Dynamic
viscoelastic
properties
H. BOBBAERS, J. CLEMENT, AND Laboratorium uoor Longfunktieonderzoek,
K. P. VAN DE WOESTIJNE Academisch Ziekenhuis
BOBBAEBS, H., J. CLEMENT, AND K. I? VAN DE WOESTIJNE. Dynamic viscoelasticproperties of the canine trachea. J. AppL Physiol.: Respirat. Environ. Exercise Physiol. 44(2): 137-143, 1978. -The dynamic elastance (E) and viscous resistance (R) of the canine intrathoracic trachea were determined by submitting excised tracheal segments to periodic variations of pressure at frequencies between 0,5 and 9 Hz, and by relating simultaneous transmural pressure and volume variatipns. Mean values of E and R are, respectively, 17.3 cmH,O ‘cm-’ and 0.16 cmH,O - cm-l - s, or, expressed in volume changes per cm length of the trachea, 53.5 cmEI,O- ml-’ and 0.49 cm&O .rnl-’ 9s. E and R increase at higher longitudinal tension of the trachea and under the influence of acetylcholine,
viscous resistance;
eIastance;
acetylcholine;
of the canine
arteries
AS COMPLIANT STRUCTURES, the airways are mechanically in parallel with the lung parenchyma. Mead (12) showed that, in the presence of an increased resistance to airflow, the airway compliance may influence markedly the frequency-dependent behavior of the lungs, at least if there is a direct proportionality between airway volume and transmural pressure. This requires a negligible airway wall viscous resistance. As far as we know, the wall resistance of the airw?.ys has not been determined. The measurements performed on arteries yield relatively large values (see DISCUSSION). If, likewise, the airways viscous resistance were appreciable, the airways would behave as stiff structures at higher frequencies, and their contribution to the frequencydependent behavior of the lungs would be less than estimated by Mead (12). A high viscous resistance would also stabilize the airways during a forced expiration and delay airway collapse (5). We thought, therefore, that it might be of interest to measure the viscous resistance of the airways. In the present study we investigate the dynamic viscoelastic properties of excised canine tracheas. A simple way of expressing the viscoelastic properties of a tube with symmetrical structure, e.g., a blood vessel, is to relate the amplitude of the changes in transmural pressure P to the amplitude of the changes in radius r, and to calculate an incremental pressurestrain ratio P/r. The latter is “dynamic” when the measurement is performed under conditions of pressure changing with time. If there is a phase lag p between P and r, one can separate the pressure-strain ratio into two parts, by relating P to the changes of r in phase with P (P/r cos rp), or 90” out of phase with P (P/r sin 50)~The former may be called an elastance E, the
St. Rafael,
trachea
3000 Leuven,
Belgium
latter a viscosity coefficient Ru. Both are expressed in cmH,O* cm? To obtain a resistance R, one divides RU by the angular velocity W. The viscous resistance R is thus P/r@ sin up and is expressed in cmH,O cm+ s. When instead of r, one relates the relative changes of radius (r/r,> to P (Pro/r), one obtains an incremental pressure-strain modulus (in cmH,O). The latter can be resolved similarly into an elastic (in cmH,O) and a viscous modulus (in cmH,O s). When we submitted segments of the trachea to peri .odic pressure variations, only the pars membranacea was observed to move, where& the-shape of the cartilagenous rings was not changing noticeably. Instead of a homogeneous tube with circular cross section, which is the model used to represent a blood vessel, we were dealing with a flat, stretched membrane becoming curved under the influence of transmural pressure changes. Accordingly, the technique of deter&nation of viscous resistance and elastance used for blood vessels was not applicable to the trachea. We developed another model, representing the trachea as a succession of rigid, open rings, the slit of which is closed by an elastic membrane. As shown in the APPENDIX it is possible under certain conditions (see DISCUSSION) to calculate the viscous resistance R and the elastance E of such a structure from its changes in volume and corresponding changes in pressure. The expressions of R and E are similar to those used for blood vessels l
l
P s R z-msin p WV 2 P s E = v ’ 2 cos p
(1)
where V is the amplitude of volume changes and S the surface of the pars membranacea. Although R reflects only the viscous work, E is a ‘(dynamic” elastance, i.e., a composite of elastic and inertial properties of the trachea. MATERIALS
AND
METHODS
Experiments. The intrathoracic part of the trachea of 11 healthy adult dogs was removed within 30 min after each animal was killed. All measurements were performed within 0.5 and 2.5 h after death. The dimensions of the investigated airway segments are given in Table 1. Rigid Perspex tubes of the-largest possible diameter were inserted and securely tied into each end of the trachea. The mounted trachea was suspended under minimal longitudinal tension in a small Perspex flow
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1.38 TABLE
BOBBAERS,
1. Dimensions
and uiscoelastic Tracheal
Dog No.
Wt, kg
3 4 5 6 7 8 9 10 11 MeankSD
Segment
Length,
a 2
properties
cm
Radius,
cm
of tracheal
Pars Membran, % circumf
4.4 3.1 3.8 5.2 6.4 5.0 3.5 5.6 3.5 4.0 4.8 4.48k1.02
27.3 21.9 11.3 11.3 17.3 14.4 26.9 9.6 9.9 10.5 11.7
0.55 0.67 0.79 0.91 0.92 0.83 0.71 1.05 0.84 0.68 0.75
Viscous -----~cmH,O cm “*s
P, cmH,O
0.791-tO.140
15.656.7
Radius: mean internal radius of the investigated SD); V: corresponding volume variations.
segment;
6.4* 1.9 5.9-t- 1.6 8.321.5 5.2+1.4 6.2k1.8 3.6kO.8 3.921.3 5.0+1.6 20.6*4,0 12.5 +4.0 6.3t2.1 7.6324.94 P: pressure
displacement plethysmograph (Fig. 1). The air inside the plethysmograph was kept at about 37°C and saturated with water vapor. Volume variations of the trachea were produced by a sinusoidal pump, provided with a variable leak. The frequency of oscillations varied between 0.5 and 9 Hz. Volume displacement of the trachea and transmural pressure (difference between inside and outside pressure) were registered simultaneously on a direct-writing recorder (Mingograph Elema) and displayed in X-Y coordinates on a storage oscilloscope (Tektronix 56 MB). The amplitude of the volume changes was kept small by adjusting the leak of the pump to keep pressure and volume changes within the limits of linearity of the tracheal wall (Fig. Z), The frequency characteristics of the electromanometers and the flow plethysmograph, corrected by means of box pressure (I@, were flat in phase and amplitude up to 10 Hz. It is demonstrated in the APPENDIX Computations. that unbiased estimates of R and E can be obtained from the slopes of the tan (P-W and V-P cos (p relationships, measured at w = 0 and P = 0, respectively. To estimate these slopes, a regression was fitted on the experimental data and constrained to pass through the origin; indeed, up and V are necessarily zero when w and P equal zero. To allow for nonlinearities and for the effect of inertia, second-order terms were introduced into the regression functions, i.e.
1.16 0.54 0.41 0.57 1.24 0.42 0.49 0.96 1.61 0.56 1.04
+ p&
+ YIP0
(2)
V = ap(P cos cp) + &(P cos cp)’ + ygo(P cos cp)
(3)
+ &&(P cos cp) The slopes at the origin are given by the regression coefficients cyl and az; R and E can thus be estimated from R = (a&,>* (S/Z) and E = az4/2, respectively. The standard errors of the estimates of R and E are derived from the standard errors of al and az given by the regressions. RESULTS
Influence
of freauencv
and alinearitv.
To the extent
Resistance
VAN
DE
WUESTIJNE
0.16+0.12 to which
Elastance
--
cmH# *ml-’ es
0.06 0.17 0.49 0.17 0.14 0.12 0.17 0.15 0.06 0.13 0.07
0.818*0.404 variations
cm&O.
