Mitigation of airways responsiveness by deep inflation of the lung

Abstract

Stretching activated strips of airway smooth muscle (ASM) significantly affects both active force and stiffness due to a temporary reduction of the proportion of cycling myosin cross bridges that are bound to their actin binding sites. For the same reason, stretch applied to ASM in situ by a deep inflation (DI) of the lungs is one of the most potent means of reversing bronchoconstriction. When the DI is sufficiently large, however, and is applied while bronchoconstriction is in the process of developing, the subsequent depression in airway resistance is more persistent than can be attributed simply to temporary detachment of ASM cross bridges. In the present study, we use a computational model to demonstrate that this DI-induced ablation of airway responsiveness can be explained by a dose-dependent reduction in the number of cross bridges available to bind to actin when the ASM in the airway wall is stretched above a critical threshold strain and that this disruption of the contractile apparatus recovers over an order of magnitude longer time scale than that of the simple reattachment of unbound cross bridges.

NEW & NOTEWORTHY The mechanisms by which deep inflation of the lung reverse bronchoconstriction and affect subsequent airway responsiveness have important potential implications for asthma, yet remain controversial. This study uses computational modeling to posit a mechanism by which sufficiently vigorous inflations applied during active bronchoconstriction not only transiently reverse bronchoconstriction, but also reduce subsequent airways responsiveness for a period of time. Fitting the model to published data in mice supports this notion.

INTRODUCTION

It has long been known that a deep inflation (DI) of the lungs is one of the most potent means of reversing bronchoconstriction, at least in normal lungs (16, 18, 19). This has led to numerous in vitro studies showing that length oscillation significantly reduces both force generation and stiffness in activated strips of airway smooth muscle (ASM) (10, 11, 26). Such observations appear to reflect the nature of ASM cross-bridge cycling dynamics; myosin cross bridges in activated ASM are continually attaching and detaching from their actin-binding sites, and increasing strain rate increases the fraction of cross bridges that are unbound relative to bound. Cessation of length oscillation then allows the unbound cross bridges to reattach with a characteristic time scale, resulting in resumption of both force and stiffness. Indeed, the classic model of Huxley (17), extended to include certain molecular complexities postulated to be peculiar to smooth muscle, has been used to account for these experimental findings (26).

We have demonstrated both in vitro (6) and in vivo (7) that if sufficiently large stretches are applied to ASM while it is actively contracting, however, the subsequent degradation in force is more persistent than can be attributed simply to temporary detachment of cross bridges. It appears as if such stretches may disrupt the contractile machinery within the ASM cell in a manner that can take minutes or more to recover (6). This might conceivably be due, for example, to disassembly of polymerized contractile protein subunits that then repolymerize over a time scale that is substantially longer than that of intact but unbound cross bridges simply reattaching to their binding sites (15, 28). Motivated by this notion, we recently developed a mathematical model in which stretch-induced ablation of ASM responsiveness is attributed to a temporary dose-dependent reduction in the total number of cross bridges available for binding (4).

While this mathematical model accounts quite accurately for in vitro data from activated ASM strips (4, 6), it remains to be seen whether it could plausibly operate within the intact lung in vivo where ASM responsiveness manifests its important physiological and clinical effects. Accordingly, the goal of the present study was to determine whether our previous in vivo experimental findings in mice (7) can potentially be explained by a model of a dynamically contracting airway in which a sufficiently vigorous DI causes a reduction in the number of viable cross bridges present in the ASM encircling the airway wall.

METHODS

Model development.

The following mathematical model incorporates aspects of a previous model we developed (8), but with some important modifications. Following our previous model, we consider an airway in cross section as a thin ring of ASM encircling a thin elastic airway wall embedded in a mechanically homogeneous network of elastic alveolar walls comprising the lung parenchyma (Fig. 1). We neglect any resistive pressure drops that might exist along the airway during breathing because these pressures are typically small compared with transpulmonary pressure (P_tp) and, thus, assume that quasi-steady-state conditions pertain throughout the breathing cycle. The radius, r, of the airway lumen at any point in time is, thus, determined by the balance between the force, F, in the airway wall (sometimes referred to as the “hoop stress”) and the transmural pressure, P_tm, across the airway wall. The force balance is provided by the Laplace law for a cylinder:

$$\begin{array}{c}{\text{P}}_{tm}=\frac{\text{F}}{r}\\ =\frac{{\text{F}}_P+{\text{F}}_A}{r}\end{array}$$ (1)

where F_P is the passive elastic recoil force of the airway wall and F_A is the active force generated by the ASM (Fig. 1).

Fig. 1.

A circular thin-walled elastic airway of radius r containing circumferential airway smooth muscle (ASM) that contracts against the outward tethering forces of elastic parenchymal attachments. F_P is the contribution to the force in the airway wall from the passive elastic recoil of the airway wall. F_A is the contribution from active ASM contraction. The transmural pressure across the airway wall when the attached parenchyma is undistorted equals transpulmonary pressure (P_tp). Parenchymal distortion increases transmural pressure by an amount ΔP_tm.

