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Respiratory system impedance at the resting breathing frequency range
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Respiratory system impedance at the resting breathing frequency range
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R E SPIR A T O R Y SY ST EM IM PE D A N C E AT T H E R E ST IN G B R EA TH IN G
F R E Q U E N C Y R A N G E
hy
M ehm et Akif Saglarn
A Thesis Presented to the
FACULTY O F T H E SCH O O L O F E N G IN E E R IN G
U N IV E R SIT Y O F SO U T H E R N C A L IFO R N IA
In Partial Fulfillment of the
Requirem ents for th e Degree
M A S T E R O F SC IEN C E
(Biomedical Engineering)
December , 1994
This thesis, written by
avm f s a g l a n a
under the guidance of Faculty C om m ittee
and a p p ro v ed by all its members, has been
presented to and accepted by the School of
Engineering in partial fulfillm ent of the re
quirements fo r the degree of
M caster of £o'»en c&
S jo m e d ■ ca ^ n«x2.rl r\^
o « „ i j v j m k .......
Facuity Com m ittee
. HE.
C o n te n ts
List O f T ables
List O f Figures
1 In trod u ction
2 L iterature R ev iew
2.1 Chest wall im p e d a n c e ...............................................................................................
2.1.1 Pathw ay im pedances of the chest w a ll..............................................
2.2 Lung and respiratory system im p e d a n c e s .....................................................
2.3 Lung and chest wall im pedances as a function of frequency and tidal
vo! ' .............................................................................................................................
2.3.1 Plastoelastic, linear viscoelastic model ..........................................
2.3.1.1 Linear viscoelastic model ......................................................
2.3.1.2 Plastoelastic m o d e l .................................................................
2.4 M o d e ls .............................................................................................................................
2..r ) Forced oscillation m ethod ...................................................................................
3 M eth o d s
3.1 D ata collection and p ro c e ss in g ............................................................................
3.2 P aram eter estim ation m e t h o d ............................................................................
4 R esu lts and D iscu ssion
4.1 D ata collected from the first subjecl...................................................................
4.2 D ata collected from the second s u b j e c t .........................................................
4.3 D ata collected from the C O P D p a t i e n t .........................................................
4.4 Sim ulations m ade on a nonlinear model of respiratory system . . .
5 C onclusion
R eferen ce List
64
L ist O f T a b les
2.1 Im pedances defined in the chest wall .................................................................. 6
2.2 Klements in D ubois’ m o d e ] ........................................................................................ 17
3.1 Resistance and com pliance values obtained by sim ulations on the lin
ear m odel............................................................................................................................. 33
3.2 Results of the experim ents on the s e t u p .............................................................. 34
iv
L ist O f F ig u r e s
2.1 Basic mechanical d e m e n ts and their P-V ch a ra c te ristic s........................... 11
2.2 Plastoelastic m o d e l ...................................................................................................... 11
2.It C om posite mechanical model proposed by Hildebjaiidt............................... 12
2.4 T hree elem ent respiratory system model ........................................................ 15
2.5 (i-element m o d e l............................................................................................................. 17
2.6 5-element m o d e l............................................................................................................. 17
2.7 T he respiratory system model proposed by Kyles ..................................... 15)
2.8 P aram eter estim ation algorithm s ........................................................................ 21
2.9 T he model given by P e s l i n ...................................................................................... 22
2.10 9-elcment m o d e l............................................................................................................. 23
2.11 Block diagram of forced oscillatory te c h n iq u e ................................................. 26
2.12 Electrical equivalent of the setup used by Micliaelson.................................. 27
3.1 Linear model of the respiratory system ............................................................... 30
3.2 Flow and pressure signals (at 0.2 II/,) on the linear m o d e l ...................... 31
3.3 Resistance and com pliance with frequency when (low at, 0.2 Hz is
input to tin* linear m o d e l ........................................................................................... 32
3.4 Experim ental s e t u p ..................................................................................................... 33
4.1 Airflow d a ta collected from the first subject when flow is constant. . . 36
4.2 Change of pressure collected from the first subject when flow is constant. 37
4.3 Heal and im aginary parts of th e transfer function between pressure
and flow (constant flow forcing)............................................................................... 38
4.4 Airflow and pressure vs tim e for the first subject when constant pres
sure is applied................................................................................................................... 39
4.5 Real and im aginary parts of the transfer function between pressure
and flow when pressure is co n stan t......................................................................... 40
4.6 Pressure and flow vs tim e (constant pressure a p p l i e d ) ............................. 42
4.7 Transfer function vs breathing frequency (constant pressure applied
at 12 bpm ) ..................................................................................................................... 43
4.8 Transfer function vs breathing frequency (constant flow applied) . . . 44
4.5) Transfer function vs breathing frequency (constant pressure at 24 bpm ) 45
4.10 Transfer function vs breathing frequency (constant flow at 24 bpm ) 46
4.1! Airway pressure and airflow vs t i m e ..................................................................... 48
4. 12 Rsophageal pressure ami transpulnionary pressure vs t i m e ................... 49
4.13 Lung im pedance vs breathing frequency (transfer function between
transpulm onic pressure and airflow)....................................................................... 50
4.14 Transfer function between airway pressure and f l o w ............................ 51
4.15 Transfer function between esophageal pressure and f l o w ............................ 52
4.16 Pressure-flow change of the nonlinear resistance.............................................. 54
4.17 Resistance of the nonlinear model vs frequency of breathing ................ 55
4.18 Resistance of a large sliding block vs frequency of forcing......................... 56
vi
C h a p te r 1
I n tr o d u c tio n
Nonlinear behavior of th e respiratory system im pedance at th e normal resting
breathing range has been observed by several investigators [1, 2] in recent years.
These observations started with the m easurem ent of respiratory system param eters,
resistance and com pliance, and continued with the p aram eter estim ations of differ
ent portions of the system ; like chest wall and lungs. Som e investigators partitioned
the chest wall and lungs into separate com ponents in order to find out the precise
location of th e nonlinearity even though this m ade th e exact estim ation of the pa
ram eters m ore difficult. Hut despite all these efforts, there is not still a consensus
w hether th e nonlinearity exists or not.
It as i rally, the nonlinearity concerns the resistance at th e breathing range between
0 and 0.5 Hz. It is quite high and with increasing frequency, decreased. On the other
hand, elastance increased with frequency.
W hy is the nonlinear behavior of the system im p o rtan t ? Q uiet breathing takes
place at frequencies like 0.2 or 0.3 Hz. If the resistance of the system is really high, it
m eans the work of breathing at th e resting level is also high. Furtherm ore, it m ay be
inaccurate to com pare m easurem ents from different studies if the system properties
are dependent on the frequency and validity of m easurem ent techniques and models
th a t assum e linearity m ust be reassessed. Such techniques include the widely used
m ethod of high frequency oscillations.
In the present study, we collected m outh pressure and flow d a ta from two healthy
subjects and a C O P D patient, and estim ated the transfer function between pressure
and flow, which is the respiratory system im pedance, by using a spectral analysis
technique.
1
First, we review the recent studies on th e respiratory system im pedance and m od
els. Most of th e system models assum e linearity and are adequate for th e frequencies
greater than 1 Hz.
T he forced oscillation m ethod, which is used to collect m outh pressure and flow
to com pute respiratory im pedance will be focused on. In the forced oscillation tech
nique, the respiratory system is forced by a piston pum p. During our experim ents,
subjects were forced at th e FR C level by a respirator.
Pressure and flow d a ta were processed by W elch’s m ethod to estim ate th e system
param eters. We validated the accuracy of this technique with sim ulations on the
linear model of the system that consist of the series connection of resistance and
compliance.
T here is alm ost no previous a tte m p t at modeling the nonlinear observations. We
conducted sim ulations on a nonlinear model of the system and our results showed
th a t this model can represent the frequency dependence of th e respiratory system
resistance at the norm al range of breathing.
2
C h a p te r 2
L ite r a tu r e R e v ie w
2.1 C h e s t w a ll im p e d a n c e
In respiratory mechanics, im pedance is defined as the complex ratio of changes in
pressure to changes ill flow. W hen chest wall im pedance is m easured, pressure
is the transm ural pressure across the chest wall, and flow is th e rate of volume
displacem ents at the body surface. On the other hand, when pressure represents the
total pressure across the system and th e flow represents the total flow, t he im pedance
is input im pedance [5]. Thus, the ratio of pleural pressure changes to total body
surface flow (A P ^ /V J,) is th e input im pedance of the chest wall (Z cu,).
T he complex ratio can be divided into real and im aginary parts. T he real part is
the pressure change in phase with flow divided by the flow am plitude and designated
as chest wall resistance, ltw. T he im aginary part is the pressure changes th a t are
90" out of phase with flow divided by the flow am plitude and is called chest wall
reactance, X cw. Elastance ( E cw) is calculated by m ultiplying the reactance by w (2tt
tim es th e frequency a t which it is m easured) ■ Ecu, can be defined as pressure change
in phase with volum e divided by the change in volume.
