Abstract
Maksym, Geoffrey N., and Jason H. T. Bates. A distributed nonlinear model of lung tissue elasticity.J. Appl. Physiol. 82(1): 32–41, 1997.— We present a theory relating the static stressstrain properties of lung tissue strips to the stressbearing constituents, collagen and elastin. The fiber pair is modeled as a Hookean spring (elastin) in parallel with a nonlinear string element (collagen), which extends to a maximum stop length. Based on a series of fiber pairs, we develop both analytical and numerical models with distributed constituent properties that account for nonlinear tissue elasticity. The models were fit to measured stretched stressstrain curves of five uniaxially stretched tissue strips, each from a different dog lung. We found that the distributions of stop length and spring stiffness follow inverse power laws, and we hypothesize that this results from the complex fractallike structure of the constituent fiber matrices in lung tissue. We applied the models to representative pressurevolume (PV) curves from patients with normal, emphysematous, and fibrotic lungs. The PV curves were fit to the equation V = A −Bexp(−KP), where V is volume, P is transpulmonary pressure, andA, B, andK are constants. Our models lead to a possible mechanistic explanation of the shape factorK in terms of the structural organization of collagen and elastin fibers.
 lung tissue mechanics
 collagen
 elastin
 stress strain
 power law
 pressurevolume curves
the elastic recoil of the lung at normal breathing frequencies is dominated by the nonlinear stressstrain characteristics of the lung tissue (35). It is commonly agreed that the main stressbearing constituents of lung tissue are collagen and elastin fibers (30). These fibers differ significantly in their mechanical properties, with collagen being three to four orders of magnitude stiffer than elastin (40). Furthermore, elastin is stretchable to >250% of its original length before rupturing, whereas collagen can stretch merely 1–3%. The origin of the curvilinear stressstrain behavior is generally thought to be one of collagen fiber recruitment. That is, elastin is responsible for load bearing at low strains when much of the collagen is “wavy” and, therefore, not contributing to the tension. As strain increases, the collagen fibers become straight and so progressively take up more load, thereby stiffening the tissue. The amounts of collagen and elastin and their spatial arrangements in the lung are altered in diseases such as pulmonary emphysema and fibrosis (39, 43), and it is thought that these alterations are largely responsible for the abnormal pressurevolume (PV) curves associated with these diseases (27). However, it is currently unclear how to precisely translate alterations in lung tissue constituents into changes in tissue mechanical properties. We have developed a model of lung tissue that exploits the collagen fiber recruitment concept by representing the collagen and elastin fibers as a series of springstring pairs. We explore how distributions of spring stiffnesses and string lengths can account for measured stressstrain curves of lung parenchyma. We also explore how our model might be used to explain the curvilinear quasistatic PV curve of whole lung.
METHODS
Analytic Model Development
Consider the model of Fig. 1. The model is constructed from a series of springstring pairs, where the springs represent elastin fibers and the strings represent collagen fibers. Each springstring pair (unit) begins at resting length, with no load, and the strings are flaccid. As the model is extended, each spring resists the applied force in a linear Hookean manner until its associated string becomes taut. The strings are inextensible, so further extension of the unit is impossible. We define the maximum extension of a unit as its “stop length.” This is thus the difference between the maximum length of the unit and the resting length of the spring. As the model is stretched, the number of units that have reached their stop length increases, thereby progressively increasing the slope of the stressstrain curve. We assume that the springstring pairs are small and numerous so that their stop lengths and spring constants can be described in terms of continuous distributions.
Distributed stop length model. A preliminary development of this model applied to lung tissue strips was previously presented by the authors (17).
