An experimental model of vitreous motion induced by eye rotations
- Andrea Bonfiglio^{1}Email author,
- Alberto Lagazzo^{1},
- Rodolfo Repetto^{1} and
- Alessandro Stocchino^{1}
Received: 10 October 2014
Accepted: 30 April 2015
Published: 12 June 2015
Abstract
Background
During eye rotations the vitreous humour moves with respect to the eye globe. This relative motion has been suggested to possibly have an important role in inducing degradation of the gel structure, which might lead to vitreous liquefaction and/or posterior vitreous detachment. Aim of the present work is to study the characteristics of vitreous motion induced by eye rotations.
Methods
We use an experimental setup, consisting of a Perspex model of the vitreous chamber that, for simplicity, is taken to have a spherical shape. The model is filled with an artificial vitreous humour, prepared as a solution of agar powder and hyaluronic acid sodium salt in deionised water, which has viscoelastic mechanical properties similar to those of the real vitreous. The model rotates about an axis passing through the centre of the sphere and velocity measurements are taken on the equatorial plane orthogonal to the axis of rotation, using an optical technique.
Results
The results show that fluid viscoelasticity has a strong influence on flow characteristics. In particular, at certain frequencies of oscillation of the eye model, fluid motion can be resonantly excited. This means that fluid velocity within the domain can be significantly larger than that of the wall.
Conclusions
The frequencies for which resonant excitation occurs are within the range of possible eye rotations frequencies. Therefore, the present results suggest that resonant excitation of vitreous motion is likely to occur in practice. This, in turn, implies that eye rotations produce large stresses on the retina and within the vitreous that may contribute to the disruption of the vitreous gel structure. The present results also have implications for the choice of the ideal properties for vitreous substitute fluids.
Keywords
Background
The vitreous humour is a transparent gel mainly composed of water (about 99 %), collagen and hyaluronic acid (HA). During eye movements the adherence of the vitreous on the retina generates motion in the gel and this, in turn, produces stresses within the vitreous and on the retina. These mechanical stresses have been suggested to possibly contribute, over long time scales, to the disruption of the vitreous gel structure, leading to vitreous liquefaction and/or posterior vitreous detachment. Once the vitreous loses its homogeneity, large localised tractions can be produced on the retina [1], which are known to be often responsible for the generation of retinal tears [2]. For the above reasons, understanding the dynamics of vitreous motion secondary to eye rotations has, in the authors’ view, great conceptual and practical importance.
Vitreous mechanical properties have been measured by several investigators [3–6], in most cases employing periodic shear tests with a rheometer. Owing to its lubricating ability and its fragile and inhomogeneous network structure, obtaining reliable data of the vitreous rheology is extremely challenging. In addition, it has been shown that vitreous properties change very rapidly after dissection, further complicating ex vivo measurements. For the above reasons the rheological properties measured by various authors have provided very sparse data.
There have been various attempts to observe and measure vitreous motion in vivo. Several researchers used techniques based on ultrasound scan measurements [7–9]. In particular, Rossi et al. [9] analysed ultrasound scan films using Robust Image Velocimetry, and obtained spatial velocity fields of the vitreous humour on planes across the vitreous chamber during single ocular saccades. Another very promising technique for measuring vitreous motion in vivo is based on magnetic resonance imaging (MRI) and has been employed by Piccirelli et al. [10, 11] to measure vitreous dynamics during low frequency periodic eye rotations. The authors used their results to indirectly estimate vitreous rheological properties by fitting the experimental data with theoretical predictions.
Even though good progress has been made in recent years in developing measurement techniques to quantify vitreous motion, our knowledge on vitreous dynamics remains incomplete.
Various theoretical models of vitreous motion have also been proposed, which significantly contributed to improve our understanding of the phenomenon. David et al. [12] first developed a mathematical model of the vitreous motion, describing the vitreous chamber as a rigid rotating sphere and the vitreous as a linear viscoelastic material. Their work was later extended by Meskauskas et al. [13], who showed that resonant excitation of vitreous motion can possibly occur during normal eye rotation conditions. Modarreszadeh and Abouali [14] recently proposed a fully numerical model of vitreous motion accounting for a realistic geometry of the vitreous chamber.
In the past, some of the present writers have also worked on model experiments of vitreous motion e.g. [15, 16]. However, our previous experimental works were all based on the use of purely viscous fluids. The aim of the present research is to extend these studies, accounting for the effects of vitreous viscoelasticity on vitreous motion, which have been shown theoretically to play a very important role.
