Dynamic nonlinearity of lung tissue: frequency dependence and harmonic distortion

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.     

Nonlinearity and harmonic distortion of dog lungs measured by low-frequency forced oscillations

The nonlinearity of lung tissues and airways was studied in six anesthetized and paralyzed open-chest dogs by means of 0.1-Hz sinusoidal volume forcing at mean transpulmonary pressures (Ptp) of 5 and 10 cmH2O. Lung resistance (RL) and elastance (EL) were determined in a 32-fold range (15-460 ml) of tidal volume (VT), both by means of spectrum analysis at the fundamental frequency and with conventional time-domain techniques. Alveolar capsules were used to separate the tissue and airway properties. A very small amplitude dependence was found: with increasing VT, the frequency-domain estimates of RL decreased by 5.3 and 14%, whereas EL decreased by 20 and 22% at Ptp = 5 and 10 cmH2O, respectively. The VT dependences of the time-domain estimates of RL were higher: 10.5 and 20% at Ptp = 5 and 10 cmH2O, respectively, whereas EL remained the same. The airway resistance increased moderately with flow amplitude and was smaller at the higher Ptp level. Analysis of the harmonic distortions of airway opening pressure and the alveolar pressures indicated that nonlinear harmonic production is moderate even at the highest VT and that VT dependence is homogeneous throughout the tissues. In three other dogs it was demonstrated that VT dependences of RL and EL were similar in situ and in isolated lungs at both Ptp levels.     

Dynamic response of the isolated passive rat diaphragm strip.

To further our understanding of the mechanisms underlying chest wall mechanics, we investigated the dynamic response of the isolated passive rat diaphragm strip. Stress adaptation of the tissue was measured from 0.05 to 60 s after subjecting the strips to strain steps of normalized strain amplitudes from 0.005 to 0.04. The tissue resistance (R), elastance (E), and hysteresivity (eta) were measured in the same range of amplitudes by sinusoidally straining the strip at frequencies from 0.03125 to 10 Hz. The stress (T) depended exponentially on the strain (epsilon) and relaxed and recovered linearly with the logarithm of time. E increased linearly with the logarithm of frequency and decreased with increasing amplitude. R fell hyperbolically with frequency and showed an amplitude dependence similar to that of E. To interpret the strong nonlinear behavior, we extended the viscoelastic model of Hildebrandt (J. Appl. Physiol. 28: 365-372, 1970) to include an exponential stress-strain relationship. Accordingly, the step response was described by T - Tr = Tr(e alpha delta epsilon - 1)(1 - gamma log t), where delta epsilon is the strain amplitude, Tr is the initial operating stress, alpha is a measure of the stress-strain nonlinearity, and gamma is the rate of stress adaptation. The oscillatory response of the model was computed by applying Fung's quasi-linear viscoelastic theory. This quasi-linear viscoelastic model fitted the step and oscillatory data fairly well but only if alpha depended negatively on delta epsilon, as might be expected in a plastic material.     

Harmonics of outer hair cell motility.

The voltage-dependent mechanical activity of outer hair cells (OHC) from the organ of Corti is considered responsible for the peripheral auditory system's enhanced ability to detect and analyze sound. Nonlinear processes within the inner ear are presumed to be characteristic of this enhancement process. Harmonic distortion in the OHC mechanical response was analyzed under whole-cell voltage clamp. It is shown that the OHC produces DC, fundamental and second harmonic length changes in response to sinusoidal transmembrane voltage stimulation. Mechanical second harmonic distortion decreases with frequency, whereas the predicted transmembrane second harmonic voltage increases with frequency. Furthermore, the phase of the second harmonic distortion does not correspond to the phase of the predicted transmembrane voltage. In contradistinction, it has been previously shown (Santos-Sacchi, J. 1992. Neuroscience. 12:1906-1916) that fundamental voltage and evoked mechanical responses share magnitude and phase characteristics. OHC length changes are modeled as resulting from voltage-dependent cell surface area changes. The model suggests that the observed harmonic responses in the mechanical response are consistent with the nonlinearity of the voltage-to-length change (V-delta L) function. While these conclusions hold for the data obtained with the present voltage clamp protocol and help to understand the mechanism of OHC motility, modeling the electromechanical system of the OHC in the in vivo state indicates that the mechanical nonlinearity of the OHC contributes minimally to mechanical distortion. That is, in vivo, at moderate sound pressure levels and below, the dominant factor which contributes to nonlinearities of the OHC mechanical response resides within the nonlinear, voltage-generating, stereociliar transduction process.     

