## Abstract

We present a systematic quantitative analysis of power-law force relaxation and investigate logarithmic superposition of force response in relaxed porcine airway smooth muscle (ASM) strips in vitro. The term logarithmic superposition describes linear superposition on a logarithmic scale, which is equivalent to multiplication on a linear scale. Additionally, we examine whether the dynamic response of contracted and relaxed muscles is dominated by cross-bridge cycling or passive dynamics. The study shows the following main findings. For relaxed ASM, the force response to length steps of varying amplitude (0.25–4% of reference length, both lengthening and shortening) are well-fitted with power-law functions over several decades of time (10^{−2} to 10^{3} s), and the force response after consecutive length changes is more accurately fitted assuming logarithmic superposition rather than linear superposition. Furthermore, for sinusoidal length oscillations in contracted and relaxed muscles, increasing the oscillation amplitude induces greater hysteresivity and asymmetry of force-length relationships, whereas increasing the frequency dampens hysteresivity but increases asymmetry. We conclude that logarithmic superposition is an important feature of relaxed ASM, which may facilitate a more accurate prediction of force responses in the continuous dynamic environment of the respiratory system. In addition, the single power-function response to length changes shows that the dynamics of cross-bridge cycling can be ignored in relaxed muscle. The similarity in response between relaxed and contracted states implies that the investigated passive dynamics play an important role in both states and should be taken into account.

- smooth muscle dynamics
- power-law force relaxation

airway smooth muscle (ASM) mechanics play a central role during airway closure in asthma. The mechanical behavior of ASM and many other biological tissues and cells has been described by power-law relaxation (3, 6, 7, 11, 15, 23, 24). That is, the response to a sudden increase in length involves a rapid increase followed by a relaxation of both stress and stiffness, which can be described as a power-law function with time (i.e., *at*^{α}, with *a* and *α* constants and *t* = time). This phenomenon occurs in both contractile and noncontractile tissues, suggesting that it may originate from noncontractile (i.e., passive) processes (23). A role for passive processes in contracted muscle is indeed supported by the observation of stiffening of ACh-stimulated muscle when cross-bridge cycling is inhibited (1, 16, 25). Consequently, the characterization of power-law relaxation is likely to be valuable in understanding the mechanics of both relaxed and contracted ASM.

Previous work has emphasized the steady-state force response in relaxed ASM tissue. The dynamic or non-steady-state behavior of ASM has received much less attention, being largely investigated through cell stretching and bead motions in cells (3, 6, 7, 11, 23, 24). The data from these studies have been analyzed in terms of power-law relaxation, but both the mechanics and the resulting behavior are not fully understood (2, 19, 22). Some models aimed at describing biological tissue mechanics, such as the generalized Maxwell model (22), fractional derivatives (22), and the soft glasses theory (19), show linear or near-linear superposition. However, Trepat et al. (23) elegantly demonstrated that the exponent of the power-law relaxation of cell stiffness increases with the amplitude of stretch in ASM cells, which is indicative of nonlinear superposition. After all, under linear superposition, an increase in amplitude of stretch merely results in an increase of the multiplier of the power-law response but not the exponent. Understanding this principle of superposition is essential in developing a descriptive model of the mechanical behavior of a system. Here, we propose an alternative superposition principle, which may better describe this nonlinear superposition. As the power law is linear on logarithmic scale, it seems likely that the superposition is also linear on the same scale. Linear superposition on a logarithmic scale (i.e., logarithmic superposition) is equivalent to multiplicative superposition on linear scale. This explains the stretch amplitude-power-law exponent relation, as the multiplication of power laws leads to a summation of its exponents.

In this study, our primary aim is to investigate whether the principle of logarithmic superposition applies to relaxed ASM and to characterize this behavior. Our secondary aim is to investigate the role of passive dynamics in both relaxed and contracted muscle.

