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Measurement uncertainty in pulmonary vascular input impedance and characteristic

impedance estimated from pulsed-wave Doppler ultrasound and pressure: clinical studies on

57 pediatric patients

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IOP PUBLISHING

PHYSIOLOGICAL MEASUREMENT

Physiol. Meas. 31 (2010) 729–748

doi:10.1088/0967-3334/31/6/001

Measurement uncertainty in pulmonary vascular

input impedance and characteristic impedance

estimated from pulsed-wave Doppler ultrasound and

pressure: clinical studies on 57 pediatric patients

Lian Tian1, Kendall S Hunter2,3, K Scott Kirby2, D Dunbar Ivy2

and Robin Shandas1,2,3,4

1Department of Mechanical Engineering, University of Colorado at Boulder, Boulder,

CO 80309, USA

2Department of Pediatric Cardiology, The Children’s Hospital, University of Colorado at Denver,

Aurora, CO 80045, USA

3Department of Bioengineering, University of Colorado at Denver, Aurora, CO 80045, USA

E-mail: Robin.Shandas@ucdenver.edu

Received 8 September 2009, accepted for publication 18 March 2010

Published 22 April 2010

Online at stacks.iop.org/PM/31/729

Abstract

Pulmonary vascular input impedance better characterizes right ventricular

(RV) afterload and disease outcomes in pulmonary hypertension compared

to the standard clinical diagnostic, pulmonary vascular resistance (PVR).

Early efforts to measure impedance were not routine, involving open-chest

measurement.Recently, the use of pulsed-wave (PW) Doppler-measured

velocity to non-invasively estimate instantaneous flow has made impedance

measurement more practical.One critical concern remains with clinical

use: the measurement uncertainty, especially since previous studies only

incorporated random error. This study utilized data from a large pediatric

patient population to comprehensively examine the systematic and random

error contributions to the total impedance uncertainty and determined the least

error prone methodology to compute impedance from among four different

methods.We found that the systematic error contributes greatly to the

total uncertainty and that one of the four methods had significantly smaller

propagated uncertainty; however, even when this best method is used, the

uncertainty can be large for input impedance at high harmonics and for the

characteristic impedance modulus. Finally, we found that uncertainty in

impedance between normotensive and hypertensive patient groups displays

no significant difference. It is concluded that clinical impedance measurement

would be most improved by advancements in instrumentation, and the best

computation method is proposed for future clinical use of the input impedance.

4Author to whom any correspondence should be addressed.

0967-3334/10/060729+20$30.00 © 2010 Institute of Physics and Engineering in MedicinePrinted in the UK 729

Page 3

730L Tian et al

Keywords:

characteristic impedance, pediatric patient

uncertainty, systematic error, random error, input impedance,

1. Introduction

Pulmonary arterial hypertension (PAH) is an important cause of morbidity and mortality in

children and adults. PAH is characterized by high blood pressure in the pulmonary circulation

that yields increases in right ventricular (RV) afterload, and is associated with arterial

remodeling and eventual failure of the RV. Current diagnosis of PAH is executed through

the measurement of pulmonary vascular resistance (PVR), which is the viscous hydraulic

opposition to the mean blood flow. However, PVR, a measure based on the assumption

of steady hemodynamics, can only provide limited information about overall pulmonary

vascular function due to its neglect of the pulsatile components of blood flow (Grant and

Lieber 1996, Weinberg et al 2004). Alternatively, pulmonary arterial (PA) input impedance

(Z), which represents the opposition to both the mean and pulsatile components of flow, has

been shown to be a much better measure of RV afterload and to better characterize pulmonary

vascular function (Milnor et al 1969, Milnor 1975, Grant and Lieber 1996, Weinberg et al

2004). Perhaps as a result, it also better predicts clinical outcomes (Hunter et al 2008).

