## Abstract

Harmonic distortion (HD) is a simple approach to analyze lung tissue nonlinear phenomena. This study aimed to characterize frequency-dependent behavior of HD at several amplitudes in lung tissue strips from healthy rats and its influence on the parameters of linear analysis. Lung strips (*n* = 17) were subjected to sinusoidal deformation at three different strain amplitudes (Δε) and fixed operational stress (12 hPa) among various frequencies, between 0.03 and 3 Hz. Input HD was <2% in all cases. The main findings in our study can be summarized as follows: *1*) harmonic distortion of stress (HD) showed a positive frequency and amplitude dependence following a power law with frequency; *2*) HD correlated significantly with the frequency response of dynamic elastance, seeming to converge to a limited range at an extrapolated point where HD=0; *3*) the relationship between tissue damping (G) and HD_{ω=1} (the harmonic distortion at ω=1 rad/s) was linear and accounted for a large part of the interindividual variability of G; *4*) hysteresivity depended linearly on κ (the power law exponent of HD with ω); and *5*) the error of the constant phase model could be corrected by taking into account the frequency dependence of harmonic distortion. We concluded that tissue elasticity and tissue damping are coupled at the level of the stress-bearing element and to the mechanisms underlying dynamic nonlinearity of lung tissue.

- nonlinear elastance
- tissue damping
- hysteresivity

dynamic lung tissue mechanics has been described as nonlinear, based on its prominent amplitude- and frequency-dependent behavior. Indeed, the relationship between stress and strain is nonlinear, even over the range of physiological deformations (12, 16, 20, 23). Nonlinear features of lung dynamics arise largely from elastic nonlinearities in lung tissue. Previous studies provide evidence of volume and pressure dependence of lung elastic modulus and viscoelasticity (5, 11, 20, 23).

Current analyses of lung mechanics based on complex Young modulus do not account for nonlinear conditions (5, 6, 26, 28). Although awake and paralyzed subjects show marked frequency dependence of linear dynamic elastance in the range of physiological frequencies (0.25–3 Hz; Refs. 8, 19), frequency dependence of elastic nonlinearity has received much less attention. The study of harmonic distortion of output and input signals of lung parenchyma strips submitted to sinusoidal deformation constitutes a simple and model-independent method to assess the degree of nonlinearity in a nondimensional way, (21, 28, 31, 35). Furthermore, the harmonic distortion approach avoids the intrinsic complexity of other nonlinear models (17, 18, 20, 31, 34, 35).

Studies show that harmonic distortion (HD) can be used to determine nonlinear lung behavior as a function of strain amplitude and operational stress (14, 21, 33). However, no studies have been performed to determine the effects of frequency on harmonic distortion of lung tissue submitted to sinusoidal input.

In line with the hypothesis that nonlinear behavior of lung tissue could be a determinant factor of elastance frequency dependence, the aim of the present work was to study changes in HD of rat lung tissue strips in relation to different oscillation frequencies at three strain amplitudes and to investigate the influence of this nonlinear behavior on the linear parameters of lung mechanics, studied according to the constant phase model (9) and the structural damping hypothesis (6). The study was carried out on lung parenchyma strips to avoid the influence of the air-liquid interface and alveolar recruitment-derecruitment phenomena. Strips were submitted to sinusoidal oscillations over a range of frequencies at three strain amplitudes. Tissue nonlinearity was accounted for by harmonic distortion.

## MATERIALS AND METHODS

All experiments were carried out in accordance with the current legislation on animal experiments in the European Union and approved by our institutional committee for animal care and research (Animal Experimentation Ethics Committee, University of Barcelona).

#### Animals and experimental procedures.

Male Sprague-Dawley rats (250–350 g) were obtained from Harlan Ibérica S.L. (Sant Feliu de Codines, Spain). Lungs were excised and stored in Krebs-Henseleith (K-H) solution as described in a previous paper (21).

A total of 17 subpleural strips (2 cm × 0.2 cm × 0.2 cm) were cut from the right or left lungs and suspended vertically in a K-H organ bath, maintained at 37°C and continuously bubbled with 95% O_{2}-5% CO_{2}. Then, strip volume (vol) and basal force were determined according to Pinart et al. (21).

