Summary: Morphoelasticity is a framework commonly employed to describe tissue growth. Its central premise is the decomposition of the deformation gradient into the product of a growth tensor and an elastic tensor. In this paper we present a 2D finite element method to solve compressible morphoelasticity problems in arterial cross sections. The arterial wall is composed of three layers: the intima, media and adventitia. The intima is allowed to grow isotropically while the areas of the media and adventitia are approximately conserved. All three layers are modeled as Holzapfel-Gasser-Ogden type anisotropic hyperelastic materials with different mechanical parameters.
This paper consists of three main contributions. First, a new energy functional that underpins the finite element method for morphoelasticity is presented. The bulk energy is associated with elastically deforming grown elements, rather than elements in the reference configuration which is the usual practice in classical hyperelasticity. A surface energy describes live pressure loading through a lumen blood pressure. We derive the equivalent weak form and show that stationary points correspond to the usual boundary value problem for mechanical equilibrium.
Second, we present the details of a displacement-based morphoelastic finite element method. The primary variable for our method is the deformation, which we separate into two groups (horizontal and vertical deformations). We arrive at a system of nonlinear algebraic equations that are organized into self-contained blocks which can be assembled independently. We present a method to compute the Jacobian of this system which is fed into a Newton algorithm. The direction of collagen fibers in each layer is computed off-line by interpolating tangent vectors from nearby boundaries and a minimum norm solution for the grown and deformed artery is found by the QR factorization. This single-field approach produces reasonable numerical results, although the locking phenomenon is clearly present. For axisymmetric meshes, our position and stress fields are validated by the solution obtained from solving a system of ordinary differential equations.
Finally, we use the code to simulate intimal thickening in pseudo-realistic arterial cross sections. We find that (i) Oscillations in the lumen-intima interface are unstable and amplify with growth (ii) Glagov remodeling occurs with lumen area initially increasing and then decreasing with growth and (iii) Maximum stresses migrate from the lumen-intima interface to the media-intima interface so that the media eventually bears the maximum stress.