The correct measurement of input impedance is thus highly important for accurate diagnosis

of PAH. Magnetic resonance imaging (MRI) has been used to evaluate the blood flow very

accurately through ensemble imaging. However, this technique does not provide beat-to-beat

measurements of flow, and thus cannot assess the biological variability. Additionally, MRI

is very expensive and time consuming, and thus is not routinely used in most hospitals or

research centers in the USA. Recently, our group has developed a new method to measure

impedance using pulsed-wave (PW) Doppler-measured instantaneous velocity and pressure

measurements; flow was derived from the PW measurement (Weinberg et al 2004). This new

method is relatively simple and easy to implement especially for pediatric patients compared

to other techniques, and has shown promise for future clinical application in evaluating the

pulmonary vascular function from the standpoint of standard of care in pediatrics.

Impedance has shown promise for routine use in clinical settings. The impedance at

the zero-frequency harmonic (Z0) correlates well with the distal vascular resistance (PVR).

Some have postulated that the first harmonic (Z1, Weinberg et al 2004) or the sum of the

first two harmonics (Z1+ Z2, Hunter et al 2008) is representative of pulmonary vascular

stiffness. Notably, this impedance sum has been shown to better predict pediatric patient

outcomes in PAH (Hunter et al 2008). On the systemic side, the increase of systemic input

impedance modulus and the shift of the first minimum impedance modulus and phase to

higher harmonic have been used to represent the stiffening of ascending aorta (Milnor 1975).

The wave reflection, which is estimated as the difference between maximum and minimum

input impedance, has been associated with left heart failure (Pepine et al 1978). Others have

proposed the use of the characteristic impedance, which provides an indirect measurement

of vascular compliance and has been used to evaluate the arterial disease of the vascular

bed (McDonald 1974, Nichols et al 1977, Lucas et al 1988, Finkelstein et al 1988). The

characteristic impedance can only be estimated through the measured input impedance due to

the presence of reflected waves (Bergel and Milnor 1965, McDonald 1974, Grant and Lieber

1996). In a word, the ventricular afterload is better defined by the impedance spectrum rather

than pressure or ventricular wall stress (Patel et al 1963, Milnor 1975, Nichols et al 1980).

Page 4

Uncertainty in impedance 731

Clearly the uncertainty associated with each of the pulmonary vascular uses noted above

is important to the clinical application of this routine PW and pressure measurement. There

are systematic errors (or biases) associated with the measuring instruments, analytical random

error introduced by the chance fluctuations in the environment or other factors from one

measurement to the next, and biological random error due to the individual’s variation around

their homeostatic state; these errors will be propagated to the input impedance. As a result,

the measured input impedance may have very large uncertainty, especially at high harmonics.

Precisely quantifying the uncertainty in input impedance can provide confidence and guidance

for its application to clinical medicine. In our previous work (Weinberg et al 2004, Hunter

et al 2008), only the random error was considered in calculating the uncertainty in input

impedance. To the best of our knowledge, no work has been published that comprehensively

studiestheuncertaintyintheinputimpedanceasmeasuredbythecombinationofPWDoppler-

approximated instantaneous flow and invasively measured pressure. Therefore, the goals of

thisstudywere(1)todevelopformulationsforthecalculationofuncertaintyininputimpedance

incorporating both systematic and random errors, (2) to investigate the uncertainty in input

impedance and determine up to which harmonic of impedance that can be used properly for

clinical application and (3) to determine if the characteristic impedance, which is calculated

from the higher harmonics of impedance, may be accurately used from these measurements.

2. Methods

We first studied the systematic uncertainty in the pressure and flow spectra moduli by

comparing each systematic uncertainty to its respective spectrum modulus for each harmonic

up to the tenth. The contribution of the systematic error to the total uncertainty in input

impedance was then investigated to see if the systematic error was negligible. Thirdly, the

percent total uncertainties in input impedance calculated from four different methods were

comparedtodeterminethemethodwiththelowestpercenttotaluncertainty. Finally,systematic

uncertainty in the pressure and flow spectra moduli and total uncertainty in input impedance

werecomparedbetweennormotensiveandhypertensive groupstoexplorehowtheuncertainty

was affected by patient condition.