Viscoelastic and dynamic nonlinear properties of airway smooth muscle tissue: roles of mechanical force and the cytoskeleton

The viscoelastic and dynamic nonlinear properties of guinea pig tracheal smooth muscle tissues were investigated by measuring the storage (G') and loss (G") moduli using pseudorandom small-amplitude length oscillations between 0.12 and 3.5 Hz superimposed on static strains of either 10 or 20% of initial length. The G" and G' spectra were interpreted using a linear viscoelastic model incorporating damping (G) and stiffness (H), respectively. Both G and H were elevated following an increase in strain from 10 to 20%. There was no change in harmonic distortion (K(d)), an index of dynamic nonlinearity, between 10 and 20% strains. Application of methacholine at 10% strain significantly increased G and H while it decreased K(d). Cytochalasin D, isoproterenol, and HA-1077, a Rho-kinase inhibitor, significantly decreased both G and H but increased K(d). Following cytochalasin D, G, H, and K(d) were all elevated when mean strain increased from 10 to 20%. There were no changes in hysteresivity, G/H, under any condition. We conclude that not all aspects of the viscoelastic properties of tracheal smooth muscle strips are similar to those previously observed in cultured cells. We attribute these differences to the contribution of the extracellular matrix. Additionally, using a network model, we show that the dynamic nonlinear behavior, which has not been observed in cell culture, is associated with the state of the contractile stress and may derive from active polymerization within the cytoskeleton.     

Changes in viscoelastic properties of rat lung parenchymal strips with maturation.

The lung extracellular matrix changes rapidly with maturation. To further our understanding of the mechanisms underlying lung tissue mechanics, we studied age-related changes in mechanical properties in lung parenchymal strips from baby (10-15 days old), young ( approximately 3 wk old), and adult ( approximately 8 wk old) rats. Subpleural strips were cut and suspended in a fluid-filled organ bath. One end of the strip was attached to a force transducer and the other to a servo-controlled lever arm. Measurements of force (F) and length (L) were recorded during sinusoidal oscillations of various amplitudes and frequencies. Resistance modulus (R) and elastance modulus (E) were estimated by fitting the equation of motion to changes in stress (T) and stretch ratio (lambda). Hysteresivity (eta) was calculated as follows: eta = (R/E)2pif, where f is frequency. Slow-cycling T-lambda curves were measured by applying a constant slow length change. Finally, quasi-static T-lambda curves were measured as stress was increased from 0 to 6 kPa and back to 0 kPa in stepwise increments. Our results showed that lung tissue from immature rats was stiffer and less hysteretic than tissue from more mature animals. In addition, tissue from baby animals behaved in a manner compatible with an increased vulnerability to plastic change.     

Mechanics, nonlinearity, and failure strength of lung tissue in a mouse model of emphysema: possible role of collagen remodeling.

