3D Homogenized Constrained Mixture Model for Eye Growth and Remodeling - A Model of Staphyloma Formation

The homogenized constrained mixture model is implemented in Abaqus through a coupled User-defined MATerial subroutine (UMAT). This computational model facilitates the simulation of various Growth & Remodeling (G&R) processes in soft biological tissues within the body. Originally designed for aneurysm growth simulations (Braeu et al. 2017 and Braeu et al. 2018), it has been adapted for cardiac G&R (Amadeus et al. 2023).

In this context, we have utilized the homogenized constrained mixture model to simulate ophthalmic G&R processes. Specifically, our focus was on investigating whether G&R theory could provide insights into staphyloma formation resulting from local scleral weakening. We simulated three different G&R scenarios by adjusting the rate of mass turnover (i.e. varying the growth parameter k_σ) and the type of growth over a duration of 13.7 years (approximately 5000 days).:

• Scenario (1): A growth parameter of k_σ = 2∙10^-4 1/days in combination with volumetric growth in thickness direction
• Scenario (2): A growth parameter of k_σ = 2∙10^-3 1/days in combination with volumetric growth in thickness direction
• Scenario (3): A growth parameter of k_σ = 2∙10^-3 1/days in combination with mass density growth

To run the different scenarios via the console, use the following Abaqus command (assuming your current directory is the repository):

abaqus job=GR_Eye_ScenarioX user=umat.f

Note: Replace "X" with the scenario number you wish to execute.

For further details and a derivation of the equations, please refer to our publication: Braeu et al. 2024.

Left column: Final shape of the posterior pole of the eye after 13.7 years of G&R. (1), (2), and (3) refer to the simulated G&R scenarios. Right column: Maximum growth stimulus (i.e. the difference between collagen fiber stress and homeostatic stress normalized w.r.t. the homeostatic stress) in the PPS over a time span of 13.7 years. Timepoint 0 represents the onset of G&R processes.