Ii.cw represents the sum of the energy dissipating properties and differs from the
usual resistance, which implies a purely linear viscous resistance. Interpretation of
It will be discussed later. T h e im aginary part, X C U f (and E ,,,,) is affected by both
elastic and inertial properties. Elastic contributions dom inate at low frequencies but
decrease as frequency rises. Inertial contributions are negligible at low frequencies
b u t increase with frequency (for E cw is calculated by m ultiplying X cw by 2 n / ) ;
they are opposite in sign to elastic contributions. Since .Yr„, at a given frequency
3
is determ ined by th e sum of elastic and inertia) forces, when these forces are of
relatively equal m agnitude in a certain frequency range, ,YruJ will be small and only
resistive forces m ust be overcom e to drive the chest wall.
2 .1 .1 P a th w a y im p e d a n c e s o f th e c h e s t w a ll
B arnas et. al. [5] showed th a t while the input im pedance of the chest wall can
be m easured, those of the separate com ponents, rib cage and diaphragm -abdom en
cannot be so easily measured because of their anatom ic overlapping.
In th e zone of apposition of the diaphragm to the rib cage, the pressure applied
to the rib rage approxim ates abdom inal pressure {P ab) rather than pleural pressure
[2.1], T hus although the displacem ent of the rib cage can be directly m easured at
th e body surface, the pressure acting on it is not sim ply Ppi. On the other hand the
volum e displacem ents at the zone of apposition cannot lie m easured directly while
the pressure is pleural pressure.
boring and Mead[21] expressed the total pressure acting on th e rib cage ( Prr) as
a weighted sum of and Pab, Prc = B .Pab + (t-B )./^ ; where B com bines the relative
surface area of th e rib cage exposed to these pressures and a pressure equivalent of
axial forcing of th e rib cage produced by the costal intersect ions of the diaphragm .
T hus th e input im pedances of the rib cage is expressed as
Zrc = [B .& P ab+ (1 - B ) A P pl}/V ,r (2.1)
Flow on the diaphragm -abdom en is defined as r.Vrc-f V i,,,, where r is th e fraction of
total rib cage displacem ent overlapping zone with abdom en and is the sinusoidal
co m p artm en t of th e flow at the belly wall surface defined as anterior, abdom inal wall
and th a t part of the lateral abdom inal wall not opposed to I lie rib cage. Therefore,
input im pedance of diaphragm -abdom en is
Z d_a = A Ppt/(r.V rc + Vhw) (2.2)
Since the input im pedances are not easy to m easure, Barnas et. al. [5] developed
the concept of pathw ay im pedances of th e rib cage and diaphragm -abdom en. They
defined the pathw ay im pedances as ZrCpoth = A PpifV rc and Z d -apath = APpi/Vbw
4
These are readily m easured and th e way they differ from the input im pedances can
be predicted.
They showed the rib cage and diaphragm -abdom en to be in parallel, therefore
following relationship exists
l / ^ u, = l / Z „ wl„ + l / Z J_a^ (2.3)
O ne result of this relationship is th a t the one which has th e larger im pedance value
lias less contribution to chest wall im pedance. T heir m easurem ents indicate th at
the pathw ay im pedance of the diaphragm -abdom en is larger than th a t of rib cage.
Input im pedances can be calculated from the pathw ay im pedances as following
if B ,r and the complex ratios of A / a t / A / 1 ,,/ mid VbwjV rc are known:
Z T C R-Pgb + (1 — B ) (2 4)
7 ,
fdT ? -v a \h . * } > {
Z d- a Vh’ iJVr
(2.5)
Both im pedances are useful but pathway im pedances have some im p o rtan t ad
vantages over input im pedances. Pathw ay im pedances are indexes of chest wall
mechanical behavior and accuracy of input im pedances are lim ited because of the r
and B param eters.
Table 2.1 sum m arizes all. Flow at the belly wall is m easured directly b u t ab d o m
inal pressure does not reflect the pressure at the belly wall. Therefore the complex
ratio A P ab/Vi,w does not represent the input im pedance of the belly wall but include
intervening abdom inal contents from the site of pressure m easurem ents and is shown
as Z hw+.
5
Com plex ratios Resistance Elastance C om m ents
Z C w = A P p ijVbs H w E w Total im pedance
Z r epmth = A Pp,/Vrr flrr Ere
Rib cage pathw ay im pedance
%d-*pa,h = APpj/Vfriu
I
a
Ed-a.
Pathw ay im pedance
of diaphragm -abdom en
ZbT ir^ . -- A I -f E bw+
” + ” indicates th e ratio is not
input imp. of belly wall
Table 2.1: Im pedances delined in th e chest wall
2.2 L u n g an d r e sp ir a to r y s y s te m im p e d a n c e s
R espiratory system im pedance {ZT>) is th e sum of chest wall im pedance (Zcw) and
lung im pedance (Zt,) [2], Lung im pedance include im pedances of airways (Zaw) and
lung tissue (Z u ). T heir relative contributions to lung im pedance depends on both
flow frequency and tidal volume.
B arnas et al. [2] described a way to roughly separate the tissue' and airways
effects on /{/,. From the increase of airway resistance with airway flow, they defined
as, ltau! — A :| + A'a.lVJm,! where ki and k? are nonnegative constants and \Vaw\ is
the m agnitude of flow. They assum ed from previous studies [17, 16] lit, decreases
hyperbolically and gave it as fit, = k3j f + k.t empirically, therefore lung resistance
can be w ritten
/?/, = (kj T kj) + A cj.|V ^ntrf + k ^ j f (2.6)
where \ Vaw \ = 2v.f.Vr and constants are calculated by linear regression and experi
m ental results. Therefore lung im pedance can be determ ined in a certain frequency
and tidal volume.
6
2.3 L u n g a n d c h e st w a ll im p e d a n c e s a s a fu n c tio n
o f fr e q u e n c y an d tid a l v o lu m e
Barnas r t al. [5, 1, 4, 3, 2] m easured lung and chest wall im pedances during relaxed
chest wall and sustained respiratory muscle contraction on hum ans and anesthetized
dogs at different frequency ranges by using the forced oscillatory technique. T heir
results showed th a t some nonlinear aspects of the respiratory system is present.
They found resistance of th e chest wall decreased when frequency was increased
and elastance increased up to 2 llz and decreased at higher frequencies. M easure
m ents of Ilantos et al.[15] are agreed with these results. Hantos concluded that
frequency dependencies were most probably due to regional differences in tissue
properties.
These results have functional significance, because quiet breathing takes place
at frequencies like 0.2 or 0.3 Hz. At these frequencies, chest wall im pedance is
substantially higher and it m eans the work of breathing is accordingly larger than
previously thought. But the energy cost would still be a small fraction of the total
energy expenditure. During exercise frequencies increase to 1 Hz where chest wall
resistance is lower and its elastance m oderately increased. Total work of breathing
in this instance would be substantially different from m any estim ates.
DuBois et al.[8] m easured the motion of the chest wall during sinusoidal forcing
at the m outh. Barnas et al. [4] repeated sim ilar m easurem ents. They m easured the
relative m agnitude and phase between the displacem ents of different parts of the
chest wall with m agnetom eters. T h e results obtained from both studies indicated
th a t the chest wall expands and deflates uniform ly up to 2 Hz. T hereafter, the
abdom en makes relatively larger excursions, and the relative m agnitude and the
phase of displacem ent at different points on the chest wall show com plex changes.
Displacem ent m easurem ents were verified by the calculation of tim e constants of
the different pathw ays [5], T he tim e constant of th e diaphragm -abdom en (TC d-a =
/2j_0/£ d _ 0) was larger than th a t of the rib cage ( T C rc) at the frequencies above
2Hz. From such differences in a two-way pathw ay model, it is expected th a t the
com ponent with the larger tim e constant to slightly lag th e one with the smaller
tim e constant.
7
B arnas et al.[5] extended their studies by m easuring Z rCpalh, Zd-apath , and Zt,w+ •
All of th e im pedances showed striking similarities with total chest wall im pedance.
Despite not knowing w hat constituent of the chest wall was responsible for plastic
like dissipation, they concluded it was a general tendency for respiratory system
tissue. However, m easurem ents m ade in th e dog[2] showed lung tissue to be linear
in the norm al range of breathing.
Large changes in R cw and E cw up to 2 Hz cannot be explained by non-uniform ity
of displacem ent within the chest wall. H ildebrandt’s [17] plastoelastic, linear vis
coelastic model can be em ployed to explain those. In H ildebrandt’ s study it is
implied there is plastic-like dissipation in the tissue which differs from viscous re
sistance in which pressure-volume hysteresis increases with flow. Total pressure
consists of flow dependent pressure (due to uew tonian resistance) and pressure to
plastic dissipation. A t low frequencies, where flow is small relative contribution of
plastic dissipation to total pressure is large. As frequency increases, pressure due to
the plastic dissipation will be constant. Since pressure due to new tonian resistance
is proportional to flow, at high frequencies R ru/ will tend to be constant.
T he model contains sliding block elem ents in addition to the standard elem ents,
dashpot and springs which mimic viscous and elastic behavior,respectively. Sliding
block elem ents account for the effect of dry (Coulom b) friction or plastic-like be
havior of the chest wall. T he hysteresis loopwidth in pressure-volum e relationship
is attrib u ted to both viscosity and dry friction. T h e viscous part of th e hysteresis
loopwidth is proportional to the m agnitude of flow. T he contribution of th e hys
teresis loopw idth by dry friction is independent of flow. Therefore, resistance, will
greatly change with frequency if dry friction is im portant. Frequency dependencies
between 0 and 2 Hz of R„v, R rc, and Rt>w+ are consistent with a com bination
of viscous and plastic-like behavior of th e chest wall and its com ponents.
