Morpho-elasticity of intestinal villi

Abstract

Villi are ubiquitous structures in the intestine of all vertebrates, originating from the embryonic development of the epithelial mucosa. Their morphogenesis has similar stages in living organisms but different forming mechanisms. In this work, we model the emergence of the bi-dimensional undulated patterns in the intestinal mucosa from which villi start to elongate. The embryonic mucosa is modelled as a growing thick-walled cylinder, and its mechanical behaviour is described using an hyperelastic constitutive model, which also accounts for the anisotropic characteristics of the reinforcing fibres at the microstructural level. The occurrence of surface undulations is investigated using a linear stability analysis based on the theory of incremental deformations superimposed on a finite deformation. The Stroh formulation of the incremental boundary value problem is derived, and a numerical solution procedure is implemented for calculating the growth thresholds of instability. The numerical results are finally discussed with respect to different growth and materials properties. In conclusion, we demonstrate that the emergence of intestinal villi in embryos is triggered by a differential growth between the mucosa and the mesenchymal tissues. The proposed model quantifies how both the geometrical and the mechanical properties of the mucosa drive the formation of previllous structures in embryos.

Figure 1.

Schematic of the intestinal wall: the inner layer called mucosa (which comprises the epithelium, the lamina propria and the muscularis mucosae), the submucosa (made of dense irregular connective tissue), the muscularis propria (oriented smooth muscles) and the outer serosa layer are evidenced. (Online version in colour.)

Figure 2.

Scanning electron micrographs of emerging villi in the jejunum of turkey embryos (reproduced with permission from [4]). The micrographs are taken at 21 days of incubations, and shown using scales of (a) 100 μm and (b) 10 μm for outlining of the morphology of the bi-dimensional undulated pattern at the free surface of the mucosa.

Figure 3.

Geometrical model of the mucosa growth process: the mapping χ transforms the position X in the reference configuration into the position x in the actual configuration . (Online version in colour.)

Figure 4.

The multiplicative decomposition of the deformation gradient F: the growth component F_g defines a natural grown state in which geometrical incompatibilities are allowed, and the elastic component F_e restores the compatibility of the tissue deformation. (Online version in colour.)

Figure 5.

Scheme of the deformation gradient F and the elastic deformation gradient F_e, after the introduction of the incremental displacement gradient Γ. The term represents the projection of the incremental nominal stress in the perturbed configuration . (Online version in colour.)

Figure 6.

Morphology of the intestinal mucosa after imposing a perturbation of the axial-symmetric solution of the elastic problem, having the form of equation (4.21). The geometrical parameters are r₀ = 2, r_i = 1.5, L = 5, m = 7, k_z = 5 and ε = 0.15. (Online version in colour.)

Figure 7.

Implementation of the numerical scheme: after a first iteration on the aspect ratio H, it follows a second iteration on the bifurcation parameter g(H). In this second cycle, the solution is numerically integrated until the condition D((g_τ(r = r_i)) is satisfied and the threshold value for the parameter g_τ is obtained. (Online version in colour.)

Figure 8.

Marginal stability curves for isotropic growth showing the isotropic growth rate g_r = g_z at different modes k_z = m = 2, 5, 10, 15. (Online version in colour.)

Figure 9.

Marginal stability curves for anisotropic growth showing the radial growth g_r (a, setting g_z = 1) and the longitudinal growth g_z (b, setting g_r = 1) thresholds, calculated at different modes m = k_z = 2, 5, 7, 10, 15. (Online version in colour.)

Figure 10.

Marginal stability curves for anisotropic growth showing the radial growth g_r (a, setting g_z = 1) and the longitudinal growth g_z (b, setting g_r = 1) thresholds, calculated at different modes m = 2, 5, 7, 10, 15 and fixed k_z = 10. (Online version in colour.)

Figure 11.

Marginal stability curves for anisotropic growth showing the radial growth g_r (a, setting g_z = 1) and the longitudinal growth g_z (b, setting g_r = 1) thresholds, calculated at different modes k_z = 2, 5, 7, 10, 15 and fixed m = 10. (Online version in colour.)

Figure 12.

Marginal stability curves showing the critical volume increase J_g at modes k_z = m = 10 for isotropic (g_r = g_z) and anisotropic (g_r = 1 and g_z = 1) growth processes. (Online version in colour.)

Figure 13.

Marginal stability curves for anisotropic growth showing the radial growth g_r (a, setting g_z = 1) and the longitudinal growth g_z (b, setting g_r = 1) thresholds at modes k_z = m = 5. The material anisotropy ratio is fixed at k₁/μ = 10, while the curves are shown at different cross-ply fibre angles α = (0, π/6, π/4, π/3). (Online version in colour.)

Figure 14.

Marginal stability curves for anisotropic growth showing the radial growth g_r (a, setting g_z = 1) and the longitudinal growth g_z (b, setting g_r = 1) thresholds at modes k_z = m = 5. The cross-ply fibre angle is fixed at α = π/4, while the curves are shown at different material anisotropy ratios k₁/μ = (0.1,1,10). (Online version in colour.)

Figure 15.

Instability thresholds in terms of volume increase due to isotropic (a) and anisotropic (b, g_r = 1) growth processes. The curves referring to the circumferential and longitudinal folding are taken from [25]. (Online version in colour.)