Pattern Selection in Growing Tubular Tissues

Tubular organs display a wide variety of surface morphologies including circumferential and longitudinal folds, square and hexagonal undulations, and finger-type protrusions. Surface morphology is closely correlated to tissue function and serves as a clinical indicator for physiological and pathological conditions, but the regulators of surface morphology remain poorly understood. Here, we explore the role of geometry and elasticity on the formation of surface patterns. We establish morphological phase diagrams for patterns selection and show that increasing the thickness or stiffness ratio between the outer and inner tubular layers induces a gradual transition from circumferential to longitudinal folding. Our results suggest that physical forces act as regulators during organogenesis and give rise to the characteristic circular folds in the esophagus, the longitudinal folds in the valves of Kerckring, the surface networks in villi, and the crypts in the large intestine.

Figure 1

Top: circumferential folding (left) and intestinal villi (right) in developing mouse intestine (adapted from Ref. [5]). Bottom: circular folding in normal and asthmatic airway (left, adapted from Ref. [7]); longitudinal folding in eosinophilic esophagitis (right, adapted from Ref. [8]).

Figure 2

Geometry of two-layered tubular tissue with inner endoderm and outer mesoderm in ungrown reference configuration ${\mathcal{B}}_0$ and grown residually stressed configuration ${\mathcal{B}}_a$.

Figure 3

Critical differential growth ratios $g_{\text{en}}^{\text{cr}}/g_{\text{me}}^{\text{cr}}$ (left), circumferential mode $m^{\text{cr}}$ (center), and longitudinal mode $k_z^{\text{cr}}$ (right) for varying aspect ratios of the inner layer $H_{\text{en}}$ at fixed $H_{\text{me}}=1.8$. Marginal stability curves are shown for elastic ratios ${\mu }_{\text{en}}/{\mu }_{\text{me}}=\{1/3,1/5,1/5.3,1/6,1/10\}$. For the sake of clarity we depict $k_z$ as a continuous parameter, assuming $L\gg R_o$.

Figure 4

Morphological phase diagrams of emerging epithelial surface patterns from numerical simulations in the postbuckling regime. The different surface morphologies are displayed for varying geometrical aspect ratios $H_{\text{en}}$ and $H_{\text{me}}$, at fixed ${\mu }_{\text{en}}/{\mu }_{\text{me}}=1/5$ (left) and for varying elastic ratios ${\mu }_{\text{en}}/{\mu }_{\text{me}}$ and geometries $H_{\text{en}}$ at fixed $H_{\text{me}}=1.8$ (right). The critical circumferential mode $m^{\text{cr}}$ and longitudinal wave number $k_z^{\text{cr}}$ correspond to the results of the linear stability analysis.