Once basal force and displacement signals were adjusted, the length between strip edges (*L*_{10}) was measured by a precision caliper. Instantaneous length (*L _{i}*) during oscillation around

*L*

_{10}was determined by adding

*L*

_{10}value to the displacement value measured at any time. Instantaneous average cross-section area (

*A*) was determined as

_{i}*A*= vol/

_{i}*L*(cm

_{i}^{2}). Instantaneous stress (σ

*) was calculated by dividing force (*

_{i}*g*) by

*A*(cm

_{i}^{2}). Instantaneous strain was calculated as Δε

*= (*

_{i}*L*−

_{i}*L*

_{10})/

*L*

_{10,}and expressed in percent. Neither amplitude dependence (<0.1% change in stiffness) nor phase changes with frequency were detected in the range from 0.01 to 14 Hz. The hysteresivity of the system (<0.003) was independent of frequency.

#### Biomechanical study.

After preconditioning procedure (21), samples were submitted to sinusoidal deformation at fixed operating stress (σ_{op} = 12 hPa) under three different strain amplitudes (Δε = 10%, 15%, and 20%) along various frequencies. In *variation A*, samples were studied at Δε = 10% along 13 frequencies (*f* = 0.03, 0.05, 0.1, 0.15, 0.2, 0.3, 0.5, 0.7, 1, 1.5, 2, 2.5, and 3 Hz). In *variations B* and *C*, samples were oscillated at Δε = 15% and 20%, respectively, along five frequencies (*f* = 0.03, 0.1, 0.5, 1, and 3 Hz). Frequencies were applied sequentially every 2 min. Sampling frequency and time were adjusted to obtain five complete cycles of 1,000 points/cycle. *Variations A, B*, and C were applied randomly. Operating stress (σ_{op}) was defined as the average stress for a given number of entire cycles. Strain amplitude (Δε) refers to peak-to-peak amplitude. The operating limits of stress were 2–25 hPa. Therefore, changes in Δε were constrained to these limits. Force and displacement signals were amplified, filtered at a corner frequency of 40 Hz, digitally converted, and recorded. Because all variations were applied through the input signal, instantaneous length of the sample was continuously monitored. Total duration of the experiment, from the beginning of preconditioning, was ∼90 min.

#### Measurement of harmonic distortion.

When a sinusoidal signal is passed through a nonlinear system, the number of harmonics of the output signal and their amplitudes depend on the degree of system nonlinearity. Therefore, the degree of nonlinearity in the output signal can be characterized by the distortion value. In general, input (strain) consists of a single fundamental sinusoid and output (stress) signal consists of a fundamental component with frequency, together with a series of higher harmonics at frequencies that are integer multiples of the fundamental frequency (28, 31).

HD of the output (stress) signal was calculated as described before (21). Briefly, HD was obtained by the discrete Fourier transform as the square root of the ratio of spectral powers sum of the ten first harmonics above the fundamental harmonic to the amplitude of the fundamental frequency (*A _{1}*), expressed as a percentage (28, 31, 35). The following equation yields HD,

*A*

_{i}is the amplitude of the harmonics. Only experiments with HD of the instrumental setting <2% were considered. Because measurement noise (background noise) can also produce power at noninput frequency in the measured output in addition to that purely due to nonlinear distortion, HD% values were corrected for background noise by linear interpolation, as described by Zhang et al. (35).

