Rheological behaviour and flow dynamics of Vitreous Humour substitutes used in eye surgery during saccadic eye movements

Five different pharmacological fluids used in eye surgery were used in this study, including: three Silicone oils (SiO), RS-Oil 1000 from Alchimia (Padova, Italy), Siluron 2000 and Siluron 5000 both from Fluoron (Ulm, Germany); Densiron 68 from Fluoron (Ulm, Germany), which is a mixture of $70\%$ of SiO with a viscosity of approximately $5\text{ Pa s}$ with $30\%$ of perfluorohexyloctane; and one Perfluorocarbons (PFLC), fluorinated perfluorodecalin (HPF10) from Alchimia (Padova, Italy). Note that all the SiOs used are composed of polymer chains of the same MW, except Siluron 2000 that is composed of $95\%$ of short molecular chains and $5\%$ of ultra-long molecular chains with a viscosity around $2500\text{ Pa s}$.

Shear measurements were performed using a DHR-2 rotational rheometer (TA Instruments) using a cone-and-plate configuration with 60 mm diameter and a $1^{\circ}$ angle.

The extensional properties of the samples were measured using an in-house miniaturised filament breakup device (Sousa et al., 2017, 2018), where the samples were subjected to the slow retraction method (SRM) (Campo-Deano & Clasen, 2010). The SRM technique allows to investigate filament thinning of fluids in extensional flow with low viscosity and/or very low relaxation times, applying a slow plate separation (instead of a step strain deformation), which allows positioning the plates close to the critical point when a stable liquid bridge still exists (Campo-Deano & Clasen, 2010; Vadillo et al., 2012). The extensional rheology of the pharmacological fluids was measured with cylindrical plates with diameter $D_{p}=2\text{ mm}$ and an initial gap of $L_{0}=0.5\text{ mm}$ for the silicone oils, and $L_{0}=0.25\text{ mm}$ for the PFLC. The SRM measurements were performed with a plate separation velocity of 5 $\mu$m/s at temperatures between $18.6\text{ }^{\circ}\text{C}$ and $23.3\text{ }^{\circ}\text{C}$. During the experiments, the evolution of the thinning fluid filament formed between the plates was monitored with video imaging using a high-speed CMOS camera (FASTCAM MINI UX100, Photron). The camera was connected to a magnifying optical lens (H) (OPTEM Zoom 70XL) allowing a variable magnification from $1\times\text{ to }5.5\times$. The video images were afterwards analysed in Matlab (MathWorks, version R2014b), in order to identify when the filament exhibits a quasi-cylindrical shape and obtain the filament diameter as a function of time. This allows subsequently determining the relaxation time for viscoelastic fluids and the filament breakup time.

Surface tension measurements were performed with a drop shape analyser (model DSA25, Kr$\ddot{\text{u}}$ss) using the pendant drop method. The measurements were performed at a temperature of ${T=20\pm 2\text{ }^{\circ}\text{C}}$.

The eye geometry in this work is the same used in our previous work and detailed information can be found in Silva et al. (2020) and in the Supplementary Material 1. Different levels of mesh refinement and different time steps were tested to ensure the accuracy of the CFD results. For mesh purposes, the eye model was split in seven different blocks: a cube in the middle and six blocks around the central cube. Meshes with $10^{3}$, $20^{3}$, $40^{3}$ and $80^{3}$ cells per block, with additional refinements in the cell adjacent to the wall (region with higher gradients of velocity) were tested. The velocity profiles, in the vitreous cavity, at different times of a saccadic movement with amplitude $A=40^{\circ}$ were compared and for the Silicone Oils no significant differences were found between the results obtained with the meshes with $20^{3}$ and $40^{3}$ cells per block, while for the HPF10 the meshes with $40^{3}$ and $80^{3}$ cells per block showed no significant differences. Based on these results, meshes with $20^{3}$ and $40^{3}$ cells per block were used to simulate the flow behaviour of the Silicone oils and the Perfluorocarbon, respectively (see Fig. 1). Two time steps were tested ($\Delta t=10^{-4}\text{ s}$ and $10^{-5}\text{ s}$), with the obtained velocity profiles showing no meaningful differences. The time step of $\Delta t=10^{-4}\text{ s}$ was thus used in all the remaining simulations.

