FREQUENCY ANALYSIS OF CATHETER SYSTEMS
USED FOR INVASIVE BLOOD PRESSURE MONITORING
by
Daniel Michael Chernoff
//
SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREES OF
BACHELOR OF SCIENCE
and
MASTER OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June, 1982
O
Daniel Michael Chernoff
1982
The author hereby grants to MIT permission to reproduce and to
distribute copies of this thesis document in whole or in part.
.....
........ ofi...................
Department of Electrical Engineering and
Computer Science, June, 1982.
Signature of Author.
C
....
M
... .,
......T
.....
Supervisor-Ac
Certified by.......~ge
Roger G. Mark, M.D., Ph.D. Thesis Supervisor-Academic
Certified by......avid
David
..*
.........
..................
...
EI11js, Thesis Supervisor (Cooperating Company)
-/
Archives
OF TECHNOLOGY
Acceptedur
..
"
m
-
a.............
- -I ......
'
" "....,"."
C*T"
*'1"92
Arthur C. Smith, Chairman
Departmental Committee on Graduate Stu ents
S1DSIRARlIFB
1U
ACKNOWLEDGEMENT
I am grateful for the guidance, support and surplus of
enthusiasm which Mr. David Ellis supplied during the research
and writing of this thesis.
truly invaluable.
His help and criticisms were
Dr. Roger Mark provided perspective on the
problem and aided in directing the focus of the research.
I wish to thank the many people at Hewlett-Packard's Waltham
Division who supplied ideas and moral support throughout this
project.
TABLE OF CONTENTS
Page
ABSTRACT..
. . . .
.•·.
·
·
iii
LIST OF TABLES. . .
. .. . .. . . . . . .
LIST OF FIGURES .
. . .. .. . . . . . . vii
.
vi
CHAPTER
a . . a . a a a . . . . .
INTRODUCTION
1.1
1.2
1.3
1.3.1
1.3.2
1.4
1.4.1
1.4.2
1.5
1.5.1
1.5.2
Brief History of Invasive Monitoring .0.
Requirements for Accurate
Waveform Reproduction. . . . . . . .
The Fluid-Filled Catheter.. . . . .
Models . . . . . . . . . . . . . .
Distortion . . . . . . . . . . . .
Frequency Response Measurement
Techniques . . . . . . . . . . . . .
Direct techniques.. . . . . . . .
Indirect techniques.. . . . . . .
Compensation . . . . . . . . . . . .
Mechanical compensation.. . . . .
Electrical compensation.. . . . .
MODELING
2.1
OF
THE
CATHETER
----
~ ~
SYSTEM.
.
.
.
1
2
.
17
. .
17
General Model - Mechanical/Electrical
Analogies.. . . . . . . . . .
Theoretical Calculation of
Line Constants . . . . . . . .
2.2.1
Longitudinal impedance . . .
2.2.2
Transverse impedance . . . .
2.3
Transmission Line Formulation.
2.3.1
Telegraph equations and
propagation constant . . . .
. . .
2.2
2.3.2
Characteristic impedance
.
.
.
.
.
.
.
.
.
.
.
.
.
. . .
. ..
2.3.3
Boundary conditions and
reflection coefficient . . . . . .
2.3.2
Natural frequencies.. . . . . . .
2.4
Lumped Model Approximation . . . . .
2.5
Effect of Trapped Air Bubbles. . . .
2.5.1
Compliance of air bubbles..
. ....
2.5.2
Relationship between bubble location
and resonant frequency ... ......
.
.
38
CHAPTER
Page
. . . . . . . . . .
3
MATERIALS .
3.1
3.2
3.3
3.5
3.6
3.7
Extension Tubing . .......
·
Transducers. . .......... ·
Flush Bag and Fluid. . ......·
Slow/Fast Flush Unit . .....
·
Bench Equipment. . ........ ·
Flow Source. ... ........
·
Tap Generator. .........
. ·
4
METHODS..AND RESULTS..
3.4
. . . . .
·
·
·
·
·
·
·
. .. .
41
·
·
·
·
·
·
·
42
42
43
44
44
45
46
. .
4.1
Determination of Line Parameters .
4.1.1
Resistance . . . . . . . . . . .
4.1.2
Inertance. . . . . . . . . . . .
4.1.3
Compliance . . . . . . . . . . .
4.2
Determination of Resonant Frequency
for the Bubble-free System . . . .
4.2.1
Experimental . . . . . . . . . .
4.2.2
Theoretical . ..
.........
4.3
Air Bubble Experiments . . . . . .
4.3.1
Resonant frequency as a function
of bubble location . . . . . . .
4.4
Tap and Flush Experiment . . . . .
4.4.1
Experiment . . . . . . . . . . .
·
·
·
·
.
·
·
·
·
·
·
·
·
.
.
.
.
. .
.
53
·
·
·
·
59
59
62
65
·
·
·
·
·
·
65
70
71
. . . . .
Tap and Flush Responses..
5.1
Initial conditions . . . . . . . .
5.1.1
Transient solution . . . . . . . .
5.1.2
5.2
Extraction of Resonant Frequency and
Damping from Fast-Flush. . . . . . .
5.3
Anticipated Usage and Clinical
Acceptability..
. . . . . . . . . .
.
49
50
51
·
·
·
·
DISCUSSION . . . . . . . . . . . . . . .
BIBLIOGRAPHY .
49
.
80
81
83
84
89
90
94
APPENDICES
Electrical/Hydraulic Analogies .
.
.
Calculation of fn and D from Step
Response . . . . . . . . . . . . . .
98
99
FREQUENCY ANALYSIS OF CATHETER SYSTEMS
USED FOR INVASIVE BLOOD PRESSURE MONITORING
by
Daniel Michael Chernoff
Submitted to the
Computer Science
the requirements
and Master of
Department of Electrical Engineering and
on May 1i, 1982 in partial fulfillment of
for the Degrees of Bachelor of Science
Science in Electrical Engineering.
ABSTRACT
The resonant behavior of fluid-filled catheter manometers
can produce severe distortion in the monitored blood pressure
waveform. Although gradual improvements in components has
resulted in catheter systems having adequate frequency response
for human blood pressure measurement, these systems are frequently compromised by the presence of occult air bubbles in
the fluid column.
A general model was developed to predict the frequency
response of catheter systems in terms of a limited number of
lumped second-order sections. This model was experimentally
verified by direct frequency response measurement and by independent measurement of component characteristics. This model
also successfully predicts the effect of bubble size and location on the frequency response.
A previously-proposed technique of measuring the frequency
response of a fluid-filled catheter system in vivo was described
and theoretically justified in terms of the lumped element system
model. This technique is believed to have significant clinical
application in dynamic in situ testing of catheter systems.
Utilization of this technique may result in higher user confidence in the catheter system.
Thesis Supervisor:
Title:
Dr. Roger G. Mark, M.D., Ph.D.
Matsushita Associate Professor of Electrical Engineering
in Medicine
LIST OF TABLES
Table
Page
4-1
Tubing resistance measurements..
. . . . .
52
4-2
Tubing compliance measurements..
. . . . .
57
LIST OF FIGURES
Figure
1-1
1-2
Page
Magnitude and phase response of a
typical catheter system . . . . . . . .
8
Waveform distortion due to nonideal
frequency response . ......
. . . . . . . .
9
. .
2-1
Modeling the catheter system .......
2-2
R' and L' as a function of frequency.
24
2-3
Relative error of predicted natural
frequency as a function of compliance
ratio
.
.. . . . . . . . . . . . .
34
. . . . .
3-1
Schematic of tap generator..
4-1
Setup for direct compliance measurement
4-2
Tubing compliance vs. frequency .
4-3
Diagram of laboratory setup for testing
of catheter system..
. . . . . . . . .
4-4
4-5
4-7
. ..
58
. .
60
61
Comparison of second-order model with
.
n
functkio
transfr
Model and experimental transfer functio
with a discrete bubble at various
locations in the fluid column . . . . .
64
67
Diagram of setup for tap and flush
experiments
4-8
48
56
Transfer function for the bubble-free
system. . . . . . . . . . . . . . . . .
experimentaml
4-6
S.
20
.
.
.
.
.
.
.
.
.
.
.
.
Transfer function with and without
a bubble midway in the fluid line .
.
.
73
74
Figure
4-9
4-10
Page
Square wave, tap, and flush time
responses for bubble-free system. . .....
Square wave, tap, and flush responses
with bubble midway in the fluid line.
5-1
5-2
75
Simulated response to pressure step
at input. . ....... . . ....... ..
. .
.
78
.
87
Simulated response to fast-flush. . .....
88
CHAPTER 1
INTRODUCTION
Present-day
invasive
accomplished using
measurement
a
site
measurement
monitoring
fluid-filled
to
an
system
catheter
externally
has
characteristics, often
existence of one or
pressure
represented
more
as
resonant
frequency
peaks
and
well
frequency
frequency components of the pressure
low-pass filtering to yield
satisfactory
with
the
phase
highest
permitting
waveform
the
resonance
beyohd. the
signal,
This
nonlinear
will usually
a
the
response
second-order,
equipment,
at
from
transducer.
shift. With properly assembled modern
occur
typically
leading
located
nonideal
is
simple
reproduction
while suppressing hig h-frequency artifact. However, a
number
of
factors, most often trapped air in the fluid line, contribute
to
a
low
resonant
fre quency
and
subsequent
waveform that cannot be corrected
by
distortion
low-pass
of
filtering.
distortion may have serious consequences, since various
of the waveform are used in clinical
diagnosis.
satisfactory frequenc y response
before
lead to the erroneous and
dangerous
even
catheter
This
features
Measurement
insertion
assumption
frequency response remains satisfactory while the
the
that
system
of
can
the
is
in
use. There is strong evidence to suggest that dynamic changes
in
the
system
-
clott ing
at
coalescence of microb ubbles,
frequency response.
the
etc.
catheter
-
can
tip,
movement
seriously
alter
and
the
2
The primary objective of this study is
to examine techniques
of determining the approximate frequency response of
system in vivo (while attached to the patient). We
catheter
a
will
examine
two direct time-domain techniques for doing-this:
(1) The fast-flush technique proposed by Gardner (1970);
(2) A
flow
impulse
produced
by
tapping
the
with
method
catheter
or
extension tubing.
We will compare these two excitations
a
(pressure
step at the catheter tip) that cannot be performed in vivo but is
an established technique for measuring frequency response.
Because the theoretical model we
choose
to
represent
the
system has a considerable influence on our interpretation of
the
above stimuli, the other major objective of this study will be to
examine
transmission-line
and
lumped-element
catheter system. Using independent evidence,
we
models
of
will
the
determine
which model is the more accurate representation and how
the
models may be reconciled. This analysis will aid considerably
our understanding of how bubble
size
and
frequency response of the system and the
position
time
affect
response
two
in
the
to
the
Invasive blood pressure monitoring of the critical-care
and
proposed excitations.
1.1 Brief History of Invasive Monitoring
post-surgical patient has become
medical practice.
The
visual
almost
display
commonplace
of
the
blood
in
modern
pressure
waveform often yields to the clinician
valuable
information
the dynamic state of the cardiovascular system.
The
monitor
sites
pressure
at
a
number
of
important
vasculature, such as in the great vessels
heart,
is
often
an
invaluable
and
ability
tool.
to
in
the
of
the
chambers
diagnostic
on
Long-term
monitoring has led to the incorporation of high- and low-pressure
alarms into monitoring systems, resulting in faster
response
of
hospital personnel to potentially life-threatening conditions.
There are at present two methods in common use for
blood
pressure
monitoring.
The
catheter-tip
relatively new device, prompted by advances
and semiconductor technology, which
invasive
manometer
is
a
in microelectronics
consists
of
strain-gauge transducer located on the tip of a
a
very, small
catheter.
These
manometers have excellent frequency response characteristics, but
suffer the disadvantages of
high-cost,
extreme
high temperature sensitivity. The older and
fragility,
more
consists of an external strain-gauge transducer
common
coupled
and
method
to
the
recording site via a hollow fluid-filled catheter, first reported
in its modern form by Lambert and Wood (1947).
generally constructed of
nylon, and is filled
system, while
having
with
a
polyethylene,
a
saline
significant
catheter-tip manometer, suffers from
response, which at times makes it
PVC,
The
catheter
is
woven
dacron,
or
solution.
cost
advantage
relatively
inadequate
measurement of the blood pressure waveform.