0.04 0.11 0.44 0.13 0.05 0.07 0.08 0.08 0.06 0.18 0.05
8.41 18.17 41.02 15.96 24.67 11.54 16.30 18.71 10.37 15.91 9.66
0.1220.12
the tracheal
cm ’
=ml
’
4.79 11.81 37.23 12.43 7.94 6.23 7.80 9.83 10.82 21.68 7.39
17.34-+9.18
segment
cmH,O
was submitted
12.5459.35 (mean
1
P in
v
water FIG. 1. Tracheal segment is enclosed in a small plethysmograph, thermostated by means of a water jacket. Volume changes are calculated from the integrated plethysmographic flow signal lvdt, to which a fraction of box pressure Pout is added (pressure correction). Transmural pressure AP is the difference between pressure measured inside tracheal segment Pin, and Pout.
l 0
tan 9 = qw
AND
segments
-_---.--_.“,“- ----- ---- --..--. . -.-._.. -.._.-_ 8.3 13.0 9.8 27.0 21.5 19.0 13.0 32.0 17.5 17.5 12.5
CLGMENT,
025
.
0.5s
l
I
0
20
40cmH20
FIG. 2. In A, record of volume V and transmural pressure changes P as a function of time. P precedes V a little. Insert shows corresponding V-P loop. In B and C, V-P loop at the same frequency, when the trachea is submitted to larger pressure variations. In A, trachea behaves as a nearly linear system. Alinearity is more marked in B and still more in C. (Dog 10.)
that the model used to calculate the viscous resistance and elastance describes the behavior of the real trachea, one expects (Eq, 4 ' of the APPENDIX): 1) an increase of tan q, the phase shift between amplitude of pressure P and volume changes V, with the angular velocity o (Eq. 2); 2) a relationship between V and the first power of P cos q (Eq. 3); 3) a quadratic relationship between both tan (p and V-P cos q with W, if the inertance of the wall Dlavs a role within the range of
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VISCOELASTIC
PROPERTIES
OF
THE
investigated frequencies. Indeed, when Eqs. 4' are expanded in series with respect to pi), terms in wZ are obtained, the magnitude of which depends on the inertia. The statistical analysis of Eq, 2 shows that its three coefficients differ significantly (P < 0.001) from zero. This is in agreement with the model’s predictions, for coeficients aI and PI. The significant interaction between P and w (coefficient rl) points to an alinearity: the relationship between tan p and m depends on the amplitude of P. Figure 3 shows that this alinearity is small within the limits of applied pressure variations. The coefficients of Eq, 3 are statistically significant, except for coefficient yp (a2, P < 0.001; &, P < 0.001; y2, P > 0.2; 8, P < 0.05). Significant coefficients atl and & were expected. The presence of a quadratic relationship between V and (P cos @ indicates alinearity. Again, the latter is small (Fig. 3). The slopes of both regressions increase with frequency. The slope of the relationship V-P cos q is an expression of the dynamic compliance (the reverse of the elastance); the dynamic compliance thus increases with frequency (significant coefficient 8, reflecting the influence of inertia). R is calculated by dividing the slopes of the two regressions: when this computation is performed at different frequencies, R does not appear to be frequency dependent between 0 and 9 Hz (t, 0.42; P > 0.5). A priori, we do not expect a linear relationship between P and V; the estimates of R and E are unbiased only at the origins of the regressions, i.e., at or) = 0, and P = 0. The values of R and E calculated under these conditions are given in Table 1. Influence of surrounding tissues. We wondered to what extent the properties of the trachea were influenced by the surrounding tissues. To approximate this influence, we performed the determinations without dissecting the esophagus away in six experiments. Five times, the resistance values were slightly higher than on the isolated trachea. The mean resistance was 0.11 and 0.15 cmH,O cm+ s, respectively, without and with l
l
139
TRACHEA
the esophagus (avg of 6 expts). The elastance was not modified. Influence of longitudinal tension. We investigated the influence of longitudinal tension on the viscoelastic properties of the trachea in two ways. In four dogs, the measurements were performed before and after elongation of the trachea, realized by moving apart the tubes over which the tracheas were tied. Thus, the tracheal segments were lengthened on the average from 4.1 to 4.9 cm. This resulted in a slight increase of R and E in three dogs, whereas R and E decreased in one dog (Fig. 4). The influence of longitudinal stresses was verified in another way. In eight experiments, we determined the mechanical properties successively of a longer and a shorter segment of the same trachea. This should not influence the value of R or E, which are independent of the length of the airway segment, In fact, a shorter segment (avg length, 2.1 cm) has a significantly (P < 0.001) larger viscous resistance and elastance (mean R, 0.43 cmHZOcm-ls; E, 29.4cmH,Ocm’) than alonger segment (avg length, 4.3 cm; R, 0.18 cmH,O . cm-l 4s; E, 18.0 cmH,O cm--l) (Fig. 4). Influence of acetylcholine. After completion of the measurements, we moistened the pars membranacea of the tracheas from the outside by a few drops of a 5% solution of acetylcholine chloride. This resulted, in the six experiments in which it was attempted, in a very pronounced contraction of the pars membranacea pulling the ends of the cartilagenous rings tightly together. We were unable to determine the mechanical properties of the trachea in the latter condition, since the volume displacements were too small to be registered with any accuracy even when the pressure variations were increased beyond 20 cmH,O. In one dog, however, in which the measurements were feasible probably because the tracheal constriction was less pronounced than in the others, acetylcholine produced a 20-fold increase of R and a 6-fold increase of E. l
DISCUSSION
Validity
of the technique. The computation
of a dy-
P=l =3 =5 FIG. 3. For 11 dogs, mean regressions between tan cpY phase shift between volume (V) and transmural pressure amplitude (P), and angular velocity cd, and between V and P cos ~0. Tnfluence of P and w, respectively, on former and latter regression is shown. BUFS are interindividual standard errors of ‘ the regression.
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140
FIG. dogs);
shorter
BUBBAERS,
4. Mean B, after segment
regression elongation of trachea
V-P cos cp: A, in control of the trachea (4 dogs); (8 dogs) (see text).
conditions ( 11 and C, for a
namic elastance E and viscous resistance R from the amplitude and phase relationships of transmural pressure and volume of the trachea is valid only if the used equations adequately describe the behavior of the trachea. The main assumptions underlying these equations are that 1) the viscous resistance and inertia are small with respect to the elastic forces and 2) the boundaries of the membrane can be approximated by an ellipse. Further conditions are that the amplitude of the pressure variations is small and that the pressure input is sinusoidal. The results show that the two assumptions are satisfied. The viscous resistance and inertia are, indeed, small with respect to the elasticity of the membrane. If the inertia were large, the elastance would decrease markedly at higher frequencies. This was not observed. The fact that the V-P cos p relationships are nearly linear over rather large pressure variations (Fig. 3) shows that the second assumption is also valid (see APPENDIX).
