A nonlinear B. SUKI
viscoelastic model of lung tissue mechanics AND
Meakins-Christie
J. H. T. BATES Laboratories,
McGill
University,
Montreal,
Quebec H2X 2P2, Canada
cal formalism involved was more troublesome than that for plastoelasticity. More recently, there have been some attempts to further develop Hildebrandt’s plastoelastic model ( 10,27). Indeed, some of these studies go so far as to address the question of whether plastoelasticity really does exist in the lungs (10). Our own perspective on the question of plastoelasticity is that its existence would be unequivocally demonstrated by the presence of static pressure-volume hysteresis in the lung. However, the requirement of an infinitely long experiment for such a demonstration means that one can only obtain a quasi-static approximation to the necessary data, which always leaves open the quesdata by use of various types of ventilation patterns similar to tion of whether the volume of the lungs was cycled slowly those that have been employed experimentally. Rti and Eti enough, to allow all viscoelastic elements in the tissues to were estimated from the simulated data by use of four different remain fully relaxed at all points during the experiment. estimation techniques commonly applied in respiratory meFurthermore, although Hildebrandt and co-workers (2, chanics studies. We found that the estimated volume depen14) reported significant quasi-static hysteresis in isodence of Rti and Eti is sensitive to both the ventilation pattern lated lungs, Saibene and Mead (22) reported the pheand the estimation technique, being in error by as much as 217 nomenon to be less important in living humans. More and 22%, respectively. recently, Shardonofsky et al. (25) studied quasi-static hysteresis in tidally ventilated paralyzed anesthetized lung tissue resistance and elastance; nonlinear systems analydogs and found it to be essentially negligible. sis; Volterra series; harmonic distortion; plastoelasticity Thus, because the question of plastoelasticity vs. nonlinear viscoelasticity cannot be addressed experimentally, it remains one of mathematical convenience. BeTHE MECHANICAL PROPERTIES of lung tissue are imporcause there has been so much emphasis recently on modtant determinants of the overall mechanical behavior of eling lung tissue mechanics in terms of plastoelasticity, the respiratory system. Indeed, it is now well appreciated we believed it important to demonstrate that a nonlinear that the mechanical significance of the tissues extends viscoelastic model is also capable of adequately accountfar beyond a simple contribution to the static recoil pres- ing for the known nonlinear and frequency-dependent sure of the respiratory system. For example, the majority aspects of lung tissue mechanics. The development of of the total lung resistance apparent at frequencies ~1 such a model is the principal subject of the present paper. Hz arises in the tissues in dogs ( 17,18) and most likely in We also show how the nonlinear and frequency-depenother species as well ( 13). Most recent studies have con- dent aspects of lung tissue mechanics can lead to quite a centrated on the manner in which lung tissue mechanics marked disparity between the estimates of lung tissue vary with frequency in or around the breathing frequency resistance and elastance at a given frequency provided by range. Such variation can be accounted for by ascribing different analysis techniques. to the tissues the property of linear viscoelasticity, which is usually represented in terms of spring-and-dashpot THEORY models (4, 11) . However, the mechanical properties of Consider a nonlinear system with input x(t) and outlung tissue are also known to vary with volume, which put y(t) . If the system has finite memory (i.e., at any cannot be accounted for by a linear model. time t the value of y depends to a finite extent on previous A significant step in the modeling of nonlinear lung values of x extending only a finite period into the past), tissue mechanics was made by Hildebrandt (15)) who then x and y can be related through a Volterra series; chose to ascribe the nonlinearities to the phenomenon of thus (24) plastoelasticity. These are modeled as collections of springs and static friction elements that exhibit static pressure-volume hysteresis. Hildebrandt did acknowledge, however, that the nonlinearities he encountered could, in principle, also be explained as arising from non2 71,7&(t 72M7J72 linear viscoelasticity, but he implied that the mathematiJ -00 J-00 B., AND J. H. T. BATES. A nonlinear viscoelastic model lung tissue mechanics. J. Appl. Physiol. 71( 3) : 826-833, 1991.-There have been a number of attempts recently to use linear models to describe the low-frequency (O-2 Hz) dependence of lung tissue resistance (Rti) and elastance(Eti) . Only a few attempts, however, have been made to account for the volume dependence of these quantities, all of which require the tissues to be plastoelastic. In this paper we specifically avoid invoking plastoelasticity and develop a nonlinear viscoelastic model that is also capable of accounting for the nonlinear and frequency-dependent features of lung tissue mechanics. The model parameters were identified by fitting the model to data obtained in a previous study from dogs during sinusoidal ventilation. The model was then used to simulate pressure and flow SUKI,
of
+r00 gt rccl
826
0161-7567/91 $1.50 Copyright
0
1991 the American
Physiological
-71Mt -
Society
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NONLINEAR
VISCOELASTICITY
(0
x
-
x(t
- 73)dT1dr2d73 + 9 .
r2)x(t
l
The gi are the kernels of the system and must satisfy the causality constraint g( n
rlt
’
l
l
q&)
=o
(2)
forany7k in Eq. 7 is a constant complex number c3. Now Eq. 7 is reduced to
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828
NONLINEAR
VISCOELASTICITY
lS
2*o /Y 0
LUNGS
where V. denotes the magnitude of V ( fo) . B,, B,, A,, A,, and A, are now the parametric constants of the model. We also note that Eqs. 12 and 13 fully describe the amplitude and frequency dependence of Rti and Eti at f. when v ( t) is quasi-sinusoidal. We have now approximated the fundamental of P (Eq. 4) as
I
s
OF
50
100
200
150
250
P(
fo) =
{Eti[V(f,)]
+
iBti[V(fo)]}V(f~)
(l-4)
Equations 5 and 6, however, show that the second and third harmonics of P both contain contributions from V( fJ in addition to their respective contributions from the second and third harmonics of V. We must therefore include additional terms in our simplified expressions for P(2f,) andP(3f,), which we choose to do as ti 0
50
100
150
200
250
4-
{Eti[V(2fo)]
P(2fJ
=
Wf,)
= {Eti[V(3f,)l
3-
21 -
I
0 0
50
I 100
Volume, I I
ml
200
150
I 250
FIG. 1. Lung tissue resistance (Rti, top), elastance (Eti, middle), and distortion factor of volume input (bottom) as a function of volume amplitude [data from Suki et al. (28)]. Values (means t SD) were obtained from 4 measurements at 5 cmH,O transpulmonary pressure with an O.l-Hz quasi-sinusoidal volume in an open-chest dog.
For the linear dynamic part of the model, G,, one may choose any viscoelastic model that adequately accounts for the frequency dependence of Rti and Eti obtained with small-amplitude forcing. We chose the following linear viscoelastic model originally proposed for the lung by Hildebrandt ( 14,15)
BWf,) = rBJ4.6
COMPUTATIONAL
f,) = A, + (0.25 + log 2rfo)B,
Parameter
14.6
+ (0.25 + log 2?rf,)Bti[V(
(12)
fo)]
METHODS
estimation.
A global optimization
procedure
(11) (8) was used to fit the model described above to experi-
This model is very simple to deal with and has been successfully used to account for the frequency dependence of Rti and Eti determined from low-frequency small-amplitude forced oscillations ( 12, 29). Now we must include the volume amplitude dependence in Eq. 9. To do this we approximated the data in Fig. 1 by simple parametric curves. We found that a second-order polynomial was satisfactory for describing Eti and that a linear function gave a good fit to Rti. We therefore replace the constants B, in Eq. 10 and A, in Eq. 11 with linear and quadratic expressions, respectively; thus
Bti[Wfo)l = ?r(B, + BJ,) Eti[V( f,)] = A, + A,& + A2Vi
Here a2, a3, b,, and b3 are additional model parameters. Equations 12-16 constitute our complete nonlinear model. Although they form a drastically reduced frequency-domain Volterra series, they should still be able to account for the experimentally observed dependences of Rti and Eti on frequency and volume amplitude, provided the various parameters in Eqs. 12,13,15, and 16 can be suitably chosen. Furthermore, because the model is a form of Volterra series, the system it represents must have a memory that decreases toward zero as one goes back arbitrarily far in time. This is the essential nature of a viscoelastic substance, the deformation of which approaches a value uniquely determined by the current stress if that stress is maintained for an arbitrarily long time. In contrast, the strain of a plastoelastic substance at constant stress remains forever determined in part by the previous stress history. Our model, therefore, is a nonlinear viscoelastic model.