We consider the model of Fig. 1 with identical spring constants for each springstring pair, whereas the stop lengths are distributed. At rest, there is no load on the model, and its resting length isL
_{r}. When the model is extended to length L >L
_{r}, all the units with stop lengths less some valuel
_{o} will have become stopped. The length of the model is then the sum of three terms
We note that Eq. 5
substituted intoEq. 4
and evaluated forl
_{o} = 0 gives the initial slope of the lengthtension curve, thus
Distributed stiffness model. An alternate model based on Fig. 1 can be developed in which the stiffnesses are distributed but the stop lengths are identical for all springstring units. LetM(k) represent the stiffness density distribution. Equation1
now becomes
Because all units of the model experience the same force, equal to the force of the unit that has just reached its stop length, we have
We note that the maximum extension (L
_{s max}) is given by
Curve Fitting to Experimental Data
We obtained a tissue strip from each of five different degassed dog lungs. The preparation and testing apparatus are described in detail elsewhere (18). Briefly, subpleural strips were taken from the left lobes, and the pleura were removed. The strips were cut to 2.80 ± 0.18 cm, with a cross section of 0.230 ± 0.025 (SD) cm^{2}. Tissue samples were placed in a tissue bath kept at 37°C in continuously circulating KrebsRinger solution bubbled with 95% O_{2}5% CO_{2}. The tissue strips were preconditioned several times by slowly cycling at 2%L
_{r}/s betweenL
_{r} and 5 kPa tension. The lengthtension curves were obtained by straining the tissue at 0.5%L
_{r}/s fromL
_{r} to >100% strain. Signals were antialias filtered to 10 Hz and sampled at 30 Hz on a 486 PC. The lengthtension curves from the five tissue strips were fitted to the function
To calculate the stop length distributions (Eq.7
) and stiffness distributions (Eq.14
) from stressstrain tissue data, the following relationship was used to convert from stressstrain to lengthtension
Computer Model Simulations
We simulated the tissue models on computer by assigning spring and string properties to 10,000 units assembled in series. The length tension curves were calculated as follows. Each unit ceases extension at the unit’s stop length, at which point the force is equal to the product of k andl, which we define as the stop force of that unit. The units are rank ordered in terms of their stop forces. The model is then extended from rest and the total extension calculated in sequence at each stop force, thus forming a piecewise linear lengthtension curve. At each point in this curve, the length of the model is equal to the resting length plus the sum of the stop lengths of all stopped units plus the extensions of the unstopped units. Note that in choosing 10,000 units, we match the areas chosen forN(l) andM(k), thus permitting direct comparison with the analytic results, as the number of units is the discrete equivalent of the areas under the continuous density distribution functions. The simulated curves were converted from lengthtension to stressstrain units by using the means of the tissue areas and resting lengths (Table1). These curves were then compared with a mean tissue stressstrain curve generated from the mean values ofG andH from the five tissue strips viaEq. 13 . Stressstrain curves were generated for the following different cases:1) stop lengths varied among the 10,000 units while the stiffnesses were all identical;2) stiffnesses varied while the stop lengths were all identical; and 3) both stop lengths and stiffnesses varied among the units.
RESULTS
Application to Lung Tissue StressStrain Curves
The five stressstrain curves from dog tissue strips are shown in Fig.2, together with their curve fits according to Eq. 16 . The parameters of the curve fits together with the root mean squared residuals for each are shown in Table 1.
Applying the distributed stop length model via Eqs.7
and
17
to the stressstrain Eq. 16
, the stop length distribution is
Similarly, applying the stiffness distribution model viaEqs. 12, 14
, and17 to Eq.16
gives
Computer Simulations
Figure 5 shows the mean experimental stressstrain curve together with curves generated by using our 10,000element numerical model. For the distributed stop length model, values for l were randomly assigned according to Eq. 18
, withk chosen according toEq. 19
. The range ofl for the distribution function was between 0 and
where F_{f} is the final force defined by ε_{f} = 2. In the distributed stiffness model, stiffnesses were distributed according toEq. 21
, withl from Eq.22
. The range of k was limited between 0 and
where again F_{f} is defined by ε_{f} = 2. The distributions for the combination model were created according toEqs. 18 and 19 , respectively, with parameters altered so as to more closely match the mean tissue stressstrain curve. Specifically, the value of k inEq. 18 and in Eq.24 decreased by a factor of 2 over the distributedl model, and the value ofl of Eqs.21 and 25 was increased by a factor of 2 over the distributedk model.