#### Linear data analysis.

Analysis and parameter estimation were performed by specific software elaborated with LabVIEW 7.0 (National Instruments, Austin, TX), as previously described (24). Briefly, the frequency response was computed from the time-domain strain (stimulus) and the time-domain stress (response), according to the transform function,
_{0} is the complex elastic or dynamic Young's modulus at the fundamental frequency; ℑ indicates the calculation of the Fourier Transform by fast Fourier Transform; σ, stress; ε, strain; and H denotes the application of the Hanning window. This result was transformed into single-sided magnitude |Ψ| and phase (ϕ) and computed as real and imaginary parts of ψ_{0}, which correspond to storage (E) and loss (G) moduli, respectively. For values of |Ψ| and ϕ corresponding to the fundamental angular frequency (ω=2πf), dynamic tissue elastance (E), tissue resistance (R), and hysteresivity (η) were calculated according to:

The following linear equation describes the dynamic complex Young modulus (Ψ) at any fundamental frequency (ω):

#### Statistical analysis.

SPSS 12.0 statistical software package was used. Data are expressed as mean ± SD unless otherwise specified. One-way ANOVA for repeated measures was used to make comparisons among different experimental conditions. Least square regressions were used (*r* = Pearson's coefficient of correlation). In all instances the significance level was set at 5% (=0.05).

## RESULTS

Figure 1 shows the percent of HD of output (stress) along frequencies at different strain amplitudes (Δε = 10%, 15%, and 20%) and fixed operating stress (σ_{op} = 12 hPa) in linear (Fig. 1, *top left*) and logarithmic (Fig. 1, *top right*) plots. Data display a similar frequency-dependence pattern of HD at the three strain amplitudes. HD of the instrumental setting (not shown) was consistently stable and virtually negligible (under 2%), indicating a good quality of the sinusoidal input signal and avoiding linear correction bias. HD showed significant power dependence on frequency that can be identified as
_{ω=1} is stress harmonic distortion at ω = 1 rad/s, and κ is the power law index of this relationship. Parameters defining frequency-dependence of HD, HD_{ω=1} and κ, showed significant but opposite dependence on strain amplitude, HD_{ω=1} increases while κ decreases with Δε (Fig. 1, *bottom*).

To identify a potential influence of nonlinearity on frequency-dependent behavior of tissue elastance, HD was used as independent variable to analyze the correlation between HD and E among frequencies. Figure 2 shows the linear relationship between E(ω) and HD(ω) for each sample at all frequencies under the three strain amplitudes. A positive correlation was observed in all cases. It is worthy to note that regression slope (B) decreased significantly as strain amplitude increased (Fig. 2, *bottom right*). Interestingly, extrapolation of the individual regression lines to zero HD (not shown in the figure) tends to converge to an elastance value inside relatively narrow limits, between 30 and 60 hPa (dashed lines have been fitted by eye to the edges of the scatterplot, following individual trends). Consequently, for a particular amplitude and operating stress, the linear relationship between tissue elastance and HD as a function of frequency [E(ω) and HD(ω), respectively] may be expressed as:
_{R} represents nonlinear-dependent elastance. E_{0} (extrapolated elastance at HD=0) showed nonsignificant amplitude-dependence (Fig. 2, *bottom left*), having an average value of 42.96 ± 16.23 hPa (mean ± SD). Consequently, HD brings in both the frequency and amplitude dependence of stiffness. The combination of the relationships in *Eqs. 3* and *4* allows describing E_{R} as a function of κ (the power law exponent of HDσ vs. frequency):
_{R,ω=1} is the value of E_{R} at ω=1 rad/s. Slope B is defined as the ratio ΔE_{R}/ΔHD and presents significant Δε dependence, as shown in Fig. 2 (*bottom right*). As can be deduced from Fig. 2, slope B was linearly related to E_{R,ω=1} at the three levels of strain amplitude: 10% (*r* = 0.828), 15% (*r* = 0.821), and 20% (*r* = 0.928).

Lung tissue damping (G) is defined as the product of tissue resistance (R) times angular frequency (ω) or dynamic elastance (E) times hysteresivity, according to *Eq. 2*. Tissue damping showed low variation with frequency [the coefficient of variation in our data was 7.43 ± 3.49% (mean ± SD)]. Figure 3 shows the relationship between the average value of G and the amount of nonlinearity at ω = 1 (HD_{ω=1}) for each strain amplitude.