2.1. Clinical data acquisition

After institutional review board approval and informed consent and assent had been obtained,

clinical data were obtained during routine cardiac catheterization of patients as part of their

regularevaluationandtreatmentattheChildren’shospitalinDenver, CO.Atotalof57patients

(medianage6.46years,range0.33–21years,25females)wereconsidered,ofwhom13patients

were with normal mean PA pressures (median age 5.69 years, range 0.92–16 years, 9 females)

and 44 patients with PAH (median age 6.68 years, range 0.33–21 years, 16 females). The

data used for calculation in this paper were obtained at a room air (baseline) condition for all

patients, i.e. no vasoreactivity data were used.

ThepressurewithinthemiddlesectionofmainPAwasmeasuredwithstandardfluid-filled

catheters (Transpac IV, Abbott Critical Care Systems, Abbott Park, IL, USA). PW Doppler

velocity at the midline of the middle section of main PA was measured with a commercial

ultrasound scanner (Vivid 5, GE Medical Systems Inc., Waukesha, WI, USA) from a

parasternalshort-axisviewasdescribedpreviously(Weinbergetal2004). FourdifferentVivid

5systemswereused,dependingonclinicalavailability;allhadnearlyidenticaldataacquisition

characteristics. By performing anatomic assessment of the MPA as well as color Doppler flow

imaging prior to PW Doppler imaging, the angle between flow in the MPA and the ultrasound

Page 5

732 L Tian et al

beamline (i.e. the Doppler angle) was minimized. For the purpose of error estimation, we

assume that the Doppler angle was always less than 5◦for all clinical measurements; this

results in less than 0.4% error in the Doppler signal, which was not considered in our study.

Cardiac output (CO) at room air condition was measured three times for each patient by

Fick’s method with measured oxygen consumption in cases where intracardiac shunts were in

place and by thermodilution otherwise (Calysto IV, Witt Biomedical, FL), and the mean CO

was used for calculations. We recall that the instantaneous flow rate Q(t) was calculated by

multiplying the Doppler-measured velocity V(t) by a constant area correction factor:

Q(t) = AcorrV(t),

where the constant Acorris obtained from

Acorr= CO/¯V,

inwhich¯V isthetime-averagedPWDopplervelocity. Notethattheabovecalculationassumes

that the instantaneous cross-section area is constant over time and the spatial profile of the

velocity is nearly constant across the vessel (Womersley 1955), which together have been

shown previously to be acceptable approximations (Weinberg et al 2004). The recorded

pressure and calculated flow time histories were then separated and collected into individual

cardiac cycle-based electrocardiographic (ECG) gating; more detail may be found elsewhere

(Hunter et al 2008).

(1)

(2)

2.2. Basics of error propagation

Consider a general data reduction equation:

y = y(x1,x2,...,xL),

with associated standard random errors

?δS

standard total error δyand uncertainty uyin y under the first-order Taylor series approximation

are (ANSI/ASME 1997, Coleman and Steele 1999)

?

j=1

uy= tv,95δy,

where the superscripts R and S refer to random and systematic errors, respectively, δjk= (δj)2

for j = k and is the covariance of the errors in xj and xk for j ?= k, tv,95is the t value

from Student’s t distribution with v degrees of freedom for a 95% confidence level and v is

the number of degrees of freedom associated with δyand is calculated from the generalized

Welch–Satterthwaite formula that can accommodate the correlated components of uncertainty

(Willink 2007).

Note that if the standard systematic error for all the N measured results of xjis the same

as δS

However, if the standard systematic error for each xn

n, the standard systematic error in xjis then estimated as (Coleman and Steele 1999)

?

n=1

with (N−1) degrees of freedom.

(3)

?δR

1,δR

2,...,δR

L

?

and standard systematic errors

1,δS

and?vS

2,...,δS

1,vS

L

?for each variable, and the associated degrees of freedom?vR

1,vR

2,...,vR

L

?