DISCUSSION
Tissue Models
We have described a model for interpreting static elastic behavior of the lung tissue strip. Collagen is considered to be infinitely stiff compared with elastin. This is based on the observation that the incremental Young modulus of collagen at typical strains is 6.8 × 10^{8}dyn/cm^{2} to 1.2 × 10^{11} (9, 14), whereas Young’s modulus of elastin is 1 × 10^{6} to 8 × 10^{6}dyn/cm^{2} (10). In other words, collagen is 100 to 10,000 times stiffer than elastin. We assumed a linear lengthtension relationship for elastin as has been used in previous studies (1, 7, 10). This is justified because elastin exhibits its nonlinear behavior outside the range of physiological strains (0.0 ∼ 0.7) (5, 29). Other components of the lung tissue that may contribute to its mechanics are the ground substance, smooth muscles, and other cells. Brown et al. (4) proposed that the deformation of the ground substance may be a significant influence on the mechanical properties of tissue, although this remains to be tested. Therefore, since this effect is not known and since these substances have much smaller elastic constants than collagen and elastin (10), we do not explicitly incorporate the effects of ground substance in our model.
The idea that elastin determines the stressstrain behavior of lung parenchyma at low strains, whereas collagen takes over as strain increases, is well known, first having been introduced by Setnikar (31) and extended later by Mead (20). More recently, there has been a few studies that explore this idea in a quantitative fashion. For example, Fung (10) used a linear expression for the constitutive equation for elastin together with an exponential function for collagen. The two fiber types were assumed to be distributed probabilistically according to observed distributions of fiber diameter and curvature. This gave a tissue strain energy function from which the incremental material constants were calculated according to the theory of continuum mechanics. Suwa et al. (37) ascribed the origin of the curvilinear stressstrain relationship of lung parenchymal strips solely to the reorientation of the constituent fibers, where a fiber does not support load until oriented in the direction of the applied strain. Models based on the gradual recruitment of kinked or “wavy” fibers were proposed by Soong and Huang (34) for human alveolar wall and by Decraemer et al. (6) for various biological soft tissues. However, these models consist of only one fiber type, elastic collagen, and assumed a specific functional form for the distribution of fiber lengths rather than estimating the distribution from the available data. Brown et al. (4) studied ligamentum propatagiale from birds and discussed a model similar to ours in which the collagen fibers had distributed initial waviness. However, they did not develop it mathematically and concluded that it may be an important factor in distensible tissues but probably was not the predominant mechanism in ligamentum propatagiale. Recently, Denny and Schroter (7) constructed a model for the alveolar duct including collagen and elastin fibers of three different diameters for different alveolar locations, based on the densities of collagen and elastin fibers found in dog lungs (25). They assumed a particular stressstrain relationship for the collagen fiber bundles, which they assumed arose from gradual recruitment of tensed collagen fibers within a bundle. Our approach, however, has a significant advantage over all the above studies in that it allows us to determine functional forms for the distributions of elastin or collagen fiber properties from experimental stressstrain data, rather than having to assume these forms a priori.
The distributions of elastin fiber diameters were measured in human pulmonary alveolar walls of human lungs by Sobin et al. (33) and in alveolar mouths and ducts by Matsuda et al. (19). The distribution of elastin diameters would be expected to relate to our distribution of elastin stiffnesses, given that the stiffness of a fiber or fiber bundle is proportional to its crosssectional area for a fixed Young’s modulus. Thus a plot of the stiffness distribution functions vs. the square root of stiffness should resemble a fiber diameter distribution function. This is borne out by the mean diameter distribution from our five tissue strips shown in Fig. 6, which demonstrates a striking similarity to the data of Sobin et al. (33) (Fig. 7). Distributions of collagen fiber curvatures in the lung (19, 33) are also similar in form to our stop length distributions, although it is not entirely clear how these two quantities relate.
These morphological observations lend support to the model, but this does not imply direct correspondence between the strings and springs of our model and the individual components of actual tissue. The arrangement of the fibers in lung tissue is not only serial but is also parallel and interwoven, forming complex interconnections (22). In fact, our stop length distribution may span multiple length scales. That is, the shorter collagen fibrils may correspond to the smallest stop lengths, but, as many fibers become taut, a stop length may be formed from a cluster of collagen fibers (connected in parallel and in series), all reorienting and straightening against the load. Thus the distribution of stop lengths in our model may represent how the collagen matrix is arranged in a fractal sense.