Though we kept the ‘amplitude of P small to stay within the limits of linearity of the system, the regressions fitted on the experimental data suggest alinearities. The latter may result from nonlinear mechanical characteristics of the tissues, from an inertia effect, or from an inadequacy of the model used to represent the trachea. These various factors should not introduce errors in our estimates of E and R, however, since we avoided their influence by determining the ‘slopes of the regressions, from which E and R were computed, at o = 0 and P = 0. To keep the trachea around its resting volume and to avoid large changes of P with varying frequency, we provided the pump used to cycle the trachea with a large leak. This resulted in a distortion of the pressure input and was no longer an exact sinusoid (Fig. .2>. To investigate to what extent this distortion introduced errors in the computation of E and R, we compared the values, calculated from P and V measured on the actual signals, with those obtained from the fundamental frequency of the oscillations. The latter was computed by means of a Fourier analysis of the pressure and volume cycles. Five to nine comparisons were performed in each experiment at various frequencies and in differ-
CLEMENT,
AND
VAN
DE
WQESTIJNE
ent conditions (stretched and nonstretched trachea, short and long segment, with and without esophagus, constricted pars membranacea or not). Figure 5 shows that both the elastance and resistance values are underestimated when they are computed from the distorted signal produced by the pump. The error is small, however, and unimportant for the lower values of E and R: those are the values observed in the unstretched, unconstricted trachea. Comparison with properties of blood vessels. A comparison of the viscoelastic properties of tubes of various sizes is only possible if one takes the initial size of the tube into account. In studies on blood vessels, the stress-strain relationships are converted into a modulus of elasticity and viscosity by relating the pressure changes to the relative changes in radius. Because of the asymmetry of the trachea a modulus cannot be computed. Therefore, we limited ourselves to a comparison between the trachea and blood vessels of similar size, i.e., the descending thoracic and the abdominal aorta, all data being expressed as elastance and viscous resistance. Table 2 shows that the values measured on the aorta are markedly higher. However, the latter measurements were performed at a pressure of about 140 cmH,O. The elastic characteristics of arteries are strongly alinear; accordingly, the elastance values cannot be compared as such with our data on the trachea determined around the resting position (transmural pressure, 0 cmH,O). Under the same conditions, the elastance of the thoracic and abdominal aorta, measured statically, are respectively about 167 and 222 cmH,O 4cm-l (1). The static elastance is lower than that determined in dynamic conditions (i.e., at frequencies of 2 Hz or more), and this difference increases for more muscular arteries (2); after allowance for this effect, the dynamic elastance can be estimated as 178 cmH,O *cm-l for the unstretched thoracic aorta, or 263 cmH,O *cm-l for the abdominal aorta. The influence of mean pressure on the viscous resistance is less marked; according to Hardung (9), the latter is but little affected by the mean stress. We conclude that the elastance and viscous resistance of the aorta are more than 10 times larger than those of the trachea, even if one takes the differences in pressure level into account. Our study shows, on the other hand, that the dynamic elastance and resistance of the trachea are qualitatively similar to those of arteries. 1) The viscous resistance is markedly smaller than the elastance; the ratio R/E is about 1% for both the trachea and the aorta (Table 2). 2) The elastance and resistance are larger when the trachea is stretched or a shorter segment is investigated. This has been observed in arteries (3) and reflects the interaction of longitudinal on circumferential stresses (16). Maybe a similar interaction exists between longitudinal and transverse stresses in the pars membranacea. 3) Constriction of the trachea results in an increase of resistance and dynamic elastance. The viscoelastic properties of arteries are modified similarly (3, 9). 4) As in blood vessels (17), the influence of wall inertia seems to be small. We observed a slight decrease of E at higher frequencies which might be due to the inertance of the wall (Fig. 3). A recent paper of Knudson
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VISCOELASTIC
PROPERTIES
OF
THE
141
TRACHEA
? :mH201cmls
Y= 0.085+0.84(+0.04)X
l
R( Four ier) 0
Q50
1.50
1.00
cmH~/cm/s FIG. 5. Comparison of R and E values determined from the P-V relationships on actual P-V records (Y axis) and on fundamental frequency of the same records (X axis). Continuous lines are calculated regression lines, dashed lines, the lines of identity.
60
SO
40
30
Y =1.886+0.89(-t
20
0.03) X
-1 10
0
10
20
30
40
50
et al. (10) showed that airway wall inertia does not delay significantly the dynamic collapse of the trachea. Pressure-volume relationships. The viscoelastic properties of the trachea can be expressed in cmH,O .ml-l or cmH,O rnP.0 s, instead of cmH,O .crn+ or cmH,O burn-’ 0s simply by dividing the latter expresl
SO cm H20/cm
sions by the half surface of the pars membranacea (Table 1). The obtained R and E values are inversely proportional to the length of the tracheal segment. Expressed per cm length, the mean viscous resistance is 0.49 k 0.43 (SD) cmH20-ml-1*~. The corresponding value for the elastance is 53.5 t 34.7 (SD) cmH,O ml-l. l
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142
BUBBAERS,
TABLE 2. Dynamic viscoelastic the aorta and trachea
properties
Thoracic
aorta
Abdominal
17 (dog) 2 (dog) 14 (dog) 11 (man) 8 (dog) 15 (dog) 17 (dog) 2 (dog) 14 (dog) 11 (man) 8 (dog) Present work
aorta
Trachea Values frequency
(dog)
of aortic elastance and resistance of 2 Hz (not specified in (17)).
Viscous Resistante cmH,O * cm-l . s
3,943 783 1,118 682 772 918 7,578 2,422 4,082 1,120 2,791 17 were
140.0 6.0 6.5 4.8 3.3 222.9 18.6 9.7 26.8 0.2 determined
at a
The latter figure is in satisfactory agreement with the value determined by Coburn et al. (7) in dogs in quasistatic conditions: 32.9 t 12.1 (SD) cmH,O ml+ per cm length of trachea, but is higher than the values found in cats by Olsen et al. (13). For a 12-cm-long trachea we estimate R to be 0.04 cmH,O ml-l s or 40 cmH,O 1-l s and E 4.5 cmH,O ml? These figures apply only to the trachea; they do not allow us to calculate the viscous resistance of the whole bronchial tree since it is unlikely that the successive bronchial generations, which differ markedly in structure, have identical viscoelastic properties. It is striking, however, that the ratio R/E for both the trachea and the aorta (Table 2) as well as for other arteries (8) is similar. Accordingly, it is not unreasonable to assume that this ratio also applies for the total bronchial tree. The elastance of the airways during tidal breathing is about 0.25 cmH,O*ml+ (12) or 0.2 cmH,O~ml+ (4). The viscous resistance of the bronchial tree may thus be estimated to be abdut 0.0025 cmH,Om ml-l 9s or 2.5 cmH,O 1-l s. l One may wonder to what extent our figures of E and R of the trachea are influenced by the active tension of the airway smooth muscle. Coburn and Palombini (6) reported that the in vivo canine trachea demonstrates a marked stress relaxation, related to the trachealis muscle active tension. In the presence of stress relaxation, the elastance increases with frequency. This is observed in blood vessels up to 2 Hz, and to higher frequencies when the active tension of the vessels is larger (3, 9). Our observation of a nearly constant value of E with frequency may be due to a complete lack of smooth muscle active tension in our preparation. In one case, we determined E under conditions of maximal contraction of the trachealis muscle; E increased with frequency up to 4 Hz. Accordingly, it is possible that we underestimate somewhat E and R, compared to the in vivo situation. Even so, R remains small with respect to E. Therefore, it is likely that R does not slow down l
l
l
l
l
l
AND
VAN
DE
WOESTIJNE
of the airway walls noticeably the movements the influence of changes in transm ural pressure.
of
Elastance cmHzO.cm-l
Reference
CLEMENT,
under
APPENDIX Model
of the Trachea
The pars membranacea is assimilated to a plane rectangular elastic membrane attached to cartilagenous rings. The deformations of the latter, under the influence of transmural pressure changes, are considered to be negligible with respect to the deformations of the pars membranacea. Accordingly, we consider only the mechanical properties of the pars membranacea. The Cartesian coordinates of any point of the membrane in the transverse and longitudinal directions are defined by x and y, respectively, the origin being taken at the center of the membrane. The transverse and longitudinal tensions in the membrane are indicated by F, and F,, respectively. If a transmural pressure is exerted on the membrane, each of its points will be displaced perpendicularly over a distance z. The elastic counterpressure on each point will be, by virtue of Laplace’s law, directly proportional to the radii of curvature in both x and y directions at that point, i.e., to the partial derivatives 6%/6x2 and 8%/6y (as long as the first derivatives &Z/&X and 62/6y are (I and w being the half lengths, respectively, of the large and small axes of the ellipse) Eq. 1’ can be solved. Indeed, then z = a2(g2 + aofy2 + aoo the coefficients
aij satisfying
the set of equations
2F sa2O + 2 F,ao2 = P l2 - ao2 ~ = 0
a20
W2
aozZ2 + aoo = 0 and the corresponding
volume
changes
v=
sp
4
become
[u.‘2+F 1 Fx
Fv
in which S is the total area of the membrane. For ellipsoidal boundaries, there is thus a linear relationship between the static transmural pressure and the corresponding volume change. Under other conditions, e.g., for rectangular boundaries, the V-P relationship cannot be linear and must be approximated by an infinite series, e.g., of the type V = aP + PP’ + . . . .