2,...,vS

L

?. Assuming there are no random error/systematic error correlations, the

δy=

??δR

y

?2+?δS

y

?2=

?

?

?

L

?

L

?

k=1

∂y

∂xj

∂y

∂xkδR

jk+

L

?

j=1

L

?

k=1

∂y

∂xj

∂y

∂xkδS

jk,

(4)

(5)

j, the standard systematic error in xjis δS

jwith infinite degrees of freedom (Dieck 1997).

jis?δS

j

?

nwhich is different for different

δS

j=

?

?

?

N

?

?δS

j

?2

n

?

(N − 1),

(6)

Page 6

Uncertainty in impedance733

We define the contribution of the systematic error to the total uncertainty as

?δS

yy

If we have another function, z = z(x1,x2,...,xL), then the covariance between y and z

on the first-order approximation is (Freund and Walpole 1987)

PS=

y

?2

?δR

?2+?δS

?2.

(7)

δyz=

L

?

j=1

L

?

k=1

∂y

∂xj

∂z

∂xkδR

jk+

L

?

j=1

L

?

k=1

∂y

∂xj

∂z

∂xkδS

jk.

(8)

2.3. Systematic uncertainty in pressure and flow spectra moduli and phases

For N time-domain measurements of pressure, P(n), the spectrum for this quantity waveform

is obtained by applying the discrete Fourier transform as

N−1

?

where the overhat indicates a Fourier-transformed quantity and k is an integer and denotes the

zero-frequency harmonic of the spectrum when k = 0 and the kth harmonic of the spectrum

when 1 ? k ? (N − 1)/2.

The systematic error in pressure arises from the offset, accuracy of the transducer

sensitivity, digital-to-analog conversion, etc. The offset of the transducer and the error in

the transducer sensitivity are both small compared to the resolution of the digital-to-analog

conversion performed by the ultrasound system (Vivid 5). Therefore, the error considered in

this study is due to digital-to-analog conversion and thus depends on the type of the digital-

to-analog converter used in the ultrasound system. The standard systematic error in pressure

in the time domain is thus constant (denoted as δS

same patient and there is no correlation between any two pressure data points. As a result,

the standard systematic errors in the real part,ˆPR(k), the imaginary part,ˆPI(k), the modulus,

|ˆP(k)| and the phase, φP(k), ofˆP(k) for a cardiac cycle with N time-domain data points are

calculated by using the theory in section 2.2 as

?

P

k ?= 0,

ˆP(k) =ˆPR(k) + iˆPI(k) =

1

N

n=0

exp

?

−i2π

Nnk

?

P(n),k = 0,1,...,N − 1,

(9)

P) for all the pressure data points in the

δS

ˆPR(k)=

δS

δS

P

?√N,k = 0

?√2N,

δS

P

δS

P

δS

ˆPI(k)=

?

0,k = 0

k ?= 0,

δS

P

?√2N,

δS

ˆPR(k)ˆPI(k)= 0,

(10)

δS

|ˆP(k)|=

?

?√N,k = 0

k ?= 0,

?√2N,

δS

φP(k)=

?

0,k = 0

k ?= 0.

δS

P

?(|ˆP(k)|√2N),

(11)

For velocity, the standard systematic error is constant (denoted as δS

datapointsinthesamepatientandthereisnocorrelationbetweenanytwovelocitydatapoints.

The systematic error in velocity considered here is a function of the pulse repetition frequency

(PRF), which is changed for each subject to maximize velocity range. Thus, the error can be

different for different patients. However, each flow data point is calculated from equations (1)

and (2). The standard systematic error of a flow is calculated by using the theory in section 2.2

as

?

?A2

V) for all the velocity

δS

Q(n)=

?

?

?

corr

?δS

V

?2+ V2(n)?δS

Acorr

?2−

⎡

⎣2AcorrV(n) · CO ·?δS

V

?2?⎛

⎝ ¯V2

M

?

j=1

Nj

⎞

⎠

⎤

⎦,

(12)

Page 7

734L Tian et al

where δS

correction factor, Acorr, δS

from the cardiac output instrumentation manufacturer (Calysto IV, Witt Biomedical, FL), Njis

the number of time-domain data points in the jth cardiac cycle and M is the number of cardiac

cycles. The covariance of the systematic error in Q(n1) and Q(n2) based on equation (8) is

⎧

Acorr=

?