Enlargement of the respiratory air spaces is associated with the breakdown and reorganization of the connective tissue fiber network during the development of pulmonary emphysema. In this study, a mouse (C57BL/6) model of emphysema was developed by direct instillation of 1.2 IU of porcine pancreatic elastase (PPE) and compared with control mice treated with saline. The PPE treatment caused 95% alveolar enlargement (P = 0.001) associated with a 29% lower elastance along the quasi-static pressure-volume curves (P < 0.001). Respiratory mechanics were measured at several positive end-expiratory pressures in the closed-chest condition. The dynamic tissue elastance was 19% lower (P < 0.001), hysteresivity was 9% higher (P < 0.05), and harmonic distortion, a measure of collagen-related dynamic nonlinearity, was 33% higher in the PPE-treated group (P < 0.001). Whole lung hydroxyproline content, which represents the total collagen content, was 48% higher (P < 0.01), and alpha-elastin content was 13% lower (P = 0.16) in the PPE-treated group. There was no significant difference in airway resistance (P = 0.7). The failure stress at which isolated parenchymal tissues break during stretching was 40% lower in the PPE-treated mice (P = 0.002). These findings suggest that, after elastolytic injury, abnormal collagen remodeling may play a significant role in all aspects of lung functional changes and mechanical forces, leading to progressive emphysema.     

Elastic properties of the bronchial mucosa: epithelial unfolding and stretch in response to airway inflation.

The bronchial mucosa contributes to elastic properties of the airway wall and may influence the degree of airway expansion during lung inflation. In the deflated lung, folds in the epithelium and associated basement membrane progressively unfold on inflation. Whether the epithelium and basement membrane also distend on lung inflation at physiological pressures is uncertain. We assessed mucosal distensibility from strain-stress curves in mucosal strips and related this to epithelial length and folding. Mucosal strips were prepared from pig bronchi and cycled stepwise from a strain of 0 (their in situ length at 0 transmural pressure) to a strain of 0.5 (50% increase in length). Mucosal stress and epithelial length in situ were calculated from morphometric data in bronchial segments fixed at 5 and 25 cmH(2)O luminal pressure. Mucosal strips showed nonlinear strain-stress properties, but regions at high and low stress were close to linear. Stresses calculated in bronchial segments at 5 and 25 cmH(2)O fell in the low-stress region of the strain-stress curve. The epithelium of mucosal strips was deeply folded at low strains (0-0.15), which in bronchial segments equated to < or =10 cmH(2)O transmural pressure. Morphometric measurements in mucosal strips at greater strains (0.3-0.4) indicated that epithelial length increased by approximately 10%. Measurements in bronchial segments indicated that epithelial length increased approximately 25% between 5 and 25 cmH(2)O. Our findings suggest that, at airway pressures <10 cmH(2)O, airway expansion is due primarily to epithelial unfolding but at higher pressures the epithelium also distends.     

Increasing strain and strain rate strengthen transient stiffness but weaken the response to subsequent compression for articular cartilage in unconfined compression

Strain amplitude and strain rate dependent nonlinear behavior and load-induced mechanical property alterations of full-thickness bovine articular cartilage attached to bone were investigated in unconfined compression. A sequence of test compressions of finite deformation (ranging from 0.9% to 34.5% nominal strain) was performed at strain rates ranging from approximately 0.053%/s to 5.8%/s. Peak and equilibrium loads were analyzed to determine strain amplitude and strain rate dependence of linear versus nonlinear responses. The test protocol was designed to reveal changes in mechanical properties due to these finite deformations by interspersing small-amplitude witness ramps of approximately 1.1% deformation and approximately 0.44%/s strain rate between the test ramps ("witness" meaning to assess any mechanical property changes). We found that peak loads displayed high nonlinearity, stiffening with both increasing compression amplitude and more so with increasing strain rate. The response to witness ramps suggested that mechanical weakening occurred when compression amplitude reached 1.9-2.9% strain and beyond, and that weakening was much more significant at higher strain rate. These findings delineate regimes of linear versus nonlinear behavior of cartilage, and indicate the types of loads which can cause mechanical property alterations. Biological implications of this study are that strain amplitude and strain rate dependent stiffening may be essential to bear physiological loads and to protect cells and matrix from mechanical damage. Structural changes reflected by mechanical weakening at small compression could also initiate remodeling or disease processes.     

