A Distribution-Moment Approximation for Coupled Dynamics of the Airway Wall and Airway Smooth Muscle

Abstract

Asthma is fundamentally a disease of airway constriction. Due to a variety of experimental challenges, the dynamics of airways are poorly understood. Of specific interest is the narrowing of the airway due to forces produced by the airway smooth muscle wrapped around each airway. The interaction between the muscle and the airway wall is crucial for the airway constriction that occurs during an asthma attack. Although cross-bridge theory is a well-studied representation of complex smooth muscle dynamics, and these dynamics can be coupled to the airway wall, this comes at significant computational cost-even for isolated airways. Because many phenomena of interest in pulmonary physiology cannot be adequately understood by studying isolated airways, this presents a significant limitation. We present a distribution-moment approximation of this coupled system and study the validity of the approximation throughout the physiological range. We show that the distribution-moment approximation is valid in most conditions, and we explore the region of breakdown. These results show that in many situations, the distribution-moment approximation is a viable option that provides an orders-of-magnitude reduction in computational complexity; not only is this valuable for isolated airway studies, but it moreover offers the prospect that rich ASM dynamics might be incorporated into interacting airway models where previously this was precluded by computational cost.

Figure 1

Underlying bistability response of a (static-force) peripheral airway to an increase in force (dashed gray line), as well as sample trajectories for each model (black lines) in the absence and presence of activated (dynamic) ASM.

Figure 2

(A) Predicted monotonic increase in closure points for both models due to increasing static external pressure. The DM shows great quantitative and qualitative agreement to the full MoC model for this static case, with a 2 to 5% difference, as well as having absolute differences in the range 0–1.2. (B and C) Comparison of the sum of the attached myosin populations for both models at P₀ = 5 cmH₂O and P₀ = 25 cmH₂O, respectively. These distribution plots were taken at the first instant after the closure threshold, $\overline{r}$, was passed. At low external pressure, the moment matching of the DM is a good approximation to the exact distribution of the MoC (B), but it deviates as the external pressure is increased, as seen in (C), which leads to qualitative differences in ${\lambda }^{\text{∗}}$.

Figure 3

(A) Individual radius trajectories for each model for fixed-external-pressure oscillations; the transition from trajectories that remain in the open state to those that transition to closed state occurs at ${\lambda }^{\text{∗}}$. (B) Comparison of the sum of the attached myosin populations for both models at frequency f = 0.33 Hz, amplitude α = 25 cmH₂O, and minimal external pressure P_min = 2 cmH₂O. The distribution plot was also taken at the first instant the closure threshold, $\overline{r}$, was passed.

Figure 4

Percentage error in closure points (${\lambda }_{\text{DM}}^{\text{∗}}$ versus ${\lambda }_{\text{MoC}}^{\text{∗}}$) for a range of external-pressure waveforms within the three-parameter space, $\{$amplitude$\}$ × $\{$frequency$\}$ × $\{$minimal external pressure$\}$. (A–F) Selected slices of the parameter space. The locations of these slices are noted in the main figure. To see this figure in color, go online.

Figure 5

ASM-only solutions, without the airway wall. A length-controlled protocol of isometric contraction was followed by a period of 1% length oscillations at 0.33 Hz, and then increased length oscillations at 4%. (A and B) Shown are the bulk observables of force, stiffness, and phosphorylation for DM and MoC, respectively. (C and D) Breakdown of the state-population transitions for the DM and MoC models, respectively.