Hysteresivity (η = R_{ω=1}/E_{ω=1}) was not related to the absolute amount of nonlinearity, but to the rate of change of the logarithm of HD with the logarithm of frequency or κ. Figure 4 shows the correlation between η and κ for all subjects at all Δε. The slope of the regression of η as a linear function of κ was not significantly different from 1 (0.985 ± 0.094, mean ± SD) and the intercept was 0.035 ± 0.0037 (mean ± SD). Therefore, this relationship can be roughly written as η = κ + 0.035.

## DISCUSSION

Lung tissue shows nonlinear behavior even under physiological conditions. Although linear models for lung tissue analysis have been largely developed, nonlinear approaches are less available and usually complex. The present study characterized the frequency-dependent behavior of harmonic distortion of stress (HD), a simple approach to nonlinear phenomena, in lung parenchyma strips of healthy rats. Instrumental (or input) harmonic distortion was consistently stable and small (under 2%) in all experimental conditions, indicating a high quality input signal, thus avoiding bias on the estimation of system nonlinearity. The main findings in our study can be summarized as follows: *1*) HD showed a positive frequency and amplitude dependence following a power law with frequency (Fig. 1); *2*) HD correlated linearly with the frequency response of dynamic elastance, and the regression seemed to converge to a limited range at an extrapolated point where HD = 0 (Fig. 2); *3*) the relationship between tissue damping and HD_{ω=1} was linear and partially explained the interindividual variability of damping (Fig. 3); and *4*) hysteresivity depended linearly on κ (the power law exponent of HD with ω; Fig. 4).

#### Frequency dependence of lung tissue nonlinearity.

We have found that HD increased with frequency following a power law. Two parameters described the frequency dependence of HD: the value of HD at angular frequency of 1 rad/s (HD_{ω=1}), related to the absolute level of nonlinearity; and the power function exponent κ, reflecting the frequency dependence of HD. HD is phenomenological and, therefore, cannot be interpreted in terms of specific structural mechanisms. However, as a measurement of the degree of lung tissue nonlinearity, it can be interpreted in the light of theories accounting for lung tissue nonlinear phenomena. The positive amplitude-dependence depicted by HD_{ω=1} is in agreement with our previous hypothesis that strain stiffening of lung tissue occurs largely through the progressive recruitment of viscoelastic elements (mainly collagen fibers) to bear stress under increasing strain (21). Let us, therefore, consider a recruitment-based nonlinear model (22) to explain our results, and especially the meaning of κ. This model has the advantage of not presupposing any specific structural composition of the material. The model assumes that, in dynamic conditions, the behavior of the Stress Bearing Element (SBE) in lung tissue corresponds to the cumulative effect of the behavior of a recruiting material (F) acting in parallel with a recruited module (M). For our purposes, we will assume each single element as well as M to be standard linear solids (SLS; Fig. 5) also known as Kelvin bodies. In this model, every single unitary element has linear viscoelastic properties and is recruited when the stretch reaches a threshold level. When, after preconditioning, the strip is stretched to obtain the initial conditions (in this case an operational stress of 12 hPa), a certain number (N_{0}) of elements have been recruited into M. If the strip is then submitted to a sinusoidal deformation around the operating length, recruitment and derecruitment of individual elements in F occurs and, consequently, the stress-strain relationship at SBE level changes continuously, inducing nonlinear behavior. If no recruitment occurs, the system would be linear. Recruitment threshold for a given element is expected to depend on *1*) intrinsic factors that define the state of the element (such as spatial orientation, configuration, mechanical properties) and *2*) the interaction of the element with its environment and with neighboring elements.

As frequency increases, the recruited module (M), having the properties of an SLS, characteristically decreases resistance and increases elastance with frequency. Concurrently, the participation of recruiting elements in the elastic behavior will also increase with frequency, thereby increasing SBE nonlinearity. Furthermore, because of frictional interactions, time also becomes a factor to be considered in the broadening of recruitment threshold and time constants distribution, and this can influence the rate of recruitment with frequency. According to these considerations, the combination of deterministic order and arbitrary disorder in the molecular structure, as well as the interaction of collagen fibers with other matrix molecules and ground substance, may contribute to the existence of a broad distribution of recruitment threshold and time constants in the extracellular matrix (4, 28). Therefore, the κ parameter will be related to SBE viscoelastic behavior, but also to the ratio of recruitment related to the interactions within and between the elements and their environment (30). This conceptual approach helps to explain some of our findings without entering into a discussion of the suitability of the model or its structural correlates. In fact, a similar rationale could be applied to several recruitment-based models (2, 18, 22).