Saccadic movements with angular displacements, $A$, between 10 and $40^{\circ}$ were computed using the fifth order polynomial equation presented by Repetto et al. (2005), based on the experimental measurements of Becker (2006). Detailed information can be found in our previous work (Silva et al., 2020) and in Supplementary Material 2. Those details are not repeated here for briefness.

During this work the whole eye model moves as a solid body, with a prescribed motion. In order to include the effect of the dynamic mesh, the convective terms in the governing equations needed to be corrected with the grid velocity. In this moving mesh framework, the continuity equation can be written as

where u is the velocity vector.

The momentum equation can be written as

where $\rho$ is the fluid density, $p$ the pressure, $t$ is time, $\bm{\tau}$ is the extra-stress tensor, and $\textbf{u}_{g}$ the velocity at which the grid surface is moving. Besides, the space conservation law (SCL) also needs to be taken into account for moving grids,

where $V$ and $S$ are the volume and surface of the grid.

The fluids HPF10, Densiron 68, RS-Oil 1000 and Siluron 5000 were simulated as Newtonian fluids in accordance with the rheology measurements presented in Section 3.1. The Newtonian stress tensor $\bm{\tau}_{s}$ is given by

where $\eta_{s}$ is the solvent viscosity. The Siluron 2000 fluid exhibits a weakly elastic behaviour. To assess if such a low value of elasticity affects the fluid flow when compared with the Newtonian fluid behaviour, Siluron 2000 was simulated as a Newtonian fluid or as a viscoelastic fluid using an Oldroyd-B model. This viscoelastic model is frequently used in the simulation of Boger fluids, and predicts a constant shear viscosity ($\eta_{0}=\eta_{s}+\eta_{p}$, where $\eta_{p}$ is the polymer viscosity) and a quadratic increase of the first normal-stress difference with shear rate in a steady viscometric flow (Morrison, 2001). For Siluron 2000, the flow history is important and the total extra-stress tensor is split in a polymeric contribution $\bm{\tau}_{p}$ and a solvent contribution $\bm{\tau}_{s}$: $\bm{\tau}=\bm{\tau}_{s}+\bm{\tau}_{p}$. The polymer contribution $\bm{\tau}_{p}$ is solved based on the upper-convected Maxwell constitutive equation (Owens & Phillips, 2002)

where $\lambda$ is the relaxation time and $\overset{\triangledown}{\bm{\tau}_{p}}$ is the upper-convected time derivative of $\bm{\tau}_{p}$, defined as $\overset{\triangledown}{\bm{\tau}_{p}}=\frac{\partial\bm{\tau}_{p}}{\partial t}+\triangledown\cdot\left[{\bm{\tau}_{p}}\left(\textbf{u}-\textbf{u}_{g}\right)\right]-{\bm{\tau}_{p}}\cdot\triangledown\textbf{u}-\triangledown\textbf{u}^{\text{T}}\cdot{\bm{\tau}_{p}}$ (Pimenta & Alves, 2017).

For Siluron 2000, two different solvent viscosities ratios were tested: $\beta=\eta_{s}/(\eta_{s}+\eta_{p})=\eta_{s}/\eta_{0}=0.98$ and $0.95$. The viscosity values used in the simulations for all the SiOs are presented in Table 1.

For the fluids and flow conditions under study, the maximum Reynolds number reached in the simulations can be calculated as $\text{Re}=\frac{\rho U_{max}R}{\eta_{0}}$, where $R$ is the eye radius, $U_{max}$ is the maximum velocity reached for each degree of movement, and $\eta_{0}$ is the zero shear viscosity of the fluid. The maximum Reynolds number reached for the SiOs fluids is below $\text{Re}=2$, while the maximum value for HPF10 is Re$\approx 717$.