T he increases in E cw is also explained by the model. It is a result of viscoelasticity.
'Phe decreases above 2 Hz is thought due to the effect of inertia. Especially for the
abdom en, because of its large mass, inertial effects are obvious. Sim ilar changes was
observed for E rc and Barnas et al.[5] concluded inertial effects were also present in
the rib cage.
8
T here are several controversies concerning the change of respiratory system
im pedance w ith tidal volume. Brusasco et al.[6] found Ep to decrease w ith in
creasing Vp in intact dogs lungs. This was reversed after the lung was excised. Suki
et al.[28] reported there were no Vp dependence in hum an lungs. H ildebrandt[l6]
found R ,t and Ep to decrease with increasing Vp in excised cat lung. T he most
recent study was done by Barnas et al. [2] in the dog, T heir results will be reviewed
later.
A lthough H ildebrandt’s model can explain the changes of R and E in a certain
frequency range, it cannot fully account for the effects of f and Vp at the sam e time.
We will discuss these after we have focused on the model.
2 .3 .1 P la s t o e la s t ic , lin e a r v is c o e la s tic m o d e l
H ildehrandt conducted experim ents in 196!) in isolated cat lungs which was forced
both sinusoidally and stepwise at a series of am plitudes and frequencies. O ne year
later, he derived a linear viscoelastic model in th e form of a transfer function[17].
Since th e linear model could not explain whole hysteresis of the lungs, he purposed
a plastoelastic model and combined it with t he former one in the sam e study.
2.3.1.1 Linear visco ela stic m o d el
To sum m arize experim ental d a ta quantitatively, H ildebrandl defined the following
em pirical equation
= ) (2.7)
w here A and B are constants and Vp is the size of the step (tidal volume). The
transfer function (or Laplace transform ) corresponding to this equation is
T(s) = P(s)/V(. s) = A + 0.25 B + Btog(s) (2.8)
W hen the input is sinusoidal, s can be replaced with jw and j = e*V/2) ■ th e following
can be obtained:
r ' - F u S - + 0'25B + + 7 { H } (2.9)
9
H ildebrandt defined He {T m} as elastance and /m { 7 ’*} as th e loss m odulus (which
is not a function of frequency).
As pointed out earlier, the effect of inertia goes up with increasing frequency. It
was included in the model. Since the inert.mce is defined from P = l.v, its effect
on the transfer function is P(s) = l.s2.V{.s) or s — ► j w , P(jtv) = — l w 2V(jw).
Because inertia brings a real term , its effect is seen on th e elastance.
H ildebrandt com pared the experim ental results (on elastance, phase angle, and
P-V loop area) with the ones obtained from the linear viscoelastic model. T he
model failed basically on the explanation of energy loss and reduction of elastanre
with increasing tidal volumes.
2 .3 .1 ,2 P l a s t o e l a s t i c m o d e l
By using idealized mechanical elem ents, he purposed a plastoelastic model and com
bined it with the viscoelastic model.
T h e mechanical elem ents are the Hooke, the Newton, and St. V enant bodies
and their characteristics are shown in Figure 2.1. T he Hooke body dissipates no
energy, which m eans it is perfectly elastic, while the Newton body does. These two
elem ents are treated as linear elements. But the St. V enant body (or dry friction)
is nonlinear.
Figure 2.2 shows tin; series plastoelastic model Kach unit in the model is called
Prandtl bodies, a com bination of elastic and dry friction elements. For simplicity,
all the elastic elem ents in parallel with dry friction elem ents are identical and each
has a com pliance c. Total com pliance of series elastic elem ents are C. T he total
num ber of the St. V enant elem ents in m otion (N) depends on pressure (P) since they
have a threshold for pressure as seen in Figure 2.1. To simplify the com putations,
H ildebrandt em pirically chose the following distribution:
N = a P 0 (2.10)
where a and 0 are constants. Total com pliance of th e system is
Hooke
Newton
-►
P
i
St. Venant
S J L
Fast
slow
V
Figure 2.1: llasic mechanical elem ents and their P-V characteristics
/
/
/
/
/
/
/
- n n s c h
xv
- n R T O 0
xv
- n r t s u ^
X V
-n r $ C H
Figure 2.2: Plastoelastic model
11
's - n m -
S lS T ft P - i
- t t -
J V S O -
- I E -
v i s c o e l a s t i c
-'W ZH
- E -
■ < - n s w -
. W
-'TTfrD'-
PLASTOELASTIC
_ ^ T
Figure 2.3: Com posite mechanical model proposed hy H ildebrandt
and this yields
v = r.7j + i>r0+l
( 2 . 12 )
where 6 = etcf[ft + 1) .
T he com posite model is shown in Figure 2.3. Artificially separation of plastoe
lastic and viscoelastic com ponents should he noted.
In their latest study, Manias et al. [2] m easured lung and chest wall im pedances
in 0.2-2 Hz frequency and . ri0-300 ml tidal volum e ranges in the dog. We will shortly
focus on these results.
Lung imprdnttcc. They observed similar / dependence of E l in hum an lungs. It
increased up to I Hz and decreased at higher frequencies. As they pointed out,
the increase is consistent with viscoelastic m aterial. T he decrease was a ttrib u te d to
inertia again. E l did not change with tidal volume. Hl between 0.2 and 0.6 Hz
decreased. This is also consistent with viscoelasticity. T hey concluded from previous
12
studies th a t this p art reflected Rn behavior. W hen frequency was greater than 0.2
Hz, R l increased w ith increasing Vt . They assum ed this nonlinear behavior was due
to the contribution of R aw to R From these results, lung tissue was described as a
linear viscoelastic m aterial but there was nonlinear properties com ing from airways.
Chest wall impedance. It has been reported th a t R C U 1 decreased with frequency
and tidal volume. Frequency dependence is due to viscoelasticity and tidal volume
dependence is due to plastoelasticity. Therefore there is some nonlinear processes
in th e chest wall. C hange of E cw was sim ilar to the hum an. But it decreased with
increasing tidal volumes, Plastoelastic model accounts this part largely but not
totally. T hey found dependence of Ecu, on Vt to be accentuated al higher f com pared
with low f. This implies th a t the effect of / depends on the am plitude of forcing
and th e viscoelastic (/-d ep e n d en t) processes depend on plastoelastic (am plitude-
dependent) processes. In Ilild eb ran d t’s model these two processes were mechanically
independent.
Total respiratory system impedance. T he change of E ra and R rt looked like th a t of
E C U I and R cw. A lthough the effects of f and Vt on E cw were different from those on
E l , the relative contributions of E c v > and £ / to E rt were alm ost equal.
They concluded th a t the behavior of the total respiratory system was nonlinear
in the norm al range of breathing. N onlinearity of elastance sources from chest wall
and, th a t of resistance has two origins: chest wall and airways. T heir relative
contributions to total respiratory system resistance depend on m inute ventilation
which is the product of frequency and tidal volume. At low m inute ventilation, the
chest wall dom inates and at higher levels, airways do.
H antos [14] also m easured the mechanical im pedance of the lungs (Z/.) in open-
cliest dogs between 0.125 and 5 Hz at m ean transpulm onary pressure of 0.2, 0.4,
and 0.8 k P a and fit experim ental d ata to two different models, one of them using
H ildebrandt’s linear viscoelastic model. T h e change of elastance and resistance
looked liked the results given above. T he models will be reviewed in th e next section,
but shortly; viscoelastic model provided b etter fit. This is another verification for the
appropriateness of the model. W hen the transpulm onary pressure was increasing,
they found resistance to increase and elastance to decrease. This observation is the
opposite of tidal volum e dependence.
13
Since m ost of the studies have quite parallel results, we can conclude th a t in
the norm al range of breathing, total respiratory system behavior is nonlinear. Even
though this conclusion is reached by the d a ta obtained from dogs, we do not expect
great difference between dog and hum an lungs, sim ilar tidal volum e dependence
should exist in hum an. This is because chest wall im pedance m easurem ents m ade
in dog and hum an [4, 5] led to alm ost the sam e change w ith frequency.
C urrently, H ildebrandt’s plastoelastic, linear viscoelastic model gives the best
explanation for the experim ental d a ta even though the viscoelastic and plastoelastic
processes are artificially separated. As pointed out in his study, it is for the pur
pose of simplified analysis. However, a mode] th a t combines both viscoelastic and
plastoelastic properties, in other words / dependence and Vr dependence, would
obviously give a much better explanation not only for th e chest wall im pedance d ata
at low frequencies but also for the nonlinearity of th e respiratory system , since the
chest wall is its m ain source.
2 .4 M o d e ls
T here are m any atte m p ts in the literature to fit electrom echanical m odels to respi
ratory im pedance d a ta obtained by using forced oscillations either at discrete fre
quencies [8, 19, ‘ 26, 25] or forced random noise [9, 10] .
T h e m ain purpose of the modeling of respiratory im pedance d a ta is to extract
detailed clinical and physiological inform ation about the state of th e respiratory
m echanical system . Models allow com putation of p aram eter values th a t have phys
iological interpretability, it is assum ed th a t param eters are functionally related to
anatom ic regions, therefore modeling is useful in th e com parison of disease states or
in diagnosis. Since it is noninvasive and requires alm ost no subject cooperation, it
has a attractiv e advantages over m any other pulm onary function tests [22].