## MATERIALS AND METHODS

### Tissue Preparation, Solutions, and Chemicals

Porcine tracheas (Yorkshire, age 9 mo) were acquired from a local abattoir and transported to the laboratory in Hanks' balanced salt solution (composition in mM: 5.3 KCl, 0.44 KH_{2}PO_{4}, 137.9 NaCl, 0.34 Na_{2}PO_{4}, 2.33 CaCl_{2}, 0.79 MgSO_{4}, 10 dextrose, and 10 HEPES buffer) on ice. Connective tissue and overlying tissues including epithelium were removed to permit isolation of ASM strips (1–2 × 1 × 7–13 mm). Each strip was suspended vertically, using silk thread ties, in a tissue bath (constant volume, 10 ml), and immersed in control physiological saline solution (composition in mM: 110 NaCl, 0.82 MgSO_{4}, 1.2 KH_{2}PO_{4}, 3.4 KCl, 2.4 CaCl_{2}, 25.7 NaHCO_{3}, and 5.6 dextrose) bubbled with 95% O_{2}-5% CO_{2} and pH maintained at 7.4. Potassium depolarization was achieved using a modified physiological saline solution with elevated K^{+} (replacement of 137.9 mM NaCl with equimolar KCl). Contractures were elicited by addition of 10^{−6} M ACh (Sigma). The protocols were exempt from formal approval in accordance with the Auckland University of Technology and Mayo Clinic College of Medicine's animal ethics guidelines.

### Experimental Apparatus and Protocols

Force and length were controlled using a Cambridge Technology 300B Dual Mode Servo motor. Signal data were acquired and sent with a National Instruments DAQ board, which was controlled using a custom National Instruments LabVIEW program. All data were acquired at 3 kHz, but analysis of power-law functions was conducted on downsampled data of 100 Hz for the fitting process. The temperature was set at 23°C using a thermal circulator connected to the tissue bath water jacket except for 1 set of experiments undertaken at 37°C. The ASM strip was equilibrated at constant length in the control saline solution for 20 min. To determine a reference length (*L*_{ref}), the ASM strip underwent repeated cycles of K^{+}-induced depolarization, washing out in the control solution, and stretching (5% of current length). This cycle was repeated until the active force (the difference between maximal contracted force and prior relaxed force) peaked or reached a plateau. The associated length is defined as *L*_{ref}. All subsequent contractions were elicited with ACh (10^{−6} M). The first few ACh-induced contractures showed a gradual increase in isometric force, so that at least 2 contractures were evoked before starting the experiments. All experiments were punctuated by a 300-s isometric ACh 10^{−6} M contracture followed by a 500-s recovery in the control solution to achieve a stable force level. Tissues were rejected as nonviable if the contractile force during these contractions did not recover to >80% of the initial value in fresh tissue. A total of 36 ASM strips from 13 animals were used for these experiments.

#### Step protocols.

Several length-change protocols were applied to the ASM to examine power-law force relaxation. This is defined as a force response to sudden length changes according to F(*t*) = *at*^{α} where *t* = 0 immediately after the length change and *a* and *α* are constants. Four types of step length-change protocol were used as illustrated in Fig. 1. All step length changes were applied as ramped length changes with a total ramp time (*t*_{r}) of 1 ms unless stated otherwise. Each is described in detail below.

*PROTOCOL A*: SINGLE-LENGTH STEP.

To test for the existence of power-law force relaxation and the amplitude dependency of multiplier and exponent, single-length steps with step amplitudes of 0.25, 0.5, 1, and 2% *L*_{ref} were applied in two directions (i.e., stretching and shortening). *t*_{step} Was taken as 200 s. To assess the effect of the speed at which the length change (Δ*L*) is applied, single-length steps (Δ*L* = −0.5% *L*_{ref}) were applied with *t*_{r} of 0.001, 0.01, 0.1, 1, and 10 s.

*PROTOCOL B*: STAIRCASE FUNCTIONS.

Length dependence of stiffness and power-law exponents for different amplitudes of stretch was investigated using staircase functions of total amplitude *L*_{2} − *L*_{1} = 2% and −2% *L*_{ref}, length step amplitude Δ*L*_{s} = 0.25, 0.5, and 1% *L*_{ref}, and step duration *t*_{step} = 10 s.

*PROTOCOL C*: SUPERIMPOSED SQUARE-WAVE STAIRCASE.

To assess whether the data from *protocol B* were dependent on the order and duration of applied length changes, staircase functions with Δ*L*_{s} = 0.5 or 1% *L*_{ref} and *L*_{2} − *L*_{1} = −2 or 2% *L*_{ref} and *t*_{step} = 200 s with a superimposed square wave of 0.25% *L*_{ref} amplitude and a period of 2 s were applied. The force response to the square wave at different times after a larger length change can indicate whether power-law exponents are dependent on other factors than length.

*PROTOCOL D*: TWO-PHASE LENGTH STEP.