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VISCOELASTIC
PROPERTIES
OF
THE
143
TRACHEA
If the pressure amplitude is small, however, the higher powers of P become negligible, i,e., the V-P relationship tends to be linear. In the limit, the slope of the observed V-P relationship at the origin P = 0 yields an estimate of the “elastance” of the membrane. In the same limit, the displacements z in all regions near the boundaries are close to zero, and since ellipsoidal boundaries are the only ones which ensure a linear relationship between V and P, the rectangular boundaries can be approximated by the nearest ellipsoidal ones, i.e., by an ellipse with the same surface as the rectangle. We can now introduce m into Eg. 1’. The displacement z becomes periodic z = A(x, Substitution differential
in the equations
y) sin wt + B(x,
equilibrium Eq. in the unknown
forces F,, F, >> Rwb,, = 0, &azo be considered as for all coefficients 2) Boundary equal powers are
w’ago - ao,P
may write a,,, utL2, bAo, bzz = 0 and Thus the equilibrium conditions will satisfied (i.e., w sufficiently small) 2. which all remaining coefficients of
= 0
w2bz0 - bo,Z2 = 0
a()()+ a(),z2 = 0 The
amplitude
y) cos tit
1’ yields function
Ru, IO’), one = 0 . . . , etc. approximately of powers 2 conditions in equated
V of the total
boo + b,,Z’
volume
changes
v = cr// 4x, Y) dx dY12+ r/i e,
a set of two A andB
partial
and its phase
shift
Y>dx dY12~1’2
J/ Bb, Y>dx dy (1 bis’)
Solution
(3’)
(3’)
y> dx dy
of the set of Eqs. 2’ and 3’ yields Rw 2F x + 2 W2 I”
tan cp =
2F
A and B as a series A =
is then
cp tan ’ = [f A(x,
We write
(2’)
= 0
S v = 2 (P cos p) 2F 2
2 2 aijxif j=o j=o
+ Iw”
(4’) 1
-
+ --+ 2;
+ ImZ
W2 B
=
2
2
i=()
j=o
bij&’
The viscous
By virtue of the symmetry of the membrane, only the even powers in x and y are retained. If the boundaries are ellipsoidal and if w is small, so that the viscous resistance and inertial forces are small with respect to the elastance, all powers in x and y greater than 2 can be neglected, leaving the following system of equations. 1) Equilibrium conditions of Eq. 1’ bis in which the coefficients of power zero in x and y are equated 2F
#20
+
2
Fara*2 - Roboo
2 FrbeO + 2 F,b,, If the coefficients
of the second
+ Rw,~ power
12F&40+ F&2 12 F,bdO + F,& But
with
elastance
forces
far
- Idaoo - Wboo
in x: are equated - IoZ&o
= 0
+ Roazo
- Mb,,
= 0
greater
than
Rob20
resistance
coefficient
R is
(5’) and the elastance
coefficient
E
(6’)
(2’)
In the case of rectangular boundaries, Eqs. 4’ can be approximated only if P is suficiently small, i.e., if the displacements z in all regions near the boundaries are close to zero. Accordingly, R and E should be estimated in the limit, from the slopes of the tan cp-o and P cos cp-V curves at their origins, o = 0 and P = 0.
(2’)
This kundig
= P = 0
resistance
one obtains
or inertial
Received
study was supported by a grant Wetenschappelijk Onderzoek.” for publication
28 February
of the “Fonds
voor
Genees-
1977.
REFERENCES 1. BERGEL, D. H. The static elastic properties of the arterial wall. J. PhysioZ., London 156: 445-457, 1961. 2. BERCEL, D. H. The dynamic elastic properties of the arterial wall. J. PhysioZ., London 156: 458-469, 1961. 3. BERGEL, D. H. The properties of blood vessels. In: Biomechanics. Its Foundation and Objectives, Y. C. Fung, N. Perrone, and M. Anliker. Englewood Cliffs, N. J.: Prentice-Hall, 1972, p. 105139. 4. BOBBAERS, H., J. CLEMENT, J. PARDAENS, AND K. P. VAN DE WOESTIJNE. Simulation of frequency dependence of compliance and resistance in healthy man. J. AppZ. PhysioZ. 38: 427-435, 1975, 5. CLEMENT, J., J. PARDAENS, AND K. P. VAN DE WOESTIJNE. Expiratory flow rate, driving pressures and time-dependent factors. Simulation by means of a model, Respiration PhysioE. 20: 353-369, 1974. 6. COBURN, R. F., AND B. PALOMBINI. Time-dependent pressurevolume relationships of the in vivo canine trachea. Respiration Physiol. 16: 282-289, 1972. 7. COBURN, R. F., D. THORNTON, AND R. ARTS. Effect of trachealis muscle contraction on tracheal resistance to airflow. J. AppZ. Physiol. 32: 397-403, 1972. 8. Gow, B. S., AND M. G. TAYLOR. Measurement of viscoelastic properties of arteries in the living dog. Circulation Res. 23: lll122, 1968. 9. HARDUNG, V. Dynamische Elastizitgt und innere Reibung muskularer Blutgeftisse bei verschiedener durch Dehnung und
10.
11.
12.
13.
14.
15.
16. 17. 18.
tonische Kontraktion hervorgerufener Wandspannung. Arch. Kreislaufforoch. 61: 83-100, 1970. KNUDSON, R. J., J. MEAD, AND D. E. KNUDSON. Contribution of airway collapse to supramaximal expiratory flows. J. Appl. Physiol. 36: 653-667, 1974. LEAROYD, B. M., AND M. G. TAYLOR. Alterations with age in the viscoelastic properties of human arterial walls. CircuZation Res. 18: 278-292, 1966. MEAD, J. Contribution of compliance of airways to frequencydependent behavior of lungs. J. AppZ. PhysioZ. 26: 670-673, 1969. OLSEN, C. R., A. E. STEVENS, AND M. B. MCILROY. Rigidity of tracheae and bronchi during muscular constriction. J. AppE. Physiol. 23: 27-34, 1967. PATEL, D, J., F. M. DE FREITAS, J. C. GREENFIELD, JR., AND D. L. FRY. Relationship of radius to pressure along the aorta in living dogs, J. AppZ. Physiol. 18: 1111-1117, 1963. PATEL, D. J., J. S. JANXCKI, R. N. VAISHNAV, AND J. T. YOUNG. Dynamic anisotropic viscoelastic properties of the aorta in living dogs. Circulation Res. 32: 93-107, 1973. PATEL, D. J., A. J. MALLOS, AND D. L. FRY. Aortic mechanics in the living dogs. J. AppZ. Physiol. 16: 293-299, 1961. PETERSON, L. H., R. E. JENSEN, AND J. PARNELL. Mechanical properties of arteries in vivo. CircuZation Res. 8: 622-639, 1960. STXNESCU, D. C., P. DE SUTTER, AND K. P. VAN DE WOESTIJNE. Pressure-corrected flow body plethysmograph. Am - Rev. Respirat. Diseases 105: 304-305, 1972.