A lthough the system is a com plicated branching distributed system , the experi
m ental im pedance d a ta have been interpreted using lum ped p aram eter m odels with
linear elem ents using resistive, inertial, and com plaint effects. T he sim plest model,
proposed by Dubois [8], consists of a resistance (R ), inertance (I), and com pliance
(C) in series [8] (Figure 2.4 ). This model has provided quite good fit to experim ental
14
I R
5 ^ V A
Figure 2.4: T hree elem ent respiratory system model
d a ta of normal subjects [10, 1 1, })], it fails to explain im pedance d a ta from subjects
with obstructive lung disease.
T he fit of a model to the experim ental d ata depends on the frequency range
studied. T he wider the frequency range, the more com plicated m odels are needed.
This m ight cause another problem since the reliability of the estim ated param eters
in a com plex model is questionable [22]. Dubois et al.’s [8] three-elem ent model
fits only to the d a ta in the 3-16 Hz. Jackson et al. [18] reported dog respiratory
im pedances between 4 and 64 Hz. They applied the three-elem ent model only to
d a ta between 4-32 Hz. T he reason they did not include 32-64 Hz range is th a t the
effective resistance was markedly frequency dependent, which is a phenom enon th at
can not occur with the three-elem ent model. T here are models th a t can explain
frequency dependence [10, 2.r )].
T he criterion th a t have been used during the param eter estim ation are sum-
squared differences, sum of root-m ean-squates differences or weighted sum of root-
m ean-squares differences. They are as following,respectively:
J = (2-13)
1=1
j . = [ ( l / " ) E | Z , n ( / , ) - Z < ( /,) I T '1 (2.14)
1=1
j , = { ( 1 / - ) D |Z ™ ( / . ) - 2 . ( / . ) l 7 l ^ ( / . ) | !|} '/2 (2.15)
1=1
where n is th e num ber of th e d a ta points, Zm(/,) is the im pedance value at /;
frequency predicted by the model and Ze(/,) is th e experim ental im pedance value.
15
Param eters of the model are determ ined by minim izing these functions. B ut when
the model is sim ple enough, param eters can be calculated w ithout any iteration by
setting partial derivatives of J function with respect to R, I, and E (1 /C ) to zero
[29], T he solution to these equations represents an optim um set of p aram eter values
for th e forced oscillatory resistance (R), inertance (1), and elastance (E). W hen
U(fi) and V{fi) are real and im aginary parts of of the m easured im pedance at the
frequency /,, Tsai et al. [29] found the following equations for three-elem ent model:
rt = - X > ( / < ) (2.16)
n i=,
I = )/2>r/, - “ E 2 * / i W i ) ( 2 -0 )
r . c t
«=i t=i
F = - £ 2jtItV U i) - - & * / i V ( / i ) (2.18)
c , c ,
1=1 t=l
where
a = ^ l / ( 2 7 r / I)2 (2.19)
t=i
* = £ ( 2 * / . ) 2 (2.20)
■=i
c = n i - a b . (2.21)
As m entioned previously, estim ation of the param eters in complex models is not
straight forward, and iterative m inim ization algorithm or em pirical curve fitting
techniques m ust be em ployed [9].
In 1956, Dubois et al. [8] proposed a 6-element model, shown in Fig (1.2).
Physiological equivalents of th e elem ents in the model are in Table 2.4.
Eyles et al. [11] gave a 5-element model in 1981 and in another study [9], they
com pared this model with 3-clernent model, obtaining respiratory im pedance d ata
from patients with obstructive lung disease and norm al subjects in 5-35 Hz frequency
range. Their model is in Figure 2.6.
They interpreted the elem ents as follows: /j, H\ and C\ represent the central
airways inertance, central airway resistance, and airway com pliance. Ri was pe
ripheral resistance and Cj was interpreted as lung and chest wall com pliance. They
16
R1 I I R2 12
Figure '2.5; 6-eleinent model
Symbols Physiological equivalent
Airway resistance
ft
Airway inertance
c x Alveolar air capacitance
ft2
Tissue resistance
h
Tissue inertance
C-2
Tissue com pliance
Table 2.2: Elem ents in D ubois’ model
R1 II R2
A A A — / T ^
A / W
C l C2
Figure 2.6: 5-element model
17
used sum -squared error function and in normal subjects, 5-elem ent model failed to
provide a b etter fit than three elem ent model. B ut in the patients, it was reported
th a t it fit the experim ental data. According to their interpretation, estim ated values
of inertanee and resistance of central airways, airway com pliance and peripheral re
sistance* were good, because values of central airways inertanee and resistance were
very close to inertanee and resistance in three-elem ent model and it is a known fact
th a t obstructive lung disease does not effect th e airways. Value of estim ated pe
ripheral resistance was twice of th a t of resistance in the three elem ent model. It
was also reasonable since the disease elevates the peripheral resistance markedly.
T he variability of all param eters but (which represents com bined com pliance of
the lung tissue and chest wall) was acceptable. They a ttrib u ted this to the relative
insensitivity of the m easured im pedance to the physiological effects represented by
(■2.
To shorten the iteration tim e in the 5-element model, Myles and Himmel [10]
converted the problem to one based on linear regression analysis. They calculated the
param eters again and values of all param eters except C 2 showed excellent agreem ent.
They concluded th a t the problem of estim ation of f ’j required th e im pedance d ata
obtained below 5 II/ and in this frequency range the criterion function had low
sensitivity to (72.
However, there are some contradictions regarding the interpretation of th e and
( ' 2 param eters in the 5-eleinent model. As m entioned Kyles and Pim m el thought
( \ represented airway com pliance and C 2 represented total tissue com pliance. But
Dubois and IVslin interpreted th a t C\ represented alveolar gas com pression com pli
ance. Jackson et al,[19] reported both of these interpretations were not correct. T he
first one did not seem reasonable since they found airways m ore com plaint than tis
sues, which is unlikely. On th e other hand, they calculated the thoracic gas volume
from ( ’1 (TCiV = f ’i . atm ospheric pressure) and resultant value was not physiolog
ically appropriate. However, in six-element model calculated value was reasonable.
Therefore it was concluded Ci represents alveolar gas com pressibility for th a t model.
In five-element model, th e frequency-dependent behavior is due to shunt com pli
ance [9], which is supposed to represent either alveolar gas compression or airway
wall com pliance. These structures are in th e lung but at low frequencies, chest
18
Rc Ic
Figure 2.7: T he respiratory system model proposed by Eyles
wall is responsible from the frequency dependence of the respiratory system . T here
fore Jackson et al. [19] concluded th a t it would be m ore appropriate to model the
frequency-dependent decrease with the model th a t includes parallel in homogeneities
th a t could represent in homogeneities in parallel structures of the chest wall, possibly
the rib cage and diaphragm -abdom en pathw ays, this model is nine-elem ent model
and we will briefly go over it later.
Besides models, param eter estim ation algorithm s are significantly im p o rtan t in
th e calculation of th e param eters. Eyles et al. [10] used four distinct algorithm s
on three different models; two of them are 5-elem ent model discussed above and
(i-element model proposed by Dubois. T he third one is in Figure 2.7. fic and lc
represent viscous and inertial properties of central airways. Tw o parallel resistancc-
com pliance com binations and /?2-C 2) represent parenchym al com partm ents.
Flowchart of each algorithm is shown in Figure 2.8. In th e one-stage gradientdescent
algorithm , a set of starting param eter values are identified by a random search sub
routine. It was reported this subroutine to use a com bination of the Gauss-N ew ton,
N ew ton-H aphson, optim um gradient, and gradient projection m ethods. T he two-
stage gradient descent algorithm contains a second random search. T h e one and
two-stage sim plex algorithm s are identical except th a t they replaced gradient de
scent algorithm s by a simplex subroutine. It was concluded th a t the com bination of
5-elem ent model and two stage simplex algorithm provided th e best fit to th e exper
im ental data. Even in this com bination, C 2 was unstable. None of the algorithm s
19
gave a tolerable error function for second and third model, inclusion of sixth ele
m ent (inertanee or resistance) com plicated the criterion function and th e numerical
m inim ization process.
T h e adaptive simplex used in the study above can be outlined as follows: A
function of n variables is to be minimized. ( jV + I ) points
(p 0l r,,...pn)
in the n dim ensional param eter space define the current "sim plex” with P, th e vector
of param eter values
[Pl l '..P„.}r
and V th e vector of function values
at the P,s. We seek to minimize th e values of th e com ponents of Y by using an
adaptive search m ethod th a t consist of three operations denoted as reflection, con
traction, and expansion. W ith each of these operations, th e set of param eter values
(/*,) yielding the highest value for the criterion function V '', (or J) will be replaced
with a b etter estim ate. T he criteria for convergence is user selected and can consist
of com paring the ratio of the m inim um and m axim um function values, com paring
param eter coordinates or using the distance between th e low and highest as
defined by their function values against some required m inim um .