1

¯V2

?δS

CO

?2+?CO2

¯V2

?δS

V

?2??M

j=1Nj

?is the standard systematic error in the area

COis the standard systematic error in cardiac output and is obtained

δS

Q(n1)Q(n2)= V(n1)V(n2)?δS

As a result, the standard systematic errors associated with flow spectrum quantities can not be

reduced to simple forms as pressure as shown in equations (10) and (11).

With the standard systematic errors for pressure and flow spectra moduli, we defined the

mean percent systematic uncertainty in the spectrum modulus at the kth harmonic for a patient

as

Acorr

?2−

⎨

⎩[V(n1) + V(n2)]A3

corr

?δS

V

?2?⎛

⎝CO ·

M

?

j=1

Nj

⎞

⎠

⎫

⎬

⎭. (13)

1

M

M

?

j=1

2δ|fj(k)|

|fj(k)|,

(14)

where |fj(k)| denotes the modulus of pressure or flow spectrum at the kth harmonic and the

number 2 in the above equation is to transfer the standard systematic error which is for a 68%

confidence level to the systematic uncertainty for a 95% confidence level.

2.4. Input impedance and the uncertainty

The input impedance is defined as

Z(k) =

ˆP(k)

ˆQ(k).

(15)

Given M cardiac cycles of pressure and flow data, we determined four different methods to

calculatetheinputimpedancemodulus|Z(k)|andphaseφ(k)andtheassociateduncertainties.

Since the random analytical error and random biological error cannot be separated, these two

errors are combined together and denoted as the random error. Itis noted that the random error

in cardiac output was not considered since the three measured values of cardiac output are

quite close. The four methods of calculation and the corresponding data reduction equations

f(x,y,...) are as follows.

Method 1. Average of the absolute values or arguments of the complex ratio:

|Z1(k)| =

1

M

M

?

?

j=1

?????

ˆPj(k)

ˆQj(k)

?????= f(x1) = x1,

ˆQj(k)

x1=

?????

ˆP(k)

ˆQ(k)

?????,

(16)

φ1(k) =

1

M

M

j=1

Arg

?ˆPj(k)

?

= f(x2) = x2,x2= Arg

?ˆP(k)

ˆQ(k)

?

.

(17)

In the notation, Arg(x) denotes the principal argument of the complex number, x, and is

between −π and π.

Page 8

Uncertainty in impedance 735

Method 2. Absolute value or argument of the average of the complex ratio:

??????

x = Re

ˆQ(k)

⎛

M

j=1

|Z2(k)| =

1

M

M

?

?

j=1

ˆPj(k)

ˆQj(k)

??????

= f(x,y) =

?ˆP(k)

⎞

?

?

x2+ y2,

?ˆP(k)

,y = Im

ˆQ(k)

,

(18)

φ2(k) = Arg

⎝1

M

?

ˆPj(k)

ˆQj(k)

⎠= f(x,y) = Arctan

?y

x

?

,

(19)

where Arctan(y/x) is the inverse tangent of y/x and is restricted to the range of [−π,π].

Re(x) and Im(x) denote the real and imaginary parts of x, respectively. Note that there is no

covariance of random errors in x and y, but the covariance of systematic errors in x and y is

considered.

Method 3. Ratio of the averages of the absolute value or difference of the average phases:

?⎛

j=1

x1= |ˆP(k)|,

|Z3(k)| =

1

M

M

?

j=1

|ˆPj(k)|

y1= |ˆQ(k)|,

⎝1

M

M

?

|ˆQj(k)|

⎞

⎠= f(x1,y1) =x1

y1,

(20)

φ3(k) =

x2= Arg(ˆP(k)),y2= Arg(ˆQ(k)),

where C is a constant (0, 2π, or −2π) which assures the calculated phase angle φ3(k) to be

within the principal argument range of [−π, π]. Note that there is no covariance of random

or systematic errors in x1and y1or in x2and y2for x1and y1or x2and y2are independent.