The consistent change of HD with frequency, following a power law, could be associated, for instance, to the microstructural distribution of fiber thickness, skewed with a long “tail” similar to a power law (25, 27). Power law can be related to general theories associating this particular behavior to intrinsic complexity of bodies (15, 30), fractal self-similarity (7), or self-organized criticality (1–3). This appears to be another argument in favor of interpretation of the κ parameter as an expression of the intrinsic complexity of lung tissue and, in particular, of the connective extracellular matrix.

#### Frequency dependence of nonlinearity and structural damping hypothesis.

In 1989, Fredberg and Stamenovic (6) presented the hypothesis that dissipative and elastic processes within lung tissue are coupled at the level of the stress bearing element, and introduced hysteresivity (η), an intensive property of the tissue, defined as the ratio of dissipated energy to stored energy (energetic efficiency). Hysteresivity is a material property at the macroscopic level that depends on tissue composition and microstructure, being frequency independent and showing low variability both interindividually and across species. This hypothesis, known as structural damping hypothesis, implies that energy dissipation is always a constant fraction of the elastic energy stored. Consequently, the elastically deformed element is also responsible for the energy dissipated, and loss modulus is a constant fraction of the elastic or storage modulus. The linear relationship between κ and η.

In our data, η showed an interindividual coefficient of variation of 17%, for all three Δε conditions together (η = 0.073 ± 0.012, mean ± SD, *N* = 51, Fig. 4). This variability, similar to that observed in previous works (13, 21, 32), is usually neglected. According to our data, nonlinearity accounts for a large part of the interindividual variability of η. Therefore, hysteresivity and κ are substantially related magnitudes, both of them linked to the intimate structure of the tissue and its mechanical efficiency. We have already hypothesized that κ could reflect not only the viscoelastic behavior but also the intrinsic complexity of the SBE. The relationship between both parameters can therefore be interpreted as a reasonable relationship between the structural inhomogeneity and complexity of the extracellular matrix and its mechanical efficiency (ratio of dissipated to stored energy). Our results also suggest that, in the hypothetical absence of frequency-dependent nonlinearity (if κ = 0, then HD would be frequency invariant), lung tissue hysteresivity would obtain a minimal value of 0.035 under the present experimental conditions.

#### Nonlinearity, tissue stiffness, and oscillation frequency.

According to the model in Fig. 5, the elastic modulus defined by extrapolating the linear relationship between dynamic elastance and HD to HD = 0 (Fig. 2), or E_{0}, would represent the elastic modulus of the recruited module (M) at initial conditions. We have hypothesized that the recruited module (M) would have virtually a linear behavior defined, according to the structural damping hypothesis (6), by Young's complex modulus (Ψ_{M})
_{0} would correspond to the linear elastic modulus, η the hysteresivity, and j =

According to the model in Fig. 5, and in agreement with our previous results (21), when stretching the tissue at a given rate dynamic elastance results from the addition of E_{0} and the instantaneous change of stiffness under the effect of recruitment. When frequency is progressively increased, the change in elastance (E_{R}) depends on the viscoelastic response of M and the progressive recruitment with frequency under the effect of threshold or time constants distribution in F. Indeed, as can be inferred from the linear relationship between HD and E represented in Fig. 2, dynamic elastance is the addition of initial elastance E_{0} and recruited elastance E_{R}, as expressed in *Eq. 4*. In agreement with the recruitment hypothesis illustrated in Fig. 5, both E_{0} and E_{R} are dynamically linked to the stress-bearing element.