P_tm is provided by the tethering forces exerted by the parenchymal attachments to the outside of the airway wall. When the parenchyma is uniformly expanded throughout the lung, these attachments serve to transmit P_tp across the airway wall. When an airway constricts, however, it causes local distortion of the parenchyma to which it is attached. This increases P_tm by an amount ΔP_tm that grows with the degree of distortion, which we model according to Lai-Fook (20). These assumptions together lead to

$${\text{P}}_{tm}={\text{P}}_{tp}\left[1.43\frac{r_{isoTLC}}{r}{\left(\frac{{\text{P}}_{tp}}{\text{P}{}_{tpTLC}}\right)}^{\frac{1}{3}}-0.43\right]$$ (2)

where r_isoTLC is the airway radius at total lung capacity (TLC) defined as the nominal maximum volume of the lung achieved at P_tp = 30 cmH₂O, and P_tpTLC is P_tp at TLC (see appendix a). Table 1 provides a list of all model variables and parameters.

Table 1.

Model variables and parameters

Quantity	Unit	Definition
F(t)	cmH2O/mm	Total force in the airway wall
FP(t)	cmH2O/mm	Passive force in the airway wall
FA(t)	cmH2O·mm−1·s−1	Active force in the airway wall
Ptm(t)	cmH2O	Transmural pressure across the airway wall
Ptp(t)	cmH2O	Transpulmonary pressure
Q(t)	cmH2O/mm	Integrity of the ASM contractile apparatus
r(t)	mm	Airway radius
Raw(t)	cmH2O·s·ml−1	Airway resistance
t	s	Time
v	mm/s	Velocity of ASM contraction
α	arbitrary length/s	Dependence of cross-bridge detachment on shortening velocity
EL	cmH2O.s−1·ml−1	Elastance of the lung
Ew	cmH2O/ml	Elastance of the airway wall
k		Fraction of airway wall that is inextensible
kbreak	s−1	Rate constant of disruption of contractile machinery in ASM
kfix	s−1	Rate constant of repair of contractile machinery in ASM
koff	s−1	Rate constant of unbinding of attached cross-bridges
kon	s−1	Rate constant of binding of unattached cross-bridges
Nmax	arbitrary length	Maximum possible number of bound cross-bridges
Pbr	cmH2O	Active force per attached cross-bridge
PtpTLC	cmH2O/mm	Transpulmonary pressure at TLC
Qmax	arbitrary force	Maximum integrity of the ASM contractile apparatus
r0	arbitrary length	Relaxed radius of isolated airway
RawTLC	cmH2O·s−1·ml−1	Airway resistance at TLC
rcrit	arbitrary length	Critical radius for disruption of contractile machinery
risoTLC	s	Relaxed radius of airway in isotropic parenchyma at TLC
rpTLC	arbitrary length	Relaxed radius of airway in parenchyma at TLC
rTLC	arbitrary length	Radius at TLC
Rw	cmH2O·s−1·ml−1	Resistance of the airway wall
vmax	arbitrary length/s	Maximum ASM shortening velocity

ASM, airway smooth muscle; TLC, total lung capacity.

F_P in Eq. 1 is determined by the passive elastic properties of the airway wall. In previous modeling studies (1, 8), we showed that the dependence of airway responsiveness on lung volume can only be explained when one assumes the airway wall to be stiffer than the parenchyma. In one of these studies (8), we assumed the elastic behavior of the airway wall to be qualitatively identical to, but quantitatively stiffer than, that of the parenchyma in terms of pressure-radius behavior. This, however, can lead to a situation in which the rate of change of r with F_A becomes infinite at certain values of r, at which point the airway becomes mechanically unstable (see appendix b). Airway instability of a somewhat similar type has been elegantly modeled by Anafi and Wilson (3), but whether such instability actually manifests in reality remains an open question. In the present study, therefore, we assume the airway remains stable. Stability is conveniently achieved by an airway wall that behaves like a Hookean spring (appendix b), which fortuitously also produces a P_tm-r relationship in compression (i.e., during bronchoconstriction) that exhibits a reasonable degree of strain stiffening. However, in expansion (i.e., during a DI) the Hookean spring assumption gives rise to unrealistic strain softening, so we replace it with the mirror image of the compressive P_tm-r relationship (see derivation in appendix b).

We can further ensure dynamic stability of the airway wall by imbuing it with a viscous resistance to length change so that rates of change of r can never be of unlimited magnitude. A finite tissue resistance in the airway wall also makes physiological sense because biological soft tissue is highly viscoelastic, meaning that elastic recoil never occurs in the absence of dissipation. This assumption, together with the elastic behavior described above (appendix b), gives rise to a passive force in the airway wall described by

$$\begin{array}{c}{\text{F}}_P=2\pi E_w\left(r-r_0\right)+2\pi R_w\frac{dr}{dt};r\le r_0\\ =\frac{2\pi E_w\left(r-r_0\right)r}{2r_0-r}+2\pi R_w\frac{dr}{dt};r>r_0\end{array}$$ (3)

where E_w is a stiffness constant, R_w is a resistive constant, and r₀ is the unstressed wall radius. We assigned R_w a value 20% that of E_w to give a viscoelastic time constant of 0.2 s, which is in keeping with the known characteristics of lung tissue (5).