(10)
and Eti(
+ iBti[V(2f~)]}Wf~) t15) + (a2 + ib2)V2(f0) + iBwwf,)l}w3f,) (16) + (a, + ib,W3(f0)
(13)
mental data from four dogs. The data were taken from the study of Suki et al. (28), where the experimental methods involved are described. The evaluation of the model parameters was performed in two steps. First, the parameters Ai (i = 0,l) and Bi (i = 0, 1,2) were obtained by fitting Eqs. 12 and 13 simultaneously to estimates of Rti and Eti obtained from quasi-sinusoidal volume oscillations. This involved minimizing the following error function ( 8, 12, 29) FE, = {% i j=l
[Rti,
j - Rti(Voj)12 , + [Et&j
(17) - Eti(Vo,j)]2}“2
where Rti, j and Eti, j are the resistance and elastance values shown in Fig. 1 measured with a f. = 0.1. Hz quasisinusoidal signal with six different amplitudes, which are
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NONLINEAR
VISCOELASTICITY
denoted by V0 j. Rti(V, j) and Eti(V,J , were obtained from Eqs. 8, lb, and 13: In the second step, the estimated values of the Ai and Bi were used with experimentally measured values of andV(3f,)topredictP(f,),P(2f0),and v(fo),wfoL P (3 fo) , respectively, using Eqs. 14-M. The values of the parameters ai and b, were chosen SO that the predicted P matched the experimentally determined ones as closely as possible. This involved minimizing the following error function FE, = m
5 {WP(jfo)
- wjfo)l}2
j=l
+ (WRjf(J
- Wjfo)] j2Y2
(18)
where the operations Re [ ] and Im[ .] denote the real and imaginary parts of their arguments, respectively, and P denotes an experimentally determined pressure harmonic. The solutions to the above optimization problems were all unique, because the estimated parameters were in each case linearly related to the error functions. Simdation studies. Having identified the model parameters in the above manner, we proceeded to perform simulation studies in which the model was used to generate p ( t) and v(t) data and some standard estimation methods applied to the data to reestimate Rti and Eti. In this way we were able to test the effects of the assumptions made in the estimation methods in a controlled manner. Simulation studies were performed with the model using four classes of v(t) with spectra that fell into classes denoted by ISl, IS2, IS3, and IS4. The fundamental frequency in each class was 0.1 Hz. In classes IS1 and IS2, the amplitudes from the second to the sixth harmonics decreased as 1 / 0, and 1 /f, respectively. This left the amplitude at 0.2 Hz free to be chosen so that K, had a value as follows: when the amplitude of the fundamental frequency was 10,40,80,120, and 160 ml, K, had a value of 1,4,8, 12, and 16%, respectively. In the spectral class IS3 the fundamental component had the same amplitude values as classes IS1 and IS2 and the amplitudes of the harmonics decreased as 1 /fl, but K, was fixed at 5%. Thus the nature of a particular v ( t) was essentially completely determined by its class and the amplitude of its fundamental. The only aspect not determined was the phase distribution of its harmonic components. Therefore, within each of the classes ISl, IS2, and IS3 and at each amplitude, we generated 25 individual v(t) signals with randomly selected component phase angles, so that our simulation results would not be biased by the choice of any specific phase angle combination. Finally, the fourth class IS4 contained only 6 v(t) signals with the same volumes at the fundamental frequency as the first three classes above and with spectra that were of a triangular wave truncated above 1.1 Hz. This resulted in a K, of 12% for each of the v(t) in this class. The relationships between each of the v(t) described above and the corresponding p ( t) produced by the model were evaluated by four techniques that have been commonly used in previous studies of respiratory mechanics to assess the apparent amplitude dependence of Rti and Eti. The first method (Ml) was similar to that used by Hildebrandt (14)) in which Rti was obtained as the ratio l
OF
829
LUNGS
of the area of the p(t) - v(t) curve during a complete breathing cycle to the mean squared flow. Eti was calculated as the ratio of the pressure difference between points of zero flow to tidal volume. The second method ( M2) was based on the linear single-compartment equation of motion = Etiv(t)
p(t)
+ Rtiv(t)
(19) Here p (t), v(t) , and v(t) were obtained as the inverse Fourier transforms of model output, input, and time derivative of input, respectively. Rti and Eti were estimated by multiple linear regression according to Bates et al. (5). The third method (M3) consisted of fitting the equation of motion of a two-compartment linear model to the time-domain data. This equation, derived by Sato et al. (23), is p(t)
= k,+(t)
ts +k3
+ k,v(t)
v(T)dT
+ k4
0
ts
P(dd~
(20)
0
The integrations in Eq. 20 were carried out numerically, and the constants ki were again estimated using multiple linear regression. The resistance and elastance of the model are (23) Rtl
.
=
k,w2 - k2h u2 + k;
-
k,
and . k,k402 + k202 - kSk4 Et1 = u2 + k;
(22)
where o = 2nf is the angular frequency. Finally, the fourth method of analysis (M4) was to calculate the complex elastic modulus at the fundamental frequency as E ( f,) = P ( fo) /V( f,) from which Eti = Re[E(f,)] and Rti = Im[ E ( fo) ] /2rfo. This method is often called Fourier analysis because it relates the Fourier transforms of the input and output at the frequency of interest, and therefore the estimation of Rti and Eti iS expected to be less influenced by the extent of the distortion of the input signal than with the other methods. RESULTS
The model developed here was fitted to data from four dogs taken from Suki et al. (28)) who used method M4 to estimate Rti and Eti (one example is shown in Fig. 1). The parameters describing nonlinearity of Rti and Eti are summarized in Table 1. The interindividual variabilities of the parameters are small except for B,, which determines the degree of amplitude dependence of Rti. The parameters accounting for the harmonic production in the lung are given in Table 2. Fairly consistent values of a2 and b, were also obtained from recordings measured with smaller input volumes (down to 60 ml), but with decreasing volume their standard deviations increased rapidly. Reliable estimates for a3 and & could not be obtained at volumes 420 ml, because with smaller V. the third harmonic in P was at most only l-2% of the funda-
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830
NONLINEAR
VISCOELASTICITY
OF
8
1. Parameters describing the amplitude dependence of lung tissue resistance and elastance TABLE
Dog No.
&I9
cmHzO/l
B,,
cmH,O A2
AtI9
49
LUNGS
A29
cmH,O/l
cmH20/12
cmH,O /I3
1 2 3 4
2.61 2.33 2.50 2.72
-0.09 -1.08 -1.30 -1.53
14.3 12.5 13.5 13.9
-27.4 -22.8 -28.5 -22.6
53.3 62.5 63.9 53.0
Mean *SD
2.54 kO.17
-1.00 to.63
13.6 kO.75
-25.3 t3.1
58.1 t5.9
mental component, resulting in an unacceptably low signal-to-noise ratio. Therefore, in the simulation studies we used only the average ai and bi obtained at V0 = 240 ml. Figure 2 demonstrates the performance of the model in predicting the pressure distortion (&) from a given Kv. If only the linear part of the model is used, achieved by setting all coefficients except A, and B, to zero, then the KP predicted by the model resembles K, instead of the measured Kp. The complete nonlinear model, however, performs much more satisfactorily and predicts the values of Kp over the volume range shown in Fig. 2 to be close to the measured ones. The volume dependences of Rti are evaluated by the four different signal-processing techniques Ml-M4 are compared in Fig. 3 and Table 3. The effect of volume on Eti is also given in Table 3. Rti and Eti obtained by M4 were entirely independent of K, and the phase angles of the components, with the percentage decrease of Rti between V0 = 10 and 160 ml being 9.2% and that of Eti being 14.8%. Methods Ml and M2 were very similar in their abilities to estimate Rti and somewhat different in their estimates of Eti. Furthermore, their results depended markedly on the input waveform. Method M3 was much more robust in its estimations of Rti with the different v(t) classes and provided Eti estimates similar to those of M2 and M4. These results show clearly that the estimation of Eti and Rti is highly dependent on the estimation method used. DISCUSSION
Most analyses of the mechanical behavior of the respiratory system and its components to date have been based on linear models of the system, understandable enough in view of the power and utility of linear systems analysis. At some level of detail in any system, however, one is eventually forced to consider the nonlinear aspects
6\0
6
Volume,
0 50 100 150 200 250 2. Distortion factors (mean t SD) of volume input (0) and pressure output ( l ) as a function of volume amplitude obtained in the same dog as in Fig. 1. Continuous and dashed lines, predictions of a nonlinear and a linear viscoelastic model, respectively. FIG.
of the system. In the case of lung tissue mechanics, the currently available experimental data clearly demonstrate the existence of important nonlinear phenomena, compelling us to invoke some kind of nonlinear analysis in our attempts to understand them. One way to analyze a nonlinear dynamical system is to establish a set of nonlinear differential or integral equations and to solve it either analytically if possible or, more usually, by standard numerical procedures. This approach must be based on some a priori knowledge about the structure of the system being modeled. In most biological systems, however, such information is invariably lacking or deficient to a degree that the system can only be considered as a black box, so that one is left trying to arrive at an empiri-
0 z E
02 2.0
0
TABLE 2. Parameters accounting for the harmonic production of lung tissue Dog No.