This
for
poor
recording
over
the
frequency
high-fidelity
1.2 Requirements for Accurate Waveform Reproduction
A number of studies have been published
which
recommend
a
certain minimum bandwidth for faithful reproduction of the
blood
pressure waveform. Geddes (1970)
these
reports,
in
which
it
is
provides
apparent
response" depends bot• on the nu:ure
degree
of
accuracy
standardized. Bruner
required,
(1981)
and
he
that
of
neither
summary
of
"adequate
frequency
tie
waveform
of
which
and
the
has
been
has correctly observed that there
still little consensus on the :aini:hIu:
these systems,
a
offers
an
frequency
excellent
requirements
analysis
of
is
of
the
practical difficulties which have prevented such a consensus from
being reached. The pressure waveform has also been
subjected
fourier analysis to determine the number of harmonics
neccessary
to achieve a. certain fidelity in a reconstructed waveform.
analyses
all
show
the
magnitude
dropping off rapidly with
showed that the
the
amplitude
waveform had fallen to 11.8%
of
the
harmonic
of
the
by
the
fourier
number:
components
sixth
These
components
Hansen
of
to
(1949)
an
arterial
harmonic;
McDonald
(1960) found the amplitude of the fifth harmonic from a number of
pressure recordings to be less than 20% of the fundamental. These
findings tend to support the view that the
higher
harmonics
not contribute significantly to the arterial pressure
it should be stressed
that
limited number of harmonics
reconstruction
is
not
of
equivalent
through a distorting measurement system,
since
a
to
the
wave,
wave
from
passing
do
but
a
it
underdamped
nature of the system causes nonlinear gain and phase shift within
the passband.
5
Hone of these studies directly analyzed
the
effect
of
an
underdamped second-order system on the pressure waveform. Gardner
(1931)
has done this, specifying an
approximate
range,
in
the
form of a chart, of resonant frequencies arid damping coefficients
which
yield
acceptable
reproduction
of
waveforms. This represents a practical if
"demanding"
incomplete
effort
defining an acceptable frequency response in terms of
the important features of a waveform (particularly
pressure
at
preserving
systolic
diastolic values). This type of analysis is clearly needed,
and
along
with a clear definition of the waveform features to be preserved,
if an objective evaluation of the adequacy of a given *transducing
system with known frequency response is to be made.
1.3 The Fluid-Filled Catheter
1.3.1 Models
The mechanical properties of fluid-filled
which lead to inadequate
frequency
response
great deal of study. Hansen and Warburg
as a harmonic oscillator (system with
have
systems
undergone
one
the
degree
of
freedom),
The coefficients of
frequency
response
this model could be shown to be related to the compliance of
elements of the system, the physical dimensions of the
mechanical lumped-element system consisting of
system
a
of
the
catheter,
and the mass of fluid filling the system. This is analogous to
and dashpot in series, or an electrical
a
(1949) modeled the system
generally extending the work of Frank (1903).
the second-order equation governing
catheter
mass,
composed
a
spring,
of
an
inductance, capacitance, and resistance.
Tnis early
in .odeling
work of a number
(1963),
Shapiro
of
researchers.
and
Krovetz
Krovetz et al (1974),
(1980),
was extended by tha experimental
.1or
Fox et
These
(1970),
al
Falsetti
(1978),
and
Yanof
et
their analysis. Another
Latimer (1968),
group
of
the
and Li et al
basic equations of fluid flow in
tubes,
study
in
pulse
wave
transmission
the
al
but
model
for
Vierhout
started
derived
et
detail
including
(1978),
al
(1974),
Shinozaki
fundamental
workers,
et
al
who evaluated the frequency response in more
retained the second-order system as
(1966),
include
from
the
originally
arterial
to
system,
and
developed a transmission line model for the catheter system. This
model has
been
shown
to
second-order model in
be
more
determining
accurate
the
than
location
the
of
simple
the
first
resonant frequency from the physical constants of the system
in predicting the presence of higher order
since the primary goal of
many
workers
resonances.
has
been
to
However,
fit
observed frequency response in the lower frequency range
model,
and
not
to
determine
the
response
a
and
the
with
priori,
a
the
second-order system has been the more commonly cited model.
1.3.2 Distortion
An ideal pressure measurement system is one which
frequency
response
to
well
beyond
interest, and either zero or linear
shift corresponds to a
pure
time
the
highest
phase
shift
delay).
This
has
flat
harmonic
(linear
of
phase
guarantees
an
output which is at worst a time-delayed but otherwise undistorted
version of
the
input.
The
physical
las
governing
pressure
transmission in a long flexible fluid column make this
extremely
difficult to achieve.
The magnitude and
phase
response
of
a
typical
catheter
system is shown in Figure 1-1. There is present a large
resonant
peak
to
which
will
amplify
harmonics
that
lie
close
the
resonance, and a subsequent falling off of the frequency response
following the resonance which will
attenuate
higher
harmonics.
Moreover, the phase shift is highly nonlinear, with a sudden
degree phase shift near the
frequency
response
is
resonant
typical
of
frequency.
This
underdamped
180
type
of
second-order
systems.
Figure 1-2 illustrates the effect this
has on a simulated
arterial
pressure
type
waveform.
shows the input to the catheter, while Figure
output from
tne
system.
The
distortion resulting from the
recording
system,
output
signal
nonideal
particularly
the
of
distortion
Figure
1-2(b)
1-2(a)
shows
exhibits
serious
characteristics
appearance
of
the
of
the
spurious
oscillations and large error in systolic pressure.
While it is theoretically
possible
signal given the output waveform and the
to
recover
frequency
the
response
the system, the latter is generally unknown. Therefore,
deal of work has gone into measuring the
frequency
input
a
of
great
response
of
catheter systems, either to perform this reconstruction or simply
to evaluate the adequacy of the system.
38 r
IH(f)
3
4a
feO)
-1'
~NI
%ill
a-
Figure 1-1.
pFmSg .
Magnitude and phase response of a typical
catheter system
BED
I
V'~+Le&
277j1
*-T-'
81
0
x. -- ---t
-.
--
ti-i
H-
r
-
I
.'
EBED
00022,
r
T
---
-
t
-!r-
-.
7.
r
17
:
K- :
I-r
ti
V-.
:F:i--
-
-- iIt
input blood pressure waveform
(a)
-•-
otu
of
.
cathetr-t.ansd
!
---..
---
s yste
·--
i
iT..:
- -i;i''
.. :
.:
; - i-i::i~li~l-ii-_::1;i·-_i~-:
_ j,~::,::-;i-
.7
r
V
.:
:::...~.
:l-~_i-~~-~~--i-,
:-i27?
LIi·
output of catheter-transducer system
(b)
Figure 1-2.
Waveform distortion due to nonideal frequency response
10
1.4 Frequency Response Measurement Techniques
A
number
of
techniques
hive
been
used
i~duce
to
the
frequency response of catheter systems. These techniques
can
separated into two classes:
involve
(1) direct techniques, which
be
analyzing the system response to a known external input; and
(2)
indirect techniques, which assume
the
a
particular
model
for
catheter (typically second-order) and make additional assumptions
about the frequency content of
class,
the
techniques
can
the
be
input
signal.
further
Within
divided
each
into
time,
frequency, and correlation-domain approaches. We will examine how
each of these techniques has been used.
1.4.1 Direct Techniques
Frequency-domain techniques all require input
having
a
known
spectrum.
If
the
system
of
is
a
signal
linear
and
time-invariant, the energy at each frequency in the output signal
is uniquely associated with the energy in
the
input
signal
that frequency, so an input which is flat in frequency and
at
phase
will produce an output which is a scaled version of the
transfer
function. The impulse function and white noise both are
flat
frequency and phase,
but
practical
considerations
noise the better choice for frequency-domain
is no evidence of white-noise excitation
direct
frequency-response
measurement
make
white
measurement.
There
having
in
possibly because other methods exist which
been
catheter
do
data to be analyzed directly in the frequency
requiring less sophisticated instrumentation.
in
not
used
systems,
require
domain,
for
the
therefore
11
An example of one such alternate method is
gain and phase shift at a number of
technique was used quite
(1968).
discrete
successfully
Swept frequency measurement
by
is
(1980),
and
Gardner
(1981).
measurements described
thus
source substituted for the
frequencies.
and
standard
used
by
an
require
appropriately
artery
requirement is difficult to meet
at
in
the
the
This
engineering
all
that
the
Latimer
Rothe
Unfortunately,
far
removed from the patient and
measure
Latimer
a
technique which has been recently been
to
the
and
Kim
of
the
system
driven
catheter
clinical
be
pressure
tip.
This
environment.
Therefore it is not surprising that these methods have
primarily
been used in experimental work.
Direct time-domain techniques for
response all
consist
of
exciting
measuring
the
system
the
frequency
with
well-described time signal, usually a step function
a
of
simple
pressure
("pop" excitation). The pressure step is achieved by pressurizing
a closed system and then suddenly relieving the pressure
catheter tip,
typically
response observed at
the
by
bursting
output
a
of
rubber
the
at
the
membrane.
The
system
can
analyzed. In order to make the analysis of the response
straightforward,
priori.
While
the
most
system
researchers
system, some (Melbin and Spohr,
1974)
have
responses
demonstrated
to
the
order
generally
have
assumed
1969; Gabe,
systems
pressure
is
which
step,
then
waveform
described
a
although
a
second-order
1972; Krovetz et
exhibit
be
al,
higher-order
only
specifically suggests that reflections from impedance
Krovetz
mismatches
(i.e. transmission-line phenomena) may be responsible. One source
(Attinger, 1969) reports that 30% of a large
number
of
systems
tested exhibited higher than second-order behavior. However,
second-order approximation
often -
is
satisfactory,
assumption makes the waveform analysis particularly
the
and
this
simple
(see
chosen,
the
Appendix A).
Even if a satisfactory model for the system is
"pop" technique remains unsuitable for dynamic
analysis
of
catheter in vivo, since it requires
catheter
tip
that
the
available. Because catheter systems can collect
clotted, or otherwise be degraded in
use,
a
bubbles,
the
become
satisfactory
test
result from a pop test performed prior to insertion
may
false sense of security. This
q'uestion
concern
raises
the
be
give
of
whether excitations applied elsewhere in the system may elicit
response from which the
appropriate
transfer
function
may
a
a
be
deduced. There have been no studies done to answer this question,
although Gardner
(1981)
has
described
a
technique
which
is
created
by
claimed to be an acceptable excitation: a flow
step
opening and then
present
releasing
the
monitoring setups, where the
flush
flush
valve
source
is
in
located
most
at
the
transducer end of the fluid column.
1.4.2 Indirect Techniques
A number of indirect
distortion-measuring
techniques
have
been described. These all assume a second-order model and attempt
to determine,
from
the
patient
location and magnitude of the
examined
the
magnitude
pressure
resonance.
of
the
signal
Brower
signal
itself,
et
the
al
(1975)
spectrum
after
preconditioning (bandpass
filtering
and
differentiation).
presence of a peak in the preconditioned spectra
with distortion, and approximate formulae given
the degree of damping.
components of the
Doherty
incoming
(1981)
pressure
normalized magnitude and phase in
a
was
associated
for
determining
determined
signal,
the
and
regression
Fourier
applied
equation
coefficients were optimized to detect resonance within a
critical range. Jackson
et
al
(1978)
used
linear
of
a
complex
certain range of frequencies was taken to indicate
whose
predictive
pair
pole
the
certain
analysis (a correlation technique) to model the spectrum
input signal: the presence
The
zt
of
the
witnin
a
presence
of resonance.
These indirect techniques all evolved because of the need to
perform dynamic analysis on the catheter system and to
eliminate
the need for manual intervention by hospital personnel, either in
testing
the
system
or
in
compensating
the
response.
These
advantages over direct methods make indirect techniques extremely
desirable. On the other hand,
the
validity
of
these
indirect
techniques rely heavily on both the assumed catheter system model
and on the assumed spectrum of the patient waveform. If the blood
pressure power spectrum appears "resonant" in the sense of having
a local maximum, as may happen in
recordings
from
the
smaller
arteries (the arterial system itself behaves as an assemblage
branched transmission lines),
then
these
techniques
erroneous results. An additional problem, although one
grown smaller as the cost of computation has
computational complexity of indirect analysis.
decreased,
Although
can
of
give
that
has
is
the
initial
14
experiments using these techniques appear
promising,
they
Iave
not yet been subjected to extensive testing using the full
range
of clinically observed waveforms.
1.5 Compensation Techniques
Various
approaches
have
been
taken
to
compensate
the
frequency response of catheter systems. These can be divided into
two classes, mechanical and electrical.
Mechanical
compensation
may be considered a problem in impedance matching, although
researchers
regard
it
as
merely
increasing
the
some
damping
coefficient. Electrical compensation involves active filtering of
the signal. We will now examine each of
these
methods
in
more
detail.
1.5.1 Mechanical Compensation
By adding additional damping to the hydraulics,
system which previously produced highly distorted
a
catheter
waveforms
be made to have a much wider useful bandwidth. Damping by
a constriction at the patient end of the catheter has
for many years. A set of
(1974),
experiments
by
LaPointe
adding
been
and
can
used
Roberge
using needle valves as resistance elements, has confirmed
the utility of the technique, and Latimer (1968) has justified it
in terms of matching the source and line impedance of an acoustic
transmission
line.
Unfortunately,
the
large
amount
of
constriction neccessary to achieve appreciable damping makes this
technique unsuitable for use with flush devices, and it is rather
sensitive
as
well.
A
more
promising
technique
is
parallel
damping, reported by van der Tweel
(1957) and Crul
(1962).