A recruitment-based rheological model for mechanical behavior of soft tissues

When lung tissue is subjected to finite deformations, phenomena appear that can only be described using nonlinear models. This paper considers the lung as a material composed of two elements, a continuous phase that acts uninterruptedly and a second phase composed of fiber elements that are recruited progressively into the mechanical process. Each individual fiber participates in the mechanical response of the set only when the deformation is above a certain value. A nine-parameter model was designed adopting standard viscoelastic elements both for the matrix and for each of the fibers. The mechanical behavior of the lung can be reproduced by a fitting process with standard numerical procedures in both dynamic-mechanical measurements and stress relaxation processes. Mechanical stress relaxation tests and dynamic-mechanical measurements have been carried out on subpleural parenchymal strips from rat lung. The model permits the reproduction of lung behavior in both types of measurements. The results show a recruitment ratio that decreases with deformation and the nonparticipation of the parallel matrix fraction in the lung's mechanical response so that a uniaxial transmission of force in the lung occurs via the recruited elements and the matrix series.     

A model for the nonlinear elastic response of large arteries

The nonlinear elastic response of large arteries subjected to finite deformations due to action of biaxial principal stresses, is described by simple constitutive equations. Generalized measures of strain and stress are introduced to account for material nonlinearity. This also ensures the existence of a strain energy density function. The orthotropic elastic response is described via quasi-linear relations between strains and stresses. One nonlinear parameter which defines the measures of strain and stress, and three elastic moduli are assumed to be constants. The lateral strain parameters (equivalent to Poisson's ratios in infinitesimal deformations) are deformation dependent. This dependence is defined by empirical relations developed via the incompressibility condition, and by the introduction of a fifth material parameter. The resulting constitutive model compares well with biaxial experimental data of canine carotid arteries.     

Effects of collagenase and elastase on the mechanical properties of lung tissue strips

The dynamic stiffness (H), dampingcoefficient (G), and harmonic distortion (k_d)characterizing tissue nonlinearity of lung parenchymal strips fromguinea pigs were assessed before and after treatment with elastase orcollagenase between 0.1 and 3.74 Hz. After digestion, data wereobtained both at the same mean length and at the same mean force of thestrip as before digestion. At the same mean length, G and H decreasedby ~33% after elastase and by ~47% after collagenase treatment.At the same mean force, G and H increased by ~7% after elastase andby ~25% after collagenase treatment. The k_dincreased more after collagenase (40%) than after elastase (20%)treatment. These findings suggest that, after digestion, the fractionof intact fibers decreases, which, at the same mean length, leads to adecrease in moduli. At the same mean force, collagen fibers operate ata higher portion of their stress-strain curve, which results in anincrease in moduli. Also, G and H were coupled so that hysteresivity(G/H) did not change after treatments. However,k_d was decoupled from elasticity and wassensitive to stretching of collagen, which may be of value in detectingstructural alterations in the connective tissue of the lung.

     

Response to a pure tone in a nonlinear mechanical-electrical-acoustical model of the cochlea

In this article, a nonlinear mathematical model is developed based on the physiology of the cochlea of the guinea pig. The three-dimensional intracochlear fluid dynamics are coupled to a micromechanical model of the organ of Corti and to electrical potentials in the cochlear ducts and outer hair cells (OHC). OHC somatic electromotility is modeled by linearized piezoelectric relations whereas the OHC hair-bundle mechanoelectrical transduction current is modeled as a nonlinear function of the hair-bundle deflection. The steady-state response of the cochlea to a single tone is simulated in the frequency domain using an alternating frequency time scheme. Compressive nonlinearity, harmonic distortion, and DC shift on the basilar membrane (BM), tectorial membrane (TM), and OHC potentials are predicted using a single set of parameters. The predictions of the model are verified by comparing simulations to available in vivo experimental data for basal cochlear mechanics. In particular, the model predicts more amplification on the reticular lamina (RL) side of the cochlear partition than on the BM, which replicates recent measurements. Moreover, small harmonic distortion and DC shifts are predicted on the BM, whereas more significant harmonic distortion and DC shifts are predicted in the RL and TM displacements and in the OHC potentials.     