In 1975, Pcslin et al,[26] m easured respiratory system im pedance between 3 and
70 Hz by oscillating pressure at the chest and m easuring flow at th e m outh. (T he
reason for th a t was to m inim ize th e influence of the cheeks, and later this im pedance
was called as transfer im pedance of the respiratory system [25]). A model was given
in the form of linear differential equation. 'They assum ed th a t passive mechanical
respiratory system was m ade up of three com partm ents: the tissues, alveolar gas,
and th e gas flowing through the airways. T he configuration above was given for
respiratory system transfer im pedance where Z|, Z aui, and Zg indicate im pedances
of tissue, airways, and alveolar gas, respectively.
20
A B C D
G R A D IEN T
DESCEN T
SIMPLEX SIMPLEX
G RA D IEN T
DESCENT
RANDOM
SEARCH
RANDOM
SEARCH SEARCH
RANDOM
SEARCH
RANDOM
G RAD IEN T
D ESCENT
SIMPLEX
(selected
parameters
varied)
RANDOM
SEARCH
(selected
parameters
varied)
RANDOM
SEARCH
Figure 2.8: P aram eter estim ation algorithm s
21
law
Figure 2.9: T he model given by Peslin
As seen from Figure 2.9, airways have resistance and inertanee, tissues consist
of resistance, inertanee, and com pliance, and alveolar gas has com pliance elem ents,
and th e following equations can be w ritten:
Z, = Ilt + j'M i - 1/toCt) (2.22)
= flaw + j w l aw (2.23)
Z9 = - j / w C g (2.24)
T he following differential equation was given in th e study:
- pH = M 0pau, + Mi V (' > oli, + ... + M n V {n)au, (2.25)
where ( Pu, — / J w) is input pressure and M0,...,M n are param eters related in some
way to the properties of the com partm ents. In 3-20 Hz, 3rd and in 3-50 Hz frequency
range 4th order equations provided acceptable error values. For 4th order equation,
the relationship between the param eters and coefficients :
Mo = 1 f C t (2.26)
Ml = fit + n 9(l + Cg/Ct) (2.27)
22
R1 II Cl
Rc Ic
Cc
L v W V — / 3 r c T 'H h
R2 12 C2
Figure 2.10: 9-elem ent model
(2.28)
A/3 — Cg{RtInw + Rgll) (2.29)
M 4 = CyItIal (2.80)
In 1985, Peslin el, al. [25] m easured input and transfer im pedances of respiratory
system in hum ans and fit these im pedances to the 6-elem ent model. T hey predicted
the im pedances in term s of the model param eters as following:
Z ra,in — Z aw + Z t- Z gf { Z t + Z g )
Zra,tr — Z au, + Zf + Z aw.Z( / Z g
(2.31)
(2.32)
From these equations they calculated the tissue and airway im pedance and com pared
these values with the ones obtained from th e model. M onoalveolar (or 6-element)
model could not account for the frequency dependence of Re(Zt) and p attern of
I m ( Z t). Therefore, they divided the tissue co m p artm en t into two parts. T he re
sultant 9-elem ent model is shown in Figure 2.10. (This model reportedly provided
quite good fit to Z , ).
23
Jackson et al. [18] m easured respiratory im pedance between 4 and 64 Ilz in
dogs and fit th e d a ta to 5-element, 6-eleinent and 9-elem ent models separately. The
five elem ent model proposed by Kyles and Pirnmel fit th e experim ental d a ta only
when resistance decreased with frequency b u t even in this region it was reported
param eters were not physiologically realistic. T hey reported the six-elem ent model
of Dubois et al. [8] to provide fit to the d a ta when its real p art decreased and
increased, but with different set of param eters. 9-elem ent model above was the only
one th a t fils with sam e set of param eters in 4-64 Hz frequency range.
In a recent study, Hantos et al. [14] m easured the mechanical im pedance of lungs
(Z/J in open-chest flogs in the frequency interval 0.12,r>-5 11/ at different values of
mean transpulm oriary pressure (/%>) and interpreted these d a ta on the basis of two
models, including th e sam e airway co m partm ent with resistance and inertanee, but
representing the mechanical properties of the pulm onary tissue in different ways.
They found the real part of the im pedance to decrease and the im aginary part to
increase asym ptotically.
In the first model, the tissue part was m odeled by a com pliance(CT i) shunting an
in-series connection of a resistance {Ri) and a com pliance (C?), this part corresponds
to a series connection of a spring with a parallel dashpot-spring com bination (Voigt
body), bung im pedance ran be w ritten as :
Zi. = R aw + ju 'L u , ~ j ( t + Jiv I h ( - i ) / x v ( C t + C 2 + j x o R t C i C t ) (2.33)
In the second model, Zi, was given as following (H ild eb ran d t’s form ulation on
the basis of stress relaxation d a ta is used for tissue p art of the equation) :
Zi, = Raw + j w / au, 4- fin /4.6m — j ( A + 0.25fi + filogw)/w (2.34)
They calculated the error functions defined in E quation 2.13 and 2.14 and the
second model gave sm aller values despite the sm aller num bers of param eters in the
first model.
24
2.5 F o r ced o s c illa tio n m e th o d
This technique was first used by DuBois et al.[8] in 1956. It has been applied most
often to spontaneously breathing subjects from which m outh pressures and flows are
processed in a variety of m anners to com pute a respiratory im pedance.
DuBois used a pum p th a t generates a transthoracic pressure, pressure difference
between th e m outh and outside of chest. T he am ount of m otion produced by sine
waves of pressure at different frequencies perm its analysis of th e m echanical prop-
erties of the chest. T he norm al respiratory p attern of airflow is not necessarily a
sine wave. However, it is possible to analyze any regularly recurring m otion into
an equivalent sum of sinusoidal movements; therefore a study of the response of the
chest-lung system to sinusoidal pum ping at various frequencies provides th e infor
m ation for understanding the response to any other form of pressure wave in the
chest. C om posite wave forms of lim ited num ber of sinusoids [20], impulses [27], and
band-lim ited gaussian w hite noise (O W N ) [21] have been used by investigators since
DuBois.
T he original setup consists of a pneum otachograph to record airflow at the m outh
of the subject, a pu m p th a t generates transthoracic pressure fluctuations and capac
itance m anom eter to m easure th e pressure. In current studies, a loudspeaker is used
to generate sinusoidal oscillations and these oscillations are recorded at th e m outh
by m eans of a pressure transducer. T he block diagram of the system is shown in
Figure 2.11.
In 1975, Michaelson [24] introduced the m ethod of spectral analysis for im pedance
m easurem ents. Since then, some investigators have used band-lim ited O W N as the
forcing function and have com puted the im pedance estim ate using cross-spectra.
In this m ethod, pressure and flow are recorded at the m outh of th e subject. He
breathes quietly via a pneum otachograph in a side tubing and oscillated by a loud
speaker. A constant bias flow reduces the build of C O 2 in the inspired air. Electrical
equivalent of th e setup is in Figure 2.12. Ea and Za represent the generator and
m easuring system . P{t) and V are th e pressure and flow at th e m outh of th e subject.
Fourier transform s of these signals, Sp and Sy are calculated. T he corresponding
power spectra Gpp = Sp . Sp and G yy = . Sy (the asterisk denoting the complex
conjugate) and th e cross-power spectrum G yp — S y . S f , are derived next.
25
Pneumotachograph
press, transducer
To subject
Pne umotac hograph
Mouth
pressure
trans. “
Pressure
amp.
Flow
amp.
Pressure generator
(loudspeaker)
Figure 2.11: Block diagram of forced oscillatory technique
Za Zr
Figure 2 . 1 2: Klectrical equivalent of the setup used by Michaelson.
T h e respiratory im pedance Z r and the squared coherence function 7 2 are esti
m ated from;
Zr = G p p /O yp (2.35)
7J = , 1 ° * ^ ( 0 < T ; <1) (2.36)
tl pp U y y
Coherence function is used to estim ate th e am ount of noise or alinearity in the rela
tion of pressure versus flow and its value is determ ined by th e investigator. Hantos
et ai.[15] m easured respiratory system im pedance at low frequencies. According to
the way they estim ated th e im pedance, when {/, Prt, Pw, Vrt are the pseudorandom
signal which drives the* loudspeaker, the pressure difference between m outh and a t
mosphere, pressure drop across the chest wall, and central airflow, respectively, 7 prtn
and 7 ^ u functions were used for respiratory system im pedance; 7 ^ v and 7 were
used for chest wall im pedance. If the product of the two coherence functions was
greater than 0.95, the spectrum was accepted.
T he system shown in Figure 2.11 has some drawbacks: T h e pneum otachograph
requires an em pirical calibration and transducers should m atch for th e various fre
quencies of the m easurem ents. Fran ken et. al. [12] replaced th e pneum otachograph
by a long rigid tube. They m easured th e pressures at the entrance and the outlet
of the tube, instead of m easuring m outh pressure and the flow. T h e ratio of these
pressures, representing th e transfer function of th e tube, depends on the im pedances
of both the tu b e and the subject ’s respiratory system . If the former im pedance is
known, th e latter can be calculated. Since the im pedance characteristics of gas
flowing through a rigid cylindrical tu b e can be predicted accurately from classical
27
physics , there is no need for an em pirical calibration. T he only requirem ent for
correct m easurem ents is th a t the transducers used for m easuring inlet and outlet
pressures have identical characteristics within the range of investigated frequencies
m -
T he following formula was given by them for th e respiratory im pedance:
* = <2'37>
where L is the length of the tube , c is th e propagation coefficient, Zc is its character
istic im pedance, and I F is its transfer function (7'F -- ( 7 pBv,)- The squared
coherence funct ion is com puted from the following equation
7 a = r )- ’P' r / (*2.38)
t n \p , ( ’r„r0
The breathing signal of the subject introduces system atic and random error on
the obtained im pedance values. It effects bot h pressure and flow signals ; the noises
on them produced by breathing are not independent. If th e num ber of im pedance
m easurem ents approaches infinity, it can be assum ed th a t there is no correlation
between the respiratory and loudspeaker signals . These errors can be corrected
[12]. T here is also errors coming from th e equipm ent.