This
can be described as another form of impedance matching, this time
matching the load (transducer) impedance to the
There are commercial devices now available
damping. One which we have observed is
which
is
placed
in
transducer. Gardner
parallel
(1981)
the
with
the
valve)
perform
Sorenson
fluid
in
series
parallel
Accunamic,
line
order
to
with
create
parallel impedance. The impedance match obtained
adjusting this device to
impedance.
at
the
has described this device, in which
fixed compliance (bubble) is placed in
resistance (needle
to
line
minimize
step
a
an
by
response
a
variable
adjustable
empirically
overshoot
is
An advantage of these mechanical compensation techniques
is
crude but nonetheless fairly effective.
that, by effectively increasing the damping to
flatten
out
the
resonant peak, they tend to extend the useful range of the system
out to approximately the resonant frequency. This can
amount
to
twice the usable bandwidth of the uncompensated system. Moreover,
no exotic electronics or processing techniques are required.
The
chief disadvantage of mechanical compensation is that it requires
an external step
input
to
observe
when
critical
achieved. Relying on the patient waveform to adjust
damping
the
is
damping
is a dubious procedure at best.
1.5.2 Electrical Compensation
If an approximate transfer function for the catheter
system
is known, inverse filtering (convolving the output signal with
network having the inverse transfer function) can greatly
a
extend
16
the bandwidth. Melbin and Spohr
(1969) describe an analog circuit
to perform inverse filtering. M1ore recently, Brower et al
(1975)
and Ciccolella (1976) have described digital filtering to perform
the same function.
If the transfer function
approach is
to
low-pass
is
not
known,
filter
the
signal,
the
most
common
with
a
high
enough
to
retain the significant harmonics of the pressure signal. This
is
frequency lower than the assumed resonance
but
cut-off
the method most manufacturers include in the monitors at present.
Frequently,
simultaneously
however,
-
the
these
resonant
conditions
peak
cannot
overlaps
be
an
met
appreciable
portion of the signal spectrum. Aggressive low-pass filtering (12
Hz cutoff and below) has been practiced by some manufacturers
an attempt to prevent resonances occuring at
higher
from causing systolic overshoot, but at
cost
the
in
frequencies
of
extremely
limited bandwidth. Low-pass filtering can be useful in preventing
high-frequency artifact from appearing in the output signal,
it is extremely limited in
its
having low resonant frequencies.
ability
to
compensate
but
systems
17
CHAPTEi
2
MODELING OF THE CATHETER SYSTEM
A
theoretical
understanding
of
the
transducing system is an important step
predict the
changes
in
system
blood
towards
pressure
being
characteristics
able
to
under
various
conditions (e.g. altering of component stiffness, tubing
length,
the presence of occult bubbles or leaks).
system model, it
is
possible
to
Under
specify
an
the
appropriate
most
desirable
characteristics for catheter, tubing, and transducer in terms
of
producing a faithful reproduction of the
In
pressure
waveform.
addition, an accurate model for the system may suggest methods of
compensation
of
the
frequency
response
involving
additional
components, such as impedance matching devices. This section will
develop a general model for
well-established
theory
of
the
wave
transducing
propagation
system
in
lines, and then proceed to establish conditions under
using
the
transmission
which
model may be simplified to a lower order lumped-parameter
the
system
with little loss in accuracy and large gain in ease of analysis.
2.1 General Model - Mechanical/Electrical Analogies
There
can
be
little
doubt
as
to
the
validity
of
a
transmission line model for the fluid-filled pressure tubing. The
presence of phase delay, attenuation, 'and acoustic impedance have
been experimentally demonstrated by many researchers, but pernaps
most elegantly by Latimer
and
Latimer
(1969),
who
determined
values for wave speed and attenuation at a number of resonant and
antiresonant frequencies. It is
interesting
to
that
the
tubes
was
rather
for
pulse wave transmission in the arterial tree. The principles
are
theory
of
acoustic
wave
originally developed not for
transmission
catheter
in
elastic
systems
largely the same but the catheter system is
note
in
but
fact
easier
to
analyze due to the limited number of reflecting sites, consistent
internal diameters and better-understood wall properties.
Figure 2-1(a) shows a physical model for the
of transducer system, represented
by
codstant internal diameter coupled
to
a
simplest
liquid-filled
a
transducer
type
tube
of
through
fluid-filled dome. An increase in.pressure initiated at the
a
left
causes liquid to flow to the right through the tubing
and
which in turn causes a deflection of
diaphragm.
This deflection is sensed by a strain
electrical
signal
is
amplified
and
the
transducer
gauge
and
processed
the
to
dome,
resulting
produce
a
shown
in
pressure recording.
An electrical model for this mechanical system is
Figure 2-1(b).
The tubing and transducer dome/diaphragm will each
be examined in turn. In each infinitesimally long segment of
the
tubing, fluid motion has associated with it friction due to shear
stresses in the fluid and inertia due to the mass and velocity of
the fluid. There are also compliances associated with the
wall and (to a lesser extent) in the
storage
of
potential
energy.
fluid
Finally,
itself,
the
wall
tubing
leading
to
exhibits
viscoelasticity which causes
energy
losses
due
mechanical
to
hysteresis effects. These physical "line constants" are
replaced
by the analogous electrical symbols in Figure 2-1(b), where:
R' = resistance/unit length due to viscosity of the fluid;
L'
=.inertance/unit length due to the mass of the fluid;
C' = compliance/unit length due to compressibility of the
fluid and stretch of the tube walls;
G' = conductance/unit length due to energy loss in the
tube walls;
dx = incremental length.
These so-called
"line
constants"
are
in
fact
all
frequency
dependent to some extent as the theoretical
development
next section will show. The transducer dome
and
exhibit similar-. resistive,
inertial,
wall
of
diaphragm
loss,
and
treating
also
elastic
effects, but the short length of the transducer relative
wavelengths we will be considering justifies
the
to
the
these
as
lumped elements.
In summary, then, we have
inertial
and
resistive
effects
associated with the fluid. These present a longitudinal impedance
to
flow
and
thus
are
represented
as
series
elements.
The
compliance and wall loss are properties of the plastic
materials
used, present a
represented as
transverse
parallel
infinitesimal
sections
characterized
by
the
impedance
elements.
then
to
An
produces
telegraph
flow,
and
thus
infinite
sum
of
these
transmission
line
a
equations
engineering theory. Prior to determining the
of
frequency
are
electrical
response
of this model, however, we will need to develop the equations
calculate val-ues for the line constants.
to
dome
d i annbrqan
''~
" DI
plastic tubing
Pin
transducer
(a)
R' dx
L'dx
R'dx
L'dx
R'dx
dx
L'dx
G'dx
(b)
Figure 2-1.
Modeling the catheter system
Rtr
C'dx
Ltr
Ctr
21
2.2 Theoretical
Calculation of Line Constants
2.2.1 Longiud nal Ipedance
The development of
(R and L)
theory
describing
laminar
oscillat.ory
fluid flow through narrow tubes has been made by Lambossy
and Womersley (1956).
The most significant result of this
(1956)
.theory
is the prediction of a "skin effect" phenomenon, which causes
increase
in
resistance
and
decrease
in
inertance
at
an
high
frequency due to alteration of the fluid velocity profile. across
the tubing cross-section. If the tubing is assumed to
be
rigid,
straight, and of circular cross-section Womersley shows that
Q= r
j
-
ej
(1)
where
Q
= volume flow
A
= amplitude
W
= circular frequency = 2rf
Aejwt= pressure gradient = -
.p
dp/dx
= fluid density
a
= r1/TU
u
= p/p = kinematic viscosity
= Womersley coefficient (dimensionless)
J0 and J1 are the zero and first-order Bessel
functions
of
the
tubing
are
complex argument
j3/2
; j =
T
= phase shift of
Tr/4 radians
Although the requirements of rigid and straight
22
not szric;ly met by
2ctheter
systems,
many
authors
(Latimer,
Jager) have applied these equations to calculate the longitudinal
impedance per unit length Z' with
good
results.
Following
the
analysis of Jager et al (1965):
Z' =
dp/dx
=
j
2J1 {j
Saj
r(Jooj
3/2j
3/2}
o
(2a)
3/2
if we write this as
ZV
2.
j
L'(w)
+ R'(w)
(2b)
where L' and R' denote resistance/unit length and
inertance/unit
length, respectively, then
P
2
S7Tr M{
1-
2
0a
_ 2J
;
aj 312Jo
MI'o= modulus 1
For the case
familiar
of
-
steady
~1o= phase
flow
these
sinle
irr4Mio
S1i -
(3a,b)
2J1
j 31/2Jo0
equations
reduce
to
the
Poiseuille equations for R' and L':
L = 4/3
W
_8 11
R'
W
7r
(4)
P2
7r
The significance of the W'omersley calculations for R' and L'
is that
effective
as
the
frequency
inertance
decrea ses
increases. In the limit
drops to 75% of
its
of
of
d.c.
oscillation
and
the
infinite
value,
and
is
increased,
effective
frequency,
the
the
resistance
the
resistance
inertance
becomes
23
infinite. A plot
of
dimensions used in
R'(w)
this
and
study
L'(Q)
is
vs.
shown
w
in
for
the
Figure
tubing
2-2.
The
viscosity and density of saline have been taken as 0.01 poise
0.998 gm/cm 3
respectively
(approximate
values
at
20
nsr
degrees
centigrade).
2.2.2 Transverse impedance (C and G)
A calculation of compliance and wall loss requires
detailed
knowledge both of the physical properties of the plastic used for
the pressure tubing and the mode(s) of wave
propagation
fluid and in the wall. These are
quite
generally
specify precisely. Equations exist which specify
in
the
difficult
the
of a tube of uniform cross-section as a function of the
to
compliance
internal
and external radii, Young's modulus and the Poisson ratio for the
tubing material. These assume a linear, isotropic medium with
no
losses, but may be applied to plastics so long as the results are
not expected to be quantitatively precise. Using the equation for
the compliance of a thick-walled tube (see reference 25) we have:
C' :
2nr
1+r /r 2
E
1-rf/r 2
,,,
i
+ s
(5)
e
where
r.
= internal diameter;
re
= external diameter;
E = Young's modulus;
s = Poisson's ratio.
No such simple equation is known describing the wall loss G,
and
24
'i
SL'(Pa-sec/m
x10,
4)
Ii
12
12
l,,,.r
L'(0)
c
S_
i
X1 0 'O
R'
5
(Pa-sec/m4 )
-
m
3
3
(
R'(O)
I
11
Ii3 41
M
7
Ii11
Figure 2-2, R' and L' as a function of frequency
11M
vw1
in any case the properties of
a
given
sample
of
plastic
are
generally not so well specified as to allow direc; calculation of
compliance and wall loss. In other studies similar to
this
one,
experimental measurements are invariably substituted
for
in the calculation of transverse
(1949)
impedance.
Hansen
theory
has
found that G' is proportional to frequency, implying a m3chanical
hysteresis loss per cycle, but it is not known
whether
this
generally true for plastic materials. More research in this
is clearly needed but is beyond the scope of the
We will use
experimentally
models, and assume
G'=O.
determined
As
long
as
present
values
for
G'/wC'
is
C'
is
area
study.
in
small,
our
this
assumption should cause negligible error in predicted location of
resonant frequency.
2.3 Transmission Line Formulation
2.3.1 Telegraph Equations and Propagation Constant
We will take the following circuit representation to
be
an
adequate model of the transducing system:
Ps
s
Transmission Line
k=propagation constant
Z=characteristic
impedance
Ctr
•
Rtransducer= 0
L transducer-O
o0
where the parameters Rtr and Ltr have been set to zero.
This
is
justified when the transducer dome radius (or effective radius iin
the case of a non-cylindrical
dome)
is much
larger
than
the
tubing radius, due to the strong inverse dependence of both R and
L on the radius. This condition is nearly always met in
catheter
systems. The telegraph equations governing this line are:
dP/dx = -(R'+jwL')Q
Q
= -(G'+jwC')P
where P represents pressure
solution which is the
waves:
(6a)
sum
(6b)
and
of
Q
is
forward
flow.
and
If
we
backward
assume
a
traveling
27
P(x,t)
=
P+e
-kx
+ Pe
+kx
}e
jwt
(7)
then we can solve for k:
k =
= V(R'+jwL')(G'+jwC')
+ j
: m/LvV' C'/(V+j)(W+j)
(
(8)
where
S= attenuation constant
B = phase constant
:WVL"C'
0o
V = R'/wL'
W = G'/wC'
squaring the above equation:
k2
=c 2
_
+2jaB
-2
= a21(VW-1)+j(V+W)
(9)
separating real and imaginary terms and solving, we have:
a = 0.5F5BV+W)
(10)
a = Bo/F
(11)
where
(12)
T{(1+V1) (1+Wz)+I-VW}
F is the correction factor, important primarily at very
frequencies, in the form derived by Latimer and
Latimer
low
(1968).
This correction factor will rarely need to be used in this study,
since by the first resonant frequency F will nearly
reached its
high-frequency
limit
of
1.
It
is,
have
always
however,
theoretical interest because if the condition V = W occurs
of
(i.e.
28
then F = 1 and a Heaviside "distorotionless line" is
R'/L'=G'/C'),
obtained, with constant attenuation and phase shift
proportional
to frequency. This condition, unfortunately, is never encountered
in practice except at isolated frequencies, because R',
L',
G',
and C' are all frequency dependent to some extent.
