On the mechanics of growing thin biological membranes

Abstract

Despite their seemingly delicate appearance, thin biological membranes fulfill various crucial roles in the human body and can sustain substantial mechanical loads. Unlike engineering structures, biological membranes are able to grow and adapt to changes in their mechanical environment. Finite element modeling of biological growth holds the potential to better understand the interplay of membrane form and function and to reliably predict the effects of disease or medical intervention. However, standard continuum elements typically fail to represent thin biological membranes efficiently, accurately, and robustly. Moreover, continuum models are typically cumbersome to generate from surface-based medical imaging data. Here we propose a computational model for finite membrane growth using a classical midsurface representation compatible with standard shell elements. By assuming elastic incompressibility and membrane-only growth, the model a priori satisfies the zero-normal stress condition. To demonstrate its modular nature, we implement the membrane growth model into the general-purpose non-linear finite element package Abaqus/Standard using the concept of user subroutines. To probe efficiently and robustness, we simulate selected benchmark examples of growing biological membranes under different loading conditions. To demonstrate the clinical potential, we simulate the functional adaptation of a heart valve leaflet in ischemic cardiomyopathy. We believe that our novel approach will be widely applicable to simulate the adaptive chronic growth of thin biological structures including skin membranes, mucous membranes, fetal membranes, tympanic membranes, corneoscleral membranes, and heart valve membranes. Ultimately, our model can be used to identify diseased states, predict disease evolution, and guide the design of interventional or pharmaceutic therapies to arrest or revert disease progression.

Keywords: biological membranes; finite elements; growth; membrane; mitral valve; shell; surface growth.

Figure 1

Creep and relaxation tests in growing thin biological membranes. Evolution of total area stretch ϑ, elastic area stretch ϑ^e, and growth area stretch ϑ^g under force control, left, and displacement control, right. Horizontal dashed lines represent the elastic stretch limit ϑ^crit beyond which growth is activated, and the maximum area growth ϑ^max to which growth is limited. Forces, left, and displacements, right, are increased linearly up to the vertical dashed lines and then held constant. Under force control, for a constant elastic stretch, the total stretch increases gradually as the growth stretch increases, left. Under displacement control, for a constant total stretch, the elastic stretch decreases gradually as growth stretch increases, right.

Figure 2

Pinched thin-walled cylinder: Irreversible nature of growth. As the cylinder is deformed, it is primarily subject to bending, while its lateral walls experience a moderate area stretch. This area stretch gradually causes the membrane to grow. Membrane growth varies regionally with ϑ^g = 1.05 corresponding to 5% area growth at the lateral wall and ϑ^g = 1.00 corresponding to no area growth at the top and bottom walls. Once the load is removed, the cylinder relaxes but the growth remains. The inhomogeneous growth pattern induces residual stresses, which prevent the cylinder from returning to its original circular configuration.

Figure 3

Stretched bilayered thin film: Irreversible nature of growth under isotropic and anisotropic conditions. As the film is biaxially stretched, the bottom layer, shown here, gradually grows up to ϑ^g = 2.2 corresponding to more than 100% area growth, while the top layer, not shown, deforms elastically at to ϑ^g = 1.0 and stores elastic energy. Once the load is removed, the bilayered film deforms out of plane. Isotropy of the elastic layer results in a symmetric out-of-plane coiling, top row. Anisotropy of the elastic layer results in an anisotropic out-of-plane coiling, bottom row.

Figure 4

Pressurized thin membrane: Regional variation of growth under isotropic and anisotropic conditions. As the membrane is inflated, it locally grows up to ϑ^g = 1.05 corresponding to 5% area growth, while other regions deform elastically at ϑ^g = 1.0 and do not grow. The isotropic membrane shows significant area growth and develops spherical protrusions, top row. Under the same pressure, the anisotropic membranes with vertical fibers, middle row, and with horizontal fibers, bottom row, develop anisotropic protrusions.

Figure 5

Chronically Stretched Mitral Leaflet: Mitral leaflet adaptation to annular dilation and papillary muscle displacement as seen in ischemic cardiomyopathy. Anterior mitral leaflet in the reference configuration at end diastole with the mitral annulus superimposed in white, top row. Acute elastic area stretch ϑ^e, middle row, in response to annular dilation of λ=1.2 and λ=1.5, in combination with asymmetric displacement of the posterior-medial papillary muscle, first and second column, and symmetric displacement of both papillary muscles, third and fourth column, by δ=5mm. Chronic area growth ϑ^g, bottom row, regionally exceed values of ϑ^g = 1.5 indicating more than 50% area growth upon leaflet adaptation.