 7.8
D°buéamin 7.7 12.5 16.7 5.2 3.0 3.0 8.2 12.8 15.9 6.6 5.3 7.7
Esmolol 5.0 11.0 11.3 2.0 0.7 1.4 2.8 2.2 3.2 2.4 3.6 2.3
””3323” 14.0 30.6 34.7 5.4 8.0 7.9 7.3 6.2 6.0 9.1 11.0 10.7

Xylazine 6.1 12.9 13.5 2.5 4.0 3.3 3.2 2.7 2.0 2.9 4.7 4.1

Saline 11.5 20.9 24.7 3.7 3.5 3.6 4.8 4.7 5.7 4.6 6.5 7.0

Hemorrhag
e

Verapamil 7.9 15.0 16.4 3.7 6.5 6.5 2.6 2.5 3.2 3.3 4.7 4.6

6.5 15.5 15.9 2.4 2.6 1.2 2.7 3.6 3.0 3.0 6.2 5.2

High Rate 7.6 13.3 21.2 4.0 2.7 4.3 2.6 2.6 4.7 3.8 6.0 6.5
Pacmg
V35°§ress' 6.9 13.7 16.6 3.7 3.4 5.5 3.6 3.0 2.4 2.7 4.0 4.6
L°WRate 6.9 15.4 16.7 4.0 2.7 4.9 3.0 2.3 2.1 2.5 3.8 3.8
Pacrng

 

Overall 8.6 16.9 19.9 4.1 5.3 5.3 4.8 6.2 6.9 5.1 6.5 6.5

 

125

 

Dobutamine
14o .-. PAP -. 140 ATP

103

 

  
     

   
  

 

 

 

a 5": "o z: “'0.

6 - 5 -
E 150 150.4 150.8 1’ 50 150.4 150.8
g, 140 GTFARX 140 PATFTUBE

 

 

 

 

 

   

 

 

   

 

 

6150 150.4 150.8 6150 150.4 150.8
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ATF
100
80
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m .
:g 1800 1800.64 1 801.28 1800 1800.64 1801 .28
E. 100
80 f
60
1800 1800.64 1801 .28 1 800 1800.64 1801 .28
Time [see]

Figure 6.2 Example segments of the reference AP (solid), measured PAP (dot), and
derived AP (dash) waveforms by the new ATF technique and previous autoregressive
exogenous input-based generalized transfer function (GTFARX) and tube model-based

partially adaptive transfer function (PATFTUBE) techniques during the dobutamine and
nitroglycerin conditions.

126

J— tilu r: “1'11
-

 

 

The AP and HR levels (mean:SD) varied widely over the different experimental
conditions. Mean arterial pressure (MAP) ranged from 62:3 to 134:1:15 mmHg, SP, from
78:8 to 162:22 mmHg, PP, from 18:1 to 55:11 mmHg, and HR, from 73:20 to 197:16
bpm.

In general, the PAP waveforms were signiﬁcantly different from the reference AP
waveforms, especially in terms of SP and PP. The overall TW, SP, and PP RMSE values
of the time-aligned waveforms were 8.6, 16.9, and 19.9 mmHg, respectively. The level
of discrepancy between the unprocessed PAP and reference AP waveforms likewise
varied over the different experimental conditions. The RMSE values were by far the
smallest during the nitroglycerin condition (average of 2.7 mmHg) and largest during the
norepinephrine condition (average of 26.4 mmHg).