Substituting Eq. 3 into Eq. 1, equating the result to Eq. 2, and then solving for $\frac{dr}{dt}$ gives

$$\begin{array}{c}\frac{dr(t)}{dt}=\frac{\begin{array}{c}{\text{P}}_{tp}(t)\left[1.43r_{iso}{}_{TLC}{\left(\frac{{\text{P}}_{tp}(t)}{{\text{P}}_{tpTLC}}\right)}^{\frac{1}{3}}-0.43r\right]\\ -{\text{F}}_A(t)-2\pi E_w\left[r(t)-r_0\right]\end{array}}{2\pi R_w};r\le r_0\\ =\frac{\begin{array}{c}{\text{P}}_{tp}(t)\left[1.43r_{iso}{}_{TLC}{\left(\frac{{\text{P}}_{tp}(t)}{{\text{P}}_{tpTLC}}\right)}^{\frac{1}{3}}-0.43r\right]\\ -{\text{F}}_A(t)-2\pi E_w\left[\frac{2\pi E_w\left(r(t)-r_0\right)r(t)}{2r_0-r(t)}\right]\end{array}}{2\pi R_w};r>r_0\end{array}$$ (4)

In our previous study (8), we imposed contraction dynamics on this model by having the ring of ASM in the airway wall shorten, according to the classic Hill-type force-velocity relationship that relates F_A to a single-valued hyperbolic function of shortening velocity $v=-2\pi \frac{dr}{dt}$. Here, we replace this relationship with a relationship between F_A and v, based on a coarse-grained version of the Huxley model of skeletal muscle dynamics that we recently developed (4) to avoid the assumption of perfect alignment between the contractile proteins and the net direction of force that is inherent in the Huxley model, an assumption that is inappropriate for smooth muscle (33). This new relationship replaces the partial differential equation of the classic Huxley model with an ordinary differential equation (4) and has the added advantage of being more straightforward to integrate numerically. Expressing this relationship in the context of the present model gives

$$\begin{array}{c}\begin{array}{c}\frac{{d\text{F}}_A(t)}{dt}=\left({\text{P}}_{br}{N}_{\max }{k}_{on}-\alpha \right)-k_{on}{\text{F}}_A(t)\\ +\frac{k_{on}}{v_{\max }}v(t){\text{F}}_A(t)-\frac{k_{off}}{v{}_{\max }}\left|v(t)\right|{\text{F}}_A(t)\end{array}\\ =Q-k_{on}{\text{F}}_A(t)-\frac{2\pi k_{on}}{v_{\max }}\frac{dr(t)}{dt}{\text{F}}_A(t)-\frac{2\pi k_{off}}{v{}_{\max }}\left|\frac{dr(t)}{dt}\right|{\text{F}}_A(t)\end{array}$$ (5)

where P_br is the mean force in each attached cross bridge, N_max is the maximum number of cross bridges available to generate force in the muscle, k_on is the rate constant of attachment of unbound cross bridges, k_off is the rate constant of detachment of bound cross bridges, v(t) is the velocity of ASM shortening (defined as positive when ASM length is decreasing), and v_max is the maximum possible shortening velocity. The constant α quantifies the rate at which the number of attached cross bridges decreases as the magnitude of velocity increases.

Inspection of Eq. 5 shows that Q contains N_max as a factor in its first term. As in our previous modeling study (4), we assume that excessive ASM strain has the effect of reducing N_max in a dose-dependent fashion, and we account for this by letting the value of Q decrease below its maximum value, Q_max, when ASM length exceeds an injury threshold, 2πr_crit, according to

$$\begin{array}{c}\frac{dQ(t)}{dt}=k_{fix}\left[Q_{\max }-Q(t)\right]-k_{break}\left[2\pi \left(r(t)-r_{crit}\right)\right]Q(t);r(t)\ge r_{crit}\\ =k_{fix}\left[Q_{\max }-Q(t)\right];r(t)<r_{crit}\end{array}$$ (6)

where k_break is a time constant governing the rate at which tensile strain disrupts the contractile machinery and k_fix is a time constant governing the rate at which the disrupted machinery reassembles. We expect k_break >> k_fix because disruption of the contractile machinery can occur within the space of a single brief stretch, but its restoration appears to take a much longer period of time (6).

Airway resistance, R_aw(t), is assumed to vary inversely with the 4th power of r(t) (corresponding to Poisseuille flow in the airways), so it is defined relative to its value at TLC, R_awTLC; thus,

$$R_{aw}(t)=R_{awTLC}\frac{r_{pTLC}{}^4}{r{(t)}^4}$$ (7)

Equations 4–7 together define the model. These equations were integrated using the first-order Euler scheme at a frequency of 100 Hz. The model was driven by an imposed P_tp(t) waveform, itself determined by a specified volume waveform V(t) acting on the volume-independent elastance of the lung, E_L. The variables and parameters of the model are listed in Table 1.

Experimental data.