It is perhaps worthy to note that the average value of the linear elastic modulus (≈43 hPa) is narrowly similar to the average Young's modulus of a single alveolar wall estimated by Cavalcante et al. (4) of ∼50 hPa in an hexagonal network model of nonlinear elasticity submitted to 2D deformations. Amplitude independence of linear (chord) elastance was observed by Navajas et al. (20). Although the physiological significance of both parameters is probably different, we observed a significant linear relationship between E_{0} and chord elastance at ω=3.14 rad/s (R = 0.491; *N* = 51; *P* < 0.001).

#### Harmonic distortion and linear models: the constant phase model.

Frequency dependence of dynamic elastance and tissue resistance follow a power law. We usually express these relationships in the following way:

Accordingly, the model below describes the linear behavior of lung tissue stress/strain relationship of the complex dynamic Young's modulus (Ψ) as a function of angular frequency (ω = 2πf):

For *Eq. 8* to be equivalent to the constant phase model, proposed by Hantos et al. (9, 10), the sum of coefficients α and β has to be equal to the unit and, consequently, hysteresivity has to be strictly independent of frequency. Therefore, the expression 1 − (α+β) will reflect the amount of fitting error in the constant phase hypothesis but also the agreement with the structural damping hypothesis. To correct for deviations of this model, a constant Newtonian tissue resistance has sometimes been included in the model (24, 32), but the correction is not complete, and in normal nonconstricted lung strips, the value of this Newtonian resistance is usually zero (21, 24).

As the α parameter is related to η according to the following equation (30),
*Eqs. 4* and *6*
*Eq. 8*, *top* plot) and α′ (by accounting for nonlinearity, *Eq. 11*, *bottom* plot) and 1-β. These relationships support the hypothesis that most of the error of the constant phase model is attributable to nonlinearity and can be corrected by taking into account HD.

Again, experimental results seem to agree with the hypothesis that not truly linear system can be observed in biological tissues. However, linear approach can be useful for small deformations, as HD decreases with amplitude of strain (21, and Fig. 1), in agreement with the classical statement in lung mechanics. In fact, much of the linear analysis has been performed using composite waveforms containing many frequencies. In this case, the energy per frequency line is surely much lower than when using sine waves at a single frequency. Even if the system is intrinsically nonlinear, perhaps for low strain amplitudes the error of using a linear model could be much lower than the tolerance. We offered a way to identify the error and eventually to correct it by taking into account nonlinearity.

#### Conclusion.

We interpreted HD with the conceptual help of a phenomenological model that does not presuppose any specific structural or architectural arrangement of extracellular matrix constituents. Therefore, the same rationale can be applied to several other models in which recruitment is the basis of the nonlinear behavior of dynamic stress-strain relationships. We have introduced the parameter κ that defines the frequency dependence of harmonic distortion and is narrowly related to hysteresivity and to the error of the constant phase hypothesis. Parameter κ reflects an intrinsic property of the stress bearing element, related to its nonlinear behavior.

It has not been our purpose to validate a particular model of tissue nonlinearity [the limitations of the model in Fig. 5 have been discussed previously (22)] but to formulate the hypothesis that nonlinearity, tissue elasticity, and tissue damping are related phenomena, intrinsically linked to the behavior of the stress-bearing element in the extracellular matrix (mainly collagen fibers), and that recruitment can account for tissue nonlinearity, as has been previously argued (2, 18, 22). We conclude that tissue elasticity and tissue damping, coupled at the level of the stress bearing element, are also coupled to the mechanisms underlying tissue nonlinear behavior and its frequency dependence through the κ parameter. Moreover, the error in the application of the linear constant phase model is related to the structural dynamic nonlinearity of the tissue.

## GRANTS

This study was supported by Fondo de Investigaciones Sanitarias de la Seguridad Social (FISS; PI 04/0671), Spain. D. S. Faffe is supported by the National Council for Scientific and Technological Development (CNPq), Brazil.

## ACKNOWLEDGMENTS

We express our gratitude to B. Johnson for skilful editing assistance. We acknowledge the scientific editorial board of *Journal of Applied Physiology* and particularly the referees for help in improving the quality of the manuscript.

- Copyright © 2011 the American Physiological Society