In our model, the properties of collagen and elastin are considered independently but their interaction is defined by their pairwise coupling and serial connection. Mercer and Crapo (22) showed that elastin and collagen fibers are interwoven especially in the alveolar entrance ring region, implying mechanical coupling of some kind. In fact, Mijailovich et al. (23, 24) considered this interaction to be responsible for the observed dynamic viscoelastic properties of lung parenchymal strips in a model based on sliding filaments. It may thus be that interaction along the lengths of the fibers is responsible for dynamic tissue properties, whereas interconnections between the two fiber matrices with fiber properties distributed in a fractallike manner are responsible for the static mechanics.
We note that for the large majority of the data (>95%) the distributions of Figs. 3 and 4 are nearly powerlaw in form, manifest in the nearly linear tails, whereN(l) is proportional to 1/l ^{2} andM(k) is proportional to 1/k. There are other examples of inverse power laws exhibited by the lung. For example, the stress response of lung tissue strips to a step change in strain exhibits powerlaw dependence with time. Also, when strips are mechanically cycled over a range of frequencies, the impedance amplitude is an inverse powerlaw function of the cycling frequency (3). The lung inflation process has also been found to follow power laws both as the magnitude of decreases in airway resistance measured from the terminal alveolar space and as the times between airway opening events (36). Complex systems with many varied elements interacting over a wide range of length scales tend to exhibit some measure or multiple measures in an inverse powerlaw sense of the form 1/f ^{ρ}, with 0 < ρ < 2, and fbeing usually a distributed frequency, amplitude, or spatial measure. In particular, the inverse powerlaw function often arises as the result of a cascade of smaller events that occur in a chain reaction (2, 41). We hypothesize, therefore, that the bulk stressstrain properties of lung tissue are due in large part to some kind of fractallike structure arising from biological selforganization. This is responsible for the 1/f character found for our collagen and elastin distributions, which is a manifestation of general complexity.
In each of our models, the parameter of the element in parallel with the distributed parameter is identical for the entire model. That is, for the distributed stop length model the stiffnesses are fixed, whereas in the distributed stiffness model the stop lengths are fixed. This is clearly not realistic considering natural physiological variability in real tissue. We thus wondered how robust our models are to variability in the fixed parameters of each model. We therefore generated an ldistributed model as before (Eq. 18 ) but with the stiffnesses chosen randomly from uniform distributions having meank chosen as in Eq.19 and with ranges from 0 to 100% of the mean. We found that the stressstrain curve changed only slightly, showing visible weakening from the mean tissue curve for ranges >50% of the mean (Fig. 8, longdashed lines). Similarly, we generated a stiffness distributed model as before according to Eq. 21 but with uniformly distributed l of the same mean as the fixed l model and with increasing ranges from 0 to 100% of the mean. Again, this caused little change in the stressstrain curve, with some stiffening observed at ranges >50% of the mean (Fig. 8, dotted lines).
We see therefore that the model is robust to the introduction of variability in the fixed parameters. Nevertheless, it is obvious that both fiber types will be distributed in real tissue, as indicated by the data of Matsuda et al. (19) and Sobin et al. (33). We presumed that the forms of the collagen and elastin distributions are those given inEqs. 18 and 21 . However, each of these distributions accounts for the entire stressstrain characteristics independently. Thus, to invoke both simultaneously, we softened their individual contributions by doubling all stop lengths and halving all stiffnesses. This produces a stressstrain curve that matches the mean experimental curve fairly well, dissociating only at high strains (Fig.5, dotteddashed line).
Lung Tissue Diseases
It is interesting to try to apply the foregoing to the behavior of the whole lung in vivo. In analogy to the tissue model developed above, we consider the lung as a series of elastic volume elements (elastance E), each of which extends to a maximum stop volume (ν). In doing so, we presume that the nature of the tissue expansion in three dimensions is not unlike that in tissue strips. That is, elastin fibers stretch elastically under the applied load whether uniaxially or from lung inflation, whereas collagen fibers straighten until taut. Thus in no way does a stop volume unit correspond to a welldefined physical structure such as an alveolus. It merely represents the action of collagen fiber straightening within the lung tissue in terms of volume units. Indeed, Mercer and Crapo (21) showed using serial reconstruction that the pressurevolume characteristics of alveoli are much the same as for the whole lung. With our model, we explore how the mechanism of mechanical stops in parallel with elastic springs can describe the deflation limb of PV curves and how changes in PV curves in disease are reflected in changes in the model constituents, the elastin and collagen fiber matrices. In particular, we examine how changes in static deflation PV curves may be related to changes in the underlying tissue structure with tissue diseases such as emphysema and fibrosis.