We compared the predictions of the model described above to experimental data in mice that we published previously (7). Briefly, anesthetized, tracheostomized mice were administered an intravenous bolus injection of the ASM agonist methacholine, while R_aw was measured continuously for the following 16–20 s using the forced oscillation technique applied with the Flexivent computer-controlled small animal ventilator (SCIREQ Scientific Respiratory Equipment, Montreal, QC, Canada). In some animals, the procedure was repeated with the lungs maintained at different volumes by applying 1, 3, or 6 cmH₂O positive end-expiratory pressure (PEEP). In other animals, the procedure was repeated at PEEP = 1 cmH₂O with DIs of the lungs applied either from 2 to 3 s, 5 to 6 s, or 8 to 9 s after the methacholine injection. The DIs consisted of 1 ml of air being delivered and then withdrawn with the Flexivent piston. All experimental procedures, as well as details of ethics approval for the experiments, are explained fully in our previous publication (7).

Model fitting.

The model was fit to the experimental data using a sequentially refined grid search procedure, as follows. First, a 5 × 5 × 5 × 5 × 5 grid of equally spaced values of the parameters Q_max, k_on, k_off, v_max, and E_w was defined such that ranges of values for each parameter encompassed the eventual best-fit values. R_aw(t) profiles were then simulated by the model using the parameter values defined by each point on this grid, in each case with P_tpTLC = 30 cmH₂O and r_isoTLC = 1 (arbitrary length unit). During each simulation, the model was driven with a V(t) signal corresponding to the data set being fit, which corresponded in one case to separate runs at PEEP levels of 1, 3, and 6 cmH₂O for 20 s, while in the other, it corresponded to a single run at a PEEP level of 1 cmH₂O for 16 s with a DI applied at the appropriate time point (see results). Each data simulation was preceded by a 30-s run-in period, during which F_A = 0 with the initial condition r(0) = 0.5, which allowed R_aw to achieve a steady-state value before the simulation of the experimental data. The simulated R_aw(t) profiles obtained at each point on the parameter grid were scaled so as to minimize the root mean squared residual (RMSR) between the experimental baseline data (prereaction to methacholine) and the corresponding simulated values of R_aw. The values of Q_max, k_on, k_off, v_max, and E_w that produced the smallest value of RMSR were then used as the center points on another parameter grid with point spacings half that of its predecessor. This process was repeated a further three times for a total of five grid searches, resulting in a final grid point spacing that was 1/16 that of the initial grid. The RMSR was reduced by less than 1% with further grid refinements.

RESULTS

Figure 2 shows R_aw following methacholine injection at t = 0 in mice maintained at PEEP levels of 1, 3, and 6 cmH₂O. Superimposed on these data are the best-fit model predictions. These predictions were obtained by having activation of the ASM ramp up continuously from 0 to its maximum level of Q_max over the first 5 s. This activation ramp represents the delay and dispersion of the methacholine as it transits from the injection site to the ASM and was determined empirically on the basis of providing a good fit to the data. Also, because these data were obtained with the lung held at fixed lung volumes corresponding to modest levels of PEEP, and the forced oscillations used to estimate R_aw were small, we assumed that the ASM in the airway wall was never strained to the point of sustaining damage. Accordingly, we avoided having to invoke Eq. 6 for this model fit by keeping Q(t) fixed at Q_max, which obviated the need to estimate k_break, k_fix, and r_crit. The best-fit model parameter values were Q = 6.8 (arbitrary force unit), k_on = 0.91 s⁻¹, k_off = 1.53 s⁻¹, v_max = 0.110 s⁻¹, and E_w = 4.86 (cmH₂O per arbitrary length unit). The RMSR for the fit was 0.138 cmH₂O·s⁻¹·ml⁻¹.

Fig. 2.

Experimental measurements of airway resistance (R_aw) vs. time (mean: open circles; SD: error bars) following a bolus intravenous injection of methacholine in mice given with the animals apneic at three different levels of positive end-expiration pressure (PEEP). Also shown are the fits of the mathematical model (solid lines).

Figure 3 shows the relationship between r and P_tm for the airway wall predicted by Eqs. 1 and 3 under passive conditions (F_A = 0) using the numerical value of E_w found for the fits shown in Fig. 2. The vertical axis in this plot has been normalized to the relaxed radius r₀, so that the relationship can be mapped onto an airway of arbitrary size. In particular, if r₀ is set equal to ~2 mm, the plot in Fig. 3 is quite reminiscent of experimental data reported for an isolated airway by LaPrad et al. (23).

Fig. 3.

Relationship between airway radius (r), normalized to the relaxed radius (r₀), and transmural pressure (P_tm) from Eq. B11 with E_w = 4.86 cmH₂O (the best-fit value from Fig. 2). This plot compares favorably to the experimental data shown in Fig. 8 of LaPrad et al. (23).

Figure 4 shows r (Fig. 4A) and R_aw (Fig. 4B) over the 16 s following methacholine injection in a separate series of experiments when the mice were held at a PEEP of 1 cmH₂O. The model was fit to the data using the same approach as in Fig. 2, with the same initial 5-s ramp up in ASM activation, except this time, it was necessary to impose a value for E_w a priori since R_aw time-course data at only a single PEEP level do not contain the information required to determine airway wall stiffness. Accordingly, we set E_w = 4.86 as found for the fit in Fig. 2. The best-fit values of the remaining free model parameters were Q = 5.6 (arbitrary force unit), k_on = 0.96 s⁻¹, k_off = 2.18 s⁻¹, and v_max = 1.30 s⁻¹. The RMSR for the fit was 0.038 cmH₂O·s⁻¹·ml⁻¹.