As m entioned before, Barnas et al. [4] used forced oscillatory technique during
the experim ents m ade to m easure respiratory system im pedance a t low frequencies.
In these studies, subject was required to hold his breath at FRO level. Therefore,
there was no breathing signal. Forced oscillatory experim ents arc m ade at FR C level
because there should not be any muscle contraction. R espiratory system im pedance
was m easured during sustained respiratory muscle activity[l] and it was found th at
resistance and elastanrc increased with frequency (this is due to th e stability of the
system ). Barnas et al. [4] m easured the im pedance while the subject sat in a volumc-
displacem ent plethysm ograph. A piston p um p driven by a linear m otor was used
to generate sinusoidal volum e signals. They m easured transthoracic pressure with a
latex fingercut balloon inserted into th e esophagus and attach ed to a transducer via
a catheter.
28
C h a p te r 3
M e th o d s
3.1 D a ta c o lle c tio n a n d p r o c e s sin g
W r m ade m easurem ents in two healthy adult males and a C O P D (C hronic O bstruc
tive Pulm onary Disease) patient. These m easurem ents were m ade by using a res
pirator. T he subject was seated with a applied noseclip and voluntarily held his
breath. lie was also supported his cheeks with both hands. As he relaxed at FR C
level, (in order to prevent any muscle contraction which would affect the param eters
estim ated), his respiratory system was oscillated by the respirator.
Two types of oscillations were performed on the healthy subjects: constant flow
and constant pressure. M outh pressure and m outh flow was m easured by a pressure
transducer (Validyne) and a Kleisch pneum otachograph equipped with a sim ilar
transducer. M easurem ents were repeated at the breathing rate of 12-13 b reath s/m in
for the first subject and 12 and 24 b re a th /m in for th e second. Sam pling rate was 20
Hz and 30 Hz during the first and second su b je c t’s d a ta collection, respectively. In
both cases, m easurem ents last 60 seconds.
D ata collection procedure of the patient was similar to th a t of th e normals. Hut in
the case of patient, during expiration forcing pressure (P R E P ) was 5cjn lf20 instead
of zero. We had airway pressure (Pa), esophageal pressure (Pe), transpulm onary
pressure (which is Pa-PK), and airflow d a ta from the patient. Sam pling frequency
was 30 Hz and d a ta collection lasted two m inutes.
29
p(t)
AAA
PC
Figure 3.1: Linear model of th e respiratory system .
3 .2 P a r a m e te r e s tim a tio n m e th o d
Resistance and com pliance values at a frequency can he determ ined from th e com
plex transfer function between sinoisoidal flow and pressure at th a t frequency. T he
com plex ratio of m outh pressure to flow is the respiratory system im pedance. Real
and im aginary parts of the im pedance are related to the resistance and com pliance
of th e respiratory system . Power spectrum of the im pedance was estim ated by using
Welch m ethod. In this m ethod, flow and pressure sequences of N points are divided
into K sections M points each. Using an M -point FK successive sections are Hanning
windowed, F F T ’d and accum ulated. We chose M as 256.
He fore collecting d a ta from hum an subjects, this technique was validated by
sim ulations on the linear model which consist s of resistance and com pliance in series
as shown in Figure 3.1.
In this model pressure is
p(i) = V ft + Pr (3.1)
and Pc is th e solution of the following differential equation:
V = C d - § (3.2)
R and C values were chosen as 2 c m //20 / / p s and 0.2 Z ,/cm //20 , respectively.
Half sinusoidal flow signals were g e n e r a t e d and pressure signal p(f) were found
from E quations (3.1) and (3.2). Sam pling frequency was 100 Hz and E uler’s m ethod
30
1
0
2
3.
5 10 15 20 25 30 35 40 45 50 0
t
6
4
w
D _
2
0,
20 25 30 35 45 40 50
t
Figure 3.2: Flow and pressure signals (at 0.2 Hz) on the linear model
was em ployed to solve the differential equation. Airflow signals were at the frequen
cies of 0.2, 0.4, 0.5, 0.8, and 1 Hz and there were 5000 sam ples which m eans for
the signal at the m inim um frequency, there were a t least 10 cycles. As an exam ple,
input flow signal at 0.2 Hz and o u tp u t pressure signal are shown in Figure 3.2.
A fter generating pressure and flow signals, transfer function was calculated by
using Welch m ethod. Im pedance was read at the closest frequency to the frequency
of the input signal. Mean value of resistance and com pliance are 2.0345 cm/Z^O and
0.2061 L fc m lliO and standard deviations around these values are 0.062 and 0.006,
respectively. Values of resistance and com pliance with frequency are shown in Table
3.1. C hange of resistance and com pliance with frequency are shown in Figure 3.3.
These changes were obtained from the com plex transfer function between pressure
31
tr 2
0.2 0.4 0.6 0.8
frequency
0.4
0.3
o
0.2
0.1
0.2 0.4 0.6 0.8
frequency
Figure If.3: Resistance and com pliance with frequency when flow at 0.2 Hz is input
to the linear model
and (low at 0.2 II/.. It should he noted th a t resistance and com pliance values at the
frequencies of the first and second harm onics (0.6 Hz and 1 Hz) are very close to
the ones at 0.2 Hz. T he com puter program for the estim ation of the param eters is
given in A ppendix A.
A nother validation of th e param eter estim ation technique was on th e d a ta col
lected hy the setup shown in Figure 3.4.
In this setup, airflow signals are produced by th e piston pum p. This pu m p could
generate sinusoidal airflow signals between 0 Hz and 0.833 Hz (50 cycles/60 s).
Stroke volum e of the respiration p u m p could also be adjusted. Pressure and airflow
signals were low-pass fdtcrcd before sam pled by the com puter. Cut-off frequency
of the filter was 50 Hz. To sam ple th e d ata Labtech N otebook program was used
32
Input frequency (Hz) Resistance Com pliance read at (Hz)
0.2 2.145 0.197 0.195
0.4 2.016 0.203 0.390
0.5 2.012 0.204 0.488
0.8 1.997 0.212 0.830
1 2.000 0.213 0.976
Table . ‘1,1: Resistance and com pliance values obtained by sim ulations on th e linear
model.
Resistance
Bottle
(complianc
0
Pressure
transducer
Piston pump
Flow transducer
Low pass filter
flow
pressure
Figure 3.4: Fjxperimental setup
Computer
33
Input frequency(Hz) Resistance Com pliance read at (Hz)
10/60 50.3083 0.0270 0.1562
15/60 46.6588 0.0236 0.2344
20/60 47.1130 0.0211 0.3125
25/60 48.0442 0.0191 0.3906
30/60 48.9703 0.0188 0.5078
35/60 50.4893 0.0180 0.5859
40/60 51.7727 0.0175 0.6641
45/60 52.8463 0.0173 0.7422
50/60 54.2462 0.0168 0.8594
Table . ‘ 1.2: Results of the experim ents on the setup
and there were two channels, pressure and flow. Sam pling frequency was 20 Hz.
C alibration was done by an oscilloscope.
Resistance and com pliance values obtained by th e transfer function between flow
and pressure are shown in 'Fable S.2.
In order to check these values, resistance and com pliance were m easured by
direct m easurem ents of flow, pressure, and volum e change. Resistance was found as
55 crnH^O/lps and com pliance was 0.03 f /rm /Z jO , T h e differences between these
values and the ones at the table might be due to th e assum ption of linearity of the
im pedance over the frequency range.
34
C h a p te r 4
R e s u lts a n d D is c u ss io n
4.1 D a ta c o lle c te d from th e first s u b je c t
As m entioned in the previous chapter, respiratory system was oscillated when either
flow or pressure was constant, m outh pressure and flow d a ta collected and processed
by using M ATLAB. W hen constant flow was applied, its change with Lime is shown
in Figure 4.1. Breathing frequency was approxim ately 0.2 Hz. M outh pressure versus
tim e is shown in Figure 4.2.
T he transfer function between flow and pressure was calculated by using W elch’s
m ethod. T h e real part of the transfer function was interpreted as respiratory system
resistance and im aginary part related to com pliance (or elastance).
T he real and im aginary parts of the transfer function is shown in Figure 4.3.
At 0.2 Ilz, resistance value is 11 cm -w ater/lps (as seen from th e figure) while the
com pliance is around 0.13 L /cm -w ater (C = 1/2 * n * / * Im ag(Il) where Im ag(H )
is the im aginary part of the transfer function at / = 0.2Ilz). Note th a t in this
spectrum , im pedance values are reliable only up to 1 Ilz where the second harm onic
of th e input signal exists. Resistance value decreased with frequency and its value
at 0.2 Hz is two times higher than the resistance a t 1 Hz.
For the constant pressure oscillations, change of flow and pressure versus tim e is
shown in Figure 4.4.