We generate static elastic PV curves by having both stop volumes and unit elastances distributed according to Eqs.7
and
14
with F,L, k, and l substituted for P, V, E and ν, respectively, where V is lung volume and P is transpulmonary pressure. The equation most commonly used for the static PV curve is that first introduced by Salazar and Knowles (28)
Equations 27 and 28 represent the stop volume distribution model, whereas Eqs. 29 and 30 represent a distribution of elastances. It is likely that the collagen fiber matrix of lung tissue and the elastic constitutive elements have distributed stop volume and stiffness properties, respectively. We therefore recognize that the two models represent the extremes of possibility, and so consider these two extremes to quantitatively assess how changes in PV curves may reflect pathological changes in lung tissue.
We apply our model Eqs.27 30 by using the values for the parameters of the Salazar and Knowles (28) equation (Eq. 26 ) obtained from the PV curves of representative emphysematous and fibrotic lungs (Table2). The changes of the model parameters expressed as a percentage of normal are shown in Table3. For emphysema, τ and ν are elevated, whereas E and γ are decreased below normal. The stop volume distribution for an emphysematous lung is thus shifted toward large stop volumes compared with normal (Fig.9). This is consistent with the nature of the disease, which consists of destruction of tissue with associated remodeling of both elastin and collagen networks as well as an increase in airspace volumes (12, 43). By contrast, in fibrosis, the changes in the tissue involve widespread inflammation, an increase in the concentration of interstitial fiber constituents, and air spaces are reduced or even filled in with inflammatory products (8). Correspondingly for fibrosis, parameters τ and ν are greatly decreased below normal while E and γ are elevated, which results in reduction in vital capacity and a stiffening of the tissue. Note that τ and ν have units of volume (and are in fact proportional toB), and thus control the vital capacity of the model, whereas E and τ have units of elastance and thus, with τ and ν, affect the shape of the PV curve.
A possible mechanism for the dominance of changes in elastancerelated properties over changes in volumerelated properties via our model is shown with the aid of Fig. 10. Here we consider the collagen and elastin fibers to be composed of the parallel contribution of multiple fibrils. The stiffness of a single unit is equal to the sum of the stiffnesses of all the individual elastin fibrils in that unit, whereas the stop length of the unit is the minimum collagen fibril length. A decrease in the number of elastin fibrils would manifest itself as a proportional decrease in the stiffness of the unit, whereas an increase in stop length would occur only if the shortest fibril were removed. The situation is similar for the addition of collagen or elastin fibrils: the elastance increases in proportion to the number of added elastin fibrils, whereas the stop length shortens only if a collagen fibril shorter than the shortest is added. Thus the elastanceassociated parameters are more likely to be affected than volumeassociated parameters under conditions that randomly affect both collagen and elastin, through tissue destruction (emphysema) or tissue building (fibrosis). Possible support for the model of Fig. 10 was demonstrated by Mercer and Crapo (22), who observed parallel collagen fibrils meandering in a wavelike fashion. They measured the maximum possible degree of fiber extension to be as much as 16.1 ± 3.2% (SE, n = 6) near the alveolar septal edge in human lungs. This is comparable in magnitude to the changes expected to occur between functional residual capacity and total lung capacity (21).
A significant result of the application of the model to the PV curves of diseased lungs is that it may provide a mechanistic description for the origin of the shape parameter K(from Eq. 26 ) in terms of the structural constituents of lung tissue and their organization. The parameter K has been found to be increased in those patients with emphysema (12, 26) and to be decreased in fibrosing alveolitis (38), as shown in Table 2. Within the context of our model, these changes in K are interpreted in terms of alterations to the collagen and elastin networks (Table 3). In fact, K is inversely proportional to the stiffness arising from the elastinrelated parameters E and τ and inversely proportional to ν and τ, which define the maximum lung vital capacity through the distension of collagen. We know of no other quantitative model that addresses how changes in tissue constituents affect the shape of the PV curve.