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viscoelastic
properties
H. BOBBAERS, J. CLEMENT, AND Laboratorium uoor Longfunktieonderzoek,
K. P. VAN DE WOESTIJNE Academisch Ziekenhuis
BOBBAEBS, H., J. CLEMENT, AND K. I? VAN DE WOESTIJNE. Dynamic viscoelasticproperties of the canine trachea. J. AppL Physiol.: Respirat. Environ. Exercise Physiol. 44(2): 137-143, 1978. -The dynamic elastance (E) and viscous resistance (R) of the canine intrathoracic trachea were determined by submitting excised tracheal segments to periodic variations of pressure at frequencies between 0,5 and 9 Hz, and by relating simultaneous transmural pressure and volume variatipns. Mean values of E and R are, respectively, 17.3 cmH,O ‘cm-’ and 0.16 cmH,O - cm-l - s, or, expressed in volume changes per cm length of the trachea, 53.5 cmEI,O- ml-’ and 0.49 cm&O .rnl-’ 9s. E and R increase at higher longitudinal tension of the trachea and under the influence of acetylcholine,
viscous resistance;
eIastance;
acetylcholine;
of the canine
arteries
AS COMPLIANT STRUCTURES, the airways are mechanically in parallel with the lung parenchyma. Mead (12) showed that, in the presence of an increased resistance to airflow, the airway compliance may influence markedly the frequency-dependent behavior of the lungs, at least if there is a direct proportionality between airway volume and transmural pressure. This requires a negligible airway wall viscous resistance. As far as we know, the wall resistance of the airw?.ys has not been determined. The measurements performed on arteries yield relatively large values (see DISCUSSION). If, likewise, the airways viscous resistance were appreciable, the airways would behave as stiff structures at higher frequencies, and their contribution to the frequencydependent behavior of the lungs would be less than estimated by Mead (12). A high viscous resistance would also stabilize the airways during a forced expiration and delay airway collapse (5). We thought, therefore, that it might be of interest to measure the viscous resistance of the airways. In the present study we investigate the dynamic viscoelastic properties of excised canine tracheas. A simple way of expressing the viscoelastic properties of a tube with symmetrical structure, e.g., a blood vessel, is to relate the amplitude of the changes in transmural pressure P to the amplitude of the changes in radius r, and to calculate an incremental pressurestrain ratio P/r. The latter is “dynamic” when the measurement is performed under conditions of pressure changing with time. If there is a phase lag p between P and r, one can separate the pressure-strain ratio into two parts, by relating P to the changes of r in phase with P (P/r cos rp), or 90” out of phase with P (P/r sin 50)~The former may be called an elastance E, the
St. Rafael,
trachea
3000 Leuven,
Belgium
latter a viscosity coefficient Ru. Both are expressed in cmH,O* cm? To obtain a resistance R, one divides RU by the angular velocity W. The viscous resistance R is thus P/r@ sin up and is expressed in cmH,O cm+ s. When instead of r, one relates the relative changes of radius (r/r,> to P (Pro/r), one obtains an incremental pressure-strain modulus (in cmH,O). The latter can be resolved similarly into an elastic (in cmH,O) and a viscous modulus (in cmH,O s). When we submitted segments of the trachea to peri .odic pressure variations, only the pars membranacea was observed to move, where& the-shape of the cartilagenous rings was not changing noticeably. Instead of a homogeneous tube with circular cross section, which is the model used to represent a blood vessel, we were dealing with a flat, stretched membrane becoming curved under the influence of transmural pressure changes. Accordingly, the technique of deter&nation of viscous resistance and elastance used for blood vessels was not applicable to the trachea. We developed another model, representing the trachea as a succession of rigid, open rings, the slit of which is closed by an elastic membrane. As shown in the APPENDIX it is possible under certain conditions (see DISCUSSION) to calculate the viscous resistance R and the elastance E of such a structure from its changes in volume and corresponding changes in pressure. The expressions of R and E are similar to those used for blood vessels l
l
P s R z-msin p WV 2 P s E = v ’ 2 cos p
(1)
where V is the amplitude of volume changes and S the surface of the pars membranacea. Although R reflects only the viscous work, E is a ‘(dynamic” elastance, i.e., a composite of elastic and inertial properties of the trachea. MATERIALS
AND
METHODS
Experiments. The intrathoracic part of the trachea of 11 healthy adult dogs was removed within 30 min after each animal was killed. All measurements were performed within 0.5 and 2.5 h after death. The dimensions of the investigated airway segments are given in Table 1. Rigid Perspex tubes of the-largest possible diameter were inserted and securely tied into each end of the trachea. The mounted trachea was suspended under minimal longitudinal tension in a small Perspex flow
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1.38 TABLE
BOBBAERS,
1. Dimensions
and uiscoelastic Tracheal
Dog No.
Wt, kg
3 4 5 6 7 8 9 10 11 MeankSD
Segment
Length,
a 2
properties
cm
Radius,
cm
of tracheal
Pars Membran, % circumf
4.4 3.1 3.8 5.2 6.4 5.0 3.5 5.6 3.5 4.0 4.8 4.48k1.02
27.3 21.9 11.3 11.3 17.3 14.4 26.9 9.6 9.9 10.5 11.7
0.55 0.67 0.79 0.91 0.92 0.83 0.71 1.05 0.84 0.68 0.75
Viscous -----~cmH,O cm “*s
P, cmH,O
0.791-tO.140
15.656.7
Radius: mean internal radius of the investigated SD); V: corresponding volume variations.
segment;
6.4* 1.9 5.9-t- 1.6 8.321.5 5.2+1.4 6.2k1.8 3.6kO.8 3.921.3 5.0+1.6 20.6*4,0 12.5 +4.0 6.3t2.1 7.6324.94 P: pressure
displacement plethysmograph (Fig. 1). The air inside the plethysmograph was kept at about 37°C and saturated with water vapor. Volume variations of the trachea were produced by a sinusoidal pump, provided with a variable leak. The frequency of oscillations varied between 0.5 and 9 Hz. Volume displacement of the trachea and transmural pressure (difference between inside and outside pressure) were registered simultaneously on a direct-writing recorder (Mingograph Elema) and displayed in X-Y coordinates on a storage oscilloscope (Tektronix 56 MB). The amplitude of the volume changes was kept small by adjusting the leak of the pump to keep pressure and volume changes within the limits of linearity of the tracheal wall (Fig. Z), The frequency characteristics of the electromanometers and the flow plethysmograph, corrected by means of box pressure (I@, were flat in phase and amplitude up to 10 Hz. It is demonstrated in the APPENDIX Computations. that unbiased estimates of R and E can be obtained from the slopes of the tan (P-W and V-P cos (p relationships, measured at w = 0 and P = 0, respectively. To estimate these slopes, a regression was fitted on the experimental data and constrained to pass through the origin; indeed, up and V are necessarily zero when w and P equal zero. To allow for nonlinearities and for the effect of inertia, second-order terms were introduced into the regression functions, i.e.
1.16 0.54 0.41 0.57 1.24 0.42 0.49 0.96 1.61 0.56 1.04
+ p&
+ YIP0
(2)
V = ap(P cos cp) + &(P cos cp)’ + ygo(P cos cp)
(3)
+ &&(P cos cp) The slopes at the origin are given by the regression coefficients cyl and az; R and E can thus be estimated from R = (a&,>* (S/Z) and E = az4/2, respectively. The standard errors of the estimates of R and E are derived from the standard errors of al and az given by the regressions. RESULTS
Influence
of freauencv
and alinearitv.
To the extent
Resistance
VAN
DE
WUESTIJNE
0.16+0.12 to which
Elastance
--
cmH# *ml-’ es
0.06 0.17 0.49 0.17 0.14 0.12 0.17 0.15 0.06 0.13 0.07
0.818*0.404 variations
cm&O.