2.3.2 Characteristic impedance
The characteristic impedance Z of the transmission
line
is
defined as
Z -
Q
dP/dx
where Q is the
volume
rate
of
flow
and
dP/dx
the
pressure
differential. Z can be expressed in terms of V and W as follows:
(13)
(R'+jwL') (G'+j wC')
SL'iC'/(V+j)/(W+j)
= IZI
Z je
(14)
2 +1)}.2s
(15)
e
= 0.5(cot-1V - cot-1W)
(16)
Zo
=
Zo{(V 2 +1)/(W
-L'/C'
alternatively, Z may be expressed
in
terms
of
resistance
reactance:
Z = :Z:(cose + jsine )
2.3.3 Boundary Conditions and Reflection Coefficient
The solution for P(x,t) and Q(x,t) can be expressed as
(17)
and
29
P(x)
= P+e- xe j-
Q(x)
= -- {P+e -cxe -jIx - Pe +cax e+jix }
e
+ Pe
(13a)
1
(18b)
Z
where
the
dropped
have
we
equations. The parameters a and
length and phase
shift
per
dependence ejw t
time
from
the
8 denote the attenuation per unit
unit
length
respectively
of
the
transmission line.
To solve equation (18)
we
need
to
specify
the
conditions at each end of the transmission line. We have
a zero-impedance pressure source
and
represented as a pure compliance.
The
a
load
(the
boundary
assumed
transducer)
conditions
boundary
are
therefore
P(x=O) = P + P = PO
Q(x=l)
Pefining
(19a)
= jwCt{P +e -k+Pe+kI
the
reflection
= Z{P e-k; -Pe
coefficient
+k
r =P+/P_
}
(19b)
and
solving
equation (19):
P+=Po
-21
1+fe 21
;
P =P
+Fe-2k £
and
P(x)
=
+e-21va!-
(e
1+re)
-kx
+
'e-2kekX)
(20)
where
1-jWC
r=
tr Z
(21)
1+j CtrZ
At the transducer, the pressure is
Poe-kk
-
P(k) =
l+re -
2k
(1+r)
(22)
I
a relation whicn cescribes the attenuation and phase shift of the
pressure wave from input to output as a function of frequency.
2.3.4 Natural Frequencies
The transmission line equations (18a,b) may also
for the natural resonant frequencies of
P(x=O) = 0. Equation (19)
the
system
be
by
solved
setting
then becomes
Z-tankt = -I/jwCt r
(23)
a transcendental equation in complex Z
and
solved in closed form. However, we can
find
k
wh-ich
cannot
approximate
be
values
for the natural frequencies by making the assumption w>>R'/L' and
w>>G'/C'
(generally
valid
at
and
above
the
first
frequency), allowing us to make the approximations k
Z
Lc'
wCtr
resonant
jwVL'C'
in which case equation (23) becomes
r
-L'/C
r
'C1)
i4
which can be solved graphically or numerically
frequencies wn"
(24)
for
the
natural
31
2.4 Lumped Model Approximation
While the transmission line model in theory is probably
system,
most accurate representation of the catheter
several practical reasons why lumped-parameter
the
there
are
are
more
models
commonly invoked to explain the resonance phenomenon. First among
these is the fact that higher-order (i.e. three-quarter-wave
above) resonances almost invariably occur at a
frequency
the range of significant blood pressure harmonics,
and
beyond
making
their
presence inconsequential. A second factor tending to minimize the
importance of the higher
resonances
is
the
Womersley
which causes the resistance (and therefore dampinS)
markedly with
frequency,
thus
minimizing
the
to
effect,
increase
height
resonant peaks. Thirdly, the presence of trapped air
of
the
bubbles
(a
common circumstance) introduces large lumped compliances into the
system, tending to make
a
lumped
circuit
representation
more
tractable than the corresponding transmission line model.
A simple way to approximate the transmission line model with
lumped-elements is to take the circuit representation
of
2-1(b) but let each section represent a finite length of
rather than taking
the
limit
dx->O.
Li,
Van
Figure
tubing,
Brummelen,
Noordergraaf (1978) have performed this analysis for
N=1,2,
and
and
infinity, where N is the number of lumped sections. These authors
also considered the effect of
different
element
( 7 vs. inverted-L) on the calculated frequency
findings indicate that sizable errors
in
the
configurations
response.
location
Their
of
the
first resonant frequency are introduced for N=1, but that as
the
32
lengrth of each section becomnes small
wavelength
(highest frequency)
becomes v.ry good.
of
relative
interest,
They also found the
to
the
the
shortest
approximation
configuration,
7
involves slightly less lumping per section, to be
more
which
accurate
for a given N than the inverted-L model.
We will adopt a slightly different tack in this study. First,
will examine the two limiting cases Ctu<<C
tu
tr
and
Ctu>>C
tu»tri
we
and
demonstrate how each can be represented by a second-order circui;
with appropriate correction factors. Then, a simple equation will
be constructed which
is
precise
for
the
introduces only a small calculable error
in
limiting
cases
resonant
and
frequency
for intermediate cases where neither compliance dominates.
Ctu << Ctr
In
this
case
the
tubing
is
rigid
compared
with
the
transducer, and the wavespeed is so high that propagation effects
may be ignored. A second-order equation is therefore valid, with
W0
tr
= i/'iLCt
(25)
Ctu >> Ctr
In this case the transducer looks like an open circuit,
we may solve equation (24)
Wm
Ctu
= 7/2LC-
and
for Ctr=O:
(26)
tu
Ctr
Let us construct the following equation which
will
satisfy
33
both limiting cases:
WU
(27a)
1
=
/L((2/w)
/LC
eq
aD
.where
r
Ctu+Ctr)
tu tr
(27b)
z
: •Jl-~D
Ceq = Ctr + (2/r)
(27c)
2Ct
u
For Ctu(<Ctr:
wo
->
I/C/L-C
tr
and for Ctu>>Ctr
->
2"o
tu
as desired.
4e
can
separately
solve
equations
(24)
and
(27)
for
intermediate values of Ctr. If we define
K
then
we
can
frequency (1 -
plot
Ctr/Ctu
the
wo )/wo
relative
error
as a function
of
in
K.
predicted
This
is
natural
shown
in
Figure 2-3. The maximum error is only 2.44%, so the approximation
introduced by equation (27(a)) seems acceptable. If
the
values of Ctr
be
and Cu
are
known,
F.igure
2-3
can
precise
used
to
Thus we have succeeded in reducing the transmission line
to
determine a correction factor for equation (27(a)).
a.0
2.5
g
1.5
1.0
.5
as
f izz
.5
1.08
1.5
2.
2.5
< = Ctr/Ctu
Figure 2-3. Relative error in predicted natural frequency
as; a function of compliance ratio K
a simple second-order circuit in terms of preserving the location
of
the
first
resonant
frequency.
Equation
(27)
implies
an
times
the
4/r2
equivalent lumped compliance ;where a fraction
total tubing compliance shunts the transducer:
Rtu
S
;i;i
H(jQ)
tu7 T r
s-
r-rnsfer
Ltu
(4/7r2-)C
c tr
Ceq
eq
4
,
Cu
+
Cr
tu
tr
function
=
Po(jw)
=
Pi(jw)
1/(LCeq)
-w
(28)
2 +(R/L)jw+(1/LCeq)
If the resonant frequency of a catheter system is calculated
using this lumped model, a useful check on the legitimacy of
the
assumptions used in constructing the model is
tthe
to
calculate
loss term
R'(w r )/L'(w r )
to verify
the
assumption
w >>R'/L'
(we
still
assume
Dividing equation (3b) by (3a):
R'(w)
u
-
pr
L'(w)
a'tanelu
2
(29)
so the inequality
Ar
>>
pr z
a 2 tane 'o
0
becomes our check on the model assumptions.
(30)
G'=0).
I
36
.Air Bubbles
2.5 Effect of Trappe
It has previously been noted that the
presence
fluid line remains tne single most coinmon
of
c-.use
pressure monitoring. This is due to the high
air
of
in
the
low-quality
compressibility
of
air relative to water, causing even very small bubbles to greatly
increase the total compliance of the system
and
thereby
the resonant frequency. The problem of including air
reduce
bubbles
our models is exacerbated by the unpredictability of bubble
in
size
and location in the clinical setup, and by the strong
dependence
of bubble compliance on temperature and pressure. The
alteration
of the normal fluid velocity profile in the vicinity of a
bubble
may
model.
also
violate
Nevertheless,
we
the
can
plane-wave
examine
assumption
specific
of
our
situations
which
are
amenable to straightforward analysis and thereby possibly develop
some intuition towards the more general situation.
2.5.1 Compliance of Air Bubbles
The compressibility of air depends on temperature,
pressure,
and molar quantity. The compliance of an air bubble may therefore
be expected to vary as pressure waves are propagated in the fluid
line. To specify the variation precisely, we
need
thermodynamic state of the bubble at all times
vs. adiabatic compression
cycle).
We
also
(e.g.
need
temperature variation of air solubility in water
time constants to know whether pumping of air
to
know
isothermal
to
and
into
solution with pressure variation is significant. In
the
know
the
associated
and
this
out
of
study,
we will be content to note the primary effect of static
pressure
on compliance and ignore all higher-order effects.
If air is assumed to be a ideal gas, then
PV = nRT.
For a bubble, we will assume n and T are constant. Then
PV = K
V
or
K
K
(31)
If the pressure changes by an infinitesimal amount dP,
then
the
volume changes by an amount dV, with the relation
PV = K = (P+dP)(V+dV)
= PV + VdP + PdV + dPdV
Cancelling like terms and ignoring the higher-order term dPdV, we
have
dV/dP= -V/P
(32)
Using the definition of compliance:
C =
-(dV/dP)
and using equations (31)
C =
and (32),
this becomes
K/P 2
(33)
so we see that the compliance is a strong nonlinear
pressure. If the pressure
excursions
inside
range from 700 mmHg to 1100 mmHg (-60 to +340
atmospheric
pressure),
between 46% and
then
the
bubble
the
function
fluid
mmHg
may
to
vary
118% of its value at atmospheric pressure.
There are several lessons to be learned from this
First, the inclusion of bubbles in the
fluid
column
significant nonlinearities in the frequency response
pressure
column
relative
compliance
of
excursions
are
present.
Second,
at
analysis.
may
cause
when
large
higher
-static
compliance
pressures the effective bubble
therefore the resonant
pressure
is
increased.
Henry
increase
may
frequency
et
al
as
(1967),
phenomenon, have even suggested checking the
detecting bubbles in the fluid column.
as
if
a
for the bubble, considerably simplifying
analysis.
will
take
in
constant
2nalyzing
this
response
means
a
the
excursions are kept small, we may assume
third approach which we
static
the
frequency
Third,
and
noting
pressures
of the system at high and low static
smaller
becomes
of
pressure
compliance
It
is
tuis
systems
with
bubbles in Chapter 4.
Published
values
of
air
compressibility
temperatures and pressures exist. At
twenty
at
degrees
various
centigrade
and 760 mmHg:
dV/dP
1.0126x10 - 5 Pa - 1
and
C = V dV/dP
(m 3 /Pa)
(34)
This is the equation we shall use to calculate bubble compliance.
2.5.2
Relationship
between
Bubble
Location
and
Resonant
Frequency
We will now consider how to include a bubble of known volume
and location in our
lumped
element
model.
The
inertance
and
resistance of the bubble will be taken as negligible, and we will
assume therefore that the bubble
may
be
modeled
compliance shunting the transmission line (we
do
distinction between bubbles clinging to the wall
as
not
of
a
lumped
make
the
any
tubing
and bubbles completely occluding
reason to do so).
the lumen,
although
We may now view the system as consisting of two
transmission lines in series: the first terminating
compliance
there may be
(the
bubble)
and
compliance (the transducer).
the
second
in
in
a
another
lumped
lumped
The analysis of section 2.4 may
now
be applied, yielding the following model:
(Z-x) Ctu
PS
where x=distanze from source to bubble and 2 is the total
length
of the system. The transfer function governing this circuit is
H(jw)
=
A
4-jBw'-Cw 2 +jDw+l
a fourth-order equation in w, where
A = x(Z-x)CIC 2 L2
B = 2x(X-x)RLC1 C2
C = [x(x-x)R 2 CIC +L(C +xC
2
2
1 )]
D = R(xC 1 + C )
2
c = ( 4 /r 2 )xCtu+Cbu
C2
(4/r
2 )(-x)
Ctu+Ctr
(35)
40
If we assume that Cbu is much larger than Ctu or Ctr,
zhn C,>>
and the two second-order circuits tend to be decoupled,
resonate independently.
The
resonant
section is approximately equal to
the bubble is advanced
in
(increasing x) the
primary
importance of this
result
the
tubing
the
of
towards
the
frequency
that
located in the catheter will not degrade the
system
as much as a bubble located
the
up
first
in
as
transducer
decreases.
recognition
farther
T.h
t1:ey
, so we see that
1//xLCI
resonant
is
frequency
i.e.
a
The
bubble
performance
system.