Extensibility changes of calcified soft tissue strips from human aorta

Some degree of calcification was noted in more than half of the 59 aortas of individuals aged from 15 to 88 we have examined at autopsy. The calcification, which is determined by x-raying the opened and flat aorta, is in patches. We have studied the influence of calcification on stress versus strain, breaking strength, and modulus of elasticity of strips of aorta to determine its importance in vascular disease. Strips of aortic wall 5 x 30 mm were cut with orientation parallel or perpendicular to the vessel axis. Elongation versus load was measured with an Instron tensile testing machine. The true stress and true strain were calculated for both calcified and uncalcified strips from the thoracic and abdominal regions in both orientations. From the stress-strain curve the following values were selected: strain, stress, and slope at 80 mmHg equivalent pressure (1 mmHg = 133.3 Pa); maximum stress, strain, and slope; and breaking stress, strain, and slope if the sample broke. There were statistically significant differences in 13 of the 36 categories between calcified and uncalcified strips. The breaking strength and strain is lower in the calcified strips. The stress-strain curve for the uncalcified strip was mathematically transformed by reducing the amount of elongation so that the curve coincided with that of the calcified strip for eight matched pairs from the same individuals. The calcification appears to immobilize part of the strip, probably causing the boundary of the calcified tissue to be a region of high stress where tissue breakdown can occur.     

Progressive adaptation in regional parenchyma mechanics following extensive lung resection assessed by functional computed tomography

In adult canines following major lung resection, the remaining lobes expand asymmetrically, associated with alveolar tissue regrowth, remodeling, and progressive functional compensation over many months. To permit noninvasive longitudinal assessment of regional growth and function, we performed serial high-resolution computed tomography (HRCT) on six male dogs (～9 mo old, 25.0 ± 4.5 kg, ±SD) at 15 and 30 cmH(2)O transpulmonary pressure (Ptp) before resection (PRE) and 3 and 15 mo postresection (POST3 and POST15, respectively) of 65-70% of lung units. At POST3, lobar air volume increased 83-148% and tissue (including microvascular blood) volume 120-234% above PRE values without further changes at POST15. Lobar-specific compliance (Cs) increased 52-137% from PRE to POST3 and 28-79% from POST3 to POST15. Inflation-related parenchyma strain and shear were estimated by detailed registration of corresponding anatomical features at each Ptp. Within each lobe, regional displacement was most pronounced at the caudal region, whereas strain was pronounced in the periphery. Regional three-dimensional strain magnitudes increased heterogeneously from PRE to POST3, with further medial-lateral increases from POST3 to POST15. Lobar principal strains (PSs) were unchanged or modestly elevated postresection; changes in lobar maximum PS correlated inversely with changes in lobar air and tissue volumes. Lobar shear distortion increased in coronal and transverse planes at POST3 without further changes thereafter. These results establish a novel use of functional HRCT to map heterogeneous regional deformation during compensatory lung growth and illustrate a stimulus-response feedback loop whereby postresection mechanical stress initiates differential lobar regrowth and sustained remodeling, which in turn, relieves parenchyma stress and strain, resulting in progressive increases in lobar Cs and a delayed increase in whole lung Cs.     

Correlations between tissue-level stresses and strains and cellular damage within the guinea pig spinal cord white matter

Strain magnitude, strain rate, axon location, axon size, and the local tissue stress state have been proposed as the mechanisms governing primary cellular damage within the spinal cord parenchyma during slow compression injury. However, the mechanism of axon injury has yet to be fully elucidated. The objective of this study was to correlate cellular damage within the guinea pig spinal cord white matter, quantified by a horseradish peroxidase (HRP) exclusion test, with tissue-level stresses and strains using a combined experimental and computational approach. Force-deformation curves were acquired by transversely compressing strips of guinea pig spinal cord white matter at a quasi-static rate. Hyperelastic material parameters, derived from a Mooney-Rivlin constitutive law, were varied within a nonlinear, plane strain finite element model of the white matter strips until the computational force-deformation curve converged to the experimental results. In addition, white matter strips were subjected to nominal compression levels of 25%, 50%, 70%, and 90% to assess axonal damage by quantifying HRP uptake. HRP uptake density increased with tissue depth and with increased nominal compression. Using linear and nonlinear regression analyses, the strongest correlations with HRP uptake density were found for groups of tissue-level stresses and groups of log-transformed tissue-level strains.     