Real anti im aginary parts of th e transfer function are shown in Figure 4.5.
35
A irflo w (liters/minute)
0.5
-0.5
60 50 40 30 20
time (seconds)
Figure 4.1: Airflow d a ta collected from the first subject when flow is constant.
36
Pressure ( c m water)
30
25
20
50 60 30
time (seconds)
Figure 4.2: ( ’hange of pressure collected from the first subject when flow is constant.
37
• 1 /(w C) Resistance (cmwatef/lps)
15
10
-5 -
- 10.
frequency
-10
-20
10
frequency
Figure 4.3: I teal and im aginary parts of the transfer function between pressure and
flow (constant, flow forcing).
38
pressure cm-water
v cn
-
lr . %
ft' ^
9 ~
=1
*
a s
<
V :
3
ft
'-1
ft
ft
e - *
5
3*
ft
3
ft
O
3
W
O
ft
ft
CJi cn
3
0
O
C n
time
airflow (Ips)
o
o
u
o
o
cn
o
o >
o
$
Q C
1 Q „ _l_____________ I _____________ I _____________ I -------------------- 1 ---------------------1 -------------------- 1 -------------------- 1 -------------------- 1 ----------------
0 1 2 3 4 5 6 7 8 9 10
frequency
%
4
-10
frequency
Figure 4.5: Ileal anti im aginary parts of the transfer function between pressure and
flow when pressure is constant.
40
As seen from the figure, resistance also decreased here with increasing frequency
of breathing. Elastance, on the other hand, increased slightly which means com pli
ance decreases with the frequency of breathing. E stim ated compliance value for the
first subject is about 0.15 h/cm -w ater.
Even though the change of resistance and compliance with frequency agreed in
both experim ents, estim ated values of the respiratory system resistance differed.
During constant pressure forcing, resistance value at 0.2 Hz was lower, it is around
6 cin-w ater/lps.
4 .2 D a ta c o lle c te d from th e se c o n d s u b je c t
The protocol here was the same as the previous. T he subject was connected to a
respirator and ronstant pressure and constant flow was applied at the m outh. But
th e forcing signal’s frequency was 24 b reath /m in as well as 12, resting breathing
frequency. Change of pressure and flow with tim e is shown in Figure 4.6 when the
pressure is constant.
W hen constant pressure at 12 breaths/m in is the input signal, real and imaginary
parts of the transfer function are shown in Figure 4.7. Resistance value at 0.2 Hz
was around ( S Jin n 1120 lips and compliance value was 0.35 L fc m H 20
W hen the flow at 0.2 Ilz is constant, the change of transfer function with fre
quency is shown in Figure 4.8. Resistance at 0.2 Hz was very close to lc tn ll20 /l p s
and com pliance was 0.35L /c m I I 20 . T he change of resistance agreed with the pre
vious experim ents, it was quite high at 0.2 Ilz and has a tendency to decrease with
frequency when either pressure or flow is constant. As we have seen in the first sub
ject, resistance values were also slightly higher than the ones when constant pressure
is applied. On the other hand, compliance is not dependent on the type of the input
signal, constant pressure or flow.
W hen the input forcing signal was constant pressure or flow at 24 hpm , transfer
functions of the system are shown in Figure 4.9 and Figure 4.10, respectively.
This kind of change of respiratory system im pedance is quite different from the
one obtained on the first subject. Here, we would like to point out th at
41
1.5
S. 0.5
§
-0.5
60 50 40 30
time
20
15
£ 10
60 50 40
20
time
Figure 4.6: Pressure and flow vs tim e (constant pressure applied)
42
-1 /w C Resistance
30
20
-10
20
frequency
-10
-15
frequency
Figure 4.7: Transfer function vs breathing frequency (constant pressure applied at
12 bpm )
43
-1 /W C Resistance
-10
-15
frequency
frequency
Figure 4.8: Transfer function vs breathing frequency (constant How applied)
44
-1 /v v C Resistance
-10
frequency
10
5
0
5
10 15 0 5
frequency
Figure ■ 1.!): Transfer function vs breathing frequency (constant pressure at 2 * 1 hptn)
45
Resistance
frequency
4
2
0
2
4
6.
0 15 5 10
frequency
Figure 4.10: Transfer func tion vs breathing frequency (constant flow at 24 bpin)
46
1. R eactance values estim ated from the prrssure-flow d a ta collected from the sec
ond subject during constant pressure forcing at 12 bpm and, constant pressure
forcing at 24 bprn could not be interpreted since they were positive instead of
being negative. This was also the case for th e resistances of th e second subject.
These experim ents require subject cooperation. As m entioned earlier, subjects
m ust be relaxed at FR C level. Any muscle contraction m ight affect th e esti
m ated param eters. Barnas et al[l] m easured th e im pedance of the chest wall
at the range of 0-4 Hz during sustained respiratory muscle contraction and
found th a t change of chest wall im pedance with frequency was quite differ
ent. Resistance values were generally higher. Since at this range, chest wall
is the main com ponent of the respiratory system im pedance, we believe th at
unexpected change of the param eters is due to lack of subject cooperation.
2. In both subjects, during constant flow forcing, resistance values were higher.
This is especially obvious in th e d a ta collected from the first subject. We feel
th e ones obtained during constant flow forcing are m ore correct because they
are m ore realistic. As pointed out above, in some cases; we found negative
com pliance during constant pressure forcing which has no meaning. This also
gives rise to th e feeling that constant flow forcing would be a m ore appropriate
way in these kind of experim ents. T h e other m ight be m aking it harder for
subjects to stay in the relaxed state. Furtherm ore, they also told th a t they
were m ore com fortable during constant flow forcing.
3. Change of respiratory im pedance for the two healthy subjects are as follows:
Real part of it at the resting frequency, which is th e respiratory system resis
tance, is at least two tim es higher than its values at the frequencies higher than
1 Hz. Nonlinear behavior of the resistance can be explained by dry and viscous
friction. Klastances increase slightly with increasing breathing frequency and
this is thought to be due to viscoelastic behavior of th e system [5, 4,3].
4 .3 D a ta c o lle c te d fro m th e C O P D p a tie n t
As m entioned in the previous section, we also had pressure-airflow d a ta from a
C O PD (O hronic O bstructive Pulm onary Disease) patient. T h e d a ta collection
47
40 60
time
80 100 120 20
2.5
2
i
° 1 e
1.5
< 5
1
°'50 20 40 60 80 100 120
time
Figure* ■1.11: Airway pressure and airflow vs tim e
m ethod m ight have differed from th e protocol defined above, but processing of the
d a ta was the same.
We had airway, esophageal, and transpulm onary pressure d a ta as well as airflow
data. D ata collection procedure in th e patient was slightly different; during expi
ration forcing pressure is not zero. Instead, it is fixed a t a constant pressure level
(here 5crnll^O ). Airway pressure, esophageal pressure, transpulm onary pressure,
and airflow vs tim e are shown in Figure 4.11 and Figure 4.12.
Transfer functions between these pressures and airflow are shown in the following
figures.
Transfer function between flow and ti anspulm onic pressure is the lung
im pedance. As seen from th e figure, it is alm ost constant in th e frequency range of
0 — I l f z . bung im pedance consists of lung tissue im pedance and airway im pedance.
i n k
m
mnrV/fi
48
transpulmonic pressure esophageal pressure
15
10
5
0.
100 120 60 80 40 0 20
time
-10
100 120 60
time
80 40 20
Figure 4.12: Esophageal pressure and transpulm onary pressure vs tim e
49
a >
$
I S )
S
cc
frequency
40
20
* 3
®
-20
-40,
frequency
Kigurr 4.1 ‘t: bung im pedance vs breathing frequency (transfer function between
transpulnioiiic pressure and airflow).
50
elastance
frequency
Resistance
cn o cn cn
Resistance
1
0
1
2
0 5 10 15
frequency
■ S
II
4)
10.
frequency
Figure 4.15: Transfer function between esophageal pressure and flow
52
Both of these im pedances are breathing frequency dependent. In norm als, Ilantos
et al [15] reported th a t tissue resistance decreased w ith frequency while airway resis
tance increased. T h e latter was also tidal volum e dependent. At the 0 1 Hz, airway
im pedance dom inate, which m eans lung resistance is supposed to increase. In the
p a tie n t’s d ata, we did not see m arked change of the lung im pedance. Actually, air
way resistance is so high th a t even at higher frequencies, where tissue im pedance
dom inate, we do not expect to see m uch of a decrease.
On the other hand, the transfer function between pleural pressure and the airflow
is associated with chest wall im pedance. Chest wall resistance has been reported to
be the main origin of the respiratory system nonlinearity. In this patient, however,
chest wall resistance would not dom inate at low frequencies of breathing, and would
not cause nonlinearity of the system resistance. Because first of all, it is quite low
and secondly high values of airway resistance would sweep all nonlinearity. Total
respiratory system resistance would be followed by airway resistance and any change
with frequency is not expected.
4 .4 S im u la tio n s m a d e o n a n o n lin e a r m o d e l o f
r e sp ir a to r y s y s te m
T he nonlinear behavior of the respiratory system in norm als can be modeled with the
model in Figure 3.1 which incorporates nonlinear resistance. Pressure-llow change
of this resistance is shown in Figure 4.16.