Fig. 4.

A: airway radius (r) vs. time in the model following activation of the airway smooth muscle (ASM) at PEEP = 1 cmH₂O (solid line) and when a deep inflation (DI) to 30 cmH₂O is applied between 2 and 3 s (dashed line). The horizontal dotted line indicates the critical airway radius (r_crit), above which disruption of the contractile machinery in the ASM occurs according to Eq. 6. B: experimental measurements of airway resistance (R_aw) vs. time following a bolus intravenous injection of methacholine in mice at positive end-expiration pressure (PEEP) = 1 cmH₂O (upper trace of symbols and error bars: mean (SD)] and when a DI was given between 2 and 3 s [lower symbols and error bars). Also shown are the fit of the model to the non-DI data (solid line), the model prediction with a DI when disruption of the contractile machinery (Eq. 6) was not included in the model (long-dashed line), and the prediction with a DI when disruption was included (short-dashed line).

Also shown in Fig. 4B is R_aw measured in the same mice when a 1 ml DI was imposed at 2 s after methacholine injection and withdrawn at 3 s (R_aw is shown only from 4 s because the length of the data window employed to calculated impedance precluded the estimation of earlier post-DI values). The model parameters obtained with the non-DI data were then used to simulate the R_aw time course with a DI to 25 cmH₂O at 2–3 s using a value for r_crit that was larger than any radius value reached during the DI (i.e., there was no possibility of strain-induced damage to contractile machinery). The DI in this case reverses the increase in R_aw in the model only briefly; following the DI, R_aw rapidly recovers toward the profile obtained without a DI (long-dashed line in Fig. 2B). However, we were able to obtain a very good fit to the DI data (short-dashed line in Fig. 2B) when r_crit was lowered to 0.22 (a value well below the maximum r reached during the DI shown in Fig. 4A) with k_break = 7 s⁻¹ and k_fix = 0.08 s⁻¹. These parameter values were found by trial and error and are not unique since both r_crit and k_break can be either increased or decreased appropriately to produce similar results.

Figure 5 shows experimental data and model predictions for DIs given between 2 and 3 s, 5 and 6 s, and 8 and 9 s after injection of methacholine. The parameter values used for the model predictions are the same as those used to generate the predictions in Fig. 4B, but with one exception. We found that using exactly the same parameter values for the DIs at 5–6 and 8–9 s produced R_aw profiles that were depressed relative to the experimental traces. However, we obtained very good fits to these last two data sets (Fig. 5) if we reduced the magnitude of the DI from 25 to 16 cmH₂O. We presume this accounts for the fact that the experimental DIs were applied using a fixed 1-ml displacement of the Flexivent piston (7). Part of this volume would have gone into compression of the gas remaining in the piston and the conduits leading to the tracheal opening. The volume shunted into gas compression in this way depends on the stiffness of the lung, which likely increased substantially with bronchoconstriction due to increased ASM tone in the airway walls and the development of time-constant heterogeneities throughout the lung. Also, activating the ASM increases the effective stiffness of the airway wall, which would have favored transit of the imposed DI into the more compliant distal parenchyma. The net result of these effects, thus, appears to have been the imposition of a significantly reduced stretch of the airway during the later two DIs when bronchoconstriction was already well under way (Fig. 5).

Fig. 5.

Effects of a deep inflation (DI) given at various times after methacholine injection. Symbols show mean experimental data without a deep inflation (DI) (○), with a DI at 2–3 s (△), with a DI at 5–6 s (▽), and with a DI at 8–9 s (□). The fit from the model to the non-DI data is shown as the upper solid line. The three lower solid lines are model predictions for the three differently timed DIs.

DISCUSSION

The principal novelty of the model developed in this study derives from its incorporation of a mechanism (Eq. 6) by which the responsiveness of the ASM in an actively contracting airway is impaired following a sufficiently large DI, and the manner in which it recovers from such a DI. Without this mechanism, the model predicts that recovery from a large DI during active bronchoconstriction is much too rapid (Fig. 4B), implying that the model is missing something important. In contrast, with the mechanism represented by Eq. 6 in place, and with appropriate choice of r_crit, k_break, and k_fix, the model is able to account rather accurately for the temporary ablating effects on airway responsiveness of a DI applied, while active bronchoconstriction is under way (Figs. 4B and 5). This supports the notion that the same mechanism potentially explains the stretch-induced reduction in ASM activation that we observed previously in tissue strips (4) may also be operative in the intact lung in vivo.