Logarithmic superposition was investigated by applying a series of two-phase length steps of which the ratio (*L*_{3} − *L*_{2})/(*L*_{2} − *L*_{1}) was varied from 0.2 to 5 and −0.2 to −5, both for *L*_{3} > *L*_{1} and *L*_{3} < *L*_{1}. For all two-phase length steps, *L*_{3} − *L*_{1} = 2% *L*_{ref}, *t*_{step} = 200 s, and *t*_{2–3} = *t*_{3–4} = 1 s. Curve fitting of consecutive length steps can indicate whether linear or logarithmic superposition is most applicable to ASM.

##### SINUSOIDAL OSCILLATIONS PROTOCOL.

To compare the dynamic response of relaxed and contracted ASM, sinusoidal length oscillations were applied to relaxed ASM as L(*t*) = *L*_{ref} + Δ*L*[sin(2π*Ft* + 0.5π) − 1], with amplitude Δ*L* = 0.5, 1, 2, 4, and 8% and frequency of 0.2 Hz. The form of the length oscillations was chosen to assure that the peak length achieved during oscillation was equal to *L*_{ref}, hence any adaptive effects from length changes above *L*_{ref} were avoided. Oscillations were maintained for 100 s followed by a recovery period of 100 s at constant length. A second series of sinusoidal oscillation experiments were applied at 2% *L*_{ref} at frequencies of 0.2, 1, 5, and 10 Hz. The same protocol was then applied to contracted ASM, 600 s after applying ACh. All oscillations were compared for asymmetry and hysteresivity.

#### Curve fitting and analysis.

Hysteresivity *η* (a measure of the hysteresis of the response to sinusoidal length changes corrected for the asymmetry of the force-length loop) has been proposed to indicate the extent of cross-bridge cycling activity. It is defined as *η* = tan{sin^{−1}[4*A*/(πΔ*F*Δ*L*)]} with *A* being the hysteresis area of the force length loop and Δ*F* and Δ*L* the force and length range, respectively (8). The degree of asymmetry of the force-length loop of sinusoidal length changes was used to indicate the nonlinearity of the force response. Asymmetry was calculated as the area ratio

Curve fitting was undertaken in MatLab with a customized version of EzyFit 2.3 (Open Source), which uses the least-squares fitting method to generate curve fits. As the signal-to-noise ratio is dependent on the measured absolute force, the quality of fit parameter *r*^{2} was normalized by dividing by the expected *r*^{2} value of the signal noise on the fitting curve. Signal analysis showed the noise to be Gaussian, which was quantified by the SD of the difference between consecutive samples. This corresponds to
*σ*_{n}). The expected value of the fit (*f*) + signal noise, *E*(
*y* is the raw data, *N* is the number of samples fitted, _{i} refers to a sample, and – is the mean value over all samples fitted.

Residual sum of squares (RSS) was used as a fitting quality statistic to compare curve fits based on different superposition principles. Instantaneous stiffness (*k*) of step length changes was calculated as *k* = (F_{post} − F_{prior})/Δ*L*, with F_{post} and F_{prior} the forces just after and just before a length change, respectively, and Δ*L* as a percentage of *L*_{ref}.

### Statistical Analysis

All data are given as mean values ± SD where *n* equals the number of ASM strips. Significance was assessed using the Student's *t*-test (paired) and for multiple protocol comparison ANOVA. The level for acceptance was set at the probability *P* < 0.05.

## RESULTS

### Power-Law Assessment and Characterization in Relaxed Muscle

To determine whether power-law force relaxation applies to ASM, rapid length changes were applied (*protocol A*: single-length step) and fitted with power functions using least mean squares fit in the log-log domain. The force response to a single-step length change was fitted by a function of the form F(*t*) = *at*^{α} (normalized *r*^{2} = 0.996 ± 0.007, *n* = 6) for both positive and negative length changes. Figure 3 shows that the power-law exponents derived from different length step amplitudes (both positive and negative) correlate with the ratio of F_{post} to F_{prior} according to *α* = *a*_{2} × log(F_{post}/F_{prior}). The average value for *a*_{2} was −0.058 ± 0.0093 with *r*^{2} = 0.996 ± 0.003 (*n* = 12). Because of the short length step duration, data from staircase functions did not confirm this correlation, except for the first length step of a staircase section after which the response progressively deviated. At large length-change amplitudes, a greater deviation from the logarithmic correlation was observed.

The effect of varying the ramp speed *t*_{r} during single-length steps (*protocol A*: single-length step) was investigated to determine the dependence of power-law force relaxation on the speed of length change. Each of the step responses converged to the same power-law response. That is, they converged to a single power-law multiplier [1-way ANOVA, *P* = 0.29 and 0.20 for positive and negative length changes, respectively (*n* = 6)] and exponent [1-way ANOVA, *P* = 0.49 and 0.4 for positive and negative length changes, respectively (*n* = 6)].