0.04 0.11 0.44 0.13 0.05 0.07 0.08 0.08 0.06 0.18 0.05
8.41 18.17 41.02 15.96 24.67 11.54 16.30 18.71 10.37 15.91 9.66
0.1220.12
the tracheal
cm ’
=ml
’
4.79 11.81 37.23 12.43 7.94 6.23 7.80 9.83 10.82 21.68 7.39
17.34-+9.18
segment
cmH,O
was submitted
12.5459.35 (mean
1
P in
v
water FIG. 1. Tracheal segment is enclosed in a small plethysmograph, thermostated by means of a water jacket. Volume changes are calculated from the integrated plethysmographic flow signal lvdt, to which a fraction of box pressure Pout is added (pressure correction). Transmural pressure AP is the difference between pressure measured inside tracheal segment Pin, and Pout.
l 0
tan 9 = qw
AND
segments
-_---.--_.“,“- ----- ---- --..--. . -.-._.. -.._.-_ 8.3 13.0 9.8 27.0 21.5 19.0 13.0 32.0 17.5 17.5 12.5
CLGMENT,
025
.
0.5s
l
I
0
20
40cmH20
FIG. 2. In A, record of volume V and transmural pressure changes P as a function of time. P precedes V a little. Insert shows corresponding V-P loop. In B and C, V-P loop at the same frequency, when the trachea is submitted to larger pressure variations. In A, trachea behaves as a nearly linear system. Alinearity is more marked in B and still more in C. (Dog 10.)
that the model used to calculate the viscous resistance and elastance describes the behavior of the real trachea, one expects (Eq, 4 ' of the APPENDIX): 1) an increase of tan q, the phase shift between amplitude of pressure P and volume changes V, with the angular velocity o (Eq. 2); 2) a relationship between V and the first power of P cos q (Eq. 3); 3) a quadratic relationship between both tan (p and V-P cos q with W, if the inertance of the wall Dlavs a role within the range of
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VISCOELASTIC
PROPERTIES
OF
THE
investigated frequencies. Indeed, when Eqs. 4' are expanded in series with respect to pi), terms in wZ are obtained, the magnitude of which depends on the inertia. The statistical analysis of Eq, 2 shows that its three coefficients differ significantly (P < 0.001) from zero. This is in agreement with the model’s predictions, for coeficients aI and PI. The significant interaction between P and w (coefficient rl) points to an alinearity: the relationship between tan p and m depends on the amplitude of P. Figure 3 shows that this alinearity is small within the limits of applied pressure variations. The coefficients of Eq, 3 are statistically significant, except for coefficient yp (a2, P < 0.001; &, P < 0.001; y2, P > 0.2; 8, P < 0.05). Significant coefficients atl and & were expected. The presence of a quadratic relationship between V and (P cos @ indicates alinearity. Again, the latter is small (Fig. 3). The slopes of both regressions increase with frequency. The slope of the relationship V-P cos q is an expression of the dynamic compliance (the reverse of the elastance); the dynamic compliance thus increases with frequency (significant coefficient 8, reflecting the influence of inertia). R is calculated by dividing the slopes of the two regressions: when this computation is performed at different frequencies, R does not appear to be frequency dependent between 0 and 9 Hz (t, 0.42; P > 0.5). A priori, we do not expect a linear relationship between P and V; the estimates of R and E are unbiased only at the origins of the regressions, i.e., at or) = 0, and P = 0. The values of R and E calculated under these conditions are given in Table 1. Influence of surrounding tissues. We wondered to what extent the properties of the trachea were influenced by the surrounding tissues. To approximate this influence, we performed the determinations without dissecting the esophagus away in six experiments. Five times, the resistance values were slightly higher than on the isolated trachea. The mean resistance was 0.11 and 0.15 cmH,O cm+ s, respectively, without and with l
l
139
TRACHEA
the esophagus (avg of 6 expts). The elastance was not modified. Influence of longitudinal tension. We investigated the influence of longitudinal tension on the viscoelastic properties of the trachea in two ways. In four dogs, the measurements were performed before and after elongation of the trachea, realized by moving apart the tubes over which the tracheas were tied. Thus, the tracheal segments were lengthened on the average from 4.1 to 4.9 cm. This resulted in a slight increase of R and E in three dogs, whereas R and E decreased in one dog (Fig. 4). The influence of longitudinal stresses was verified in another way. In eight experiments, we determined the mechanical properties successively of a longer and a shorter segment of the same trachea. This should not influence the value of R or E, which are independent of the length of the airway segment, In fact, a shorter segment (avg length, 2.1 cm) has a significantly (P < 0.001) larger viscous resistance and elastance (mean R, 0.43 cmHZOcm-ls; E, 29.4cmH,Ocm’) than alonger segment (avg length, 4.3 cm; R, 0.18 cmH,O . cm-l 4s; E, 18.0 cmH,O cm--l) (Fig. 4). Influence of acetylcholine. After completion of the measurements, we moistened the pars membranacea of the tracheas from the outside by a few drops of a 5% solution of acetylcholine chloride. This resulted, in the six experiments in which it was attempted, in a very pronounced contraction of the pars membranacea pulling the ends of the cartilagenous rings tightly together. We were unable to determine the mechanical properties of the trachea in the latter condition, since the volume displacements were too small to be registered with any accuracy even when the pressure variations were increased beyond 20 cmH,O. In one dog, however, in which the measurements were feasible probably because the tracheal constriction was less pronounced than in the others, acetylcholine produced a 20-fold increase of R and a 6-fold increase of E. l
DISCUSSION
Validity
of the technique. The computation
of a dy-
P=l =3 =5 FIG. 3. For 11 dogs, mean regressions between tan cpY phase shift between volume (V) and transmural pressure amplitude (P), and angular velocity cd, and between V and P cos ~0. Tnfluence of P and w, respectively, on former and latter regression is shown. BUFS are interindividual standard errors of ‘ the regression.
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140
FIG. dogs);
shorter
BUBBAERS,
4. Mean B, after segment
regression elongation of trachea
V-P cos cp: A, in control of the trachea (4 dogs); (8 dogs) (see text).
conditions ( 11 and C, for a
namic elastance E and viscous resistance R from the amplitude and phase relationships of transmural pressure and volume of the trachea is valid only if the used equations adequately describe the behavior of the trachea. The main assumptions underlying these equations are that 1) the viscous resistance and inertia are small with respect to the elastic forces and 2) the boundaries of the membrane can be approximated by an ellipse. Further conditions are that the amplitude of the pressure variations is small and that the pressure input is sinusoidal. The results show that the two assumptions are satisfied. The viscous resistance and inertia are, indeed, small with respect to the elasticity of the membrane. If the inertia were large, the elastance would decrease markedly at higher frequencies. This was not observed. The fact that the V-P cos p relationships are nearly linear over rather large pressure variations (Fig. 3) shows that the second assumption is also valid (see APPENDIX).
