In this project we will consider the mathematical modelling of the cornea, which is the clear front portion of the eye. The cornea is multiscale in nature and the structural composition is directly related to its functionality.  The corneal microstructure comprises collagen fibres and water which are constituents of the stroma which is enclosed by a thick capsular layer.  The cornea's fluid content and the structure and arrangement of its collagen fibres affects its biomechanics, morphology, and transparency.  Due to the constituents being both solid and fluid, the cornea is a perfect candidate to be modelled via the Theory of Poroelasticity. This theory is the modelling approach used to describe the interaction between fluid flow, pressure and bulk solid deformation within a linear porous medium [1].

The cornea can be described as a poroelastic composite and therefore can be modelled via the effective governing equations developed for poroelastic composites presented by Miller & Penta (2020) [3].   This work derives the quasi-static governing equations for the macroscale behaviour of a linear elastic porous composite comprising a matrix interacting with inclusions and/or fibres, and an incompressible Newtonian fluid flowing in the pores. This model is derived via the use of the asymptotic homogenization technique [2]. 

The asymptotic homogenisation technique can be applied when there is a distinct length scale separation between the length of the cornea and the size of the fibres and the pores.  The technique derives the new macroscale model by upscaling a fluid–structure interaction problem between the elastic constituents and the fluid phase of the corneal stroma. The resulting system of partial differential equations (PDEs) is of poroelastic type and encodes the properties of the microstructure in the coefficients of the model.  This is the perfect choice of structure and modelling approach to capture the constituents of the cornea.  

Due to the cornea being enclosed by a thick capsule an undrained analysis is important.  This means that it is assumed that the fluid is behaving as a suspension that supports the fibre arrangement and does not exit the system during deformation. Therefore, even though the fluid has almost no velocity it still plays a crucial role in the mechanical behaviour. 

Defects in the capsule are an important key feature and can be caused due to injury, disease or ageing. The defect plays a crucial role in the occurrence of many eye diseases and therefore it must be accounted for when understanding the diseased cornea.  This can be implemented as a macroscale boundary condition on a poroelastic composite material.  This defect alters the fluid properties as there is now a leakage of the internal fluid and this has an important influence on the pressure profiles and risk of perforation of the eye.  

This defect can be modelled using a wound healing approach such as those found in Dale et al 1994 [7] and Machado et al 2011 [6] and similar works. The defect must heal to maintain the integrity of the eye and therefore a scar forms over the defect containing many collagen fibres.  These fibres may not be well aligned to maintain the transparency of the cornea.  In the case that the transparency of the cornea is severely affected then there is ongoing work on regenerative treatments or surgical transplantation.  

This defect also leads to a further area for investigation and modelling as this can be the entry point for bacterial invasion.  The bacteria invade the porous stroma and modify the pore structure.  This means that diseases such as corneal ulceration are incredibly difficult to medically manage hence motivating modelling approaches to better understand the behaviour and predict patient outcomes. 

During this project, you will have the opportunity to develop their skills in mathematical and numerical modelling. They will become familiar with the theory of poroelasticity, multiscale modelling and homogenisation.  This includes but is not limited to continuum mechanics, geometry, PDEs and asymptotics.  The project will also involve numerical simulations using mainly Comsol Multiphysics and Matlab, however there is no requirement that you should know how to use this software prior to the project.  Prior biological experience is not necessary; however, you should have a keen interest in learning about the physiology of the eye.  

References

[1] Raimondo Penta, Laura Miller, Alfio Grillo, Ariel Ramírez-Torres, Pietro Mascheroni, and Reinaldo Rodríguez-Ramos. "Porosity and diffusion in biological tissues. Recent advances and further perspectives." Constitutive modelling of solid continua (2020): 311-356.
[2] Raimondo Penta and Alf Gerisch. "An introduction to asymptotic homogenization." Multiscale Models in Mechano and Tumor Biology: Modeling, Homogenization, and Applications. Springer International Publishing, 2017.
[3] Laura Miller, and Raimondo Penta. "Effective balance equations for poroelastic composites." Continuum Mechanics and Thermodynamics 32.6 (2020): 1533-1557.
[4] Misson, Gary P. "Circular polarization biomicroscopy: a method for determining human corneal stromal lamellar organization in vivo." Ophthalmic and Physiological Optics 27.3 (2007): 256-264.
[5] Misson, Gary P. "The theory and implications of the biaxial model of corneal birefringence." Ophthalmic and Physiological Optics 30.6 (2010): 834-846.
[6] Machado, Maria JC, et al. "Dynamics of angiogenesis during wound healing: a coupled in vivo and in silico study." Microcirculation 18.3 (2011): 183-197.
[7] Dale, Paul D., Philip K. Maini, and Jonathan A. Sherratt. "Mathematical modeling of corneal epithelial wound healing." Mathematical biosciences 124.2 (1994): 127-147.

Further information

Please note that this opportunity will commence 1 October 2025.

You will also have the opportunity to interact and visit the Department of Civil and Environmental Engineering at the Politecnico di Milano, Italy to interact with experts in eye biomechanics.