All three techniques were able to derive the AP waveform with considerably greater
accuracy than merely time aligning the PAP waveform over all the experimental
conditions. Each of the techniques also afforded improved accuracy over the
unprocessed PAP waveforms for each condition, except for nitroglycerin in which the
measured PAP and AP waveforms were already in close agreement and dobutamine for
the GTFARX technique only. The ATF technique was the most accurate over all the
conditions. The overall TW, SP, and PP RMSE values of the AP waveforms derived by
the new technique were 4.1, 5.3, and 5.3 mmHg, respectively. The corresponding RMSE
values for the GTFARX and PATFTUBE techniques were similar to each other and, on
average, 5.0, 6.4, and 6.7 mmHg. The ATF technique achieved its most signiﬁcant
improvements in accuracy over the two previous techniques during the conditions of

dobutamine (average reduction in TW, SP, and PP RMSE values by 5.7 mmHg), saline

127

 

(2.0 mmHg), hemorrhage (1.9 mmHg), phenylephrine (1.6 mmHg), nitroglycerin (1.4
mmHg), esmolol (1.4 mmHg), and norepinephrine (1.3 mmHg). However, the GTFARX
and PATFTUBE techniques did derive the AP waveform with similarly greater accuracy

than the new technique during the conditions of baseline (average reduction in TW, SP,

and PP RMSE values by 2.3 mmHg) and verapamil (2.1 mmHg).

6.4 Discussion

Pressure waveforms simultaneously measured from the central aorta and a peripheral
artery show striking differences in both morphology and level (see, e.g., Figure (6.2)).
Most importantly from a clinical point of view, peripherally measured SP and PP are
generally larger than their centrally measured counterparts. This counter-intuitive
ampliﬁcation of the pressure waveform with increasing distance from the central aorta
arises from wave reﬂections in the arterial tree. It is therefore the pressure in the central
aorta that truly indicates cardiac afterload and myocardial perfusion. Perhaps, as a
consequence, previous studies have shown that centrally measured arterial pressure can
offer superior clinical information to more distally measured pressure [Safar 2009,
Wadell 2001]. Even so, the PAP waveform is much more commonly measured in
practice due to the relative ease and safety of its measurement.

In this study, we developed a technique to mathematically transform a PAP waveform
so as to obtain the AP waveform conveniently and safely. The technique was inspired by
the following investigators: l) Stergiopulos et a1. [Stergiopulos 1998] who proposed a
physical basis for the transfer ﬁmction relating PAP to AP through an arterial tube model;
2) Sugimachi et al. [Sugimachi, 2001] and Westerhof et al. [Westerhof 2007] who

employed the tube model to establish a partially adaptive transfer function for deriving

128

 

the AP waveform, and 3) Cohen who thought to compute the total aortic ﬂow waveform
from a PAP waveform by ﬁnding a black-box, rather than physical model-based, transfer
function that maps the PAP waveform to zero during diastole (personal communication in
2000 and now described in [Cohen 2008] as a result of this study). Our technique
similarly deﬁnes the transfer function relating PAP to AP through a tube model and then
estimates the unknown model parameters ﬁ'om the PAP waveform as well as a one-time
non-invasive measurement of the wave propagation delay time between the aorta and
peripheral artery measurement site by likewise exploiting the fact that aortic ﬂow is
negligible during diastole (see Figure (6.1)). In this way, in contrast to the conventional
generalized transfer ﬁmction and perhaps all other previous techniques for
mathematically deriving the AP waveform, the new technique is able to fully adapt to the
inter-subject and temporal variability of the arterial tree with only a single PAP
waveform available for analysis.

To demonstrate the feasibility of the new ATF technique, we measured a PAP
waveform from a femoral artery and the reference AP waveform from the ascending
aorta of six healthy beagles of similar size during twelve different hemodynamic
interventions (see Table (6.1)). Thus, unlike most previous related efforts, we were able
to assess the efﬁcacy of the technique over a diverse set of signiﬁcant perturbations of
known effect. On the other hand, our measurements did not allow us to investigate the
technique in the context of a diverse population of subjects.