The instantaneous stiffness was analyzed from staircase functions (*protocol B*: staircase function). For positive length changes, the stiffness correlates linearly with F_{prior} (*r*^{2} = 0.96 ± 0.02, *n* = 6), whereas for negative length changes, it correlates linearly with F_{post} (*r*^{2} = 0.98 ± 0.01, *n* = 6; Fig. 4*A*). Similar correlations are found for staircase functions with superimposed square waves (*protocol C*: superimposed square-wave staircase; *r*^{2} = 0.989 ± 0.006 and 0.988 ± 0.005, *n* = 6; Fig. 4*B*).

### Logarithmic Superposition

To examine whether logarithmic superposition adequately describes the force response to consecutive length steps in relaxed muscle, 2-phase length steps with varying amplitude ratios were applied (*protocol D*: 2-phase length steps). It was predicted that the force response directly after a length step could be described by the multiplication of the individual force response to both length steps. To assess whether logarithmic superposition was more likely to apply than linear superposition, the following 2 curve-fitting procedures were compared on quality of fit.

#### Logarithmic superposition.

The force response was fitted piecewise with F(*t*) = *at*^{α1} for 0 < *t* < *t*_{2–3}, where *t*_{2–3} is the time between the 2 length changes, *a* and *α*_{1} are fitting parameters, and *t* is the time set to 0 directly after the first length change. For *t* > *t*_{2–3}, F(*t*) = *at*^{α1}*b*(*t* − *t*_{2–3})^{α2}, with *b* and *α*_{2} the fitting parameters.

#### Linear superposition.

The force response was also fitted piecewise, with F(*t*) = F_{prior} + *at*^{α1} for 0 < *t* < *t*_{2–3}, with *a* and *α*_{1} the fitting parameters, and for *t* > *t*_{2–3,} F(*t*) = F_{prior} + *at*^{α1} + *b*(*t* − *t*_{2–3})^{α2},with *b* and *α*_{2} the fitting parameters.

Figures 5 and 6 show comparisons of quality of fit (RSS) using logarithmic and linear superposition for all two-phase length steps of opposite sign. These figures illustrate that in most cases logarithmic superposition resulted in significantly better fits to the data (Figs. 5*A* and 6*A*). However, when a negative length change was followed by a much smaller positive (stretch) length change, the assumption of linear superposition gave a closer fit. For consecutive length changes of equal sign, there was no statistical difference in the quality of fit.

### Hysteresivity and Asymmetry in Contracted vs. Relaxed ASM

To test whether cross-bridge cycling contributes to the force response in relaxed ASM, the effect of sinusoidal length oscillations was compared in both contracted and relaxed muscle (sinusoidal oscillation protocol). Two indicators of the force response were used: hysteresivity and asymmetry. Hysteresivity was significantly larger at oscillation amplitudes exceeding 2 and 4% (Fig. 7*A*) and was attenuated at frequencies exceeding 1 and 5 Hz in contracted and relaxed ASM (Fig. 7*B*), respectively. Absolute values of the hysteresivity were significantly larger in relaxed than in contracted muscle (25 ± 14%, *n* = 6). Both relaxed and contracted ASM show an increase in asymmetry with increasing oscillation amplitude and frequency (Fig. 7, *C* and *D*). Besides a dependence on amplitude, frequency, and state (relaxed or contracted), 2-way repeated-measures ANOVA also indicated interaction between these variables for hysteresivity but none for asymmetry. Steady-state force before oscillations in relaxed muscle was 15 ± 5% of the force in contracted muscle (*n* = 6).

### Temperature Effects

The power-law relaxation was also tested at 37°C to assess whether it applies at core body temperature. The muscle was contracted and subsequently washed with control solution at both 23 and 37°C. Meanwhile, the length was perturbed by a continuous square wave similar to the superimposed square wave of *protocol C*. Force and stiffness in relaxed muscle increased after the temperature change by 18 ± 14 and 21 ± 13%, respectively (*n* = 6). Both returned to the values before the temperature change after a single contracture at 37°C. The power-law exponent, force, and stiffness during contracture were not significantly changed at 37°C.