1

M

M

?

j=1

Arg(ˆPj(k)) −

1

M

M

?

j=1

Arg(ˆQj(k)) + C = f(x2,y2) = x2− y2+ C

(21)

Method 4. Ratio of the absolute values of the complex average or phase difference of the

complex averages:

??????

φ4(k) = Arg

M

j=1

?y1

Note that there is no covariance of random errors in any two variables of x1, y1, x2and y2, but

the covariance of the systematic error in x1and y1or in x2and y2is calculated.

Mathematically, the above four methods apply the non-associative operators of average,

absolute or argument, and ratio (division) to obtain modulus or phase. As a result of the

operator’s non-associativity, the propagation of error is different for each method. The above

|Z4(k)| =

x1= Re(ˆP(k)),

1

M

M

?

⎛

j=1

ˆPj(k)

??????

???????

1

M

M

?

j=1

ˆQj(k)

??????

?

− Arctan

= f(x1,y1,x2,y2) =

x2= Re(ˆQ(k)),

M

ˆQj(k)

⎠+ C

?

y2= Im(ˆQ(k)),

x2

x2

1+ y2

2+ y2

1

2

y1= Im(ˆP(k)),

⎞

(22)

⎝1

M

?

ˆPj(k)

⎠− Arg

⎛

⎝1

M

?

j=1

⎞

= f(x1,y1,x2,y2) = Arctan

x1

?y2

x2

?

+ C.

(23)

Page 9

736 L Tian et al

four methods represent all possible ways to calculate the impedance given the definition of

impedance by equation (15). For the first two methods, the complex impedance for each

cycle is first calculated. Then, in method 1, the impedance modulus and phase of each cycle

are calculated before the average values are obtained. In contrast, first the average complex

impedance is obtained for all the cycles in method 2 and then the impedance modulus and

phase are calculated. In methods 3 and 4, the average spectra moduli and phases for pressure

and flow are calculated separately before the impedance modulus and phase are obtained. In

method 3, the pressure and flow spectra moduli and phases are calculated before the average

spectra moduli and phases are obtained, while in method 4, the average pressure and flow

spectra are obtained for all the cycles before the pressure and flow spectra moduli and phase

are calculated.

We define the percent total uncertainty in impedance modulus as the total uncertainty

divided by the modulus, as shown previously. However, use of this definition with phase

will lead to a huge percentage because the phase can be very small. Thus, the percent total

uncertainty in impedance phase is defined as the total uncertainty divided by 2π.

2.5. Characteristic impedance

Many methods exist in the literature to estimate the characteristic impedance (Bergel

and Milnor 1965, McDonald 1974, Finkelstein et al 1988, Weinberg et al 2004, Segers

et al 2007) from experimental measurements.

method has not been established.The characteristic impedance is estimated here by

averaging the impedance moduli from the first minimum up to the eighth harmonic,

|ZC| = (|Z1st minimum| + ··· + |Z8|)/N, where N is the number of impedance moduli in

the calculation. The uncertainty in characteristic impedance is calculated through the error

propagation for each method.

However, the superiority of any single

2.6. Statistical analysis

All the data are presented as mean ± SD unless specified otherwise. The one-way ANOVA

test was used to compare the mean contributions of systematic uncertainty to the total

uncertainty in impedance from the four methods, and to compare the mean percent total

uncertainty in impedance from the four methods. The paired t-test was performed to identify

differences of the means of the percent total uncertainty in impedance between any two

methods. The two-sample t-test and equivalence test were used to compare the uncertainties

between normotensive and hypertensive groups. The confidence level was set at 95% for all

tests. Finally, we chose 20% error as our maximum acceptable error, and showed this 20%

demarcation as a line in many results.