%9
cmH,O A2
b
a39
cmH:b/12
cmH20/13
ml
0
IS3
IS4
b
cmH&13
1 2 34
1.00 1.20 1.24
5.29 4.27 5.34 5.49
0.39 0.02 0.44
2.87 2.46 2.77 3.30
Mean tSD
1.26 to.25
5.10 to.56
0.24 to.21
2.85 to.35
0
50
100
150
Volume,
0
50
100
150
ml
FIG. 3. of resistance of nonlinear viscoelastic model determined Amplitude dependence by 4 evaluation techniques. Dashed, dotted, and continuous lines, evaluation methods Ml /M2, M3, and M4, respectively. Curves in the 4 panels were obtained as average responses of nonlinear viscoelastic model to volume inputs from signal classes ISl, IS2, IS3, and IS4.
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NONLINEAR
VISCOELASTICITY
3. Average %V, dependence of Rti and Eti between i/OS of 10 and 160 ml
TABLE
Ml
IS1 IS2 IS3 IS4
M2
M3
M4
Rti
Eti
Rti
Eti
Rti
Eti
Rti
Eti
29.2 19.7 9.4 8.9
11.6 13.0 13.8 12.0
29.2 19.7 9.4 8.9
14.0 14.2 14.7 14.5
7.1 8.1 9.2 8.5
14.2 14.4 14.7 14.5
9.2 9.2 9.2 9.2
14.8 14.8 14.8 14.8
VO, volume
amplitude
at the fundamental.
cal relationship between the measured inputs and outputs. In the present study our goal was to derive some kind of nonlinear viscoelastic model of low-frequency lung tissue mechanics. A rigorous mathematical derivation of the conditions under which a nonlinear system can be expanded in a Volterra series as in Eq. 1 was given by Palm and Poggio (21)) who attempted a simple expression of these conditions by saying that “Systems which do not respond critically to certain changes in input show a smooth dependence of the output on the input.” Given that pulmonary pressure-volume data do not show any features that could be described mathematically as discontinuities, the mechanical properties of lung tissue would seem to qualify for description by a Volterra series. In contrast, a system containing a single plastoelastic element (Prandtl body) does not qualify because it always shows a discontinuity in p(t) when v(t) changes direction, while viscoelastic models do not show such discontinuities. However, a plastoelastic model with a smooth continuous distribution of yield stresses might also not show discontinuous behavior. On the other hand, the phenomenon of viscoelasticity is associated with a finite memory of the tissues to previous deformations, whereas plastoelasticity involves infinite memory. Because the orthogonalized version of the Volterra series, called Wiener series, is able to account for any timeinvariant nonlinear system with finite memory that is not “explosive” (24)) we believed that the most general approach to our goal would be to base our model on the Volterra representation. The Volterra series has already been applied in a number of biological situations (19, 20), although the input involved has invariably been Gaussian white noise. In the present study, we had to consider sinusoidal inputs, because these were used to gather the experimental data on which our analysis is based. We also had to drastically simplify the general Volterra expansion to uniquely identify model parameters from the available data. In making this simplification we assumed that the amplitude dependences of Rti and Eti are independent of frequency, which seems reasonable in view of some recently published data (5,7) demonstrating that Rti and Eti obtained with different tidal volumes differ by roughly a constant factor over the frequency range O-2Hz. Another important assumption was that the softening of the tissues is essentially determined by the amplitude of VO. This approximation appears to be useful when either Kv or V0 is small. It is likely to break down, however, when both V, and Kv become large, in which case we would not
OF
LUNGS
831
be able to neglect those terms in Eq. 3 accounting for nonlinear cross-talk between frequency lines. These terms would cause the tissue to soften with amplitude at the fundamental frequency if there are strong higherorder harmonics in v(t). This means that Rti and Eti determined with sinusoidal and composite v(t) signals could be different. Indeed, Sato et al. (23) found systematically lower resistances during regular ventilation than during sinusoidal oscillation at the equivalent tidal volumes. The former resulted in a v ( t) signal having a Kv > 48% (unpublished observation). Recently, Fredberg and Stamenovic (10) introduced the concept of structural damping in the lung by postulating that elasticity and resistivity within the tissues are “coupled at the level of the stress bearing elements.” They argued that such a mechanism implies that plastic deformations could take place at the microstructural level and suggested four processes by which this may occur: 1) the detachment /attachment of cross-bridges in the contractile elements, 2 ) the adsorption / desorption of surface-active molecules in the alveolar surface film, 3) the movement of the collagen and elastin fiber network, and 4) the recruitment /decruitment of alveoli. Indeed, there is strong evidence both from in vitro (26) and in vivo (3) studies that lung surfactant exhibits significant hysteresis, particularly over large volume excursions. Given the importance of the surfactant layer in determining overall lung mechanical properties, these data seem to imply that plastoelasticity should be an important feature in a model of lung mechanics. Another phenomenon that can certainly give rise to static pressure-volume hysteresis at low lung volumes is gas trapping w
Despite the above compelling evidence in favor of plastoelasticity in the lung, experimental evidence shows that quasi-static hysteresis in the living lung undergoing modest volume excursions above functional residual capacity is very small, if not negligible ( 25). The lung still behaves nonlinearly over this volume range, however, so truly plastic phenomena may become manifest only over very large volume excursions. In any case, there are examples in the literature of nonplastoelastic models accounting for complicated tissue mechanical behavior. Lanir (16) showed theoretically that during loading of a network of elastic fibers the progressive unfolding and reorientation of the fiber macromolecules in the direction of the applied force would result in a constitutive law for the material called quasi-linear viscoelasticity, originally described by Fung ( 11) . This law is quasi-linear in that it includes only the first integral term in Eq. 1, but the input x(t) in the integral is replaced by some nonlinear function of x(t) . One of the prominent features of quasi-linear viscoelasticity as proposed by Fung ( 11) is that it can account for the observation that the area of the force-length loop of dynamically stressed tissue is independent of frequency over a wide range of frequency, which originally gave rise to the plastoelastic concept with regard to the lung proposed by Hildebrandt (15). The nonlinear viscoelastic model developed in this paper accounts for two key properties of lung tissues: the decrease of both Rti and Eti with increasing amplitude of v ( t) and a frequency-independent p (t) - v(t) loop area.
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NONLINEAR
VISCOELASTICITY
OF
LUNGS
real lung tissue, it enables us to investigate the consequences of this behavior under precisely controlled conditions through computer simulation experiments. Some important conclusions can even be drawn analytically. For example, if we define p(t) to be the output of the respiratory system and v(t) to be the input, the impulse response of the system would be obtained by measuring the p(t) resulting from the application of an impulse in v(t) . It can be seen from Eq. 1 that the impulse response of a general nonlinear viscoelastic system is p(t)
Pressure
(arbitrary
units)
4. Pressure-volume loops simulated by nonlinear viscoelastic model using data of dog 3 in Tables 1 and 2. When plotted in absolute units of pressure and volume, loops are of very different magnitudes and, therefore, difficult to compare. We have therefore scaled loops so that they each have unity volume excursion. Tidal volumes were 240 ml (solid line), 60 ml (dashed line), and 15 ml (dotted line). FIG.