The model also mimics the key dependencies of R_aw on time and PEEP observed in mice following intravenous injection of methacholine (Fig. 2). These dependencies are accounted for by an ordinary differential equation (Eq. 5) that accounts for the binding and unbinding dynamics of ASM cross bridges and another such equation that accounts for the force balance between airway wall stiffness, airway wall resistance, and parenchymal tethering (Eq. 4). These two equations are, for the most part, derived from existing expressions in the literature, as described in the model derivation section of methods and in appendix a. The passive elastic properties of the airway (appendix b) are not exactly the same as assumed for some other models (8, 21, 23), but they are, nevertheless, physically reasonable. In particular, the airway model used here exhibits strain stiffening both in compression and expansion, as one would expect of any tube comprising biological soft tissue. Furthermore, the airway exhibits elastic behavior (Fig. 3) that resembles experimental data (23).

The model predictions in Figs. 4 and 5 support the hypothesis that stretching activated ASM with sufficient vigor in vivo (7) results in some form of temporary disruption of the force-generating machinery. Nevertheless, the precise disruption and subsequent repair processes responsible remain uncertain. There is evidence from ultrastructural studies that they may involve disassembly and reassembly of the contractile proteins themselves (15, 28), thereby transiently reducing the number of cross bridges available to bind to actin and, thus, reducing isomeric force commensurately. There is even recent evidence that such an effect may be mediated by the actin-severing protein cofilin (22). In any case, this notion constitutes the motivation behind Eq. 6, but it is not the only possibility. For example, stretch might impair the mechanical linkage between the ASM cell and the tissue in which it is embedded, allowing the cell to shorten without imparting the same degree of length change to the airway wall. Repair of this damage might then occur through the reestablishment of cross-linkers between the ASM cell and adjacent fibers of the extracellular matrix in the wall (9). Whatever the actual mechanism, the rate of recovery from DI-induced disruption (governed by the estimated value of k_fix of 7 s⁻¹ in Figs. 4 and 5), appears to be of a magnitude slower than the rate at which unbound cross bridges reattach to their actin binding sites (governed by the estimated value of k_on of 0.91 s⁻¹ in Fig. 2). It should also be noted that we cannot get an accurate fix on the value of k_break in the present study because we do not know the value of r_crit; determining r_crit accurately would require experimentation with different magnitudes of sigh to determine that above which decrements in subsequent airway responsiveness start to occur. All we can say at this point is that a sigh of 25 cmH₂O in mice, which is approximately that used in our previous study (7), stretches the airway to above r_crit. In generating the fits shown in Figs. 4 and 5, we used a value for r_crit that seemed reasonable and then determined the corresponding value of k_break, but these are not the only values that provide essentially equivalent fits.

An important finding relative to the effects of a DI was that to obtain accurate fits to the experimental data in Fig. 5, we found it necessary to reduce the magnitude of the DI substantially when simulating the DIs at 5–6 s and 8–9 s compared with that at 2–3 s. This suggests that the stretch applied to the airway wall, and thus to the ASM, was correspondingly less during the later two DIs in the mice in our previous study (7). We speculate that this was because active ASM force was already well developed at 5–6 and 8–9 s after methacholine injection, in contrast to 2–3 s after injection when contraction had barely started (Fig. 5). Presumably, this resulted in the airway being stiffer at the later time points and, thus, less easily expanded by a DI, possibly as a result of more shunt compression of the gas in the ventilator cylinder or because the applied volume was redirected toward more compliant distal regions of the lung. Whatever the reason, this finding is interesting because reduced stretch of a stiffer airway wall has been suggested as a reason why a DI is often less effective at reversing bronchoconstriction in asthmatic subjects compared with normal subjects (24). Our modeling results, thus, suggest that airway stiffening during active bronchoconstriction may protect the airway against disruption of its contractile machinery.

It is important to note that the DI-induced reduction in responsiveness apparent in Figs. 4 and 5 is different from the phenomenon of bronchoreversal that has been extensively studied by others (25, 31, 32). Bronchoreversal refers simply to the expanding of constricted airways by the increased forces of airway-parenchymal interdependence occurring during a DI. In fact, bronchoreversal is precisely the phenomenon that accounts for the dramatic dependence of airway responsiveness on PEEP, exemplified in Fig. 2, which is mimicked by our model. This PEEP dependence arises from the outward tethering of the airway wall by its parenchymal attachments, which opposes the shortening of the circumferentially oriented ASM within the wall. When the ASM is stretched moderately in this way, it merely undergoes eccentric contraction, so myosin cross bridges are pulled off their actin binding sites to allow the ASM cell to lengthen, but the contractile fibers themselves are not otherwise compromised. It must be noted, however, that even bronchoreversal will not occur if the strain is too small; Norris et al. (27) recently showed that strips of ASM tissue must be strained by a little more than 3% to achieve significant detachment of cross bridges.

Thus, bronchoreversal does not involve any effect on ASM contractility per se, unlike the related phenomenon known as bronchoprotection, which refers to the reduction in responsiveness incurred when a DI is applied immediately before administration of bronchial agonist. Bronchoprotection was first demonstrated in human subjects by Malmberg et al. (25) and has subsequently been studied by other investigators in both human (31, 32) and animals (16), but its importance remains controversial since the effect tends to be rather subtle and may be reduced in certain situations, such as obesity (30). On the other hand, administering a significant stretch to the ASM, while it is actively contracting can cause a major reduction in subsequent responsiveness in ASM strips in vitro (6), and the present study provides support for the notion that a corresponding effect may occur in vivo (Figs. 4 and 5). To our knowledge, there is no published evidence of such an effect in humans. This is perhaps because it would require a maneuver that is not easily applied, namely, the sudden administration of a substantial dose of agonist to the lungs followed by a DI precisely timed to occur as the response to the agonist is in the process of developing. Nevertheless, we hypothesize that such a maneuver would disrupt the contractile machinery within the human ASM cell in a dose-dependent manner, as described empirically by Eq. 6.