Though we kept the ‘amplitude of P small to stay within the limits of linearity of the system, the regressions fitted on the experimental data suggest alinearities. The latter may result from nonlinear mechanical characteristics of the tissues, from an inertia effect, or from an inadequacy of the model used to represent the trachea. These various factors should not introduce errors in our estimates of E and R, however, since we avoided their influence by determining the ‘slopes of the regressions, from which E and R were computed, at o = 0 and P = 0. To keep the trachea around its resting volume and to avoid large changes of P with varying frequency, we provided the pump used to cycle the trachea with a large leak. This resulted in a distortion of the pressure input and was no longer an exact sinusoid (Fig. .2>. To investigate to what extent this distortion introduced errors in the computation of E and R, we compared the values, calculated from P and V measured on the actual signals, with those obtained from the fundamental frequency of the oscillations. The latter was computed by means of a Fourier analysis of the pressure and volume cycles. Five to nine comparisons were performed in each experiment at various frequencies and in differ-
CLEMENT,
AND
VAN
DE
WQESTIJNE
ent conditions (stretched and nonstretched trachea, short and long segment, with and without esophagus, constricted pars membranacea or not). Figure 5 shows that both the elastance and resistance values are underestimated when they are computed from the distorted signal produced by the pump. The error is small, however, and unimportant for the lower values of E and R: those are the values observed in the unstretched, unconstricted trachea. Comparison with properties of blood vessels. A comparison of the viscoelastic properties of tubes of various sizes is only possible if one takes the initial size of the tube into account. In studies on blood vessels, the stress-strain relationships are converted into a modulus of elasticity and viscosity by relating the pressure changes to the relative changes in radius. Because of the asymmetry of the trachea a modulus cannot be computed. Therefore, we limited ourselves to a comparison between the trachea and blood vessels of similar size, i.e., the descending thoracic and the abdominal aorta, all data being expressed as elastance and viscous resistance. Table 2 shows that the values measured on the aorta are markedly higher. However, the latter measurements were performed at a pressure of about 140 cmH,O. The elastic characteristics of arteries are strongly alinear; accordingly, the elastance values cannot be compared as such with our data on the trachea determined around the resting position (transmural pressure, 0 cmH,O). Under the same conditions, the elastance of the thoracic and abdominal aorta, measured statically, are respectively about 167 and 222 cmH,O 4cm-l (1). The static elastance is lower than that determined in dynamic conditions (i.e., at frequencies of 2 Hz or more), and this difference increases for more muscular arteries (2); after allowance for this effect, the dynamic elastance can be estimated as 178 cmH,O *cm-l for the unstretched thoracic aorta, or 263 cmH,O *cm-l for the abdominal aorta. The influence of mean pressure on the viscous resistance is less marked; according to Hardung (9), the latter is but little affected by the mean stress. We conclude that the elastance and viscous resistance of the aorta are more than 10 times larger than those of the trachea, even if one takes the differences in pressure level into account. Our study shows, on the other hand, that the dynamic elastance and resistance of the trachea are qualitatively similar to those of arteries. 1) The viscous resistance is markedly smaller than the elastance; the ratio R/E is about 1% for both the trachea and the aorta (Table 2). 2) The elastance and resistance are larger when the trachea is stretched or a shorter segment is investigated. This has been observed in arteries (3) and reflects the interaction of longitudinal on circumferential stresses (16). Maybe a similar interaction exists between longitudinal and transverse stresses in the pars membranacea. 3) Constriction of the trachea results in an increase of resistance and dynamic elastance. The viscoelastic properties of arteries are modified similarly (3, 9). 4) As in blood vessels (17), the influence of wall inertia seems to be small. We observed a slight decrease of E at higher frequencies which might be due to the inertance of the wall (Fig. 3). A recent paper of Knudson
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VISCOELASTIC
PROPERTIES
OF
THE
141
TRACHEA
? :mH201cmls
Y= 0.085+0.84(+0.04)X
l
R( Four ier) 0
Q50
1.50
1.00
cmH~/cm/s FIG. 5. Comparison of R and E values determined from the P-V relationships on actual P-V records (Y axis) and on fundamental frequency of the same records (X axis). Continuous lines are calculated regression lines, dashed lines, the lines of identity.
60
SO
40
30
Y =1.886+0.89(-t
20
0.03) X
-1 10
0
10
20
30
40
50
et al. (10) showed that airway wall inertia does not delay significantly the dynamic collapse of the trachea. Pressure-volume relationships. The viscoelastic properties of the trachea can be expressed in cmH,O .ml-l or cmH,O rnP.0 s, instead of cmH,O .crn+ or cmH,O burn-’ 0s simply by dividing the latter expresl
SO cm H20/cm
sions by the half surface of the pars membranacea (Table 1). The obtained R and E values are inversely proportional to the length of the tracheal segment. Expressed per cm length, the mean viscous resistance is 0.49 k 0.43 (SD) cmH20-ml-1*~. The corresponding value for the elastance is 53.5 t 34.7 (SD) cmH,O ml-l. l
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142
BUBBAERS,
TABLE 2. Dynamic viscoelastic the aorta and trachea
properties
Thoracic
aorta
Abdominal
17 (dog) 2 (dog) 14 (dog) 11 (man) 8 (dog) 15 (dog) 17 (dog) 2 (dog) 14 (dog) 11 (man) 8 (dog) Present work
aorta
Trachea Values frequency
(dog)
of aortic elastance and resistance of 2 Hz (not specified in (17)).
Viscous Resistante cmH,O * cm-l . s
3,943 783 1,118 682 772 918 7,578 2,422 4,082 1,120 2,791 17 were
140.0 6.0 6.5 4.8 3.3 222.9 18.6 9.7 26.8 0.2 determined
at a
The latter figure is in satisfactory agreement with the value determined by Coburn et al. (7) in dogs in quasistatic conditions: 32.9 t 12.1 (SD) cmH,O ml+ per cm length of trachea, but is higher than the values found in cats by Olsen et al. (13). For a 12-cm-long trachea we estimate R to be 0.04 cmH,O ml-l s or 40 cmH,O 1-l s and E 4.5 cmH,O ml? These figures apply only to the trachea; they do not allow us to calculate the viscous resistance of the whole bronchial tree since it is unlikely that the successive bronchial generations, which differ markedly in structure, have identical viscoelastic properties. It is striking, however, that the ratio R/E for both the trachea and the aorta (Table 2) as well as for other arteries (8) is similar. Accordingly, it is not unreasonable to assume that this ratio also applies for the total bronchial tree. The elastance of the airways during tidal breathing is about 0.25 cmH,O*ml+ (12) or 0.2 cmH,O~ml+ (4). The viscous resistance of the bronchial tree may thus be estimated to be abdut 0.0025 cmH,Om ml-l 9s or 2.5 cmH,O 1-l s. l One may wonder to what extent our figures of E and R of the trachea are influenced by the active tension of the airway smooth muscle. Coburn and Palombini (6) reported that the in vivo canine trachea demonstrates a marked stress relaxation, related to the trachealis muscle active tension. In the presence of stress relaxation, the elastance increases with frequency. This is observed in blood vessels up to 2 Hz, and to higher frequencies when the active tension of the vessels is larger (3, 9). Our observation of a nearly constant value of E with frequency may be due to a complete lack of smooth muscle active tension in our preparation. In one case, we determined E under conditions of maximal contraction of the trachealis muscle; E increased with frequency up to 4 Hz. Accordingly, it is possible that we underestimate somewhat E and R, compared to the in vivo situation. Even so, R remains small with respect to E. Therefore, it is likely that R does not slow down l
l
l
l
l
l
AND
VAN
DE
WOESTIJNE
of the airway walls noticeably the movements the influence of changes in transm ural pressure.
of
Elastance cmHzO.cm-l
Reference
CLEMENT,
under
APPENDIX Model
of the Trachea
The pars membranacea is assimilated to a plane rectangular elastic membrane attached to cartilagenous rings. The deformations of the latter, under the influence of transmural pressure changes, are considered to be negligible with respect to the deformations of the pars membranacea. Accordingly, we consider only the mechanical properties of the pars membranacea. The Cartesian coordinates of any point of the membrane in the transverse and longitudinal directions are defined by x and y, respectively, the origin being taken at the center of the membrane. The transverse and longitudinal tensions in the membrane are indicated by F, and F,, respectively. If a transmural pressure is exerted on the membrane, each of its points will be displaced perpendicularly over a distance z. The elastic counterpressure on each point will be, by virtue of Laplace’s law, directly proportional to the radii of curvature in both x and y directions at that point, i.e., to the partial derivatives 6%/6x2 and 8%/6y (as long as the first derivatives &Z/&X and 62/6y are (I and w being the half lengths, respectively, of the large and small axes of the ellipse) Eq. 1’ can be solved. Indeed, then z = a2(g2 + aofy2 + aoo the coefficients
aij satisfying
the set of equations
2F sa2O + 2 F,ao2 = P l2 - ao2 ~ = 0
a20
W2
aozZ2 + aoo = 0 and the corresponding
volume
changes
v=
sp
4
become
[u.‘2+F 1 Fx
Fv
in which S is the total area of the membrane. For ellipsoidal boundaries, there is thus a linear relationship between the static transmural pressure and the corresponding volume change. Under other conditions, e.g., for rectangular boundaries, the V-P relationship cannot be linear and must be approximated by an infinite series, e.g., of the type V = aP + PP’ + . . . .
