Optical modelling of the human retina

The human retina is a complex structure in the eye that is responsible for the sense of vision. It is part of the central nervous system, composed by several layers, among which the outer nuclear layer that comprises the cells bodies of light sensitive photoreceptor cells (rods and cones). There are a number of eye-related pathologies that can be identified by the detailed analysis of the retinal layers. These pathologies can be diagnosed more conclusively with the help of the increasingly popular noninvasive optical imaging technique – optical coherence tomography (OCT).

Problem description

Diabetes mellitus is one of the most prevalent diseases in developed countries, estimated to affect 8.5% of the European population, according to recently collected data. A resulting complication of diabetes, diabetic macular edema (DME), is a major cause of visual loss in diabetic patients.

DME is defined as an increase in retinal thickness due to fluid accumulation that can be intra- or extra-cellular. In intra-cellular edema, cells have increased fluid intake, becoming enlarged. Extra-cellular edema, in contrast, results from fluid accumulation outside the cell, generally as a consequence of the breakdown of the blood-retinal barrier and subsequent leakage into the retinal space. Distinguishing which case is present or more prevalent in a patient’s eye at an early stage is usually not straightforward.

A common method to assess the progression of DME in patients is to monitor their retinal thickness, e.g. with OCT. This technology’s working principle is analogous to ultrasound, but it uses light instead of sound to locate subtle differences in the tissue being analyzed. Discontinuities in the refractive index of the tissue give rise to light scattering, with some light backscattered to the detector. Factors such as the shape and size of the scatterer, wavelength of the incident light and refractive index differences have an impact on the amount of backscattered light. During a scan, the OCT machine directs a light beam into the retina and extracts, through interferometry, the backscattered light intensity of retinal structures and their depth location in an A-scan. By transversely moving the light beam, several A-scans can be collected into a cross-sectional image – a B-scan. Usually, several cross- sectional images are acquired by probing an azimuthal direction and combined into a volume.

Using OCT, patients with DME can be identified, when contrasted with healthy controls, as they exhibit an increased retinal thickness. However, OCT is still unable to directly assess changes at the cellular level. In this work, we aim to identify and understand the microscopic changes that lead to the differences in the OCT data between healthy and diseased cases, which are not possible to detect through direct observation.

Modelling and computational challenges

In order to better understand the information carried in an optical coherence tomography, it is crucial to study in detail the behaviour of the electromagnetic wave as it travels through the sample. Several different models have been developed to describe the interactions of the electromagnetic field with biological structures. The first models were based on single-scattering theory, which is restricted to superficial layers of highly scattering tissue in which only single scattering occurs. Simulating the full complexity of the retina, in particular the variation of the size and shape of each structure, distance between them and the respective refractive indexes, requires a more rigorous approach that can be achieved by solving Maxwell’s equations.

Our method combines a light scattering simulation using a Monte Carlo (MC) routine (a stochastic method) with a model of the outer nuclear layer (ONL). This layer was chosen as it consistently presents the characteristics of DME, already mentioned, in the groups analyzed and because spherical scatterers can adequately model it, which helps to simplify the simulation. By varying the model’s parameters, we expect to reproduce data gathered from healthy and DME eyes and potentially infer which changes at the cellular level are responsible for the OCT data differences between groups.

The parameters describing the interaction of light with the medium were estimated with a nodal Discontinuous Galerkin Finite Element Method (DG-FEM) model of Maxwell’s equations. For the time integration, we used an improved fourth order, 14-stage low-storage Runge-Kutta. Such schemes are very popular in this context as they retain the qualities of the original Runge-Kutta schemes while decreasing the memory consumption significantly. In order to avoid undesirable reflections caused by nonabsorbing boundary conditions, that invade the simulation domain and interfere with the observation of the phenomenon of interest, we consider the perfectly matched layer boundary conditions. The validation of the proposed methodology was done by comparison with Mie’s theory, considering the light scattering for a single sphere, using the same parameters as inputs for both models. The obtained results are in agreement with those obtained using Mie’s theory, with small percentage differences of 0.37% and 0.06% for the scattering anisotropy and scattering cross-section, respectively.

Preliminary achievements

In this project, we were able to identify possible changes at the cellular level able to reproduce the differences found in the ONL of DME eyes as gathered noninvasively by OCT. Although other changes at the cellular level may be responsible for these differences, in this preliminary work we discarded variations that were not in accordance with the known physiology. These findings may help to shed a light into the pathophysiology of DME at the cellular level in vivo.

Framework of the project

This is a project in the field of Biomedical Engineering that brings together researchers with multidisciplinary backgrounds from the Mathematics and Physics Departments of the Faculty of Sciences and Technology of the University of Coimbra and from IBILI (Institute for Biomedical Imaging and Life Sciences), an internationally recognized centre of excellence for research in health sciences of the Faculty of Medicine of the University of Coimbra. The project is being developed in the framework of the Laboratory for Computational Mathematics of the Centre for Mathematics of the University of Coimbra and was supported by FCT, the Portuguese funding agency for science, research and technology.

Contacts

Adérito Araújo
CMUC – Centre for Mathematics, University of Coimbra
E-mail: alma@mat.uc.pt

Sílvia Barbeiro
CMUC – Centre for Mathematics, University of Coimbra
E-mail: silvia@mat.uc.pt