An example of a series of p(t) - v(t) loops simulated with our model is provided in Fig. 4, demonstrating that the slope of the lines joining the extrema of the loops, and hence the effective elastance, is a decreasing function of tidal volume. Also, with increasing volume the curves change from elliptical to a shape showing clear secondorder nonlinearity (i.e., curved like a quadratic function) . This is in qualitative agreement with a number of experimental observations (1,2,14,22,28; J. H. T. Bates unpublished data). In contrast, the general characteristics of the p ( t) - v(t) curves simulated by a viscoplastic model (15, 27) suggest rather strong odd-order nonlinearities (i.e., sigmoidal like a cubic function) even with the smallest tidal volume. Also, the results of an analysis such as given by Stamenovic et al. (27) to discriminate between the plastoelastic and linear viscoelastic contributions to lung mechanical properties will depend to a certain degree on the particular model structure chosen. Nevertheless, none of the above discussion rules out the possibility of the plastoelastic interpretation. Therefore a decision concerning whether a plastoelastic or a nonlinear viscoelastic model is more appropriate for the lung tissues in any fundamental sense must remain unresolved ., inasmuch as the only basis for such a choice would be the existence or otherwise of static p (t) - v(t) hysteresis, which, of course, is impossible to measure. Moreover, recent work by Shardonofsky et al. (25) shows that plasticity in the lungs of a living dog during normal tidal ventilation is minimal and may indeed even be truly negligible, although continuing gas exchange in the lungs makes it impossible to be sure. This is not to say that one must not use a plastoelastic model to represent lung tissues, but we believe that mathematical convenience is the only reasonable criterion to use in deciding between a plastoelastic and a nonlinear viscoelastic representation. The utility of our nonlinear viscoelastic model is that, to the extent that it accuratelv mimics the behavior of
= g,(t)b
+ g,(t, t)b2 + g3(t, t, t)b3 + 9
l
l
(23)
where b is the impulse scaling factor. The complex impedance of the system, on the other hand, would be obtained by applying some appropriate broad-band v(t) to the system, measuring the resulting p (t), and then calculating the ratio of the transforms Z ( f ) = P ( f ) /V ( f ) . We have seen above that the results of this procedure depend on the harmonic content of v(t) in a complicated way and are not equal to the Fourier transform of Eq. 23. Thus the fundamental result of linear systems analysis, that the Fourier transform of the impulse response equals the complex impedance, does not apply, in general, to nonlinear systems. Therefore, in a nonlinear system, we cannot easily compare the results of time domain and frequency domain analyses, which might explain, at least in part, the reported inconsistencies between various lung mechanical variables derived from time and frequency domain data (15). The amplitude dependence of lung mechanical properties is usually studied by the application of quasi-sinusoidal v(t) . Unfortunately the volume generators used in such experiments are invariably imperfect to the extent that the shapes of their v(t) are not independent of the frequency of cycling or the impedance against which the pump has to produce a given volume. This means that Kv is a function of the amplitude and generally increases with volume amplitude (as in Fig. 1). This situation was represented in our simulation studies by the signal classes IS1 and IS2 in which Kv was a linear function of the fundamental component of v(t) . Depending on the evaluation technique chosen, Fig. 3 shows that this can lead to a very large apparent amplitude dependence of lung tissue mechanics, which may explain the diversity of reports in the literature regarding the amplitude dependence of Rti that we alluded to above. The most inputsensitive methods are Ml and M2, and we suggest that they should not be used to study the amplitude dependence of tissue mechanics whenever K, exceeds 3-4%. Method M3 gives reasonable results because its structure implicitly leads to frequency-dependent Rti and Eti values that are similar to those of the lung in the frequency range O-2 Hz. The most reliable is method M4 because it deals with the input harmonics separately. However, none of the four methods is completely independent of the harmonic content of the input, and when Rti and Eti depend strongly on the amplitude, even the Fourier method may give misleading results if both the tidal volume and K, are large. Finally, we note that the measure of Rti defined by Fredberg and Stamenovic (10) lies between Ml and M4, because it normalizes the p ( t) - v ( t) loop area to the fundamental harmonic of v( t) . Because
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NONLINEAR
VISCOELASTICITY
the loop area actually contains contributions from all the harmonics present in p ( t) and v(t) , their method is expected to provide a slightly smaller tidal volume dependence than M4. In summary, we have developed a nonlinear viscoelastic model that is capable of quantitatively accounting for the known amplitude and frequency-dependent aspects of lung tissue mechanics. We do not claim our model to be the only one derivable form the Volterra series that could possibly account for lung tissue mechanics. One might argue, in fact, that in deriving the model we eliminated some of the terms of the series on rather weak grounds. However, the point of the exercise was to produce a nonlinear viscoelastic model that works, because it was our intention to demonstrate that such a model exists. It is immaterial whether there is more than one. Our model thus shows that it is not essential to invoke the concept of plastoelasticity with regard to the lung tissues, although it does not show that such a concept is inappropriate. Indeed, we conclude that the question of plastoelasticity vs. nonlinear viscoelasticity is one of mathematical convenience rather than a matter of fundamental importance. Furthermore, at this stage in our understanding of these two phenomena, a model based on either mechanism can only be considered purely empirical with no particular physiological meaning accruing to any of the model parameters. We have also shown, using our model, that the nonlinear and frequency-dependent nature of lung tissue mechanics can cause the apparent volume dependence of Rti and Eti to depend very much on the types of experiments performed and the analysis procedures used. This work was supported by the Medical Research Council of Canada (MRC) and the EL / JTC Memorial Research Fund and by Hungarian National Basic Research Fund Grant OTKA 1074. J. H. T. Bates is a scholar of the MRC. Address for reprint requests: J. H. T. Bates, Meakins-Christie Laboratories, 3626 St. Urbain St., Montreal, Quebec H2X 2P2, Canada. Received 10 July 1990; accepted in final form 10 April 1991. REFERENCES 1. BACHOFEN, 2. 3. 4. 5. 6.
H. Lung tissue resistance and pulmonary hysteresis. J. 24: 296-301, 1968. H., AND J. HILDEBRANDT. Area analysis of pressurevolume hysteresis in mammalian lungs. J. Appl. Physiot. 30: 493497,197l. BACHOFEN, H., S. SCHORCH, M. URBINELLI, AND E. R. WEIBEL. Relations among alveolar surface tension, surface area, volume, and recoil pressure. J. Appl. Physiol. 62: 1878-1887, 1987. BATES, J. H. T., K. BROWN, AND T. KOCHI. Identifying a model of respiratory mechanics using the interrupter technique. Proc. Annu. Conf. IEEE Eng. Med. Biol. Sot. 9th 1987, p. 1802-1803. BATES, J. H. T., F. SHARDONOFSKY, AND D. E. STEWART. The low-frequency dependence of respiratory system resistance and elastance in normal dogs. Respir. Physiol. 78: 369-382, 1989. BEDROSIAN, E., AND S. 0. RICE. The output properties of Volterra Appl. Physiol. BACHOFEN,
OF LUNGS
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systems (nonlinear systems with memory) driven by harmonic and Gaussian inputs. Proc. IEEE 59: 1688-1707, 1971. 7. BRUSASCO, V., D. 0. WARNER, K. C. BECK, J. R. RODARTE, AND K. REHDER. Partitioning of pulmonary resistance in dogs: effect of tidal volume and frequency. J. Appl. Physiol. 66: 1190-1196, 1989. 8. CSENDES, T., B. DAROCZY, AND 2. HANTOS. Nonlinear parameter estimation by global optimization: comparison of local search methods in respiratory system modelling. In: System ModelZing and Optimization, edited by A. Prekopa and B. Strazicky. New York: Springer, 1986, p. 188-192. 9. FRAZER, D. G., AND G. N. FRANZ. Trapped gas and lung hysteresis. Respir. Physiol. 46: 237-246, 1981. 10. FREDBERG, J. J., AND D. STAMENOVIC. On the imperfect elasticity of lung tissue. J. Appl. Physiol. 67: 2408-2419, 1989. 11. FUNG, Y. C. Biomechanics. New York. Springer-Verlag, 1981. 12. HANTOS, Z., B. DAROCZY, T. CSENDES, B. SUKI, AND S. NAGY. Modeling of low-frequency pulmonaiy impedance in dogs. J. Appl. Physiol. 68: 849-860, 1990. 13. HANTOS, Z., B. DAROCZY, B. SUKI, AND S. NAGY. Low-frequency respiratory mechanical impedance in the rat. J. Appl. Physiol. 63: 36-43,1987. 14. HILDEBRANDT, J. Dynamic properties of air-filled excised cat lung determined by liquid plethysmograph. J. Appl. Physiol. 27: 286290,1969. 15. HILDEBRANDT, J. Pressure-volume data of cat lung interpreted by a plastoelastic, linear viscoelastic model. J. Appl. Physiol. 28: 365372,1970. 16. LANIR, Y. On the structural origin of the quasilinear viscoelastic behavior of tissues. In: Frontiers in Biomechunics, edited by G. W. Schmid-Schonbein, S. L.-Y. Woo, and B. W. Zweifach. New York: Springer-Verlag, 1986, p. 131-136. 17. LORING, S. H., J. M. DRAZEN, J. C. SMITH, AND F. G. HOPPIN, JR. Vagal stimulation and aerosol histamine hysteresis of lung recoil. J. Appl. Physiol. 51: 477-484, 1981. 18. LUDWIG, M. S., I. DRESHAJ, J. SOLWAY, A. MUNOZ, AND R. H. INGRAM. Partitioning of pulmonary resistance during constriction in the dog: effects of volume history. J. Appl. Physiol. 62: 807-815, 1987. 19. MARMARELIS, P. Z., AND V. 2. MARMARELIS. Analysis of Physiological Systems. The White-Noise Approach. New York: Plenum, 1978. 20. MARMARELIS, V. Z. (Editor). Advanced Methods of Physiological System Modeling. New York: Plenum 1989, vol. 2. 21. PALM, G., AND T. POGGIO. The Volterra representation and the Wiener expansion: validity and pitfalls. SIAM J. Appl. Math. 33: 195-216,1977. 22. SAIBENE, F., AND J. MEAD. Frequency dependence of pulmonary quasi-static hysteresis. J. Appl. Physiol. 26: 732-737, 1968. 23. SATO, J., B. L. K. DAVEY. F. SHARDONOFSKY, AND J. H. T. BATES. Low-frequency respiratory system resistance in the normal dog during mechanical ventilation. J. Appl. Physiol. 70: 1536-1543, 1991. 24. SCHETZEN, M. The Volterra and Wiener Theories of Nonlinear Systems. New York: Wiley, 1980. 25 . SHARDONOFSKY, F., J. SATO, AND J. H. T. BATES. Quasi-static pressure-volume hysteresis in the canine respiratory system in vivo. J. Appl. Physiol. 68: 2230-2236, 1990. 26. SMITH, J. C., AND D. STAMENOVIC. Surface forces in lungs. I. Alveolar surface tension-lung volume relationships. J. Appl. Physiol. SO:l341-1350,1986. 27 . !~TAMENOVIC, D., G. M. GLASS, G. M. BARNAS, AND J. J. FREDBERG. Viscoplasticity of respiratory tissues. J. Appl. Physiol. 69: 28.