Sim ulations m ade on the nonlinear model showed th a t th e model provides a good
fit to th e experim ental d ata over the physiological range of breathing frequency ( <
1 Hz ). To determ ine the resistance and com pliance, the sam e m ethod was employed
in the simulations: Sinusoidal airflow signals were applied to the nonlinear model,
pressure signals were calculated and th e transfer function between these two signals
was determ ined by using Welch m ethod and from the real part of the spectrum
resistance values at different frequencies were found. T he change of resistance with
53
Pressure
flow
-A
Figure 4.16: Pressure-flow change of the nonlinear resistance.
frequency of breathing is shown in Figure 4.17. T he transfer function of t,lie nonlinear
resistance can be given as
11(f) = (4/1/(tt2/ F / ) + R (4.1)
when the input airflow is d V fd t = n fV jsin (2 iT f t) where f forcing frequency and Vr
tidal (or stroke) volume [13].
This kind of change can be explained by llildebrandt[17]’s plastoelastic, linear
viscoelastic model. Ilildebrandt suggests th a t there is plastic-like dissipation (dry
friction) in the tissue which is different from viscous resistance. Pressure due to
viscous resistance is proportional to the flow. But relative contribution of plastic
dissipation to the total pressure is constant. At low frequencies, flow is small and so
is viscous pressure; contribution of plastic dissipation to total pressure is large. But
when frequency increases viscous resistance takes over.
T h e m echanical equivalent of H ildebrandt’s m athem atical model would consist
of sliding block elem ents in addition to th e standard elem ents, dashpot and springs
which m im ic viscous and clastic behavior; because, sliding block can account for the
54
resistance
3.1
2.9
2.8
2.7
2.6
2.5
2.4
2.3
0.7 0.8 0.9 0.5 0.6 0.2 0.3 0.4
frequency
Figure A. 17: Resistance of the nonlinear model vs frequency of hrcutliing
55
3.5
2.5
(0
I
E
i
C L
■ a
0 3
0.5
C.7 0.9 0.2 0.3 0.4 0.5 0.6
frequency of breathing
0.0
Figure 4.18: Resistance of a large sliding block vs frequency of forcing.
effect of dry friction or plastic-like behavior. T o verify th a t, we collected flow and
pressure d a ta by connecting a spirom eter to the setup in Figure 3.4. T he spirom eter
sim ulated a sliding large block whose resistance is expected to have a nonlinear
change. By using the sam e procedure (applying sinusoidal airflow signals at different
frequencies and processing the d a ta in th e sam e way defined in th e previous section)
we obtained the resistance vs frequency change of the block shown in f igure 4.18.
56
C h a p te r 5
C o n c lu s io n
From our experim ents on tlu* healthy subjects, we can say th a t nonlinear behavior
of th e respiratory system is obvious. Results ol our study are very similar to those
of th e investigations conducted by Hantos et al.[15], B arnas et al.[5] and Barnas et
al.[4]. T he d a ta we collected (m outh flow and pressure) do not allow us to determ ine
the location of the nonlinearity.IIantos et al.[L5] suggested th a t noniinearity was
due to the regional differences in tissue properties. However, from their relative
displacem ent m easurem ents, Barnas et al.[4] concluded large changes in resistance
and elastance cannot be explained by nonuniform ity of displacem ent. We believe,
like B arnas et al.[d], it is due to plastic dissipation, a nonlinear property of the chest
wall. As we m entioned earlier wo did not see frequency dependence on the p a tie n t’s
im pedance. If unequal tim e constants of the different regions (due to the regional
differences) was an answer to the nonlinearity, we should have seen it in the C O PD
patient as well. Therefore, our observations have given rise to the idea th a t the
nonlinearity originates from visco-plasticity of the chest wall.
In th e O O PI) patient, airway resistance was so high th a t any nonlinear change
was not observed. In the dog, Barnas et al.[2] found lung im pedance to decrease
between 0.2 and 0.0 Hz. This was attrib u ted to the tissue resistance and decrease
was consistent with viscoelasticity. At the higher frequencies, airway resistance
dom inated. In our results in the patient, airway resistance is also quite high at
th a t range and neutralize the decrease of the lung tissue resistance and th a t is most
likely why we observe lung im pedance as constant at th a t range of breathing. We
expect in the C O P D patients, effects of th e airways to th e overall respiratory system
resistance would be dom inant as well, it m eans chest wall resistance which has been
57
assum ed to have nonlinear properties would not, have th a t im p o rtan t consequences
in the C O P D patients.
In the patient, we have the system im pedance up to 1 Hz of breathing since
the input forcing signal was at 0.2 Hz. But we can easily predict th a t th e total
respiratory im pedance would be slightly increasing after 1 Hz due to the nonlinear
effect of the airway im pedance. It m eans, unlike norm als, in C O P D patients work
of breathing is sm aller at the norm al range of breathing.
In both healthy subjects, elastancc of the total respiratory system im pedance
increased with increasing frequency of breathing. This was also th e case for th e elas-
tance of the p a tie n t’s chest wall im pedance (which is th e transfer function between
esophageal pressure and airflow). Klastances of the lungs and airways do not have
th a t sharp change. At the breathing range, inertial forces which have a significant
effect on th e elastancc can be neglected. They come into account after 2 Hz and
cause elastancc to decrease. O ur observations are consistent with viscoelastic be
havior of the chest wall and it is m ost likely th e m ain origin of th e nonlinearity of
elastancc change in normals. In the C O P D patient, its effect would probably be
very im p o rtan t to the overall respiratory system elastance since chest wall elastance
cannot be neglected if com pared to the lungs’.
T here is no a tte m p t in th e literature to model the frequency dependence of the
respiratory system except Ilild eb ran d t’s plastoelastic, linear viscoelastic model. We
showed by sim ulations th a t the nonlinear model in Figure 3.1 (which has the nonlin
ear resistance) is a very good representation of the frequency dependent (viscoelastic)
process. As we went over earlier, some investigators [2, 4] have suggested im pedance
of the system is also a function of tidal volum e of the forcing signal. B arnas et al
[2] showed th a t frequency and tidal volum e dependent processes are tied to each
other. For sm aller tidal volumes, resistance at low frequencies is higher. In H'dde-
b ra n d t’s model, these two processes were m echanically independent. However in the
nonlinear model both of these processes are included in the transfer function of the
nonlinear resistance and it could probably explain th e tidal volum e dependence seen
in the healthy individuals as well.
58
R e fe r e n c e L ist
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61
A p p e n d ix A
C o m p u te r p r o g r a m w r itte n for t h e lin e a r m o d e l
form at long
sf= 100;
is f= l/s f;
t=0:isf:50;
f=0.2;
v t = l ;
w =(2*pi*f);
a fl= v t* (w /2 )* ain (w * t);
a f2 = (sq u a re (2 * p i* f* t)+ l)/2 ;
af= afl.*af2;
r= 2 ;
c=0.2;
T C = (r* c );
ic = l/c ;
N = l;
k l= (sf/(2 * f));
k 2 = 2 * k l;
P c (l)= 0 .0 ; for j= l:5 0 * f;
k lN = k l + N ;
k 2 N = k 2 + N ;
for i = N :k lN -l;
P c ( i+ l) = P c ( i) + ic*af(i)*isf;
end
P(N :k lN )= a f(N :k lN )* r+ P c (N :k lN );
for i= k lN :k 2 N ;
p (i) - 0 ;
end
m =isf;
for k = ( k lN + l) :( k 2 N )
P c { k )= (P c (k lN )* e x p (-m /{ T C )));
a f(k )= -P c (k )/r;
62
m = m + isf;
end;
N = N + k 2 ;
end;
su b p lo t(2 ,l,l);
plot(t,af);
x lab el(’t ’);
yla b el(’A F ’);
subplot(2,l,2);
pIot(t,p);
xlabel(’t ’);
ylabeI(’P re s.’);
N=512*2;
S = sp ectru m (af,p ,N );
d=real(S (:,4));
fl= sf* (0 :(N /2 )-l)/N ;
f l = f l ’;
co m = -(l./((2 * p i)* fl.* im ag (S (:,4 ))));
su b p Io t(2 ,l,l);
p lo t(fl(l:1 5 ),d (l:1 5 ));
xlabelf’frequency’);
y label(’R ’);
subplot(2,l,2);
p lo t(fl( 1:15),com ( 1:15));
x label(’frequency’);
y label(’C ’);
% This program is for linear model
****** ********************************* ******
63
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Asset Metadata
Creator
Saglam, Mehmet Akif
(author)
Core Title
Respiratory system impedance at the resting breathing frequency range
School
School of Engineering
Degree
Master of Science
Degree Program
Biomedical Engineering
Degree Conferral Date
1994-12
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
engineering, biomedical,OAI-PMH Harvest
Language
English
Contributor
Digitized by ProQuest
(provenance)
Advisor
Yamashiro, Stanley M. (
committee chair
), Khoo, Michael Chee-Kuan. (
committee member
), Maarek, Jean-Michel (
committee member
)
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c18-4155
Unique identifier
UC11357940
Identifier
1376506.pdf (filename),usctheses-c18-4155 (legacy record id)
Legacy Identifier
1376506-0.pdf
Dmrecord
4155
Document Type
Thesis
Rights
Saglam, Mehmet Akif
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...
Repository Name
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Repository Location
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Tags
engineering, biomedical