The RheoTool package (Pimenta & Alves, 2017) in the open-source finite-volume code OpenFOAM® (version 2.2.2) was used to solve the governing equations. The no-slip boundary condition was set in the whole eye surface. The time-derivatives were discretised using a second-order implicit Backward Euler scheme, and the semi-implicit method for pressure-linked equations-Consistent (SIMPLEC) algorithm was selected to ensure the velocity-pressure coupling in a segregated way. The convective terms of both the momentum and constitutive equations were discretised with the Convergent and Universally Bounded Interpolation Scheme for the Treatment of Advection (CUBISTA) (Alves et al., 2003). A zero pressure gradient normal to the walls was assumed and the polymeric stress tensor was extrapolated from the cells adjacent to the wall (Pimenta & Alves, 2017). Finally, the log-conformation tensor formulation was used to solve the constitutive equation.

The velocity magnitude contours in the plane $z=0$ for all the fluids under study are presented in Fig. 8. The vitreous cavity rotates anti-clockwise around the $z$ axis. HPF10 presents very low viscosity (at $T=37\text{ }^{\circ}\text{C}$, the viscosity is $3.57\times 10^{-3}\text{ Pa s}$), which means that the diffusive time scale is much higher than for the remaining fluids. Considering the eye as a perfect sphere with a diameter of $D=0.024\text{ m}$, the diffusive time scale can be estimated as $t_{diff}=\frac{\rho D^{2}}{\eta}$: HPF10 shows a diffusive time scale of 303 s, while for all the SiO-based fluids the diffusive time scale is lower than 1 s. Note that the values are not accurate for the real geometry under study, but they still provide an estimate of the differences in the diffusive time scales between the tested fluids. For the HPF10 fluid, the lens indentation significantly affects the flow dynamics: for times $t=0.1t_{D}$ and $t=t_{p}$ the velocity magnitude right behind the lens indentation is considerable when compared to the rest of the vitreous cavity. Finally, it is possible to observe that for $t=2t_{D}$, due to inertial effects, there is still fluid motion close to the walls.

The SiO-based fluids show distinct velocity profiles when compared to the PFLC. Due to their high shear viscosity, the diffusive time scale is low and for $t=t_{p}$ there is fluid motion across the whole vitreous cavity, except at the centre. It is possible to consider that these fluids move similarly to a rigid body, where individual fluid particles display a rotational motion, without being significantly deformed. With the increase of the SiO viscosity, the momentum propagates faster in the vitreous cavity (see the velocity contours for $t=0.1t_{D}$). For time $t=t_{D}$ there is almost no fluid motion in the vitreous cavity and the flow stops right after $t=t_{D}$. In our previous work (Silva et al., 2020) we show that for times higher than $t=t_{D}$ the VH gel phase produces considerable velocity gradients inside the vitreous cavity. Based on the results presented in Fig. 8, it seems that the lens indentation affects the velocity magnitude contours of the SiO-based fluids mostly in the beginning of the movement.

Abouali et al. (2012) also studied the flow behaviour of three different Newtonian fluids when subjected to saccadic movements with amplitudes between $A=10^{\circ}$ and $50^{\circ}$. Their geometry was a simplified vitreous cavity and did not show the complexity of the vitreous cavity studied here. Nevertheless, for a saccadic movement of $A=50^{\circ}$ the flow behaviour of water and a SiO with a shear viscosity of $0.97\text{ Pa s}$ followed a similar trend to the results presented here.