373-988, 1990. SUKI, B., 2. HANTOS,
B. DAR~CZY, G. ALKAYSI, AND S. NAGY. Non.inearity and harmonic distortion of dog lungs measured by low%equency forced oscillations. J. Appl. Physiol. 71: 69-75, 1990. 29. SUKI, B., PESLIN, C. DUVIVIER, AND R. FARRE. Lung impedance in wealthy humans measured by forced oscillations from 0.01 to 0.1 Hz. J. Appl. Physiol. 67: 1623-1629, 1989.
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viscoelastic model of lung tissue mechanics AND
Meakins-Christie
J. H. T. BATES Laboratories,
McGill
University,
Montreal,
Quebec H2X 2P2, Canada
cal formalism involved was more troublesome than that for plastoelasticity. More recently, there have been some attempts to further develop Hildebrandt’s plastoelastic model ( 10,27). Indeed, some of these studies go so far as to address the question of whether plastoelasticity really does exist in the lungs (10). Our own perspective on the question of plastoelasticity is that its existence would be unequivocally demonstrated by the presence of static pressure-volume hysteresis in the lung. However, the requirement of an infinitely long experiment for such a demonstration means that one can only obtain a quasi-static approximation to the necessary data, which always leaves open the quesdata by use of various types of ventilation patterns similar to tion of whether the volume of the lungs was cycled slowly those that have been employed experimentally. Rti and Eti enough, to allow all viscoelastic elements in the tissues to were estimated from the simulated data by use of four different remain fully relaxed at all points during the experiment. estimation techniques commonly applied in respiratory meFurthermore, although Hildebrandt and co-workers (2, chanics studies. We found that the estimated volume depen14) reported significant quasi-static hysteresis in isodence of Rti and Eti is sensitive to both the ventilation pattern lated lungs, Saibene and Mead (22) reported the pheand the estimation technique, being in error by as much as 217 nomenon to be less important in living humans. More and 22%, respectively. recently, Shardonofsky et al. (25) studied quasi-static hysteresis in tidally ventilated paralyzed anesthetized lung tissue resistance and elastance; nonlinear systems analydogs and found it to be essentially negligible. sis; Volterra series; harmonic distortion; plastoelasticity Thus, because the question of plastoelasticity vs. nonlinear viscoelasticity cannot be addressed experimentally, it remains one of mathematical convenience. BeTHE MECHANICAL PROPERTIES of lung tissue are imporcause there has been so much emphasis recently on modtant determinants of the overall mechanical behavior of eling lung tissue mechanics in terms of plastoelasticity, the respiratory system. Indeed, it is now well appreciated we believed it important to demonstrate that a nonlinear that the mechanical significance of the tissues extends viscoelastic model is also capable of adequately accountfar beyond a simple contribution to the static recoil pres- ing for the known nonlinear and frequency-dependent sure of the respiratory system. For example, the majority aspects of lung tissue mechanics. The development of of the total lung resistance apparent at frequencies ~1 such a model is the principal subject of the present paper. Hz arises in the tissues in dogs ( 17,18) and most likely in We also show how the nonlinear and frequency-depenother species as well ( 13). Most recent studies have con- dent aspects of lung tissue mechanics can lead to quite a centrated on the manner in which lung tissue mechanics marked disparity between the estimates of lung tissue vary with frequency in or around the breathing frequency resistance and elastance at a given frequency provided by range. Such variation can be accounted for by ascribing different analysis techniques. to the tissues the property of linear viscoelasticity, which is usually represented in terms of spring-and-dashpot THEORY models (4, 11) . However, the mechanical properties of Consider a nonlinear system with input x(t) and outlung tissue are also known to vary with volume, which put y(t) . If the system has finite memory (i.e., at any cannot be accounted for by a linear model. time t the value of y depends to a finite extent on previous A significant step in the modeling of nonlinear lung values of x extending only a finite period into the past), tissue mechanics was made by Hildebrandt (15)) who then x and y can be related through a Volterra series; chose to ascribe the nonlinearities to the phenomenon of thus (24) plastoelasticity. These are modeled as collections of springs and static friction elements that exhibit static pressure-volume hysteresis. Hildebrandt did acknowledge, however, that the nonlinearities he encountered could, in principle, also be explained as arising from non2 71,7&(t 72M7J72 linear viscoelasticity, but he implied that the mathematiJ -00 J-00 B., AND J. H. T. BATES. A nonlinear viscoelastic model lung tissue mechanics. J. Appl. Physiol. 71( 3) : 826-833, 1991.-There have been a number of attempts recently to use linear models to describe the low-frequency (O-2 Hz) dependence of lung tissue resistance (Rti) and elastance(Eti) . Only a few attempts, however, have been made to account for the volume dependence of these quantities, all of which require the tissues to be plastoelastic. In this paper we specifically avoid invoking plastoelasticity and develop a nonlinear viscoelastic model that is also capable of accounting for the nonlinear and frequency-dependent features of lung tissue mechanics. The model parameters were identified by fitting the model to data obtained in a previous study from dogs during sinusoidal ventilation. The model was then used to simulate pressure and flow SUKI,
of
+r00 gt rccl
826
0161-7567/91 $1.50 Copyright
0
1991 the American
Physiological
-71Mt -
Society
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NONLINEAR
VISCOELASTICITY
(0
x
-
x(t
- 73)dT1dr2d73 + 9 .
r2)x(t
l
The gi are the kernels of the system and must satisfy the causality constraint g( n
rlt
’
l
l
q&)
=o
(2)
forany7k in Eq. 7 is a constant complex number c3. Now Eq. 7 is reduced to
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828
NONLINEAR
VISCOELASTICITY
lS
2*o /Y 0
LUNGS
where V. denotes the magnitude of V ( fo) . B,, B,, A,, A,, and A, are now the parametric constants of the model. We also note that Eqs. 12 and 13 fully describe the amplitude and frequency dependence of Rti and Eti at f. when v ( t) is quasi-sinusoidal. We have now approximated the fundamental of P (Eq. 4) as
I
s
OF
50
100
200
150
250
P(
fo) =
{Eti[V(f,)]
+
iBti[V(fo)]}V(f~)
(l-4)
Equations 5 and 6, however, show that the second and third harmonics of P both contain contributions from V( fJ in addition to their respective contributions from the second and third harmonics of V. We must therefore include additional terms in our simplified expressions for P(2f,) andP(3f,), which we choose to do as ti 0
50
100
150
200
250
4-
{Eti[V(2fo)]
P(2fJ
=
Wf,)
= {Eti[V(3f,)l
3-
21 -
I
0 0
50
I 100
Volume, I I
ml
200
150
I 250
FIG. 1. Lung tissue resistance (Rti, top), elastance (Eti, middle), and distortion factor of volume input (bottom) as a function of volume amplitude [data from Suki et al. (28)]. Values (means t SD) were obtained from 4 measurements at 5 cmH,O transpulmonary pressure with an O.l-Hz quasi-sinusoidal volume in an open-chest dog.