As
a
practical aside, we note that when there is suspicion of a bubble
causing a low resonant frequency,
the
begin at the transducer dome and then
the catheter.
search
proceed
should
generally
backwards
toward
41
CHAPTER
3
MATERIALS
All experiments carried out in
utilized a single brand of
tVDes of
the
pressure
course
extension
high-quality
catheterization
components
laboratories
as
and
addition to these components, a
by
intensive
number
of
interchangeably.
from
Because
of
several
the
they
represent
many
hospital
care
units.
additional
manufacturers
relative
stiffness
plastics used in these valves, their wide bore, and
contribution
considered
to
to
the
overall
significantly
system
affect
length,
the
two
partly
system
(three
were
used
of
the
their
they
In
elements
were required in the experimental setup. Hydraulic valves
and four-way stopcocks)
study
and
chosen
because
used
This
tubing
ressure tr nsducer. These elements were
on the basis of availability and partly
typical
of
small
were
response.
not
A
pressurized IV bag and standard fluid were used to flush and fill
the hydraulic system. Several different pressure sources and flow
sources were used as
test 'inputs.
A
pressure
amplifier, CRT-
display, tape and strip chart recorders, spectrum
converter,
and computer
system dynamics.
All
facilities
of
these
described in more detail below.
were
materials
used
and
analyzer,
D/A
analyze
the
components
are
to
3.1
Extension Tubing
The tubing used in
monitoring kit
(HP
No.
this study was obtainec
14233A) mark:3ed by
tubing is constructed of translucent,
from
a
pressure
iiew1eýt- Pckard.
high-density
polyethyle.ne,
with an internal diameter of 1.18 mm, outer diameter of 1.93
and length of four feet (1.22 meter).
The ends are supplied
one male and one female luer fitting. No
specifications
are
available
characteristic compliance is
measured
tubing is relatively stiff compared
manufacturer
this
for
in
to
market. A previous study by Gardner (1981)
for commercial pressure tubing lists
a
section
the
4.1.3.
This
brands
of
with
but
of static
range
mnm,
technical
tubing,
similar
The
on
the
compliances
1.6
mm3 /100 mm Hg for six feet of tubing, with an average
to
17.1
compliance
of 7.4 mm 3 /100 mm Hg/6 ft. In comparison, the measured compliance
of the H-P tubing in these units is 2.72 mm 3 /100
mm Hg/6 ft under
static conditions and 0.34 mm3 /100 mmHg/6 ft at high frequency.
3.2 Transducers
Two transducers were used in these experiments. The first, a
Bentley Trantec Model 800, was generally used as the primary test
element in the transducing system because it could be modeled
a pure compliance over a wide
second,
a
Hewlett-Packard
range
1290A,
of
static
exhibited
phenomenon at lower static pressures which was
pressures.
a
fluid
probably
movement of fluid trapped between the plastic diaphragm
quartz
transducing
element.
Since
this
phenomenon
as
The
leakage
due
and
to
the
manifests
43
itself
as
a
frequency-dependenc
compliance at low frequencies),
primary transducer
compliance
(largor
apparent
the 1290A proved undesirable
for thas. experiments.
Instead,
used as a reference transducer to monitor
input to the extension tubing. In
this
the
as a
129a A
tha
pressure
application,
was
at
the
where
the
transducer effectively shunts the pressure source, compliance
not an important issue. Specifications for the
transducers list maximum compliances of
mmHg,
0.04
Bentley
and
is
and
H-P
3 /100
mmn
0.15
respectively.
3.3 Flush Bag and Fluid
As in a typical hospital setup, the
equipped with a means of
standard IV bag was
filling
filled
with
flushing
and
a
monitoring
standard
3ystem
with
was
fluid.
fluid
A
(described
below) and pressurized to between 200 and 300 mmHg by means of an
inflatable bag holder. A length of
pltssic
large-diameter
and three-way valve connects the flush bag to
the
rest
tube
of
the
system.
The standard solution consisted of debubbled water, prepared
by vigorous boiling. A small quantity of soap solution was
as a wetting agent and all excess air was
purged
from
added
the
before pressurizing, thus ensuring a minimum of dissolved air
the fluid. These precautions, coupled with slow, careful
of
the
hydraulic
system,
were
found
to
be
of
of
fluid
or
in
filling
the
resulted in high values of compliance. This
system
solution,
in
filling
the
utmost
importance in excluding air from the system. Lack of care in
preparation
bag
the
invariably
developed
44
after noting tha
I:as
i4t
virtu~aly
impossible
to
completely
eliminate trapped air from the system aft-er filling with
was not believed to differ significantly from
saline normally
used
in
terms
of
its
the
saline,
physiological
inertial
and
viscous
properties.
3.4 Slow/Fast Flush Unit
A Sorenson Intraflo
flush
element
was
experiment involving the system response to
Intraflo and similar
units
provide
a
included
a
high
fast
for
the
flush.
The
impedance
channel
between the flush bag and the catheter, in an attempt to keep the
catheter tip free from
blood
clots.
A
parallel
low-impedance
channel can be opened manually to provide a large bolus of
for the
same
purpose
resulting from the
(fast
release
flush).
(closure)
The
damped
of
this
fluid
oscillations
valve
we.re
of
interest in this study.
3.5 Bench Equipment
Two H-P pressure
excitation to
and
amplifiers
processed
the
(model
78503C)
resultant
provided
signals
the
from
the
transducers. The amplifiers were modified for this experiment
provide a flat
bandwidth
out
to
measurements to be made beyond the
100
Hz,
normal
allowing
12
Hz
tubing was provided by, a blood pressure simulator
601). This simulator
features
a
square-wave
frequency
bandwidth
these amplifiers. The reference input excitation to the
of
pressure
(Biotek
output
to
for
Model
step
response measurements, nine selectable pressure waveforms (stored
in
read-only
memory),
manual
controls, and provision for an
systolic
external
and
diastolic
electrical
simulator dome has two ports which connect to
the
level
input.
fluid
The
column
via luer fittings.
Frequency measurements were made by applying a
source to the external input jack of the
white
noise
approximate
was
amplitude
small-signal
limited
Biotek
to
excitation.
noise
simulator.
10
The
white
mmHg
The
RMS
to
pressure
transfer
function (magnitude and phase measurement) between the
reference
and test transducers was determined
Spectrum Analyzer. If desired, the
stored on an H-P Model 3960
using
an
amplifier
Instrumentation
visual record produced on a H-P Model 78172A
H-P
Model
outputs
could
Recorder
Chart
3582A
be
and/or
a
Recorder.
A
diagram of the setup is shown in Figure 4-3.
Computational facilities (for modeling
primarily
of
an
H-P
Model
85
desktop
purposes)
consisted
computer,
which
has
hardwired BASIC as a programming language. The H-P 85 also has an
interface bus which allowed it to be used (in conjunction with
digital plotter) to
accept
spectrum analyzer and
digital
X-Y
plotter.
to
spectral
produce
For
and
time
high-quality
transient
data
from
a
the
graphics
on
the
circuit
analysis,
a
simulation program SPICE was run on an H-P 3000 series computer.
3.6 Flow Source
For direct measurements of tubing compliance as
a
function
of frequency, a specially designed sinusoidal flow generator
used. The
flow
source
consisted
of
a
commercial
was
microliter
46
syringe
(Hamilton model 70011)
plunger and is
accurate
which utilizes a tungsten wi re
to 0.01 microliter.
A
custom-madc
adapter allowed the syringe to be tightly coupled to
the
as
luer
system
under test.
The syringe barrel was rigidly mointed in an aluminum block.
The plunger was coupled to a dc servo motor through a
mechanical
linkage, bearing, and eccentric cam. This arrangement produced
sinusoidal motion of the plunger.
be
altered
between
0
and
0.8
The stroke cf the syringe could
microliter
by
varying
eccentricity of the cam with a linear feed screw,
The maximum frequency was conservatively set
adequate
plunger
needle
and
barrel,
to
limit
but
the
and ;i frequency
range of 0.1-14 Hz could be obtained by varying the motor
between the
a
this
speed.
friction
range
was
for our purposes.
3.7 Tap Generator
In order to test the system response to a
manually
impulse, a crude form of a "mousetrap" tapper
built. A schematic representation
Figure 3-1. The tapper
consists
of
of
the
a
was
tapper
platform
applied
designed
is
on
and
shown
which
in
the
pressure tubing is secured by two clamps into a milled channel. A
tubular steel spring is
spring arm passes
manually
through
its
retracted
resting
and
position,
'tubing (causing a flow impulse as fluid is
deforms
the
repeated
considered a potential problem, the cylindrical
a
The
displaced)
returns to rest. Because tubing failure from
wrapped with plastic tape to provide
released.
small
spring
amount
and
then
taps
was
arm
was
of
shock
47
absorption.
48
top view
latform
spring
tubing
lip
plastic
side view
~
Figure 3-1. "Mousetrap" tap generator
49
CHAPTER 4
METHODS AIND RESULTS
4.1.
Determination of Line Parameters
In order to determine the most
transducing system, it
physical
parameters
is
first
which
appropriate
neccessary
best
tnhese parameters can
then
be
to
characterize
components of the system. The measured or
inserted
for
the
determine
the
model
the
separate
calculated
the
into
examination, allowing comparison between the
values
model
frequency
under
response
of the model and of the experimental, system. In some cases
fluid inertance L) there was no
convenient
of
technique
(e.g.
available
for direct measurement, and some theoretical calculations were of
neccessity substituted for direct
(e.g.
flow resistance
flow) could
be
R) the
measured
observation.
at
value
but
the
zero
a.c.
In
other
frequency
value
could
estimated by reference to pulsatile flow theory. The
use
unavailable, would have been
of
of
a
flow
transducer,
considerable
(steady
only
be
theoretical
and experimental values for each parameter are compared
possible. Clearly, the
cases
whenever
which
benefit
in
was
this
study, since it would have allowed simultaneous a.c. measurements
of / pressure
and
flow.
Nevertheless,
the
d.c.
measurements
obtained represent a positive step towards quantitative
analysis
and verification of
pressure
monitoring system.
the
theoretical
models
for
the
50
4.1.1 Resistance
Theoretical
Calculating a theoretical value for Rtu actually
specifying the frequency of interest,
has shown the flow resistance to be
(see Fig. 2-2).
However, at
zero
Womersley
since
a
requires
function
frequency
(1956)
of -frequency
Poiseuille's
Law
applies, and one can write
tu -
Taking:
8nr
4
rn=0.001 Pa-sec
(approximate for water at 20 degrees C)
Z.=1.22 meter
r=0.59 mm
We calculate
Rtu=
2.56x10 1 0 Pa-s/m'.
Experimental
To measure
Rt,
a calibrated pressure source was
to one end of the column. The rate of flow through
was measured by collecting the effluent in
a
attached
the
column
flask
graduated
and noting the quantity of fluid collected in a given period of
time. Invoking the hydraulic equivalent of Ohm's
Law,
we
see
that
R =
P/Q
where P represents the source pressure and Q the rate of volume
flow. Because there is a source resistance associated with
flush bag and tubing, the resistance measured
by
this
the
method
51
represents
the sum of the source and load (tubing)
To correct for this,
same
technique
the source
without
the
resistance
fluid
resistances.
(measured
column
these measurements are shown in Table 4-1. The
to
the
the
attached)
subtracted from the total measured resistance. The
measured in this way is very close
by
WJas
results
value
of
for
theoretical
Rtu
value
based on Poiseuille's Law.
4.1.2 Inertance
The calculation of fluid inertance
is
aiso
dependent
on
whether a parabolic flow profile (the Poiseuille assumption)
valid or not. The equation for inertance of a cylindrical
is
tube
of constant cross-section, derived in section 2.2, is
L =
=
fluid mass/(cross-sectional area) 2
pZ/ rr
2
where a multiplying factor of
4/3
frequency to account for a parabolic
must
be
flow
values for fluid volume, density of water,
included
profile.
tubing
at
low
Inserting
radius
and
length, we find that
3
2
u = 1.088x10 9 Pa-sec /m
The value of Ltu at resonance will, in general,
lie
between this value and the zero-frequency value. Since
somewhere
we
are
interested in linearizing our model by choosing values for
the
line parameters that match the true
values
at
re sonance,
we
I
R
tu
Q(mm )
Q(mm /sec)
7850
735
.0981
5500
550
.2727
3250
325
.2923
3570
357
.2521
average
.2724 +/- .0201
R(bag+tubing)-R(bag only)
.1743
+/-
.0201 mmHg-sec/mm
(2.32 +/- 0.26) *10
10
Pa-sec/m
equivalent circuit:
Rbag
Pbag
Table 4-1. Tubing resistance measurements
R(mmHg-sec/mm
53
ne :a to know thi
e locazion of ;he natural frequency of our
model
before we can specify
a
(3a).
However,
vlue
since
n;
frequency in
this
equation
value
(27),
(3a)
and
(27).
iterative procedure occurs
L
using
equation
determine
ei4C
a
we need to resort
to
technique to determine a value
equations
for
for
So
L tu
long
that
as
new
an
natural
iterative
satisfies
convergence
both
of
(unproven in theory), there
the
are
no
compliance
of
anticipated difficulties.