Methods
Experimental set-up
We performed laboratory experiments using an apparatus similar to that employed in previous works [15], but with some important modifications. The present model consists of a Plexiglas cylinder with an internal spherical cavity of radius R=1.25 cm. For the purpose of this study, we neglected the effects due to the non-sphericity of the vitreous chamber. The geometrical dimensions of the spherical cavity is approximately equal to that of the human vitreous chamber, so as to avoid any scale effects that would arise from working with a magnified eye model and would be difficult to cope with in the case of viscoelastic fluids. This is because considering for instance the case of linearly viscoelastic fluids, one should preserve between the prototype and the scaled model the values of two dimensionless parameters, e.g. the Womersley number and the ratio between the elastic and viscous components of the fluid (see for instance [17]). Once a certain fluid has been chosen for the experiments this should be accomplished by changing the model rotation frequency. However, the above two dimensionless parameters typically vary independently with the frequency, making this approach impracticable.
where ε is the angular displacement, t ^{∗} is time, A is the amplitude of eye rotations and ω is their angular frequency. In the above expression, superscript asterisks indicate dimensional variables that we will scale in the following. As explained in [18] and [19], a sinusoidal law is the simplest way to represent a sequence of saccadic eye movements in both directions with a prescribed amplitude and duration. This choice is also justified by the fact that all existing data of vitreous rheological properties have been obtained considering harmonic oscillations of the vitreous. Moreover, the theoretical models by David et al. [12] and Meskauskas et al. [13], with which we will compare our results in the following, have been developed under the assumption of harmonic oscillations of the eye. Finally, this choice allows us to investigate, in the simplest possible context, the occurrence of resonance phenomena within the vitreous. We plan to extend the present work by considering the case of real saccadic eye rotations in future research.
Main parameters of the experiments
# Solution | [A g a r] (m g/m l) | [H A] (m g/m l) |
---|---|---|
1 | 0.85 | 0.95 |
2 | 0.95 | 0.7 |
3 | 1.00 | 1.00 |
4 | 1.05 | 1.05 |
5 | 1.05 | 0.95 |
The rheological properties of the fluids were measured with an Anton Paar Physica MCR301 rheometer (torque range 10^{−5}–200 mNm with resolution 1 nNm, maximum rate 3000 rpm, angular frequency 10^{−5}–100 Hz), equipped with plate-cone system (diameter 50 mm, angle 2 °, gap 0.210 mm). The rheometer allowed us to perform tests in oscillatory regime at a fixed strain amplitude γ and variable values of angular frequency ω, ranging between π≤ω≤30π rad/s. The rheological properties of the fluid are described through the complex modulus G ^{∗}=G ^{′}+i G ^{″} that represents the ratio between the complex amplitudes of stress and strain in an oscillatory test. G ^{′} is a measure of the elasticity of the fluid (storage modulus) and G ^{″} is a measure of its viscosity (loss modulus) [22].
Particular care was taken to control and keep the fluid temperature constant during each experimental run, because the rheological properties of the fluid are highly sensitive to temperature.
Solutions used in the experiments
Exp. # | Fluid # | ε(°) | ω (H z) | T(°C) |
---|---|---|---|---|
1–4 | 1 | 2–8 | 6.28–31.41 | 24 |
5–8 | 2 | 2–8 | 5.28–31.41 | 24 |
9–12 | 3 | 2–8 | 6.28–31.41 | 23 |
13–16 | 4 | 2–8 | 6.28–50.26 | 24 |
17–20 | 5 | 2–8 | 6.28–62.82 | 22 |
Post-processing
The primary results from the PIV analysis of the recorded images are velocity fields \((v_{x}^{*}(x^{*}, y^{*}, t^{*}), v_{y}^{*}(x^{*}, y^{*}, t^{*}))\), where \(v_{x}^{*}(x^{*}, y^{*}, t^{*})\) and \(v_{y}^{*}(x^{*}, y^{*}, t^{*})\) are the velocity components in the x ^{∗} and y ^{∗} directions respectively at time t ^{∗}. Since we are interested in studying the flow field induced by a periodic forcing, we set the sampling rate of the PIV system in order to obtain between 20 to 50 vector fields within a single oscillation period T=2π/ω. The measurement of each velocity field at a specific time t ^{∗} was repeated about 40 times and, then, an ensemble average was performed. The standard deviation of the two velocity components was found to be at most 5 % of the average velocity. The resulting vector fields were interpolated onto a polar co-ordinate system (r ^{∗},θ), which is a convenient choice for the circular geometry of the domain.
We note that, as expected, in all experiments v _{ r } is invariably much smaller than v _{ θ }.
We finally produced radial profiles of the circumferential velocity V _{ θ }(r,t) by averaging v _{ θ } along the θ-direction, taking advantage of the axial symmetry of the flow.
Note that \(\overline {K}\) is always smaller than 1 in the case of a purely viscous fluid, while \(\overline {K} \rightarrow 1\) is the fluid tending to behave as a rigid body.
This formula is obtained by using the expression of the shear rate of strain in spherical co-ordinates and taking its integral over time.