Effect of glycosaminoglycan degradation on lung tissue viscoelasticity

We tested the hypothesis that matrix glycosaminoglycans contribute to lung tissue viscoelasticity. We exposed lung parenchymal strips to specific degradative enzymes (chondroitinase ABC, heparitinase I, and hyaluronidase) and determined whether the mechanical properties of the tissue were affected. Subpleural parenchymal strips were obtained from Sprague-Dawley rats and suspended in a Krebs-filled organ bath. One end of the strip was attached to a force transducer and the other to a servo-controlled lever arm that effected sinusoidal oscillations. Recordings of tension and length at different amplitudes and frequencies of oscillation were recorded before and after enzyme exposure. Resistance, dynamic elastance, and hysteresivity were estimated by fitting the equation of motion to changes in tension and length. Quasi-static stress-strain curves were also obtained. Exposure to chondroitinase and heparitinase I caused significant increases in hysteresivity, no decrement in resistance, and similar decreases in dynamic elastance relative to control strips exposed to Krebs solution only. Conversely, measures of static elastance were different in treated versus control strips. Hyaluronidase treatment did not alter any of the mechanical measures. These data demonstrate that digestion of chondroitin sulfate and heparan sulfate alters the mechanical behavior of lung parenchymal tissues.     

Nonlinear properties of two-dimensional wake waves excited by relativistic electron bunches in plasma

The properties of a nonlinear plasma wake wave excited by an axially symmetric relativistic electron bunch are studied. It is shown that the nonlinear dependence of the wake wavelength on the transverse coordinate leads to distortion of the phase front of the wake wave and to steepening and oscillations of the transverse profile of the wakefield. The magnetic field of the wake wave is nonzero and oscillates at a frequency higher than the plasma electron frequency. Because of nonlinearity, the amplitude of the excited wake wave changes with distance from the bunch. The increase in nonlinearity leads to the development of turbulence and chaotization of the wakefield and results in the switching-on of the thermal effects and plasma heating.     

A nonlinear biphasic model of flow-controlled infusion in brain: Fluid transport and tissue deformation analyses

A biphasic nonlinear mathematical model is proposed for the concomitant fluid transport and tissue deformation that occurs during constant flow rate infusions into brain tissue. The model takes into account material and geometrical nonlinearities, a hydraulic conductivity dependent on strain, and nonlinear boundary conditions at the infusion cavity. The biphasic equations were implemented in a custom written code assuming spherical symmetry and using an updated Lagrangian finite element algorithm. Results of the model showed that both, geometric and material nonlinearities play an important role in the physics of infusions, yielding important differences from infinitesimal analyses. Geometrical nonlinearities were mainly due to the significant enlargement of the infusion cavity, while variations of the parameters that describe the degree of nonlinearity of the stress–strain curve yielded significant differences in all distributions. For example, a parameter set showing stiffening under tension yielded maximum values of radial displacement and porosity not localized at the infusion cavity. On the other hand, a parameter set showing softening under tension yielded a slight decrease in the fluid velocity for a three-fold increase in the flow rate, which can be explained by the substantial increase of the infusion cavity, not considered in linear analyses. This study strongly suggests that significant enlargement of the infusion cavity is a real phenomenon during infusions that may produce collateral damage to brain tissue. Our results indicate that more experimental tests have to be undertaken in order to determine material nonlinearities of brain tissue over a range of strains. With better understanding of these nonlinear effects, clinicians may be able to develop protocols that can minimize the damage to surrounding tissue.