For the linear dynamic part of the model, G,, one may choose any viscoelastic model that adequately accounts for the frequency dependence of Rti and Eti obtained with small-amplitude forcing. We chose the following linear viscoelastic model originally proposed for the lung by Hildebrandt ( 14,15)
BWf,) = rBJ4.6
COMPUTATIONAL
f,) = A, + (0.25 + log 2rfo)B,
Parameter
14.6
+ (0.25 + log 2?rf,)Bti[V(
(12)
fo)]
METHODS
estimation.
A global optimization
procedure
(11) (8) was used to fit the model described above to experi-
This model is very simple to deal with and has been successfully used to account for the frequency dependence of Rti and Eti determined from low-frequency small-amplitude forced oscillations ( 12, 29). Now we must include the volume amplitude dependence in Eq. 9. To do this we approximated the data in Fig. 1 by simple parametric curves. We found that a second-order polynomial was satisfactory for describing Eti and that a linear function gave a good fit to Rti. We therefore replace the constants B, in Eq. 10 and A, in Eq. 11 with linear and quadratic expressions, respectively; thus
Bti[Wfo)l = ?r(B, + BJ,) Eti[V( f,)] = A, + A,& + A2Vi
Here a2, a3, b,, and b3 are additional model parameters. Equations 12-16 constitute our complete nonlinear model. Although they form a drastically reduced frequency-domain Volterra series, they should still be able to account for the experimentally observed dependences of Rti and Eti on frequency and volume amplitude, provided the various parameters in Eqs. 12,13,15, and 16 can be suitably chosen. Furthermore, because the model is a form of Volterra series, the system it represents must have a memory that decreases toward zero as one goes back arbitrarily far in time. This is the essential nature of a viscoelastic substance, the deformation of which approaches a value uniquely determined by the current stress if that stress is maintained for an arbitrarily long time. In contrast, the strain of a plastoelastic substance at constant stress remains forever determined in part by the previous stress history. Our model, therefore, is a nonlinear viscoelastic model.
(10)
and Eti(
+ iBti[V(2f~)]}Wf~) t15) + (a2 + ib2)V2(f0) + iBwwf,)l}w3f,) (16) + (a, + ib,W3(f0)
(13)
mental data from four dogs. The data were taken from the study of Suki et al. (28), where the experimental methods involved are described. The evaluation of the model parameters was performed in two steps. First, the parameters Ai (i = 0,l) and Bi (i = 0, 1,2) were obtained by fitting Eqs. 12 and 13 simultaneously to estimates of Rti and Eti obtained from quasi-sinusoidal volume oscillations. This involved minimizing the following error function ( 8, 12, 29) FE, = {% i j=l
[Rti,
j - Rti(Voj)12 , + [Et&j
(17) - Eti(Vo,j)]2}“2
where Rti, j and Eti, j are the resistance and elastance values shown in Fig. 1 measured with a f. = 0.1. Hz quasisinusoidal signal with six different amplitudes, which are
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NONLINEAR
VISCOELASTICITY
denoted by V0 j. Rti(V, j) and Eti(V,J , were obtained from Eqs. 8, lb, and 13: In the second step, the estimated values of the Ai and Bi were used with experimentally measured values of andV(3f,)topredictP(f,),P(2f0),and v(fo),wfoL P (3 fo) , respectively, using Eqs. 14-M. The values of the parameters ai and b, were chosen SO that the predicted P matched the experimentally determined ones as closely as possible. This involved minimizing the following error function FE, = m
5 {WP(jfo)
- wjfo)l}2
j=l
+ (WRjf(J
- Wjfo)] j2Y2
(18)
where the operations Re [ ] and Im[ .] denote the real and imaginary parts of their arguments, respectively, and P denotes an experimentally determined pressure harmonic. The solutions to the above optimization problems were all unique, because the estimated parameters were in each case linearly related to the error functions. Simdation studies. Having identified the model parameters in the above manner, we proceeded to perform simulation studies in which the model was used to generate p ( t) and v(t) data and some standard estimation methods applied to the data to reestimate Rti and Eti. In this way we were able to test the effects of the assumptions made in the estimation methods in a controlled manner. Simulation studies were performed with the model using four classes of v(t) with spectra that fell into classes denoted by ISl, IS2, IS3, and IS4. The fundamental frequency in each class was 0.1 Hz. In classes IS1 and IS2, the amplitudes from the second to the sixth harmonics decreased as 1 / 0, and 1 /f, respectively. This left the amplitude at 0.2 Hz free to be chosen so that K, had a value as follows: when the amplitude of the fundamental frequency was 10,40,80,120, and 160 ml, K, had a value of 1,4,8, 12, and 16%, respectively. In the spectral class IS3 the fundamental component had the same amplitude values as classes IS1 and IS2 and the amplitudes of the harmonics decreased as 1 /fl, but K, was fixed at 5%. Thus the nature of a particular v ( t) was essentially completely determined by its class and the amplitude of its fundamental. The only aspect not determined was the phase distribution of its harmonic components. Therefore, within each of the classes ISl, IS2, and IS3 and at each amplitude, we generated 25 individual v(t) signals with randomly selected component phase angles, so that our simulation results would not be biased by the choice of any specific phase angle combination. Finally, the fourth class IS4 contained only 6 v(t) signals with the same volumes at the fundamental frequency as the first three classes above and with spectra that were of a triangular wave truncated above 1.1 Hz. This resulted in a K, of 12% for each of the v(t) in this class. The relationships between each of the v(t) described above and the corresponding p ( t) produced by the model were evaluated by four techniques that have been commonly used in previous studies of respiratory mechanics to assess the apparent amplitude dependence of Rti and Eti. The first method (Ml) was similar to that used by Hildebrandt (14)) in which Rti was obtained as the ratio l
OF
829
LUNGS
of the area of the p(t) - v(t) curve during a complete breathing cycle to the mean squared flow. Eti was calculated as the ratio of the pressure difference between points of zero flow to tidal volume. The second method ( M2) was based on the linear single-compartment equation of motion = Etiv(t)
p(t)
+ Rtiv(t)
(19) Here p (t), v(t) , and v(t) were obtained as the inverse Fourier transforms of model output, input, and time derivative of input, respectively. Rti and Eti were estimated by multiple linear regression according to Bates et al. (5). The third method (M3) consisted of fitting the equation of motion of a two-compartment linear model to the time-domain data. This equation, derived by Sato et al. (23), is p(t)
= k,+(t)
ts +k3
+ k,v(t)
v(T)dT
+ k4
0
ts
P(dd~
(20)
0
The integrations in Eq. 20 were carried out numerically, and the constants ki were again estimated using multiple linear regression. The resistance and elastance of the model are (23) Rtl
.