Results
The two flow fields in the Fig. 3 show striking differences. The low frequency flow field, Fig. 3 a, is similar to the purely viscous case, with the maximum circumferential velocity located at the boundary of the chamber, see for comparison the results discussed in [15]. On the contrary, in the high frequency case, Fig. 3 a, the maximum of the circumferential velocity is no longer at the boundary, but within the core of the flow. This behaviour, which cannot occur in the case of a purely viscous fluid, is strictly related to the elastic nature of the fluid. In particular, when the container oscillates close to particular frequencies (the so-called natural frequencies of the system) resonant excitation of vitreous motion can occur. At resonance, the response of an oscillator can be much more intense than the forcing itself. In our case, this implies that particle oscillations within the core of the domain can have a larger amplitude than at the boundary.
The maximum value of \(\overline {K}\) attained in each case depends on the rheological properties of the solution. Generally speaking, the response is more intense when the ratio G ^{″}/G ^{′} (loss factor) is smaller. For instance, the maximum value of \(\overline {K}\) is much larger in the case of solution s5 than s1, and the value of the loss factor is significantly smaller in the former case.
Discussion and conclusions
In the present work, we presented an experimental model of the vitreous humour motion induced by eye rotations. Understanding the dynamics of vitreous humour is clinically relevant since indications exist that the possible occurrence of retinal tears and, eventually, retinal detachment are related to mechanical actions exerted by the vitreous on the retina. Moreover, various authors e.g. [23, 24] hypothesised that mechanical stresses within the vitreous might contribute to the disruption of the gel structure, over long times scales. This could explain the occurrence of liquefaction that is typically observed with advancing age.
We modelled the vitreous chamber as a rigid sphere and we filled it with an artificial vitreous with viscoelastic properties. In particular, we chose a fluid with rheological properties similar to those of the real vitreous, measured ex vivo by various authors. We remark, however, that there is still a lot of uncertainty concerning the mechanical properties of the vitreous, and the available measurements provide values within a very wide range. In order to keep the problem as simple as possible, yet retaining some important ingredients, we modelled eye movements as sinusoidal periodic rotations and investigated the role of frequency on the results.
Various experimental models of vitreous motion induced by eye rotations have been proposed in the past, however, to our knowledge, this is the first case in which a viscoelastic fluid has been employed. Our results show that this is a crucial ingredient. In fact, for all solutions employed to create the artificial vitreous, resonance excitation of vitreous motion was observed for values of the frequency that are typical of eye rotations. With resonance we refer to tendency of the fluid to oscillate with larger amplitudes at some frequencies than at others. Such frequencies are named “resonant frequencies”. At resonance, the system stores and easily transfers energy between two different storage modes. In the case of the motion of a fluid completely filling a closed rotating domain, which is considered in the present work, viscoelasticity is a necessary ingredient to possibly generate resonance phenomena. This is because the system needs to be able to store elastic energy during certain phases of motion, which is then transformed into kinetic energy at other times. We note that in this work we have not described the transient conditions during which the oscillation amplitude progressively grows, starting from rest, until periodic flow conditions are reached. If the eye model rotates at a frequency close to resonant, oscillations in the core of the domain can have a significantly higher amplitude than at the wall. This is a relevant phenomenon to investigate because, if resonance occurs, particularly large values of the stress are attained on the retina and within the vitreous. The order of magnitude of natural frequencies investigated in the experiments are comparable with typical frequencies associated with saccadic eye movements, as derived following the same approach as in [18, 19], where the angular frequency for sinusoidal oscillations was computed as ω=π/D, where D is the saccade duration.
Resonant excitation of vitreous motion in-vivo has not been observed, and we are not yet in the position of assessing whether it can actually occur. This is due to a lack of in-vivo data on vitreous dynamics. As mentioned in the Introduction, Piccirelli et al. [11] measured vitreous motion using an MRI based technique. In their experiments, patients were asked to rotate their eyes following a target performing a sinusoidal motion. However, they only considered low frequencies, at which our experimental model does not predict the occurrence of resonance. Rossi et al. [9] used an ultrasound technique to measure vitreous dynamics. Since in their investigation they asked patients to perform single saccadic rotations, it is difficult to assess from their data whether natural frequencies of the system exist that could be resonantly excited by periodic rotations. Moreover, in order for the vitreous to be visible at the ultrasound scan, the authors had to restrict their study to patients with a somewhat degenerated vitreous structure. This lack of information points to the need of new in vivo measurements of vitreous motion. If resonance can actually occur during normal eye movements, it would imply that the vitreous and the retina can be subjected to much higher mechanical stimuli than we would expect.
The present results have practical value also from another point of view. A lot of effort has been devoted in recent years to the identification of the ideal material to be employed as a vitreous substitute. Various authors regard fluid elasticity as being very important to avoid excessive flow within the vitreous chamber [25–27]. This is certainly true; however, if the ratio between fluid viscosity and elasticity is not large enough, resonance can occur and produce undesirably large values of the stress in the fluid and on the retina. This should be kept in mind in the design of the ideal artificial vitreous.
Declarations
Acknowledgements
This research work was partly funded by the University of Genoa - PRA 2013.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Authors’ Affiliations
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