Our results (see Table (6.2) and Figure (6.2)) showed that the average of the overall
TW, SP, and PP RMSE values of the AP waveforms derived by the ATP technique was

4.9 mmHg. This error effectively amounted to a 68% reduction in wave distortion in the

129

 

 

 

PAP waveforms. The corresponding RMSE values for the GTFARX and PATFTUBE
techniques were each about 22% higher than the new technique, despite the fact that both
of these previous techniques had the unfair advantage of being developed with or
“seeing” the entire reference AP waveform from one of the six homogenous dogs (i.e., a
baseline condition and two of the interventions). Note that, even under the most
favorable scenario for the GTFARX and PATFTUBE techniques in which the reference AP
waveforms from all six dogs were utilized for both development and testing, the RMSE
values for these techniques still remained larger than the ATF technique by about 7% on
average (results not shown). While the previous techniques would indeed be developed
in practice with data from many more subjects (e.g., hundreds), they would be applied to
an even greater number of subjects (e.g., all patients). In other words, the previous
techniques would be developed with data from a much smaller fraction of the subject
population in practice than the one-sixth proportion employed herein. We therefore
believe that the new technique will show even greater than 22% improvement in accuracy
over the previous techniques in actual practice.

Not surprisingly, the ATF technique was able to derive the AP waveform with
appreciably better accuracy than the GTFARX and PATFTUBE techniques speciﬁcally
during the conditions wherein the arterial tree was directly perturbed. The one exception
was the vasopressin condition, which resulted in a relatively small hemodynamic change
(see Table (6.1)). However, the new technique actually offered by far its greatest
improvement in accuracy over the two previous techniques during the dobutamine
condition. Evidently, intense vasodilation occurred during this positive cardiac inotropic

condition, as centrally measured PP and HR markedly increased relative to the baseline

130

 

 

 

condition without a change in MAP (see Table (6.1)). For the remaining conditions in
which the heart was primarin inﬂuenced, the three techniques showed similar overall
accuracy, with the ATF technique affording notably greater accuracy during the esmolol
condition but particularly less accuracy during the verapamil condition. The new
technique was also equally less accurate than the previous techniques during the baseline
condition. It is unclear why the ATF technique was not as effective during these two
conditions.

The parallel tube model upon which the ATF technique is based (see top panel of
Figure (6.1)) neglects arterial tapering and stiffening, as the dominant wave reﬂection
sites appear to be at the [arterial terminations due to their high resistance [McDonald
1974, Westerhof 1972]. The model also ignores the inertance of the distal arterial bed,
since it is well appreciated that inertial work is small compared to viscous work in the
high-pressure systemic arterial tree. Nevertheless, such modeling inaccuracies as well as
imperfect parameter estimation represent the sources of error of the technique. To
determine the relative contributions of the modeling and parameter estimation errors to
the derived AP waveform error, we ﬁrst found the “actual” parameter values using the
measured PAP and AP waveforms. We speciﬁcally established the actual parameter
values for each individual segment of analysis by ﬁnding the pressure—>pressure transfer
function (see middle panel of Figure (6.1)), which when applied to the PAP waveform
segment, best ﬁts the measured AP waveform segment in the least squares sense. We
then applied the transfer function with the actual parameter values to the PAP waveform
segment so as to derive the AP waveform segment. The TW RMSE value of the derived

AP waveforms therefore represents the lower bound for the ATF technique due only to its

131

 

 

modeling error. Table (6.3) shows the resulting TW RMSE values, along with the SP and
PP RMSE values (which do not represent lower bounds), for each condition and overall.
The overall TW RMSE value was 3.0 mmHg, which indicates that most (73%) of the
corresponding value for the ATF technique is due to modeling error. Finally, as a related
comment, note that the GTFARX and PATFTUBE techniques derived the AP waveform with
similar average overall RMSE values (see Table (6.2)) perhaps as a result of the
modeling error offsetting the partial individualization of the latter tube model-based
technique.

For this demonstration study, we obtained the single measurement of the wave
propagation delay time between the aorta and the peripheral artery measurement site

(Tdi) for each dog using a few beats of the measured AP and PAP waveforms during the
baseline period. In practice, Tdi may be measured non-invasively by, for example,

placing a handheld tonometer on the carotid artery and determining the time between the
onsets of upstroke of the waveform that it measures and the recorded PAP waveform
[Sugimachi 2001]. For continuous monitoring applications, such an operator-required

measurement would likely need to be made periodically (e.g., every few days).