=
k,w2 - k2h u2 + k;
-
k,
and . k,k402 + k202 - kSk4 Et1 = u2 + k;
(22)
where o = 2nf is the angular frequency. Finally, the fourth method of analysis (M4) was to calculate the complex elastic modulus at the fundamental frequency as E ( f,) = P ( fo) /V( f,) from which Eti = Re[E(f,)] and Rti = Im[ E ( fo) ] /2rfo. This method is often called Fourier analysis because it relates the Fourier transforms of the input and output at the frequency of interest, and therefore the estimation of Rti and Eti iS expected to be less influenced by the extent of the distortion of the input signal than with the other methods. RESULTS
The model developed here was fitted to data from four dogs taken from Suki et al. (28)) who used method M4 to estimate Rti and Eti (one example is shown in Fig. 1). The parameters describing nonlinearity of Rti and Eti are summarized in Table 1. The interindividual variabilities of the parameters are small except for B,, which determines the degree of amplitude dependence of Rti. The parameters accounting for the harmonic production in the lung are given in Table 2. Fairly consistent values of a2 and b, were also obtained from recordings measured with smaller input volumes (down to 60 ml), but with decreasing volume their standard deviations increased rapidly. Reliable estimates for a3 and & could not be obtained at volumes 420 ml, because with smaller V. the third harmonic in P was at most only l-2% of the funda-
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830
NONLINEAR
VISCOELASTICITY
OF
8
1. Parameters describing the amplitude dependence of lung tissue resistance and elastance TABLE
Dog No.
&I9
cmHzO/l
B,,
cmH,O A2
AtI9
49
LUNGS
A29
cmH,O/l
cmH20/12
cmH,O /I3
1 2 3 4
2.61 2.33 2.50 2.72
-0.09 -1.08 -1.30 -1.53
14.3 12.5 13.5 13.9
-27.4 -22.8 -28.5 -22.6
53.3 62.5 63.9 53.0
Mean *SD
2.54 kO.17
-1.00 to.63
13.6 kO.75
-25.3 t3.1
58.1 t5.9
mental component, resulting in an unacceptably low signal-to-noise ratio. Therefore, in the simulation studies we used only the average ai and bi obtained at V0 = 240 ml. Figure 2 demonstrates the performance of the model in predicting the pressure distortion (&) from a given Kv. If only the linear part of the model is used, achieved by setting all coefficients except A, and B, to zero, then the KP predicted by the model resembles K, instead of the measured Kp. The complete nonlinear model, however, performs much more satisfactorily and predicts the values of Kp over the volume range shown in Fig. 2 to be close to the measured ones. The volume dependences of Rti are evaluated by the four different signal-processing techniques Ml-M4 are compared in Fig. 3 and Table 3. The effect of volume on Eti is also given in Table 3. Rti and Eti obtained by M4 were entirely independent of K, and the phase angles of the components, with the percentage decrease of Rti between V0 = 10 and 160 ml being 9.2% and that of Eti being 14.8%. Methods Ml and M2 were very similar in their abilities to estimate Rti and somewhat different in their estimates of Eti. Furthermore, their results depended markedly on the input waveform. Method M3 was much more robust in its estimations of Rti with the different v(t) classes and provided Eti estimates similar to those of M2 and M4. These results show clearly that the estimation of Eti and Rti is highly dependent on the estimation method used. DISCUSSION
Most analyses of the mechanical behavior of the respiratory system and its components to date have been based on linear models of the system, understandable enough in view of the power and utility of linear systems analysis. At some level of detail in any system, however, one is eventually forced to consider the nonlinear aspects
6\0
6
Volume,
0 50 100 150 200 250 2. Distortion factors (mean t SD) of volume input (0) and pressure output ( l ) as a function of volume amplitude obtained in the same dog as in Fig. 1. Continuous and dashed lines, predictions of a nonlinear and a linear viscoelastic model, respectively. FIG.
of the system. In the case of lung tissue mechanics, the currently available experimental data clearly demonstrate the existence of important nonlinear phenomena, compelling us to invoke some kind of nonlinear analysis in our attempts to understand them. One way to analyze a nonlinear dynamical system is to establish a set of nonlinear differential or integral equations and to solve it either analytically if possible or, more usually, by standard numerical procedures. This approach must be based on some a priori knowledge about the structure of the system being modeled. In most biological systems, however, such information is invariably lacking or deficient to a degree that the system can only be considered as a black box, so that one is left trying to arrive at an empiri-
0 z E
02 2.0
0
TABLE 2. Parameters accounting for the harmonic production of lung tissue Dog No.
%9
cmH,O A2
b
a39
cmH:b/12
cmH20/13
ml
0
IS3
IS4
b
cmH&13
1 2 34
1.00 1.20 1.24
5.29 4.27 5.34 5.49
0.39 0.02 0.44
2.87 2.46 2.77 3.30
Mean tSD
1.26 to.25
5.10 to.56
0.24 to.21
2.85 to.35
0
50
100
150
Volume,
0
50
100
150
ml
FIG. 3. of resistance of nonlinear viscoelastic model determined Amplitude dependence by 4 evaluation techniques. Dashed, dotted, and continuous lines, evaluation methods Ml /M2, M3, and M4, respectively. Curves in the 4 panels were obtained as average responses of nonlinear viscoelastic model to volume inputs from signal classes ISl, IS2, IS3, and IS4.
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NONLINEAR
VISCOELASTICITY
3. Average %V, dependence of Rti and Eti between i/OS of 10 and 160 ml
TABLE
Ml
IS1 IS2 IS3 IS4
M2
M3
M4
Rti
Eti
Rti
Eti
Rti
Eti
Rti
Eti
29.2 19.7 9.4 8.9
11.6 13.0 13.8 12.0
29.2 19.7 9.4 8.9
14.0 14.2 14.7 14.5
7.1 8.1 9.2 8.5
14.2 14.4 14.7 14.5
9.2 9.2 9.2 9.2
14.8 14.8 14.8 14.8
VO, volume
amplitude
at the fundamental.
cal relationship between the measured inputs and outputs. In the present study our goal was to derive some kind of nonlinear viscoelastic model of low-frequency lung tissue mechanics. A rigorous mathematical derivation of the conditions under which a nonlinear system can be expanded in a Volterra series as in Eq. 1 was given by Palm and Poggio (21)) who attempted a simple expression of these conditions by saying that “Systems which do not respond critically to certain changes in input show a smooth dependence of the output on the input.” Given that pulmonary pressure-volume data do not show any features that could be described mathematically as discontinuities, the mechanical properties of lung tissue would seem to qualify for description by a Volterra series. In contrast, a system containing a single plastoelastic element (Prandtl body) does not qualify because it always shows a discontinuity in p(t) when v(t) changes direction, while viscoelastic models do not show such discontinuities. However, a plastoelastic model with a smooth continuous distribution of yield stresses might also not show discontinuous behavior. On the other hand, the phenomenon of viscoelasticity is associated with a finite memory of the tissues to previous deformations, whereas plastoelasticity involves infinite memory. Because the orthogonalized version of the Volterra series, called Wiener series, is able to account for any timeinvariant nonlinear system with finite memory that is not “explosive” (24)) we believed that the most general approach to our goal would be to base our model on the Volterra representation. The Volterra series has already been applied in a number of biological situations (19, 20), although the input involved has invariably been Gaussian white noise. In the present study, we had to consider sinusoidal inputs, because these were used to gather the experimental data on which our analysis is based. We also had to drastically simplify the general Volterra expansion to uniquely identify model parameters from the available data. In making this simplification we assumed that the amplitude dependences of Rti and Eti are independent of frequency, which seems reasonable in view of some recently published data (5,7) demonstrating that Rti and Eti obtained with different tidal volumes differ by roughly a constant factor over the frequency range O-2Hz. Another important assumption was that the softening of the tissues is essentially determined by the amplitude of VO. This approximation appears to be useful when either Kv or V0 is small. It is likely to break down, however, when both V, and Kv become large, in which case we would not
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be able to neglect those terms in Eq. 3 accounting for nonlinear cross-talk between frequency lines. These terms would cause the tissue to soften with amplitude at the fundamental frequency if there are strong higherorder harmonics in v(t). This means that Rti and Eti determined with sinusoidal and composite v(t) signals could be different. Indeed, Sato et al. (23) found systematically lower resistances during regular ventilation than during sinusoidal oscillation at the equivalent tidal volumes. The former resulted in a v ( t) signal having a Kv > 48% (unpublished observation). Recently, Fredberg and Stamenovic (10) introduced the concept of structural damping in the lung by postulating that elasticity and resistivity within the tissues are “coupled at the level of the stress bearing elements.” They argued that such a mechanism implies that plastic deformations could take place at the microstructural level and suggested four processes by which this may occur: 1) the detachment /attachment of cross-bridges in the contractile elements, 2 ) the adsorption / desorption of surface-active molecules in the alveolar surface film, 3) the movement of the collagen and elastin fiber network, and 4) the recruitment /decruitment of alveoli. Indeed, there is strong evidence both from in vitro (26) and in vivo (3) studies that lung surfactant exhibits significant hysteresis, particularly over large volume excursions. Given the importance of the surfactant layer in determining overall lung mechanical properties, these data seem to imply that plastoelasticity should be an important feature in a model of lung mechanics. Another phenomenon that can certainly give rise to static pressure-volume hysteresis at low lung volumes is gas trapping w
Despite the above compelling evidence in favor of plastoelasticity in the lung, experimental evidence shows that quasi-static hysteresis in the living lung undergoing modest volume excursions above functional residual capacity is very small, if not negligible ( 25). The lung still behaves nonlinearly over this volume range, however, so truly plastic phenomena may become manifest only over very large volume excursions. In any case, there are examples in the literature of nonplastoelastic models accounting for complicated tissue mechanical behavior. Lanir (16) showed theoretically that during loading of a network of elastic fibers the progressive unfolding and reorientation of the fiber macromolecules in the direction of the applied force would result in a constitutive law for the material called quasi-linear viscoelasticity, originally described by Fung ( 11) . This law is quasi-linear in that it includes only the first integral term in Eq. 1, but the input x(t) in the integral is replaced by some nonlinear function of x(t) . One of the prominent features of quasi-linear viscoelasticity as proposed by Fung ( 11) is that it can account for the observation that the area of the force-length loop of dynamically stressed tissue is independent of frequency over a wide range of frequency, which originally gave rise to the plastoelastic concept with regard to the lung proposed by Hildebrandt (15). The nonlinear viscoelastic model developed in this paper accounts for two key properties of lung tissues: the decrease of both Rti and Eti with increasing amplitude of v ( t) and a frequency-independent p (t) - v(t) loop area.