4.1.3 Compliance
Theoretical
Determining a theoretical
value
for
the
plastic pressure tubing is quite difficult,
since
exhibits viscoelasticity (wall loss) and creep
the
tubing
(plastic
flow)
which makes the compliance complex and frequency dependent. The
classical
formula
equation (5) may be
for
compliance
cautiously
of
applied
a
thick-walled
and
reasonably valid at higher frequencies, where
expected
creep
wall loss) may be neglected. Substituting in these
values:
E
r
h
1
s
=
=
=
=
=
1.72
0.59
0.40
1.19
0.46
x10 8
mm
mm
m
Pa
we can estimate Ctu as
Ctu
= 3.25x10
m'/Pa
tube,
to
(but
be
not
approximate
Exper imental
To directly measure
Ctu(as well as
frequency, the flow generator
used. The system was
CtG
described
assembled
as
as a function
in
section
depicted
in
3.6
Figure
taking extreme care to exclude air from the system and
cause
large
errors
in
compliance
4-1,
solution
measurement.
The
transducer compliance was directly measured by calculating
ratio
of
pressure
applied
(see
volume
Table
increment
4-2).
The
to
observed
compliance
change
was
average
value
of
2.68x10 - 1 5 m 3 /Pa
manufacturer specifications for this
gave some degree
of
confidence
,
well
transducer.
that
the
in
relatively
independent of frequency over the frequency range tested,
an
was
tighten
all couplings, since even microbubbles in the filling
will
of
with
within
This
extraneous
the
result
sources
of
compliance had been eliminated.
The
total
compliance
of
the
parallel
combination
transducer and extension tubing was then measured between
and
13.2 Hz,
and Ctu calculated
of
0.12
as
Ctu :C total -C tr
where
.Ctuis taken to be independent
of
frequency.
A
strong
frequency dependence was noted (see Figure 4-2) with the tubing
becoming increasingly stiff as the frequency was increased.
curve-fitting and extrapolation,
Ctufor very low and very high
frequencies could be estimated. A high frequency compliance
1.95
x10
14
m 3 /Pa
By
of
(the curve has apparently reached a limiting
value by 13.2 Hz) was used for
this
represents
resonance,
the
value
all
in
subsequent
the
models,
since
range
around
frequency
and for the purposes of this study we
inaccuracies
in
can
tolerate
component values at low frequency.
One assumption which should be justified at this point, is
that the frequency range over
which
these
measurements
made is well below the resonant frequency of the
assumption is critical because the effect of
and reflections near resonance
results
in
systeim.
tbinJ
a
change
effective compliance (i.e. the tubing can no longer be
as a lumped element).
Fortuitously,
the
compliance
tubing reached a limiting value well below the assumed
frequency of the system (.fn>50 Hz),
were
This
iLpe
n4
in
t:he
treated*
of
this
natural
so treating the tubing as a
lumped comrpliance in this measurement appears to be justified.
to
motor
syringe
3
A
to amplifier
stopper
4 ft tubing
transducer
Figure 4-1. Setup for direct compliance measurement
Stroke
(aV)
Test
transducer
only
transducer
0.04 mm
0.20 mm
Frequency
(Hz)
P
(mmHg)
C
(mm3/100mmHg)
0.20
112
0.0357
1.25
115
0.0348
1.60
115
0.0348
2.75
115
0.0348
4.50
117
0.0342
0.12
11.6
1.724
0.92
16.4
1.219
1.20
18.4
1.087
1.83
22.0
0.909
3.00
30.0
0.667
4.05
37.4
0.535
5.25
44.0
0.455
13.20
70.0
0.290
+ tubing
Table 4-2.
Tubing compliance measurements
I
58
×I014
+: measured
20.0
19.0
Ctu 18. 0
I
(m3/paP7 . 0
16.0
15. 0
14.0
JL
II
t 7..
,..2____fl!
! I
:i
I '
t
I
I I
i ,
.,
f 'I I, ...
_.
_
I
I
~~
L
,
i
brr
13. 0
t~~
L
-
I
ar
I
i
'
12. 0
11. 0
10.0
I
9.0
8.0
I
__
7.0
6.0
-I
f
.I_
I
2
3
I
_
i
5.0
4.0
3.0
2.0
1.0
0.0
0
1
4
5
6
7
8
9 10 11 12 13 14 15
f(Hz)
Figure 4-2.
Tubing compliance vs frequency
59
4.2 Determination
of
Resonant
for
Frequency
Bubble-free
the
System
4.2.1 Experimental
To test the validity of the lumped-parameter model in terms
the
of predicting the resonant frequency of
the experimental frequency
response
was
monitoring
compared
system,
with
theory
using the measured or inferred values of the line parameters. The
system shown in Figure
response measured using
4-3
the
was
assembled,
white
noise
and
frequency
the
source
and
spectrum
analyzer at a static pressure of 50 mmHg. The system was verified
as being bubble-free by remeasuring the frequency response
at
static pressure of 150 mmHg and noting no apparent change in
a
the
frequency response. Because the compliance of an
air
bubble
is
strongly
the
effect
of
dependent
on
the
applied
pressure,
increasing the static pressure on a system with bubbles will be a
decrease
in
the
effective
compliance
and
thus
a
shift
in
resonance to a higher frequency, as has been shown by Henry et al
(1967). The gain and phase of the transfer function at both values
of static pressure is shown
in Figure
4-4.
We
note
that
artifact is present at about 56 Hz. This artifact, which is
in all spectra presented in this work, is the result
noi/se generated within the pressure
amplifiers.
of
This
which was of major importance in our study.
at
a
seen
carrier
noise
unavoidable since the amplifiers are being used far beyond
design bandwidth, but fortunately did not occur
an
is
their
frequency
60
white noise
source
BIOTEK
to pressure
amplifier
reference
trans ducer
3-way
valve
pressure
amplifier
4 ft. extension
tubing
test
transducer
Figure 4-3. Diagram of laboratory setup for testing
of catheter system
61
It
IM
I
in
mmHg
as
-i•
-2
LO6
I_
5.8
438
omHg
18
L8
-I
-18'
-273
r-oww
-rlo
Figure 4-4. Transfer function for the bubble-free system
4.2.2 Theoretical
To locate the resonant frequency of The second-order
we need to si3.ultaneously solve equations (3) and
and
Ltu
(Cou
and
2.68xi0 - 1 5 m/Pa
Ctr
are
assumed
constant
and
at
for
Rtu
19.5
and
, respectively). Starting with an initial guess
at the resonant frequency of the model allows
and L,
(27)
model,
the
two
sets
of
equations
calculation
can
then
be
of
solved
iteratively until R and L (and wn) converge. Convergence has
been shown to be guaranteed, but we experienced
no
R
not
difficulties
in computing a convergent solution. The values for R, L, fn,
and
D (damping coefficient) found by this method are:
fn = 45.78 Hz
R1tu
5.561
x1010 Pa-sec/m 3
Ltu=
1.265
x10
D
Pa-sec2/m 3
= 0.0761
The frequency response
of
the
second-order
system
with
this
natural frequency and damping coefficient is shown in Figure
4-5
with the experimental curve superimposed. The agreement is fairly
good considering the many approximations involved in constructing
the model and the possible errors in measuring the compliance
the tubing (extraneous sources of compliance
measurements high). Our check on
equation
(30), becomes
r
>>
2 tan;o
pr
0
the
tend
assumption
to
of
make
low
of
our
loss,
63
287.6 rad/sec >> 43.9 rad/sec
which is not a bad approximation.
64
"r
.0
II
/,M
I
solid
dashed
90
1
der
model
1.0
I
-§0
S(Hz)
~
~24,
-279d
PUU
PjAlURW
JU%/
Figure 4-5. Comparison of second-order model with experimental
transfer function
65
4.3 Air Bubble Experiments
4.3.1 Resonant Frequency as a Function of Bubble Location
The development
of
section
2.5
has
suggested
that
the
presence of a bubble in the pressure tubing may be represented by
a parallel lumped compliance introduced at the appropriate
in the
transmission
simplification
line
was
then
model
to
of
the
system.
theoretically
transmission line into two sections, pre-
and
model each section as a second-order circuit.
cascade of two
R,L,C
circuits
fourth-order equation in w (see
desirable
separate
the
post-bubble,
This
,;hose ;ransfer
equation
A
point
and
leads
to
a
function
is
a
(28)).
The
frequency
response of this fourth-order system exhibits two resonant.peaks,
a dominant (primary) peak at a lower frequency
and
a
secondary
peak at a higher frequency. Since from a practical point of
it is the primary resonance which is of interest, the main
of this theory is that, for a bubble of given size,
resonant frequency decreases as the bubble
is
the
advanced
view
result
primary
in
the
tubing towards the load (transducer) end.
To test this theory, we measured the frequency
response
of
our standard system as a bubble of known volume was
advanced
in
the tubing by means of controlled flushing. The response function
of the system was measured using the 3582A Spectrum Analyzer
and
noise source as in the previous experiment. First, the system was
carefully
filled
with
the
standard
solution.
The
frequency
response was then measured at static pressures of 50 and 150 mmHg
and compared to verify the absence of any bubbles. Next, a bubble
66
of known volume at atmospheric
3 ) was introduced
mm
pressure (29.0
at the Biotek dome and advanced
through the fluid column by
slow
flushing. The frequency response of the system was Measured
the bubble located at various points (20, 40, 60, 80,
along the fluid column.
The
response
functions
Figure 4-6(a)-(f). To compare these results with
section
2.5.2,
plotted
in
Figure
4-6.
damping
at
higher
frequencies
the
theoretical
Note
response
that
the
tends
to
and 122 cm)
are
the
with
shown
in
theory
of
functions
are
also
higher-than-predicted
obscure
resonant peak in the magnitude plot, but the phase
the
plot
second
clearly
shows the 180 degree phase shift around the natural frequency.
For the tubing line constants, Rtu and Ltu were
allowed
to
vary with the excitation frequency according to equation (3)
and
the bubble compliance was calculated using
equation
(34) .
HIote
that the model tends to overestimate the high frequency resonance
and underestimate the damping. There is
for
the
former
result.
The
latter
no
is
obvious
most
nonnegligible wall loss (viscoelasticity) in
our
model
ignores.
The
qualitative
explanation
likely
the
agreement,
due
tubing,
to
which
however,
is
bubble
as
"decoupling" the fluid line into two sections, each of which
may
encouraging,
for
it
supports
the
view
of
the
be approximated by a lumped second-order circuit.
I Ilmlt•
II
in
dows
a8
t8
-27
no bubble
-306
I
-j
•)Ust
(a)
J.JuLI;
.
2.8
LI
0.
-275
PAi
WUVJ
x= ZO
cm
(b)
Figure 4-6.
r
lA
dashed line: theoretical
4.8
Jiu
iru
eqermentaLLLal
Model and experimental transfer functions
68
CJI
I II"fl
4.B
.artifact
2.1
LO
-273
-USI li
J%--rv
%WLL
(c)
5.8
4.8
10
rtifact
LI
1.8
-271
x=-6 0
rAa
cm
r AM= Mi
(d)
1n
r
Illff%3
4.08
3.8
2,8
artifact
/
-18
-278
-3Hi 1 -4
x=80
(e)
5.10
4.
3.0
artifact
LI
Ll
-98
-118
-278
-1ZF/C
tm
rlPim WwIY
(f)
4.4 Tap and Flush Experiment
The theoretical development of Chapter 2 has suggested
that
most catheter systems may be approximated by a limited number
of
lumped sections, the number of sections neccessary
on
bubble distribution within the
system
but
d-epending
perhaps
practically
limited to two or three. This lumped model has had the feature of
simplifying system analysis at the expense of absolute
accuracy.
In terms of deducing the system response function H(w)
from
the
time response to a known input (typically a pressure step at
the
patient end of the catheter), many
the
researchers
lumped second-order model to be adequate.
have
This
is
shown
because,
most cases, even a rough knowledge of the location of
resonant frequency and damping coefficient may be
judge the adequacy of the system or
compensation. via
impedance
even
matching.
to
the
Often,
first
sufficient
attempt
in
to
frequency
discovery
of
a
lower-than-expected resonant frequency may result in a search for
occult air bubbles rather than inverse filtering of the
waveform or other
electrical
technique
which
pressure
requires
fairly
precise knowledge of the transfer function.
The clinical unsuitability of the "pop" technique and
other-
inputs which require access to the catheter tip led us to examine
two alternate methods of exciting the natural frequencies of
the
system which do not require withdrawal of the catheter. The first
method, tapping the extension
tubing
with
a- small
device, represents an attempt to provide an impulse
mechanical
of
to the system. The other, approximating a pressure step,
pressure
is
the
"flush"
technique
technique is
quite
recommended
by
Gardner
convenient
in
that
(1970,1981).
the
flush
This
device
normally present as part of the monitoring setup. We tested
of these excitations
on
debubbled
systems
as
well
is
both
as
those
containing a bubble to determine two things:
of
the
(2) Is the primary resonance of the system, as seen from
the
(1) Is the time or
frequency
independent
response
location of the input?
normal pressure source (the catheter
tip),
sufficiently
excited by the input to be detected?