Our model obviously has numerous limitations reflecting the practical necessity of deciding which phenomena must be represented to reasonably account for experimental observation and which can be safely neglected as being of second-order importance. For example, we modeled bronchoconstriction in terms of a single airway rather than attempting to represent the myriad branches of the entire airway tree in a mouse lung. Thus, our model has no capacity to account for the effects of regional heterogeneity throughout the lung on the measurement of R_aw. This is important because deep inflations have the potential to alter the distribution of ventilation heterogeneities throughout the lung (12), something that might be testable by observing how repeatable the results of a DI maneuver are in a given subject, particularly, if any ventilation defects that develop in the lung during bronchoconstriction can be imaged at the same time. In addition, we have not considered other effects known to manifest at the level of the ASM strip, such as the dependence of force on tissue length and its capacity to adapt to length with time (13, 14, 29), as well as possible effects of ASM strain rate on responsiveness (32). With regard to the latter point, our previous experiments in mice suggest a preeminent role for strain itself (7). Nevertheless, we cannot discount the potential importance of other factors, which for now must remain subjects for future investigation.

Our findings may have clinical significance. Here, we have modeled the results of previously published DIs of the mouse lung that were rather extreme; roughly, the entire vital capacity of the animal was forced into its lungs within a small fraction of a second (7). It is difficult to imagine this maneuver ever being employed in human subjects, and even in animal models it perhaps stretches the bounds of physiological relevance. Therefore, it remains to be seen whether there exists a safe combination of ASM strain and strain rate, inducible by a DI in human subjects that could attenuate bronchoconstriction to subsequent airway challenges. If such a maneuver could be identified, it might have utility in the management of bronchospasm in asthmatic subjects.

In conclusion, we have shown via computational modeling that a putative mechanism for the strain-induced temporary ablation of contractility observed in ASM strips in vitro can potentially explain the still controversial matter of how DIs of the lung attenuate airways responsiveness in vivo. Our model suggests that applying a DI to the constricted lung results in recovery of bronchoconstriction over two time scales that differ by roughly an order of magnitude. The faster time scale reflects reattachment of myosin cross bridges pulled off their actin binding sites by the stretch of the ASM, while the slower time scale reflects reassembly of the contractile apparatus disrupted at some structural level.

GRANTS

This study was supported by National Institutes of Health Grants R01 HL-124052 and R01 HL-130847.

DISCLOSURES

J. H. T. Bates is a member of the Board of Advisors and a minor shareholder in Oscillavent, LLC.

AUTHOR CONTRIBUTIONS

J.H.B. conceived and designed research; J.H.B. and V.R. performed experiments; J.H.B. and V.R. analyzed data; J.H.B. interpreted results of experiments; J.H.B. prepared figures; J.H.B. drafted manuscript; J.H.B. and V.R. edited and revised manuscript; J.H.B. and V.R. approved final version of manuscript.

APPENDIX A: AIRWAY-PARENCHYMAL INTERDEPENDENCE AND TRANSMURAL PRESSURE

The following derivation can be found in our previous publication (8). We repeat it here for completeness.

If the mechanical properties of the airway wall were to permit the airway to passively expand at the same rate as the parenchyma when the lung is inflated, then the parenchyma would expand isotropically, in which case P_tm would equal P_tp (Fig. 1) and airway radius would have the value r_iso. In this case, the linear dimensions of the parenchyma at every location throughout the lung would scale with the cube root of lung volume. In addition, if E_L is independent of lung volume, then the linear dimensions of both the parenchyma and r_iso increase with the cube root of P_tp, which means that

$$\frac{r_{iso}}{r_{isoTLC}}={\left(\frac{{\text{P}}_{tp}}{{\text{P}}_{tpTLC}}\right)}^{\frac{1}{3}}$$ (A1)

where r_isoTLC is the airway radius at total lung capacity (TLC) embedded in isometrically expanding parenchyma, and P_tpTLC is P_tp at TLC (defined as the nominal maximum volume of the lung achieved at P_tp = 30 cmH₂O).

In general, however, the parenchyma around an airway does not inflate isotropically because the airway and parenchymal stiffnesses are not perfectly matched. Furthermore, the addition of F_A induces additional parenchymal distortion in the vicinity of the airway wall to a degree that falls off rapidly with radial distance (2, 20). The difference between the actual radius, r, and r_iso causes P_tm to be different than P_tp by an amount, ΔP_tm, (Fig. 1) that can be approximated as (20)

$$\begin{array}{c}{\text{P}}_{tm}={\text{P}}_{tp}+\Delta {\text{P}}_{tm}\\ ={\text{P}}_{tp}\left[1.43\frac{r_{iso}}{r}-0.43\right]\\ ={\text{P}}_{tp}\left[1.43\frac{r_{isoTLC}}{r}{\left(\frac{{\text{P}}_{tp}}{\text{P}{}_{tpTLC}}\right)}^{\frac{1}{3}}-0.43\right]\end{array}$$ (A2)

where the last line in Eq. A2 was obtained by substituting for r_iso from Eq. A1.

