Flow dynamics of vitreous humour during saccadic eye movements

Abstract

In this work, we reveal the flow dynamics of Vitreous Humour (VH) gel and liquid phases during saccadic movements of the eye, considering the biofluids viscoelastic character as well as realistic eye chamber geometry and taking into account the saccade profile. We quantify the differences in the flow dynamics of VH gel and liquid phases using viscoelastic rheological models that are able to model the VH shear rheology, considering different amplitudes of saccadic movements (10^∘, 20^∘, 30^∘ and 40^∘). For this purpose, the computational fluid dynamics (CFD) open source software OpenFOAM® was used. The results portray a distinct flow behaviour for the VH gel and liquid phases, with inertial effects being more significant for the VH liquid phase. Moreover, the Wall Shear Stress (WSS) values produced by the VH gel phase are more than twice of those generated by the VH liquid phase. Results also show that for different amplitudes of eye movement both the velocity magnitude in the vitreous cavity and the shear stresses on the cavity walls rise with increasing saccadic movement displacement.

Introduction

The human eye has one of the fastest muscles in the body, being able to produce rotations of 40° in approximately $100\text{ ms}$ (Enderle, 2010). In the different situations of daily life, the eyes are able to produce different, voluntary or involuntary, movements. Saccadic eye movements are the most common movements of the eye. They are very quick ballistic movements produced by both eyes simultaneously and in the same direction when fixating an object (Kumari et al., 2017; Bronzino, 2000; Liversedge et al., 2011). In fact, the eyes are never completely at rest, because they are frequently producing saccades. How these movements affect the dynamics of the eye's fluids is not yet fully understood.

The majority of the eyeball, more specifically the space between the lens and the retina, is filled with Vitreous Humour (VH) fluid, a transparent gelatinous avascular structure (Chirila et al., 1998; Nickerson et al., 2005, 2008; Sharif-Kashani et al., 2011; Silva et al., 2017), which exhibit non-Newtonian rheological behaviour (Silva et al., 2017). VH is composed of long collagen fibrils suspended in patterns of hyaluronic acid or hyaluronan (HA) molecules and other glycosaminoglycans (GAGs), which surround and stabilise water molecules. This fluid is only produced during the embryonic stage and is not replenished throughout the entire life of a person (Black and Hastings, 1998). It is known that as a consequence of ageing and/or some diseases the VH becomes progressively liquefied (Baino, 2011), with the corresponding rheological properties varying during lifetime. In fact, it is possible to distinguish two different phases in VH, a liquid and a gel phase, that coexist in the vitreous chamber (Sebag, 1987). Many of the eye's diseases are directly or indirectly associated with the morphological changes of this biofluid with age, and consequent changes in its flow behaviour and performance (Repetto, 2006; Stocchino et al., 2007).

Over the years, several studies regarding VH flow behaviour have been performed using numerical, analytical, or experimental approaches. Table 1 presents a summary of the key points of those studies. In the late 1990's, David et al. (1998) presented both analytical and numerical solutions for a time-dependent motion of the VH during saccadic motion of the eye. The authors studied analytically the motion of a viscoelastic fluid, and also tested numerically the case of a Newtonian fluid. The vitreous cavity was simulated as a perfect sphere and a sinusoidal function was used to represent the saccadic movements. In 2005, Repetto et al. (2005) presented an experimental study of the VH motion induced by saccadic eye movements (see the key points of the study in Table 1). The authors considered VH as a Newtonian fluid, and the eye as a perfect sphere. Despite the simplifications assumed, both David et al. (1998) and Repetto et al. (2005) studies were critical starting points to understand the flow patterns during saccadic movements. In the following years, Repetto et al. (Repetto, 2006; Repetto et al., 2010) and Stocchino et al., 2007, Stocchino et al., 2010 performed analytical and experimental studies, respectively, of the dynamics of the VH induced by saccadic movements. The eye models were created based on the same assumptions as in Repetto et al. (2005), but an improved non-spherical shape was considered to resemble the real vitreous chamber. Their results showed that the non-spherical shape of the cavity generates flow fields and stresses on the boundary that are significantly different from those generated in the case of motion within a sphere, with vortices forming in the area of the lenses indentation, which the authors believed might play a role in the generation of retinal detachments. Abouali et al. (2012) analysed saccadic movements with a range of amplitudes from 10 to ${50}^{\circ }$. Their eye model had the size of the real human eye, with a radius of 12 mm and different degrees of indentation resembling the lenses shape in the anterior part of the eye. All the studies mentioned so far helped enhance our understanding of the flow dynamics of the biofluid during saccadic eye movements, and/or provided insights into the importance of considering the lenses indentation. However, all of those works were performed using Newtonian fluids, while it is known that both VH phases exhibit viscoelastic rheological behaviour (Silva et al., 2017) and that their rheology is closely related to its functionality in the eye. Therefore, assuming that the biofluid has a Newtonian rheology is an oversimplification that can lead to significant differences in terms of the predicted flow behaviour during saccadic eye movements.

The first studies about eye movements considering fluids with viscoelastic properties were published in 2011 (Repetto et al., 2011; Meskauskas et al., 2011). However, both studies considered the eye as a sphere. The study of Meskauskas et al. (2011) showed that the maximum velocity inside the eye cavity can be twice the maximum wall velocity, and that resonance effects related with the viscoelasticity of the fluids generate larger wall shear stresses, and could be relevant for the occurrence of retinal detachment. Those results were later corroborated by Isakova et al. (2014), who predicted the same trend using an analytical approach, and also by Bonfiglio et al. (2015) who also obtained similar results based on an experimental study. Modarreszadeh and Abouali (2014) focused on providing a reliable numerical procedure for the study of VH as a viscoelastic substance and under oscillatory movements. The open-source software OpenFOAM® was used and the viscoelastic solver developed by Favero et al. (2009) was adapted to handle dynamic meshes. As their focus was to provide a reliable numerical tool, their paper presents mostly solver validations, and only preliminary results of the VH behaviour, considering a Giesekus model for the VH gel-like phase and a Newtonian liquefied phase under sinusoidal movement with frequency of 10 rad/s and amplitude of 3%, were presented.

In summary, there is a body of research work on the flow behaviour of vitreous humour. However, most of the studies rely on a variety of simplifications (e.g. simplified eye geometry, assumption of VH as a Newtonian fluid, considering sinusoidal movements) that may lead to significant differences in the flow behaviour when compared with the real VH in the eye. Additionally, to the best of our knowledge, there are no numerical studies that consider the viscoelastic behaviour of the VH liquid phase, as it is usually simulated as a Newtonian fluid (Balachandran and Barocas, 2011; Repetto et al., 2011; Modarreszadeh and Abouali, 2014).

The aim of this study is the numerical investigation of the flow behaviour of VH using adequate viscoelastic rheological models that are able to model the VH behaviour, using a geometry that is a better approximation of the vitreous humour chamber and real saccade profiles, to obtain a better insight about the flow dynamics of VH during saccadic movements. The remainder of the paper is organised as follows: the next section presents the numerical methodology used and the solver validations performed; the results are then presented and discussed; finally, the main conclusions are summarised in the last section.