3. Results

3.1. Pooled data

The bias for pressure, 2δS

has an average of 0.385 ± 0.009 mmHg for all the patients. The bias for velocity, 2δS

patient-to-patient due to changes in the velocity range and pulse repetition frequency settings

during acquisition of the spectral image and has an average of 1.13 ± 0.26 cm s−1for all the

patients. The bias for cardiac output, 2δS

Ingeneral,themodulusofthepressureorflowspectrumdecreasesquicklyastheharmonic

number increases for the first several harmonics and exhibits small oscillations within the

P, varies modestly due to the four different ultrasound systems and

V, varies

CO, is 10% of the cardiac output (CO).

Page 10

Uncertainty in impedance737

higher harmonic region (>fourth harmonic). As a result, the modulus can be smaller than the

systematic uncertainty at a certain harmonic, but larger than the systematic uncertainty at the

next higher harmonic. Presumably due to the biological variation and random process of

the measurement, it is also possible that moduli at a harmonic from one or several measured

cardiac cycles for a patient are smaller than the associated systematic uncertainty, but the

moduli computed for other cycles are larger than the systematic uncertainty at that harmonic.

The percent systematic uncertainties in pressure and flow spectra moduli are shown in

figure 1 plotted using the boxplot function in Matlab (similar plots for following figures unless

specified otherwise). For each box, the central mark is the median, the edges of the box are the

25th (q1) and 75th percentiles (q3). The + symbols represent the outliers that are outside the

whisker range [q1− 1.5(q3− q1)

pressure (figure 1(a)), the averages are 0.13% ± 0.06%, 0.75% ± 0.48%, 1.7% ± 1.1%, 5.7%

±4.9% and 5.6% ± 4.6% for the zero-frequency, first, second, third and fourth harmonics,

respectively. The error clearly increases from the fifth to the eighth harmonic; the median of

these errors is still below or at 20%, but select patients have errors over 20% or even 50%. For

the ninth and the tenth harmonics, the error medians are over 30% and there are many data

points that exceed 100%, with several outliers over 200%. For flow (figure 1(b)), the average

are 10.0% ± 0.0028%, 10.0% ± 0.0073%, 10.0% ± 0.035%, 10.6% ± 1.7% and 10.6% ±

0.78% for the zero-frequency, first, second, third and fourth harmonics, respectively. From

the fifth to the tenth harmonic, all the medians are smaller than 20% but with more and more

outliers over 20% as the harmonic number increases. There are only two outliers greater than

50% at each of the ninth and tenth harmonics.

The average contribution of the systematic error to the total uncertainty in impedance

modulus and phase for the grouped data as computed by the four methods is presented in

figure 2. The systematic error of the modulus (figure 2(a)) provides average contributions

of at least 70% to the total uncertainty in the zero-frequency, first and second harmonics,

respectively, for all four methods, and contributes about 40% (methods 3 and 4) to 50%

(methods 1 and 2) to the total uncertainty for harmonics up to the tenth. For the phase

(figure 2(b)), the average contribution of the systematic error increases from about 8% to

about 45–60% as harmonic increase from the first to the tenth. The mean contributions of the

systematic error show no significant differences between the four methods to compute moduli

at the zero-frequency, first and second harmonics (P > 0.4, ANOVA), but are significantly

different from the third to the tenth harmonic (P < 0.04, ANOVA) and for phase for all the

harmonics up to the tenth (P < 0.003, ANOVA).

Representative plots of input impedance and its associated total uncertainty calculated

with the four methods are shown in figure 3. In general, the impedance moduli and phases

obtained from the four methods are very similar and have very small uncertainty up to the

fourthharmonic, butdifferencesbetweeneachmethodbegintoemergeandoveralluncertainty

increases in the higher harmonics.