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VISCOELASTIC
PROPERTIES
OF
THE
143
TRACHEA
If the pressure amplitude is small, however, the higher powers of P become negligible, i,e., the V-P relationship tends to be linear. In the limit, the slope of the observed V-P relationship at the origin P = 0 yields an estimate of the “elastance” of the membrane. In the same limit, the displacements z in all regions near the boundaries are close to zero, and since ellipsoidal boundaries are the only ones which ensure a linear relationship between V and P, the rectangular boundaries can be approximated by the nearest ellipsoidal ones, i.e., by an ellipse with the same surface as the rectangle. We can now introduce m into Eg. 1’. The displacement z becomes periodic z = A(x, Substitution differential
in the equations
y) sin wt + B(x,
equilibrium Eq. in the unknown
forces F,, F, >> Rwb,, = 0, &azo be considered as for all coefficients 2) Boundary equal powers are
w’ago - ao,P
may write a,,, utL2, bAo, bzz = 0 and Thus the equilibrium conditions will satisfied (i.e., w sufficiently small) 2. which all remaining coefficients of
= 0
w2bz0 - bo,Z2 = 0
a()()+ a(),z2 = 0 The
amplitude
y) cos tit
1’ yields function
Ru, IO’), one = 0 . . . , etc. approximately of powers 2 conditions in equated
V of the total
boo + b,,Z’
volume
changes
v = cr// 4x, Y) dx dY12+ r/i e,
a set of two A andB
partial
and its phase
shift
Y>dx dY12~1’2
J/ Bb, Y>dx dy (1 bis’)
Solution
(3’)
(3’)
y> dx dy
of the set of Eqs. 2’ and 3’ yields Rw 2F x + 2 W2 I”
tan cp =
2F
A and B as a series A =
is then
cp tan ’ = [f A(x,
We write
(2’)
= 0
S v = 2 (P cos p) 2F 2
2 2 aijxif j=o j=o
+ Iw”
(4’) 1
-
+ --+ 2;
+ ImZ
W2 B
=
2
2
i=()
j=o
bij&’
The viscous
By virtue of the symmetry of the membrane, only the even powers in x and y are retained. If the boundaries are ellipsoidal and if w is small, so that the viscous resistance and inertial forces are small with respect to the elastance, all powers in x and y greater than 2 can be neglected, leaving the following system of equations. 1) Equilibrium conditions of Eq. 1’ bis in which the coefficients of power zero in x and y are equated 2F
#20
+
2
Fara*2 - Roboo
2 FrbeO + 2 F,b,, If the coefficients
of the second
+ Rw,~ power
12F&40+ F&2 12 F,bdO + F,& But
with
elastance
forces
far
- Idaoo - Wboo
in x: are equated - IoZ&o
= 0
+ Roazo
- Mb,,
= 0
greater
than
Rob20
resistance
coefficient
R is
(5’) and the elastance
coefficient
E
(6’)
(2’)
In the case of rectangular boundaries, Eqs. 4’ can be approximated only if P is suficiently small, i.e., if the displacements z in all regions near the boundaries are close to zero. Accordingly, R and E should be estimated in the limit, from the slopes of the tan cp-o and P cos cp-V curves at their origins, o = 0 and P = 0.
(2’)
This kundig
= P = 0
resistance
one obtains
or inertial
Received
study was supported by a grant Wetenschappelijk Onderzoek.” for publication
28 February
of the “Fonds
voor
Genees-
1977.
REFERENCES 1. BERGEL, D. H. The static elastic properties of the arterial wall. J. PhysioZ., London 156: 445-457, 1961. 2. BERCEL, D. H. The dynamic elastic properties of the arterial wall. J. PhysioZ., London 156: 458-469, 1961. 3. BERGEL, D. H. The properties of blood vessels. In: Biomechanics. Its Foundation and Objectives, Y. C. Fung, N. Perrone, and M. Anliker. Englewood Cliffs, N. J.: Prentice-Hall, 1972, p. 105139. 4. BOBBAERS, H., J. CLEMENT, J. PARDAENS, AND K. P. VAN DE WOESTIJNE. Simulation of frequency dependence of compliance and resistance in healthy man. J. AppZ. PhysioZ. 38: 427-435, 1975, 5. CLEMENT, J., J. PARDAENS, AND K. P. VAN DE WOESTIJNE. Expiratory flow rate, driving pressures and time-dependent factors. Simulation by means of a model, Respiration PhysioE. 20: 353-369, 1974. 6. COBURN, R. F., AND B. PALOMBINI. Time-dependent pressurevolume relationships of the in vivo canine trachea. Respiration Physiol. 16: 282-289, 1972. 7. COBURN, R. F., D. THORNTON, AND R. ARTS. Effect of trachealis muscle contraction on tracheal resistance to airflow. J. AppZ. Physiol. 32: 397-403, 1972. 8. Gow, B. S., AND M. G. TAYLOR. Measurement of viscoelastic properties of arteries in the living dog. Circulation Res. 23: lll122, 1968. 9. HARDUNG, V. Dynamische Elastizitgt und innere Reibung muskularer Blutgeftisse bei verschiedener durch Dehnung und
10.
11.
12.
13.
14.
15.
16. 17. 18.
tonische Kontraktion hervorgerufener Wandspannung. Arch. Kreislaufforoch. 61: 83-100, 1970. KNUDSON, R. J., J. MEAD, AND D. E. KNUDSON. Contribution of airway collapse to supramaximal expiratory flows. J. Appl. Physiol. 36: 653-667, 1974. LEAROYD, B. M., AND M. G. TAYLOR. Alterations with age in the viscoelastic properties of human arterial walls. CircuZation Res. 18: 278-292, 1966. MEAD, J. Contribution of compliance of airways to frequencydependent behavior of lungs. J. AppZ. PhysioZ. 26: 670-673, 1969. OLSEN, C. R., A. E. STEVENS, AND M. B. MCILROY. Rigidity of tracheae and bronchi during muscular constriction. J. AppE. Physiol. 23: 27-34, 1967. PATEL, D, J., F. M. DE FREITAS, J. C. GREENFIELD, JR., AND D. L. FRY. Relationship of radius to pressure along the aorta in living dogs, J. AppZ. Physiol. 18: 1111-1117, 1963. PATEL, D. J., J. S. JANXCKI, R. N. VAISHNAV, AND J. T. YOUNG. Dynamic anisotropic viscoelastic properties of the aorta in living dogs. Circulation Res. 32: 93-107, 1973. PATEL, D. J., A. J. MALLOS, AND D. L. FRY. Aortic mechanics in the living dogs. J. AppZ. Physiol. 16: 293-299, 1961. PETERSON, L. H., R. E. JENSEN, AND J. PARNELL. Mechanical properties of arteries in vivo. CircuZation Res. 8: 622-639, 1960. STXNESCU, D. C., P. DE SUTTER, AND K. P. VAN DE WOESTIJNE. Pressure-corrected flow body plethysmograph. Am - Rev. Respirat. Diseases 105: 304-305, 1972.
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