Numerical Modelling and Analysis of Peripheral Airway Asymmetry and Ventilation in the Human Adult Lung

Abstract

We present a new one-dimensional model of gas transport in the human adult lung. The model comprises asymmetrically branching airways, and heterogeneous interregional ventilation. Our model differs from previous models in that we consider the asymmetry in both the conducting and the acinar airways in detail. Another novelty of our model is that we use simple analytical relationships to produce physiologically realistic models of the conducting and acinar airway trees. With this new model, we investigate the effects of airway asymmetry and heterogeneous interregional ventilation on the phase III slope in multibreath washouts. The model predicts the experimental trend of the increase in the phase III slope with breath number in multibreath washout studies for nitrogen, SF 6 and helium. We confirm that asymmetrical branching in the acinus controls the magnitude of the first-breath phase III slope and find that heterogeneous interregional ventilation controls the way in which the slope changes with subsequent breaths. Asymmetry in the conducting airways appears to have little effect on the phase III slope. That the increase in slope appears to be largely controlled by interregional ventilation inhomogeneities should be of interest to those wishing to use multibreath washouts to detect the location of the structural abnormalities within the lung.

Modelling structural determinants of ventilation heterogeneity: a perturbative approach

“…The concentration of inert gas on the network is then calculated using a one-dimensional advectiondiffusion equation (S1 File §1.3) that accounts for transport into the alveolar sacs (similar to [11,12,21]).…”

“…where d i is the airway diameter and the constant C is defined as C = 1.08 for inspiration and C = 0.37 for expiration. In the acinar ducts, pure molecular diffusion is modelled but with a modified airway cross-section, as described in [12],…”

“…These ODEs are solved directly using the Eigen [22] factorisation routine 'PartialPivLU' in C++, as outlined in detail in S1 File §1.2. The concentration of inert gas on the network is then calculated using a one-dimensional advection-diffusion equation (S1 File §1.3) that accounts for transport into the alveolar sacs (similar to [23,14,15]). Transport in the acinar airways is included in this description as the details of its structure are important to accurately model gas transport [24,25,26,27].…”

“…Within each lobar region the transport equation is identical to the 'trumpet' representation introduced in [28] (S1 File §1.4), with effective diffusion in the conducting region given by Taylor-like dispersion described in [29]. In the acinar region, pure molecular diffusion is modelled but with a modified airway cross-section, as described in [15]. The transport partial differential equations (PDEs) are discretised using a finite volume method (detailed in S1 File §2), and solved iteratively using the Eigen [22] 'BiCGSTAB'…”

“…According to cast estimates there are on the order of 10 4 − 10 5 branches in the conducting zone [8,9], and on the order of 10 7 including the acinar ducts. There have been numerous approaches to resolve this problem, such as using a symmetric description with an effective diffusion coefficient to account for heterogeneous ventilation [10], compartmental models with asynchronous or asymmetric ventilation [11,12,13], modelling a single heterogeneous acinus [14,15], and replacing the acini with well mixed units or symmetric models reducing the computation to ∼ 10 4 branches [16,17,18,19,20]. Models have considered both heterogeneity in healthy lungs (such that average diameters and lengths converge to the healthy lung data) and pathological heterogeneity (where branches are constricted on average).…”