4.4.1 Experiment
The experimental system set up to answer these questions
depicted in Figure 4-7. Two sections of the H-P
were connected in series
with
bubble to be introduced into
a
the
three-way
side
pressure
valve,
port
of
is
tubing
allowing
the
valve. This was done because the side port is a typical
a
three-way
site
of
bubble entrapment and also to prevent the bubble from being swept
out of the tubing during the "fast-flush".
A
Sorenson
flush unit was connected between the transducer and
normal clinical location),
and
the
Biotek
Intraflo
tubing
pressure
(the
simulator
connected to the other end of the tubing.
The system was filled with
the
standard
solution
in
usual manner, and the transfer function measured at Pstatic =
mmHg and again at 150 mmHg to verify the absence of
trapped
(see Figure 4-8(a)). The response of the system to the
the
50
air
following
four excitations was tnen tested:
(1) Biotek square wave (the model excitation)
(2) Tap at x=20 cm
(3)
Tap at x=224 cm (1-x=20 cm)
(4) Flush
Typical response waveforms are shown in Figure 4-9(a)-(d).
The lack of accurate calibration of the force
delivered
by
the tap device, possible motion artifact of the tubing during the
tap and flush
rigorous
procedures,
analysis
of
and
these
other
problems
responses
serve
exceedingly
to
make
difficult.
Nonetheless, the following qualitative analysis may be useful:
(1) The Biotek
step
response
was
artifact free. Determination of resonant
from the peaks of the
time
response
was
consistently
frequency
the
and
relatively
most
damping
easy
and
matched the spectrum analysis determined previously;
(2) The fast-flush response
although care had to be taken
to
was
relatively
avoid
artifact-free,
disturbing
the
tubing
during the maneuver. The initial "spike" artifact in the response
could not be used for analysis;
(3)
The tap response invariably produced some high-frequency
oscillation which made
determination
of
f
n
and
D
much
more
difficult. It is not known whether this is due to motion artifact
of the tubing, phase
cancellation
resulting
from
a
secondary
wavefront (the impulse can propagate in both directions away from
the tap),
or perhaps even another mode of wave propagation in the
73
trans
ns ducer
bubble
to
pressure
amp
w
"pigtail"
valve
to flush bag
Figure 4-7. Diagram of setup for tap and flush experiments
74
in8
II"ilf
i
window= 2 sec
4,0
ndows
3.8
2.8
L8
-•8
-188
-278
ree system
-MLp#W(&)
(a)
4.8
318
2.8
L8
-18
-278
phse(de)
with bu bble
(b)
Figure 4-8.
Transfer function with and without bubble
75
*
mmHg
mmHI
160
120
80
40
0
-40
-80
-120
-160
.2
.4
.6
(a)
.8
sec
mmH
1.0
sec
(b)
mmHI
sec 0
%VFU
(c)
Figure 4-9.
(d)
Square-wave, tap, and flush responses for bubble-free
system
tube wall itself. We also observed that the tap
greater in amplitude as the tap site
was
response
moved
closer
became
to
transducer, possibly because of the attenuation produced by
the
wall
loss in the line.
A considerable change in the responses to these
occurred when a bubble was inserted into the
side
three-way stopcock connecting the
two
transfer function of
system
with
bubble
decrease
in
resonant
exhibits
Lhe
the
predicted
lengths
appearance of a second frequency peak. The
produce a response from which the lower
port
of
of
the
tubing.
The
(Figure
4-8(b))
frequency
Biotek
resonant
damping were easy to deduce (Figure 4-10(a)). The
(Figure 4-10(b)-(c)) unfortunately are dominated
excitations
and
continued
to
frequency
tap
by
and
responses
oscillation
at the higher resonance, leading to the tentative conclusion that
the tap is an inadequate excitation for the primary resonance.
The flush response has components at both the secondary
and
primary frequency, as is seen in Figure 4-10(d). However, further
experimentation revealed that the high frequency response is
not
due to the flush itself. Rather, it is due to the manner in which
the flush valve in
the
Intraflo
closes.
The
release
"pigtail" which opens the valve actually produces a flow
in the tubing as
the
rubber
membrane
which
valves
of
the
impulse
the
flow
reseats. This can be shown by repeating the flush experiment with
equal pressures in the flush bag and tubing. Upon release of
flush valve,
the
piston-like
action
produces the high-frequency response
of
shown
the
the
rubber
membrane
in
Figure
4-10(e).
77
Comparing Figures 4-10(d) and (e),
it appears that most,
if
not
all, of the high-frequency oscillation produced by the fast-flush
is due to the mechanics of valve closure in the Sorenson unit and
not from the initial conditions
set
up
Unfortunately, other makes of flush device
by
steady-state
were
not
determine if they were more suitable for this purpose.
flow.
tested
to
78
mmHg
I-
0o
Sea
8se
(b)
(a)
__
I
tap @ x=224 cm
=H
S160
120
80
40
01
IA .
-40
-80
L20
.-1
160
f
" ,,
ll
"I
0
, I,
.2
,
I
,
,
,.
,
.4
.6
(c)
,.
I
.
.
I. ,
1.0
.8
see
Figure 4-10. Square-wave, tap, and flush responses with
hnhhl_ midwn v in
t-h f11id 1in t
HmmH
mR!
see
(d)
8
vwe
(e)
8
CHAPTER 5
DISCUSSIOJ
We have gone
to
considerable
lengths
in
this
study
to
develop a lumped-element model for the catheter system that is
good
approximation
approach through
to
the
the
more
first
accurate
resonant
transmission
frequency,
experimentally demonstrate its validity. It may
and
be
a
line
then
argued
to
that
this analysis was not really neccessary to achieve our
purported
objective of finding an in vivo
frequency
method
of
measuring
response. We feel, however, that the effort put into modeling has
at least paid off in developing a rationale for understanding the
relative importance of tubing, transducer, and bubble compliances
in determining the primary resonant
frequency
given dimensions. The analysis has also
range of component (R, L, and C)
of
served
values
for
a
to
system
suggest
which
the
of
the
lumped
approximation is a good one.
Having justified the use of a lumped second-order
represent the catheter system
tested
in
this
proceeded to demonstrate how an occult bubble
model
study,
causes
we
than one frequency. This result is crucial to
of the role that location and energy
response to tap and
flush
results in more detail.
inputs.
resonances
our
distribution
We
can
now
then
decoupling
of the fluid column into pre- and post-bubble sections, with
result that the column may have significant
to
at
the
more
understanding
play
examine
in
the
these
81
5.1 Tap and Flush Responses
The results of section 4.4 strongly
excitation leads to response
surprising
for
two
artifacts.
reasons:
first,
suggest
This
the
that
is
the
not
impulse
tap
entirely
has
to
be
transmitted through the tubing wall, which may cause longitudinal
wave propagation in the wall, temporary narrowing
due to relaxation effects,
and
other
wall
of
the
related
lumen
phenomena;
second, the impulse is a signal which requires very large
forces
to transfer significant energy to the system - the large pressure
variations in the tubing make
transmission
nonlinearities
likely to be evident in the response. Even though
we
an alternate flow impulse source - the flush valve
more
discovered
-
which
eliminate some of the problems of the "mousetrap" method
may
(tubing
effects) it still seems impossible to circumvent the nonlinearity
problems.
Another
problem
with
the
tap
sensitivity. As we have seen, pre-
excitation
and
is
post-bubble
location
excitations
give qualitatively different responses. This may possibly be
the
result of reflection from the impedance mismatch existing at
the
bubble
The
and
high-frequency
impedance mismatch will
attenuation
tend
to
limit
in
the
the
tubing.
amount
of
energy
transferred through the bubble and thus isolate the two halves of
the system. The attenuation makes
appreciable
Moreover,
response
the
energy
without
is
very
it
difficult
using
extremely
localized
to
within
achieve
high
the
an
forces.
tubing,
intuitively making the lumped-element model seem unsuitable.
All
of these considerations combine to make the tap an unattractive
82
method of excitation.
On the other hand,
we have found evidence
to
the flush excitation (although perhaps not the
device tested) may give
results
technique to justify its use in
similar
the
sugges:
particular
enough
clinical
to
and
circuit shown
previous
energy
below,
catheter
considerations.
which
system
First,
represents -a
containing
a
d.c.
"pop"
We
can
using
the
consider
the
justify the similarity of the flush and pop responses
lumped model
flush
the
setting.
that
model
bubble
of
our
mid-tubing
(component values typical of our experimental system):
C1 =3.1*10
-13
C2 =6.2 10-15
Vs
S
V
S
where we have substituted V and I for P
and
Q
respectively
to
avoid confusion between electrical charge and hydraulic flow.
We can consider the pop and flush excitations as setting
certain initial conditions in the
line
system to decay. This corresponds to the
solution to the
coupled
differential
and
then
allowing
homogeneous
equations
the
(unforced)
governing
fourth order circuit:
(L 1 C1 s 2 + R1 C1 s -1)V1 + (L1 C2 S2 + R1 C2 s)V = 0
2
-V 1 + (L 2 C2 s 2 + R2 C2 s +l)V
up
2
= 0
the
83
5.1.1 Initial conditions
The initial conditions set up by the pop excitation are:
hl 1=
2=bIt = o
V1 V 2 = Vs
which means that all the energy is stored in the compliances. For
the flush, we have
I=I2=:
V
s
Ri+R 2
Is ;
Ibdt=
r 0
V2 = 21 Vx dx = Vý/12
0 .
V=2fVV 2x
0S
x = 7V2/12
S
where we have represented the pressure as a
the distance x along the fluid column and
linear
computed
function
the
square of the pressure within each section. The reason
of
average
for
this
will be evident shortly.
Now, we can determine the approximate distribution of energy
within the system under each set of initial conditions. The total
energy stored in each section is
E
EC V
EL
For the pop excitation, there is no flow,
independent of location.
i
and
1,2.
the
Since C1 >> C2 (commonly the
pressure
case
is
even
84
with small bubbles),
the majority of the energy in
stored in the bubble. Therefore the
response
is
the system
is
primarily
the
decay of the left-hand RLC circuit. Since
the
lowest
tends to be associated with
this
means
the
bubble,
resonance
that
the
response will in most cases be approximately second order.
For tihe flush, there is kinetic energy in the
and potential energy in
the
compliances.
Using
fluid
the
motion
component
values shown,
V 2L
= 0.5LI
Ekin= Ek
kin
kin 2
1
= 0.5 sL
R2
4.8xlO-13V 2
s
4
Ept
0.5C V
E pot
0.5C2 V
2
pot 1
1 1
=- (1/24)C V
1 s
= 1.3x10
(7/24)C 2 Vz = 1.8x10
We see that for the component
values
14
s
1 5V
chosen
(typical
system we have studied) the majority of the energy is
the fluid motion
and
not
in
the
compliances.
for
the
stored
Therefore
in
the
initial energy stored within each section is approximately equal.
5.1.2 Transient solution
Given
these
initial
conditions,
transient solution to the ideal pop and
we
can
flush
determine
inputs.
However,
this will require recomputing the line parameters R and L,
these are frequency dependent.
These
observed location
of
and
height
the
were
estimated
resonant
peaks
the
since
from
the
in
the
experim:ental data (Figure 4.7(b))
as:
R= 1.5x10 1 0 pa-sec/ m
10Pa-sec/rn
3
R =
2.Sx10
L=
3
7.0x108 Pa-secimn
L = 6.3x10 8 Pa-secmn3
2
anrd
values
parameter
A transient solution with these
equivalent
initial conditions determined from the
d.c.
was determined using the SPICE circuit
simulation
the
circuit
program.
The
pop
test
initial canditions is shown in Figure 5-1 and for the flush
test
the
transient solution from 0 to 400 milliseconds for
in Figure 5-2.
expected,
As
test
pop
the
almost
causes
entirely low-frequency resonating response, while the flush
yields
a
response
that
is
a
of
mixture
the
response,
is easy to distinguish,
the
particularly
since
oscillations are quickly damped. This is
because the damping increases
damping is defined
as
with
guarantees that it
oscillations, so
we
frequency
loss/cycle,
damping and faster oscillation of
will
may
decay
to
not
the
faster
reasonably
be
test
low-
high-frequency resonances. The low-frequency
an
and
however,
high-frequency
expected,
and
both
because
loss/time.
The
high-frequency
the
higher
resonance
than
the
low-frequency
expect
the
low-frequency
response to dominate. This lends some support
to
our
assertion
that the flush excitation is a satisfactory input for determining
the low-frequency resonance. Rothe and Kim (1980)
have
observed
"86
th.at the flush waveform produced by
Intraflo
valve
to-transducer
excited
part of
system including
snap
primarily
their
catheter.
the
catheter
While
we
of
the
Sorenson
extension-tube-
system,
have
the
not
not
the
entire
tested
entire
systems containing long catheters, it seems likely,
in
view
of
our test results and analysis, that it is the flow impulse caused
by
the
snap
of
the
Intraflo
valve
that
produces
high-frequency oscillations noted in this study and
in
the
the
;irk
of Rothe and Kim. The flush method itself appears to be otherwise
sound.
How may these results be extended to more general siz-Iions
(different component values, more bubbles in the tubing)?.e
have
outlined a general method for attacking this problem although
we
have only computed a
system.