However, we did not explicitly include the effects of surface tension in the model, and it may be that changes in surface tension in these diseases may contribute to changes inK. Surface tension contributes a significant portion to the elastic recoil of the lung clearly shown by the difference between PV curves for saline and airfilled lungs (32,35). Furthermore, Haber et al. (13) found thatL _{m} correlated well with K in airfilled lungs but was unrelated to K determined from salinefilled lungs, suggesting that changes in airspace size were related to surface tension effects rather than tissue properties. However, we note as per the analysis of Hoppin and Hildebrandt (15) that surface tension properties are mechanically coupled. Removal of surface tension forces (e.g., by saline filling) alters alveolar wall geometry by changing tissue distention and thus affects the recoil attributable to the tissues.
The shapes of PV or lengthtension curves can be altered by digestion of tissue constituents by proteases (16, 29). Karlinsky et al. (16) found that the chord compliance was increased in isolated hamster lungs exposed to elastase, but no change in vital capacity occurred. By contrast, collagenase caused an increase in vital capacity and a change in the shape of the PV curve. These observations are consistent with our model. However, when the measurements were done in vivo by the same authors, there was an increase in vital capacity observed in the elastasetreated lungs. This contradicts the theory that the collagen matrix is responsible for limiting the maximum extension. They attributed the difference between the in vitro and in vivo results to the repair process after elastase injury, which may disrupt the collagen matrix. Sata et al. (29) found that the stressstrain curves of microtissue specimens from hamster parenchyma could be described by two exponential functions, where the exponent of the first exponential was primarily responsible for the lowstrain portion of the curve, whereas the upper portion of the curve was due to the exponent of the second exponential. They found that elastase treatment affected only the first exponent, whereas collagenase affected only the second component, which is in agreement with our model.
In any case, we do not claim that our model accounts for all aspects of the lung PV curve but merely that it embodies the important contribution from the tissues themselves whose significance for lung recoil is well accepted. Indeed, Wilson et al. (42) invoked our mechanism to explain the regional mechanical heterogeneity they observed during expirations from total lung capacity in dogs when using parenchymal markers. Such regional variation is predicted by our model, as different lung regions would have different samplings of the underlying collagen and elastin fiber distributions and thus would follow different local PV curves.
Concluding Comments
We have introduced a model that accounts for the nonlinear stressstrain curve of lung tissue based on the fundamental differences in the structural constituent fibers. A mechanistic description is developed that quantitatively defines the relative roles of collagen and elastin fibers. The model predicts that the distributions for the collagen and elastin fibers are 1/flike for the large part of the data. We hypothesize that this reflects a fractal kind of organization of the tissue constituents. We further speculate that by extending the model to the whole lung in vivo, important structuretofunction links are made, which contrast the roles of collagen and elastin in the tissue diseases of emphysema and fibrosis. The predicted changes in the tissue constituent properties, such as changes in airspace size and tissue elasticity in these diseases, are matched by the changes in the model parameters. It is likely that Kfrom the Salazar and Knowles study (Eq.26 ; Ref. 28) may have both surface tension and tissue origins. This model provides a possible mechanistic link betweenK and both the collagen and elastin tissue properties.
Acknowledgments
This work was supported by the Natural Sciences and Engineering Research Council of Canada, Fonds pour la Formation de Chercheurs et l’Aide à la Recherche (FCAR), the Medical Research Council of Canada, and the J. T. Costello Memorial Research Fund. J. H. T. Bates is a Chercheurboursier of the Fonds de la Recherche en Santé du Québec. G. N. Maksym is supported by FCAR.
Footnotes

Address for reprint requests: J. H. T. Bates, MeakinsChristie Laboratories, McGill Univ., 3626 St. Urbain St., Montreal, Quebec, Canada H2X 2P2 (Email: jason{at}meakins.lan.mcgill.ca).
 Copyright © 1997 the American Physiological Society