## DISCUSSION

We addressed two main questions on the dynamic (i.e., non-steady-state) behavior of ASM. Is logarithmic superposition of force applicable? What is the role of passive dynamics in relaxed and contracted muscle? The main findings of this work are: *1*) logarithmic superposition was found to fit the data significantly better than linear superposition for most of the two-phase step combinations; and *2*) passive dynamics dominate the force response in relaxed muscle and may play an important role in contracted muscle as well. Furthermore, we show quantitative data on the relationships of force-stiffness and force-power-law exponent in relaxed ASM.

### Confirmation of Power-Law Force Adaptation in Relaxed Muscle

We provide direct evidence for the existence of absolute power-law force relaxation in response to sudden length changes in ASM tissues (Fig. 3). These findings are consistent with the reported power-law relaxation of stiffness in ASM cells measured by bead excitation (23). However, our experiments showed that power-law behavior also applies to sudden length reductions. Hence, the term power-law force adaptation may be a more appropriate description than force relaxation. Apparently, intercellular connections and intracellular differences between in vivo ASM and cultured ASM cells do not affect the type of dynamic response. However, experiments on intact tissues have indicated that the dynamics of tissues are more nonlinear than those of cultured cells (15). The force response to length changes in our experiments could be accurately fitted with power laws from the first 0.01 s until several minutes after a length change. A single power-function response over such a broad time scale indicates that relaxed ASM responds to length changes as a single mechanical entity. Furthermore, we have observed that the speed of length changes, and consequently the shear rate, does not affect the final force response. Although this contradicts the concept of shear thinning, the fluidization as defined in the soft glasses theory has also been shown to be dependent on the amplitude of length change rather than the rate (23). It should be noted, however, that these length changes consisted of a step length increase and decrease of the same magnitude in quick succession rather than the step and hold as applied in our research. Furthermore, our results also agree with logarithmic superposition as a ramped length change would result in convergence to the same power function independent of the rate of length change. This is because the total length change rather than the ramp speed determines this power function.

Our data (Figs. 3 and 4) are supported by findings in cultured ASM cells, which indicates that intercellular components and processes have little effect on the type of dynamic force response. For instance, the linear dependence of the stiffness on the force before a length change mirrors the linear dependence of stiffness on prestress in ASM cells (21, 27). Our finding that the power-law force adaptation exponent depended on the ratio of F_{post} to F_{prior} (Fig. 3) is consistent with the correlation seen between prestress and power-law force adaptation exponents in cells (21, 27). Furthermore, this finding implies that the force response to the peak-to-peak length change during an oscillation converges to a power-law function, which is dependent on the frequency. Indeed, whereas the speed of stretch does not affect the power-law exponent, the time to complete a stretch does affect the peak force reached. As the peak force decreases with this stretch time according to a power function, the peak-to-peak force observed during oscillations will depend on the frequency with a power law similar to that in cells (7). Notably, these findings are more likely to have a physiological significance since a similar response occurred in ASM at 37°C.

### Logarithmic Superposition

When two length steps are applied in rapid succession, the force response of the second step is more accurately fitted with logarithmic rather than linear superposition on most occasions (Figs. 5 and 6). Whereas logarithmic superposition always showed a high quality of fit, in some situations linear superposition resulted in an equal or better fit. These may be attributed to the fact that the force response was near featureless (i.e., almost linear on logarithmic scale) in these specific situations (*top* curves in Figs. 5*A* and 6*A*) resulting in only small differences between the two fits. The multiplicative nature of logarithmic superposition may cause the error of fit in the first length step to be magnified in the second length step much more than with linear superposition. Hence, when the effect of the first length step is minimized by a larger second length step, linear superposition can lead to a significantly better fit.

A further indication of logarithmic superposition comes from the power-law exponents relations. When logarithmic superposition is applied to 2 length steps of opposite sign but equal magnitude, the resultant force response appears in the form F(*t*) = F_{prior}*c*_{2}(*t* − *t*_{1–2})^{α2}*c*_{1}*t*^{α1}, with *c*_{1} and *c*_{2} constants and *t* = 0 directly after the first length change. Taking the limit of *t*_{1–2}→0, the sum of the exponents should be 0, and the multiplier *c*_{1}*c*_{2} should go to 1 to assure no residual force difference as effectively the length is unchanged. For this to occur, *c*_{2} has to equal *c*_{1}^{−1} and *α*_{1} = −*α*_{2}. The logarithmic relation between F_{post}/F_{prior} and *α* implies that when *c*_{1} = *c*_{2}^{−1}, *α*_{1} will equal −*α*_{2} because log(*x*) = −log(*x*^{−1}).