It
system
in
solution
for
one
particular
should not prove too difficult to model any specified
terms of lumped sections in
extension of this work
would
a
similar
be
:nanner.
to' determine
An
a
interesting
"worst-case"
response - one that contains resonances close enough together
make the decoupling assumption poor.
to
87
decoupled circuit
20.0
DA321-HPPPICE VERS
with pressure step at
3e .1
input
12:12P 13JAN82
0.0
0.0
100.0
TIME
Figure 5-1. Simulated response to pressure step at input
200.0
- 3
10
deco
d
u p 1e
1 i rcuitI
..i th
flush s•,urce
40.0
0.0
0
0
T IME
Figure 5-2.
Simulated response to fast-flush
89
5.2 Extraction of Resonant Frequency and
There is
some reason
(i.e. one without
valve
;c believe that
closure
into
the
a
good
artifact)
designed. What seems to be required is
force a bolus of fluid
Damnping from Fast-flush
a
fluid
flush
exists
valve
or
that
column
as
source
can
does
it
be
not
closes.
However, even if such a device is made available, there are still
a
number
of
practical
considerations
to
be
considered
attempting to calculate resonant frequency and damping
in
from
the
Foremost among the anticipated difficulties in clinical
use
flush response.
of
the
fast-flush
for
determining
resonant
separating the flush response from the blood
Gardner (1981)
frequency
pressure
has demonstrated the use of the
waveform.
Sorenson
in
clinic. The flush valve is released during the diastolic
of the cardiac cycle where the
slowly. A
recording
determination of
of
the
resonant
pressure
response
frequency
is
is
is
changing
then
the
portion
the
most
examined.
relatively
damping requires a careful separation of the
is
The
easy,
superimposed
but
flush
and patient waveforms.
We believe that the flush response analysis may be performed
by digital computer with high reliability, relieving the hospital
personnel of most of the burden of analysis. What is required
some method of patient pressure waveform prediction.
This
is
would
allow subtraction of the predicted waveform from the superimposed
patient and flush waveforms, resulting in a time-domain filtering
of the flush
response
from
the
combined
signal.
Statistical
90
methods of waveform prediction exist, so there is
believe that this
computation
could
not
be
no
performed.
algorithmic devices would have to be designed in
the
computer
calculation
reject
reason
artifacts,
order
to
Other
to
make
particularly
the
high-frequency oscillation that we have shown may occur even with
an ideal flush source.
5.3 Anticipated Usage and Clinical Acceptability
The
issue
of
clinical
acceptance
of
catheter
system
measurement/correction devices that require user intervention has
been raised (Doherty, 1981).
require no user
intervention
Distortion
at
all
Doherty and by others (Brower 1975,
have
extremely
systems
been
described
attractive,
distortion
but
described have been shown to reliably
the
by
none
automatically
of
estimate
the
methods
distortion
the entire range of catheter systems in use and patient
waveforms. Moreover,
that
Jackson et al 1980). The idea
of determining and correcting waveform
is obviously
analysis
central
assumption
methods, that the pressure waveform
spectrum
pressure
underlying
does
over
not
these
contain
local maxima simulating a resonance, is open to. some
criticism.
Whether a method which requires user
acceptable
depends in part on how and when the
interaction
method
is
is
intended
to
be
part
of
used.
Certainly the calculation of dp/dt is an important
many
catheterization
high-fidelity
procedures
system.
frequency response
will
However,
not
be
that
a
requires
catheter
suitable
an
system
for
extremely
with
accurate
poor
dp/dt
91
measurements even if it is compensated.
Mechanical
compensation
can only extend the frequency range a limited amount, and inverse
filtering methods (Ciccolella 1976) cannot be expected to perform
well far beyond resonance where the deviation
response becomes
large.
Therefore,
from
second-order
determination
of
resonant
frequency and damping is probably more important to the physician
or technician than elaborate compensation methods when
measuring
dp/dt.
preferred
Presently,
catheter-tip
manometers
are
the
instruments for this measurement, and until fluid-filled
systems
can reliably achieve
Hz
flat
frequency
beyond, this will probably remain the
response
case.
to
Even
100
and
if .debubbled
systems become the norm, this will be difficult to achieve
given
the physics of the fluid column and the need for flexible
tubing
of reasonable length.
Simple
pressure
bandwidths of dp/dt
monitoring
measurements,
does
not
making
much more attractive. The most common and
require
the
fluid-filled
noticeable
large
systems
effect
of
low resonant frequency and low damping on the pressure signal
is
systolic overshoot. If the measurement is being made in a central
artery or in the heart, systolic overshoot may falsely indicate a
valvular lesion or disease, or high peripheral resistance.
peripheral artery, the overshoot
may
falsely
trigger
alarms. It is with these "suspect" systems that a
In
a
pressure
direct
method
of determining the system resonant frequency and damping is
most
needed. In the busy clinical environment, a catheter system
that
seems to be reproducing the
will
pressure
waveform
accurately
probably not be tested for adequate frequency response using the
92
fast-flush or any other technique. However,
pressure waveform (damped or
should be
able
abnormalities
to
and
given
resonant-looking),
differentiate
between
catheter-induced
a
suspicious
the
fast-flush
real
blood-pressure
distortion
reliability. An occasional flush test does not seem
price to
pay
for
pressure waveform.
higher
confidence
in
the
with
too
displayed
high
high
a
blood
93
94
BIBLIOGRAPHY
1. Brower, R.W., et ai: A fully automatic device for
compensating for artifacts in conventional
catheter-manometer pressure recordings. Biomed. Engng.
10:305-310, 1975.
2. Bruner, J.M.R.,
et al: Comparison of direct and indirect
methods of measuring arterial blood pressure, part 1. Med.
Instrum. 15:97-101,
3. Ciccolella, S.A.:
1981.
Compensation of fluid-filled catheter
response using digital filter techniques. Master's thesis,
University of Rhode Island,
1976.
4. Falsetti, H.L., et al: Analysis and correction of pressure
wave distortion in fluid-filled catheter systems.
Circulation 49:165-172, 1974.
5. Fox, F.,
et al: Laboratory evaluation of pressure transducer
domes containing a diaphragm. Anesth. Analg. 57:67, 1978.
6. Frank, 0.: Kritik der elastichen manometer. Zschr. Biol.
44:445-61 3 , 1903.
7. Gabe, I.T.: Pressure measurement in experimental physiology.
In Cardiovascular Fluid Dynamics Vol. 1. Edited by Bergel,
D.H..
8.
New York: Academic Press, 1972.
Gardner, R.M.,
et al:
Catheter-flush system for continuous
monitoring of central arterial pulse waveform. J. Appl.
Physiol. 29: 911-913, 1970.
9. Gardner, R.M.: Direct blopod pressure measurement - dynamic
response requirements. Anesthesiology 54:227-236, 1981.
10. Geddes, L.A.: The Direct and Indirect Measurement of Blood
Pressure. Chicago: Year Book,
11.
Hansen, A.T.:
1970.
Pressure measurement in the human organism.
Acta Physiol. Scand.
12. Hansen, A.T.,
19 (Suppl. 68):1-227, 1949.
and Warburg, E.: The theory for elastic
liquid-con3aining membrane manometers. Acta Physiol. Scand.
19 .(Suppl. 65):306-332, 1949.
13. Henry, W.L.,
et al: A calibrator for detecting bubbles in
cardiac catheter-manometer systems. J. Appl. Physiol.
23:1007-1009, 1967.
14.
Jackson, L.B., Jaron, D.,
and Wood, S.L.: Compensation of
fluid-filled catheter pressure waveforms by linear
predictive analysis and digital inverse filtering. Proc.
Fifth New Eng. Bioengng. Conf., Apr.
15.
Jager, G.N., Westerhof, N.,
1977.
and Noordergraaf, A.: Oscillatory
flow impedance in electrical analog of arterial system.
Circ. Research 16:121-133, 1965.
16.
Krovetz, L.J., et al: Limitation of correction of frequency
dependent artefact in pressure recordings using harmonic
analysis. Circulation 50:992-997, 1974.
17. Lambert, E.H.,
aid Wood, E.H.: The use of a resistance wire
strain gauge manometer to measure intraarterial pressure.
Proc. Soc.
Exper. Bio. & Med. 64:186-190, 1947.
18. Lambossy, P.: Oscillations forcees d'un liquide
incompressible et visqueux dans un tube rigide et
horizontal. Helv. Phys. Acta 25:371-336, 1952.
19.
Lapointe, A.C.,
and Roberge, F.A.: Mechanical damping of the
manometer system used in the pressure gradient technique.
IEEE Trans. Biom. Engng. 21:76-77,
1974.
20. Latimer, K.E.: The transmission of sound waves in
liquid-filled catheter tubes used for intravascular
blood-pressure recording. Med. & Biol. Engng. 6:29-42,
1968.
21.
Latimer, K.E.,
and Latimer, R.D.: Measurements of
pressure-wave transmission in liquid-filled tubes used for
intravascular blood- pressure recording. Med. & Biol.
Engng. 7:143-169, 1969.
22. Li,
J.K.-J., van Brummelen, G.W., and Noordergraaf, A.:
Fluid- filled blood pressure measurement systems. J. Appl.
Physiol. 40:839-843, 1978.
23. McDonald, D.A.: Blood Flow in the Arteries. London: Arnold &
Co.,
1960.
24. Melbin, J.,
and Spohr, M.: Evaluation and correction of
manometer systems with two degrees of freedom. J. Appl.
Physiol. 27:749-755, 1969.
25. Roark, R.J., and Young, W.C.: Formulas for Stress and Strain
(p. 104).
26. Rothe, C.F.,
New York: McGraw-Hill, 1975.
and Kim, K.C.: Measuring systolic arterial blood
pressure. Critical Care Med. 8:683-689, 1980.
27. Shapiro, G.G, and Krovetz, L.J.:
Damped and undamped
frequency response of underdamped catheter manometer
systems. Am. Heart. J. 80:226-236, 1970.
97
28. Shinozaki, T.,
et al: The dynamic responses of liquid-
filled
catheter systems for direct measurement of blood pressure.
Anesthesiology 53:498-504,
1980.
29. Vierhout, R.R.: The response of catheter-manometer systems
used for direct pressure recording. Ph.D. thesis,
University of Nijmegen, Holland, 1966.
30. Yanof, H.M., et al: A critical study of the response of
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31.
Womersley, J.R.: Method for the calculation of velocity, rate
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98
APPEI1DIX A
Table of AnalIogous ElecL ri cal and Hydraulic
Electrical
'-Electrical
Voltage
Hydraulic
H1ydraulic
Pressure
V
Charge
(coulombs)
Current
I=dq/dt
(A)
Power
VI
(W)
Resistance
R=V/I
(ohm)
Inductance
L=V/(dI/dt)
Capacitance
C=I/(dV/dt) =q/V
(F)
Kinetic Energy
1/2LI
(J)
Potential Energy
1/2CV
(J)
Resistance/m
R'=R/I
(ohm/m)
Inductance/m
L'=L/1
(H/m)
Capacitance/m
C'=C/l
(F/m)
P=F/A
(N/m2 or Pa)
Fluid Displacement
X=Ax
(m3 )
Fluid Velocity
Q=dX/dt=Ax'
(m3
/ sec)
Power
PQ
(J/sec)
Resistance
R=P/Q
(Pa-sec/m 3 )
Inertance
L=m/A
(Pa-sec 2
/m 3
Compl iance
C=X/P
(m3 / Pa)
Kinetic Energy
1/2LQ
(J)
Potential Energy
1/2CP
(J)
Resistance/m
R'=R/1
(Pa-sec/m 3 )
Inertance/m
L'=L/1
(Pa-sec 2
Compliance/m
C'=C/1
(m3 / Pa)
A=cross-sectional area of the tube in mA
l:length of tubing
/ m3)
Un its
99
APPENDIX B
Calculation of fn and D From the Step Response
A system is said to be second order if its dynamic
can be described by a
second-order
second-order equation used to
differential
response
equation.
-pproxima~e the frequency
The
response
of the catheter system is
2
s +2Ds
2DS + 1
S2
n
n
P (t) = P (t)
o
i
where
o
= undamped natural frequency (rad/sec)
D = damping ratio (dimensionless)
The step response for an underdamped system (D < 1) is
P (t) =
on
0
0 =
I
e-
sin(/I-
2t + 0
Dn
arcsin(/F-D')
We can determine D and
n (or fn=
) from the
step
by measuring the time between peaks and the ratio of the
response
heights
of adjacent peaks. For the typical step response shown below:
PT
->
V-T 4
time
100
the response reaches a maximum
(2n+1)/2T
when
the
sine
. Solving for. tn and the ratio Yn/Yn+
argument
,
equals
where Yn is
the
output pressure at time t=tn, we have
t
(2n+.l)
n
-
-n
Yn+1
If we define
V =
log
Yn
then
V
D Irt~T
2fV 2
to solve for fn,
we note the time period between maxima.
This
yields
f
1/T
r
where f
is the resonant frequency of
frequency is simply
fn
=
f*r/
/
=
the
system.
The
natural