Alternatively, Tdii may be continuously approximated through, for example, a

simultaneously recorded ECG or phonocardiogram.

It is claimed that the generalized transfer function is justiﬁed when applied to a PAP
waveform from the upper limb but not the lower limb [O’Rourke 2008]. Despite this
claim, we measured the PAP waveform from a femoral artery in this study due to
experimental convenience, the observation that this waveform appears signiﬁcantly

different from the AP waveform (see Figure (6.2) and Table (6.2)), and the fact that this

132

"15'3“?

 

peripheral artery is commonly cannulated in clinical practice. Our study may therefore
be amongst, if not, the ﬁrst to demonstrate the feasibility of mathematically transforming
a PAP waveform from a lower limb to the AP waveform. In principle, due to its ability
to adapt to arterial tree changes, our technique should be applicable to pressure
waveforms obtained from any peripheral artery including the readily accessible radial
artery (but excluding distal arteries in the coronary circulation for which arterial entry
ﬂow is signiﬁcant during diastole). However, while we believe that our technique will
also reveal improved accuracy over the generalized transfer function when applied to a
PAP waveform from the upper limb, we do acknowledge the possibility that the overall

difference in accuracy may not be as signiﬁcant.

133

 

__ -_ “I“!

 

 

Table 6.3 Derived AP waveform errors using actual tube model parameter values. TW
RMSE values represent lower bound for the ATF technique.

 

 

 

 

Condition TWRMSE [:r: H 1 PP
Baseline 4.1 5.8 3.8
Phenylephrine 2.9 3.0 2.2
Nitroglycerin 2.9 3.6 3.0
Dobutamine 4.8 2.6 3.5
Esmolol 1.6 2.1 0.9
Norepinephrine 3.7 2.6 2.2
Xylazine 1.9 2.9 1.4
Volume 3.5 1.5 1.3
Hemorrhage 2.1 3.6 1.7
Verapamil 2.7 2.2 1.3
High Rate Pacing 3.1 3.9 5.3
Vasopressin 1.5 1.4 1.5
Low Rate Pacing 1.8 1.2 2.1
Total 3.0 3.2 2.9

 

134

In summary, we have developed perhaps the ﬁrst fully adaptive technique for
mathematically deriving the AP waveform from one PAP waveform and have
demonstrated its feasibility in animals over a wide range of physiologic conditions. In
the ﬁrture, it would be worthwhile to explore reﬁnements to the parallel tube model upon
which the technique is based (e.g., the inclusion of additional parameters to more
accurately represent the terminal loads) as well as continuous approximation of the wave
propagation delay time from simultaneously recorded non-invasive measurements. In
addition, validation of the technique as applied to invasive and non-invasive PAP
waveforms from the upper and lower limbs of humans is a must. If such follow-up
studies prove successful, then the technique could be employed for more precise arterial
pressure monitoring and titration of therapy in inpatient settings wherein PAP catheters
are inserted and other clinical environments such as outpatient clinics and home in
conjunction with non-invasive PAP devices. Finally, subsequent combination of the
technique with an AP waveform analysis technique that we have also recently developed
[Swamy 2009] may ultimately permit continuous monitoring of cardiac output and left

ventricular ejection fraction in addition to AP from just a single PAP waveform.

135

 

 

CHAPTER 7

CONCLUSIONS

In this dissertation we have presented several techniques to estimate clinically
relevant hemodynamic parameters from routinely measured blood pressure waveforms.