The time variation of the wall shear stress at point P, WSS${}_{\text{p}}$, for all the fluids under study is presented in Fig. 9. Point P is located in the posterior position of the vitreous chamber and represents a position close to the center of the macula (point with coordinates (-12,0,0) mm, see Supplementary Material 1). Note that for Siluron 2000 fluid the WSS include both the Newtonian and polymeric contributions. Additionally, the time and value of the maximum peak WSS${}_{\text{p}}$ reached by each fluid is presented in Table 3. For the SiO-based fluids, the peak WSS${}_{\text{p}}$ occurs at the beginning of the movement for $t<0.1t_{D}$, and these fluids present the highest values, between $10.0\text{ Pa}$ for RS-Oil 1000 and $14.0\text{ Pa}$ for Siluron 5000, according to their viscosity. All the SiOs present their lowest WSS${}_{\text{p}}$ at $t=t_{p}$, or right after that time. The HPF10 presents a peak WSS${}_{\text{p}}$ of $1.17\text{ Pa}$ for $t/t_{D}=0.12$, and the lowest WSS${}_{\text{p}}$ occurs at $t/t_{D}=0.56$.

The WSS contours of the pharmacological fluids under study for a saccadic movement of $A=40^{\circ}$ are presented in Fig. 10. Due to the high WSS values reached in the beginning of the movement with the SiO-based fluids, the WSS contours for $t=0.04t_{D}$ are also presented. Note that due to the differences in the maximum WSS values, two different scales were used. The results show that, for all fluids tested, the maximum values of WSS occur around the centre of the lens indentation. The HPF10 fluid shows an interesting WSS profile along the lens indentation, with higher values concentrated along the $z$-axis reaching the outer area of the lens indentation (see $t=0.04t_{D}$ and $t=0.1t_{D}$), whereas for all the other fluids the results show a constant WSS in the region around the lens indentation. For $t=t_{p}$, the fluid HPF10 also shows WSS contours lower in one side of the indentation ($y<0$) than in the other ($y>0$). The maximum WSS obtained for the HPF10 fluid is approximately $1.5\text{ Pa}$. From all the fluids under study, SiO-based fluids show the highest WSS values. Siluron 5000 fluid reaches a maximum WSS in the centre of the lens indentation around $30\text{ Pa}$, while RS-Oil 1000 reaches a maximum WSS around $18\text{ Pa}$.

In the study performed by Abouali et al. (2012), the WSS contours are presented for water, glycerol and a SiO fluid, and the values are in agreement with the ones presented here. However, in their study the WSS contours in the borders of the lens indentation show considerable differences when compared to the values inside and outside the lens indentation. Such phenomenon seems to be a consequence of the geometry in that area. Here, the geometry used presents a smoother transition between the lens indentation and the rest of the vitreous cavity, and no discrepancy was found between the WSS values on that region and in the neighbourhood.

The pharmacological fluids are mostly used in eye surgery to reattach the retina or close retinal tears that are created in the VH fluid as a consequence of the degradation of the collagen and HA structures. The forces generated between the fluid and the retina are key to keeping the retina in its proper position. The PFLC (HPF10) generates stresses much lower than the VH gel phase (see Silva et al. (2020)), which might compromise their effectiveness to reattach the retina or eliminate retinal tears. In fact, as mentioned in the introduction, the use of these fluids in eye surgery is rare, as SiO-based fluids are the main choice for the treatment of RD and RT. The SiO-based fluids generate stresses higher than VH gel phase, which seems appropriate to reattach the retina, as higher forces will pull and keep the retina in its proper position. However, it can be found in the literature that the use of SiO-based fluids for long periods often leads to complications such as cataracts, glaucoma, and corneal damage among others (Baino, 2011; Giordano & Refojo, 1998; Federman & Schubert, 1988). This can be related to the fact that the WSS values produced by the SiO are more than the double of those produced by VH gel phase (Siluron 2000 reaches a maximum WSS${}_{\text{p}}$ of 12.11 Pa, while VH gel phase has a maximum WSS${}_{\text{p}}$ of 5 Pa). From a mechanical point of view, a fluid with a behaviour similar to the VH gel phase, that would be able to produce stresses in the walls with similar magnitudes as the ones produced by VH gel phase, would be ideal to be used as a permanent VH substitute.