APPENDIX B: AIRWAY WALL STIFFNESS

We previously modeled the passive stiffness of the airway by assuming it to have a radius vs. transmural pressure relationship that is functionally identical to that of pure linearly elastic parenchyma, except scaled by a factor 0 < k < 1 (8). That is,

$$\begin{array}{c}\frac{r}{r_{TLC}}=k+(1-k){\left(\frac{{\text{P}}_{tm}}{\text{P}{}_{tpTLC}}\right)}^{\frac{1}{3}}\\ =k+(1-k){\left(\frac{{\text{F}}_P}{r\text{P}{}_{tpTLC}}\right)}^{\frac{1}{3}}\end{array}$$ (B1)

where the second line is obtained by substitution from Eq. 1 with F_A = 0. Solving Eq. B1 for F_P gives

$${\text{F}}_P={\left[\frac{\frac{r}{r_{TLC}}-k}{(1-k)}\right]}^{3}r\text{P}{}_{tpTLC}$$ (B2)

which, when substituted into Eq. 1, gives

$${\text{P}}_{tm}=\frac{{\left[\frac{\frac{r}{r_{TLC}}-k}{(1-k)}\right]}^{3}r\text{P}{}_{tpTLC}+{\text{F}}_A}{r}$$ (B3)

Equating Eqs. A2 and B3 and solving for F_A gives

$$\begin{array}{c}{\text{F}}_A=-{\left[\frac{\frac{r}{r_{TLC}}-k}{(1-k)}\right]}^{3}r\text{P}{}_{tpTLC}-0.43r\text{P}{}_{tp}+1.4r_{iso}{\text{P}}_{tp}\\ ={\left(a-br\right)}^{3}r-cr+d\end{array}$$ (B4)

where the various quantities in the first line are grouped for convenience into the constants a, b, c, and d in the second line, assuming that P_tp and thus r_iso are fixed. Differentiating Eq. B4 with respect to r gives the elastic stiffness of the airway as

$$\begin{array}{c}\frac{d{\text{F}}_A}{dr}=-3b{\left(a-br\right)}^{2}r+{\left(a-br\right)}^3-c\\ =-4b^2{r}^3+8a{b}^2{r}^2-5a^{2}br+a^3+c\end{array}$$ (B5)

This stiffness is a cubic function of r, so it has at least one and possibly three real roots corresponding to values of r for which stiffness is zero. Thus, there exists at least one value of r for which an infinitesimal change in F_A will give rise to an infinite rate of change in r, which is the very definition of mechanical instability.

Now let the airway behave from an elastic perspective like a Hookean spring both in extension and compression according to

$${\text{F}}_P=2\pi E_w\left(r-r_0\right)$$ (B6)

Equating Eqs. A2 and B6 and solving for F_A gives

$${\text{F}}_A=1.43r_{iso}{\text{P}}_{tp}+2\pi r_0{E}_w-r\left(2\pi E_w+0.43{\text{P}}_{tp}\right)$$ (B7)

Differentiating with respect to r, for fixed values of P_tp and r_iso, now gives

$$\frac{d{\text{F}}_A}{dr}=-\left(2\pi E_w+0.43{\text{P}}_{tp}\right)$$ (B8)

which can never be zero for P_tp > 0. In other words, an airway with the elastic behavior defined by Eq. 2 is never unstable for P_tp > 0.

The passive pressure-radius relationship of the airway wall is found from Eq. B6 via the Laplace law to be

$${\text{P}}_{tm}=\frac{2\pi E_w\left(r-r_0\right)}{r}$$ (B9)

Airway stiffness is the derivative of this expression with respect to r, so the rate of change of stiffness with r is the second derivative, namely

$$\frac{d^2{\text{P}}_{tm}}{d{t}^2}=\frac{-4\pi E_w{r}_0}{r^3}$$ (B10)

which is negative for all r. This means that the magnitude of airway stiffness increases as r decreases below r₀, which corresponds to strain stiffening of the airway wall in compression just as experimental observations suggest (23). By the same token, however, Eq. B10 also implies strain softening of the airway in expansion, which is in direct contradiction to the experiment and contrary to the behavior of biological soft tissue, in general. Therefore, to make the behavior during expansion conform more closely to experimental observation (23), we use the mirror image of the relationship during compression described by Eq. B9 to arrive at the following P_tm-r relationship for the airway during both compression (r ≤ r₀) and expansion (r < r₀). That is,

$$\begin{array}{c}{\text{P}}_{tm}=\frac{2\pi E_w\left(r-r_0\right)}{r};r\le r_0\\ =\frac{2\pi E_w\left(r-r_0\right)}{2r_0-r};r>r_0\end{array}$$ (B11)