We show that the ASM mechanics conform to a single material response, with a long mechanical memory despite little being known about the molecular processes. This response is possibly the result of a large range of mechanisms inside the cells acting at different rates. It is speculated that the large protein molecules are subject to forced reptation (4) during a stretch. With the many different protein sizes and entanglement in the cell, this might combine into a single mechanical response characterized by logarithmic superposition.

### Comparison of Contracted and Relaxed ASM

The force response to a single-length step in relaxed muscle can be accurately fitted by a power function over 5 decades of time (10^{−2} to 10^{3} s). As cross-bridge cycling is expected to occur at specific (low) speeds, the single power-law response indicates that the dynamics of cross-bridge cycling do not play an important role in the dynamic response in relaxed ASM for the length changes used here. Attached cross-bridges can still provide a basal force in relaxed ASM, but the cycling rates may be too low to be affected by length changes. Alternatively, the forces applied during these experiments may be too small to affect the cross-bridges. As passive dynamics seem to dominate the force response to length changes in relaxed ASM, similarities in the force response between contracted and relaxed muscle may indicate the importance of passive dynamics in contracted ASM. To assess the influence of passive dynamics on contracted ASM dynamics, the effect of sinusoidal length oscillations was investigated in contracted and relaxed muscle. In our experiments, sinusoidal oscillations with peak length equal to the *L*_{ref} showed no force reduction except for a small reduction for 8% *L*_{ref} oscillations in contracted ASM. Previous investigations have shown that sinusoidal length oscillations reduce the force in ASM considerably, which was attributed to disruption of the cross-bridge cycle (8–10, 17). However, as the muscles were stretched beyond the initial contracted length during oscillation, the reduction in force could have been caused by exceeding the contracted length rather than the oscillatory movement itself. A stretch beyond the contracted length has been shown to reduce the steady-state force when the tissue is returned to the contracted length (12). Similarly, in relaxed muscle, stretching above the initial length (i.e., the length of the prior isometric contraction) reduces the steady-state force at this length until a subsequent contraction (20). These observations are supported by a lack of force decline when triangular wave oscillations are applied when peak length equals contracted length (18).

To determine whether the dynamic force-length relation during oscillations is caused by disruption of cross-bridge cycling or passive processes, we assessed hysteresivity and asymmetry. Figure 7 demonstrates that these two characteristics follow similar trends in both contracted and relaxed ASM. For hysteresivity, we found a statistically significant interaction in the two-way ANOVA analysis between both the amplitude and frequency variables and the state (contracted or relaxed). This may occur because of the large difference in the absolute force levels between the two states, which may influence the relationships of the amplitude and frequency with hysteresivity. However, there was no interaction observed with respect to asymmetry. These findings may indicate that cross-bridge cycling has insignificant effect on the force response to oscillations applied in this work. Furthermore, a reduced shortening velocity after oscillations is unsupportive of increased cross-bridge cycling rates (26). Concordantly, the shortening velocity directly after oscillations as applied in this research was significantly reduced (data not shown). Taken together, these findings indicate that the fundamental processes underpinning this response in contracted and relaxed ASM are likely to be the same. During an isometric contracture, the muscle is believed to be in the slow cycling latch state after an initial fast cycling phase, as suggested by the low ATP turnover rates (5). If the cross-bridge cycling rate is unchanged during length oscillations, it is too low to have much effect on the dynamic response. However, if the muscle responds to oscillations with an increase in cross-bridge cycling rate, as predicted by the four-state latch-bridge model (13, 14, 17), the effect on ASM dynamics increases.

### Summary

This research was undertaken to improve understanding of ASM dynamics. The confirmation of the existence of logarithmic superposition and the characterization of power-law force adaptation in ASM can provide the basis for the development of a predictive mathematical model of ASM dynamics. This model can be further supported by existing data on the frequency response of ASM. As breathing dynamics have shown to be of great importance to the pathophysiology of asthma, understanding the dynamics of the dominant structural and active component of airways, ASM, is essential for understanding the disease. We anticipate that logarithmic superposition and power-law force adaptation are strongly interconnected and are a manifestation of a single viscoelastic behavior. The observed long mechanical memory of ASM implies that changes in dynamic loading of airways can have a long lasting effect on airway caliber. This may explain the importance of deep inspiration in regulating airway resistance. Furthermore, our results suggest that passive viscoelastic behavior plays an important role in the dynamics of contracted ASM as well.

## DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

- Copyright © 2010 the American Physiological Society