In chapter 2 we introduced a technique to estimate the clinically more relevant
central aortic pressure waveform from multiple, less invasively measured peripheral
artery pressure waveforms distorted by wave reﬂections. The technique is based on
multi—channel blind system identiﬁcation in which two or more measured outputs
(peripheral artery pressure waveforms) of a single input, multi-output system (arterial
tree) are mathematically analyzed so as to reconstruct the common unobserved input
(central aortic pressure waveform) to within an arbitrary scale factor. Consequently, in
contrast to previous, related efforts, the technique does not utilize a generalized transfer
function or any training data and is therefore entirely patient and time speciﬁc. We have
evaluated the technique with respect to four swine in which peripheral artery pressure
waveforms from the femoral and radial arteries and a reference central aortic pressure
waveform were simultaneously measured during diverse hemodynamic interventions.
We report that the technique was able to reliably estimate the entire central aortic
pressure waveform with an overall root-mean—squared-error of 4.7 mmHg. This error
effectively represented a reduction in the total wave distortion in the measured peripheral

artery pressure waveforms by 45%. As a result, the technique was also able to provide

136

 

 

 

similar improvements in the estimation of systolic pressure, pulse pressure, and the
ejection interval.

In chapter 3 we proposed a reﬁnement to the MBSI technique wherein the transfer
function characterizing the path between the aorta and the peripheral arteries is
represented using damped sinusoidal functions. This representation effectively lowered
the number of estimation parameters and therefore helped in reducing the reconstruction
error. This improvement facilitated ﬁtting of the reconstructed waveform to lumped
element Windkessel models for continuous estimation of beat-to-beat cardiac output.
With further successful testing, the new technique may ultimately be employed for
automated and less invasive monitoring of central hemodynamics in places where
multiple catheter systems are employed.

In the next chapter we outlined a technique to estimate left ventricular ejection
fraction (EF) by model-based analysis of the aortic preSsure waveform. In this way, in
contrast to conventional imaging, EF may be continuously monitored without making any
assumptions about the left ventricular geometry. Moreover, with the proportional
parameter estimates, relative changes in left ventricular end-diastolic volume (EDV),
cardiac output (CO), total peripheral resistance, and maximum left ventricular elastance
(Emax) may also be monitored on a beat-to-beat basis. To evaluate the technique, we
measured aortic pressure waveforms, intermittent reference EF and EDV via standard
echocardiography, and other cardiovascular variables from six anesthetized dogs during
various pharmacological and volume interventions. Our results showed overall EF and
calibrated EDV root-mean-squared-errors of 5.6% and 4.1 ml, respectively, as well as

reliable estimation of relative changes in Em,x and beat-to-beat CO. These results

137

 

demonstrate, perhaps for the ﬁrst time, the feasibility of estimating EF from only blood
pressure. With further successful testing, the technique may potentially be employed for
continuous EF monitoring in subjects instrumented with an aortic catheter such as
animals in research and patients undergoing cardiac catheterization. Future work involves
extending the technique to estimate EF and other left ventricular parameters from
measurements of peripheral artery pressure waveforms.

In subsequent chapters we utilized a transmission line representation of the
arterial path between the aorta and the peripheral arteries to derive the wave reﬂection
characteristic of the reﬂection path. This model was then extended to estimate aortic
pressure waveform from a single peripheral artery pressure waveform based on the
knowledge that aortic input ﬂow is zero during diastole. To demonstrate the feasibility of
this adaptive transfer ﬁmction technique, we performed experiments in six healthy dogs
in which peripheral artery and aortic pressure waveforms were simultaneously recorded
during twelve different hemodynamic interventions. Our results showed that the aortic
pressure waveforms derived by the technique were in superior agreement to the reference
aortic pressure waveforms (average overall error of 4.9 mmHg) than the unprocessed
peripheral artery pressure waveforms (average overall error of 15.1 mmHg) and the aortic
pressure waveforms derived by two previously proposed transfer functions developed on
a subset of the same canine data (average overall error of 6.0 mmHg).

Future work involves validating these techniques on data from human subjects
and exploring more sophisticated models for arterial tree representations. With such

further successful testing the techniques may ultimately be employed in the clinical

138

 

 

 

settings for continuous monitoring of cardiovascular status and also in home and

ambulatory environments with the use of non-invasive measurement devices.

139

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