Fig. 1.

Configurations during finite growth of the airway wall. Top row: Schematic of the evolution of the airway wall geometry with time, with the colours correspoding to tissue constituent volume fraction. Middle rows: Schematics of the airway geometry in the grown and stressed configurations, at the indicated time. At initial time $t_0$, the airway wall is defined by inner radius $R_1$, interface radius $R_{\mathrm{int}}$, and outer radius $R_2$ in ${\mathit{\kappa }}_0$, the original, stress-free reference configuration. Airway wall growth is described by the mapping, $\mathbf{G}$, from the original configuration to ${\mathit{\kappa }}_g$, the grown, zero-stress configuration, defined by grown radii ${\xi }_1,{\xi }_{\mathrm{int}}$ (not shown), and ${\xi }_2$ at time $t_1$. The airway is deformed via $\mathbf{F}$ to configuration $\mathit{\kappa }$, the current, stressed configuration, defined by radii $r_1,r_{\mathrm{int}}$, and $r_2$, subject to pressure boundary conditions $P_1$ and $P_2$. Bottom row: Schematics of transmural mechanical stress distributions, generated from the active contraction and/or the elastic deformation, and contractile agonist. The mechanical stress, $\tau$, modulates certain rates in the constitutive mass balance equations, e.g. (2.18, 2.21), to influence growth, $\mathbf{G}$