Thegroupaveragepercenttotaluncertaintyininputimpedanceforfourmethodsareshown

in figure 4. The average percentages increase with the harmonic number for both impedance

modulus and phase for all four methods. For modulus (figure 4(a)), the mean percentages for

the four methods are very close up to the third harmonic and show no significant difference

(P > 0.2, ANOVA). Method 3 has the smallest percent total uncertainty at harmonics higher

than the third, and is significantly different from the other three (P < 0.031, paired t-test)

except for the mean percentages between method 3 and method 1 at the ninth harmonic (P =

0.197, paired t-test) and the mean percentages between method 3 and method 4 at the seventh

harmonic (P = 0.112, paired t-test). For phase (figure 4(b)), the mean percentages for the four

methodsareverycloseuptothefourthharmonicandshownosignificantdifference(P>0.08,

q3+ 1.5(q3− q1)] (Tukey 1977, McGill et al 1978). For

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Figure 1. The percent systematic uncertainty in (a) pressure and (b) flow spectra moduli.

ANOVA). Methods 1 and 3 give very close mean percentages although they are significantly

different (P < 0.05, paired t-test). Methods 2 and 3 have significant differences in the mean

percentages from the fifth to the tenth harmonic (P < 0.02, paired t-test). Methods 3 and 4

show no significant difference in the mean percentages from the fifth to the tenth harmonic

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Figure2. Thepercentagecontributionofthesystematicerrortothetotaluncertaintyinimpedance:

(a) modulus and (b) phase as computed by the four methods. Bars represent the sample standard

deviation.

(P > 0.07, except for P = 0.0053 at the eighth harmonics, paired t-test), but the means and

the variations of percentages of method 3 are always smaller than those of method 4.

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Figure 3. Representative plots of input impedance: (a) modulus and (b) phase and the associated

total uncertainties calculated from the four methods. Bars represent the total uncertainty.

Thepercenttotaluncertaintyincharacteristicimpedancecalculatedfromthefourmethods

is shown in figure 5. The mean percentages of the four methods display significant differences

(P<0.05,ANOVA).Method3hasthesmallestvariationofthepercentages(25%±6.7%)and

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Figure 4. Percent total uncertainty in input impedance: (a) modulus and (b) phase for the four

methods. Bars represent the sample standard deviation.

has much lower median and mean percentages than the three other methods. The difference

in the percentage between method 3 and the other three methods is significant (P < 0.015,

except for P = 0.061 between method 3 and method 1, paired t-test).

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Figure 5. Percent total uncertainty in characteristic impedance modulus calculated from the four

methods.

3.2. Comparison of normotensive and hypertensive groups

To study the difference of the uncertainty between normotensive and hypertensive groups, we

selected 10 patients from the normotensive group with mean PA pressures less than 20 mmHg

and 26 patients from the hypertensive group with mean PA pressures over 27 mmHg and

comparedtheirimpedance errors. AllotherpatientshadmeanPApressuresof24or25mmHg

and were excluded from the study to better clarify the effects of a more severe disease state

on the uncertainty.

The percent systematic uncertainty in pressure spectrum modulus is shown in figure 6.

The percentage differences between normotensive and hypertensive groups are significant

for the zero-frequency, first, second, seventh, eighth and ninth harmonics (P < 0.025, t-

test). No significant differences are seen for other harmonics, but both the mean and median

percentages in the hypertensive group are smaller than those in the normotensive group. The

percent systematic uncertainty in flow spectrum modulus is also studied (not shown) and there

are no significant differences between the two groups for all the harmonics up to the tenth

(P > 0.06, t-test, except for the zero-frequency and the first harmonics). However, these mean

percentages at the zero-frequency and the first harmonics are nearly identical to 10.0% due to

the fixed systematic error in cardiac output with a negligible random error.

Theaveragecontributionofthesystematicerrortothetotaluncertaintyininputimpedance

for the two groups calculated from method 3 is shown in figure 7. The mean contributions

showasignificantdifferenceformoduliatthefourth,seventhtotenthharmonicsandforphases

at first to fourth, sixth, ninth and tenth harmonics (P < 0.05, t-test). Though no significant

differences were seen in the moduli and phases for other harmonics, the mean contributions

in the normotensive group are larger than those in the hypertensive group.