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VISCOELASTICITY
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real lung tissue, it enables us to investigate the consequences of this behavior under precisely controlled conditions through computer simulation experiments. Some important conclusions can even be drawn analytically. For example, if we define p(t) to be the output of the respiratory system and v(t) to be the input, the impulse response of the system would be obtained by measuring the p(t) resulting from the application of an impulse in v(t) . It can be seen from Eq. 1 that the impulse response of a general nonlinear viscoelastic system is p(t)
Pressure
(arbitrary
units)
4. Pressure-volume loops simulated by nonlinear viscoelastic model using data of dog 3 in Tables 1 and 2. When plotted in absolute units of pressure and volume, loops are of very different magnitudes and, therefore, difficult to compare. We have therefore scaled loops so that they each have unity volume excursion. Tidal volumes were 240 ml (solid line), 60 ml (dashed line), and 15 ml (dotted line). FIG.
An example of a series of p(t) - v(t) loops simulated with our model is provided in Fig. 4, demonstrating that the slope of the lines joining the extrema of the loops, and hence the effective elastance, is a decreasing function of tidal volume. Also, with increasing volume the curves change from elliptical to a shape showing clear secondorder nonlinearity (i.e., curved like a quadratic function) . This is in qualitative agreement with a number of experimental observations (1,2,14,22,28; J. H. T. Bates unpublished data). In contrast, the general characteristics of the p ( t) - v(t) curves simulated by a viscoplastic model (15, 27) suggest rather strong odd-order nonlinearities (i.e., sigmoidal like a cubic function) even with the smallest tidal volume. Also, the results of an analysis such as given by Stamenovic et al. (27) to discriminate between the plastoelastic and linear viscoelastic contributions to lung mechanical properties will depend to a certain degree on the particular model structure chosen. Nevertheless, none of the above discussion rules out the possibility of the plastoelastic interpretation. Therefore a decision concerning whether a plastoelastic or a nonlinear viscoelastic model is more appropriate for the lung tissues in any fundamental sense must remain unresolved ., inasmuch as the only basis for such a choice would be the existence or otherwise of static p (t) - v(t) hysteresis, which, of course, is impossible to measure. Moreover, recent work by Shardonofsky et al. (25) shows that plasticity in the lungs of a living dog during normal tidal ventilation is minimal and may indeed even be truly negligible, although continuing gas exchange in the lungs makes it impossible to be sure. This is not to say that one must not use a plastoelastic model to represent lung tissues, but we believe that mathematical convenience is the only reasonable criterion to use in deciding between a plastoelastic and a nonlinear viscoelastic representation. The utility of our nonlinear viscoelastic model is that, to the extent that it accuratelv mimics the behavior of
= g,(t)b
+ g,(t, t)b2 + g3(t, t, t)b3 + 9
l
l
(23)
where b is the impulse scaling factor. The complex impedance of the system, on the other hand, would be obtained by applying some appropriate broad-band v(t) to the system, measuring the resulting p (t), and then calculating the ratio of the transforms Z ( f ) = P ( f ) /V ( f ) . We have seen above that the results of this procedure depend on the harmonic content of v(t) in a complicated way and are not equal to the Fourier transform of Eq. 23. Thus the fundamental result of linear systems analysis, that the Fourier transform of the impulse response equals the complex impedance, does not apply, in general, to nonlinear systems. Therefore, in a nonlinear system, we cannot easily compare the results of time domain and frequency domain analyses, which might explain, at least in part, the reported inconsistencies between various lung mechanical variables derived from time and frequency domain data (15). The amplitude dependence of lung mechanical properties is usually studied by the application of quasi-sinusoidal v(t) . Unfortunately the volume generators used in such experiments are invariably imperfect to the extent that the shapes of their v(t) are not independent of the frequency of cycling or the impedance against which the pump has to produce a given volume. This means that Kv is a function of the amplitude and generally increases with volume amplitude (as in Fig. 1). This situation was represented in our simulation studies by the signal classes IS1 and IS2 in which Kv was a linear function of the fundamental component of v(t) . Depending on the evaluation technique chosen, Fig. 3 shows that this can lead to a very large apparent amplitude dependence of lung tissue mechanics, which may explain the diversity of reports in the literature regarding the amplitude dependence of Rti that we alluded to above. The most inputsensitive methods are Ml and M2, and we suggest that they should not be used to study the amplitude dependence of tissue mechanics whenever K, exceeds 3-4%. Method M3 gives reasonable results because its structure implicitly leads to frequency-dependent Rti and Eti values that are similar to those of the lung in the frequency range O-2 Hz. The most reliable is method M4 because it deals with the input harmonics separately. However, none of the four methods is completely independent of the harmonic content of the input, and when Rti and Eti depend strongly on the amplitude, even the Fourier method may give misleading results if both the tidal volume and K, are large. Finally, we note that the measure of Rti defined by Fredberg and Stamenovic (10) lies between Ml and M4, because it normalizes the p ( t) - v ( t) loop area to the fundamental harmonic of v( t) . Because
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NONLINEAR
VISCOELASTICITY
the loop area actually contains contributions from all the harmonics present in p ( t) and v(t) , their method is expected to provide a slightly smaller tidal volume dependence than M4. In summary, we have developed a nonlinear viscoelastic model that is capable of quantitatively accounting for the known amplitude and frequency-dependent aspects of lung tissue mechanics. We do not claim our model to be the only one derivable form the Volterra series that could possibly account for lung tissue mechanics. One might argue, in fact, that in deriving the model we eliminated some of the terms of the series on rather weak grounds. However, the point of the exercise was to produce a nonlinear viscoelastic model that works, because it was our intention to demonstrate that such a model exists. It is immaterial whether there is more than one. Our model thus shows that it is not essential to invoke the concept of plastoelasticity with regard to the lung tissues, although it does not show that such a concept is inappropriate. Indeed, we conclude that the question of plastoelasticity vs. nonlinear viscoelasticity is one of mathematical convenience rather than a matter of fundamental importance. Furthermore, at this stage in our understanding of these two phenomena, a model based on either mechanism can only be considered purely empirical with no particular physiological meaning accruing to any of the model parameters. We have also shown, using our model, that the nonlinear and frequency-dependent nature of lung tissue mechanics can cause the apparent volume dependence of Rti and Eti to depend very much on the types of experiments performed and the analysis procedures used. This work was supported by the Medical Research Council of Canada (MRC) and the EL / JTC Memorial Research Fund and by Hungarian National Basic Research Fund Grant OTKA 1074. J. H. T. Bates is a scholar of the MRC. Address for reprint requests: J. H. T. Bates, Meakins-Christie Laboratories, 3626 St. Urbain St., Montreal, Quebec H2X 2P2, Canada. Received 10 July 1990; accepted in final form 10 April 1991. REFERENCES 1. BACHOFEN, 2. 3. 4. 5. 6.
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