# Dynamics of a Gel-Based Artificial Tear Film with an Emphasis on Dry Disease Treatment Applications

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## Abstract

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## 1. Introduction

^{5}to 10

^{6}mPa·s); it decreases by a factor of 1 to 4 when the applied shear is above the yield-shear stress [15,16]. In addition, the viscosity of these solutions drops during the blink but never falls below a certain threshold (around 10 mPa·s, or ten times the viscosity of water).

## 2. Results and Discussion

^{−3}s

^{−1}), and this viscosity is strongly reduced when the shear rate increases (about 10 mPa·s at 100 s

^{−1}). A percolated and disordered suspension of individual elastic sponges that absorb the solvent is a well-known characteristic of carbomer gels [35]. The rheological behavior of these products can be modeled with the Herschel–Bulkley (HB) constitutive model [36]:

^{n}). The dotted line in Figure 1 shows an example of the fitting of the HB model to the gel with the lowest yield stress. Table 1 summarizes the obtained values of the model parameters for all the gels. The shear-thinning index, n, differs very little between the products (0.4 to 0.5). The consistency, k, varies from 2 to 20 Pa·s

^{n}. The yield stress ranges from 4.7 to 33.8 Pa. More precisely, some products based on Carbopol

^{®}980 NF Polymer have a concentration of 0.2% and yield stress of about 30 Pa, whereas for the same concentration, other gels have a yield stress of about 15 Pa. The product with a lower concentration (0.13%) has lower yield stress (7.3 Pa). In the case of Carbopol

^{®}974 P Polymer, as the products have a concentration of 0.3%, their yield stress ranges from 26 to 28 Pa. The product with the lowest concentration (0.25%) has lower yield stress (4.7 Pa). The yield stress for each category of carbomer is not entirely correlated to its concentration. Indeed, the physicochemical characteristics and hence the formulation of the solvent may modify the behavior of carbomer microgels.

#### 2.1. Influence of the Tear Gel Rheological Parameters

_{0}, on stability and tear film dynamics. Since tears present shear-thinning behavior, we considered values of n less than one. Although tear film breakup can occur at any position on the corneal surface because of TFLL structures (not considered here), the discussion below focuses on tear film breakup occurring just beneath the upper lid. Our results show that the lowest film thickness value is found near the upper eyelid, which is consistent with clinical observations [37]. The thinner the tear film, the greater its tendency to touch the corneal surface, and the higher the risk of tear breakup. In the following subsections, the graphs in the figures are flipped vertically, and 90° rotated compared with the scheme in Figure 9 in the section Materials and Methods. Hence, the moving upper eyelid is on the right side.

#### 2.1.1. Effect of Flow Behavior Index, n

_{0}=1 Pa and τ

_{0}= 4.5 Pa, see Figure 3. Figure 3a–d indicates that the Newtonian tear films exhibit the lowest tear film thicknesses close to the upper eyelid than the gel-based tear films. Figure 3a–d suggests that gel-based tears can reduce the risk of tear film rupture near the moving upper eyelid. Figure 3a–d also indicates that the more the gel-based formulation exhibits a shear-thinning behavior (low n), the higher the minimum film thickness near the upper eyelid, and the more the risk of tear film breakup is reduced. The influence of shear-thinning is amplified by increasing the consistency parameter, k (see Figure 3a,c). Figure 3d shows that yield shear can compensate for the low shear-thinning effect (n = 0.7); see Figure 3c,d. It can be noted that when τ

_{0}= 1 Pa and k = 0.07 Pa·s, the thickness profile tends to have a quasiuniform thickness over the center of the cornea flat. This result is interesting because any local changes in tear film thickness will result in an irregular air/tear interface, thus introducing aberrations to the eye’s optical system, which may cause the blurry vision commonly encountered in dry eye patients [38].

#### 2.1.2. Effect of Consistency Index, k

_{0}is fixed at 0, 0.2, 1, and 4.5 Pa. Figure 4a shows that the film thickness value decreases significantly near the lower fixed eyelid when the yield stress τ

_{0}is null. The value of the consistency number is small, i.e., when the gel-based formulation does not exhibit elastic behavior (no yield stress) and the shear-thinning behavior of the gel formulation is not amplified. In this condition, the risk of tear film breakup is high near the lower eyelid. This thinning of the tear film near the fixed eyelid is alleviated by enhancing the elastic behavior (high value of τ

_{0}); see Figure 4d. The influence of the consistency parameter k on the film thickness near the upper moving eyelid is relatively small compared with the case of Newtonian tears when the values of the yield stress τ

_{0}are null or low; see Figure 4a,b. The effect of the consistency parameter k is noticeable at higher values of the yield stress τ

_{0}because the elastic behavior is enhanced (high yield stress) and the shear-thinning behavior is amplified (high k); see Figure 4c,d. It is worth highlighting that the uniformity of the gel film thickness is better when the yield stress is null (Figure 4a). This means low values of yield stress help the gel to spread uniformly over the cornea.

#### 2.1.3. Effect of the Yield Stress τ_{0}

_{0}on the tear film thickness profile. The flow index and the consistency index values are fixed at n = 0.5 and k = 0.07, 0.6, and 2.5 Pa·s. Figure 5a shows that for low consistency value (k = 0.07 Pa·s), increasing yield stress τ

_{0}increases the minimum film thickness near both eyelids. On the other hand, for consistency index k fixed at 2.5 Pa·s, we observe that τ

_{0}= 1 Pa is a cutoff value. Above this cutoff value, the minimum film thickness decreased significantly, which means that higher yield stress and the amplification of the shear rate contribution to the gel viscosity can break the tear film. When k = 2.5 Pa·s and τ

_{0}= 4.5 Pa, the gel film is thinner in the lower part of the cornea and thicker in the upper part of the cornea. This non-uniform distribution of the gel film can blur the vision. Table 2 summarizes our results indicates that the following value of n = 0.5, k = 0.6 Pa·s, and τ

_{0}= 1 Pa results in the highest value for the minimum gel-based tear film. One can iterate these values to design optimal gel-based artificial tears to alleviate the tear film breaking phenomenon.

#### 2.1.4. Evolution of Local Velocity at the Eyelids

_{0}are low; see Figure 6 and Figure 7a,b. This result mains a low gel viscosity eases its flow.

#### 2.1.5. Shear Stress of the Tear Film near Eyelids

## 3. Conclusions

_{0}, flow behavior index n, and the consistency index k on the spreading gel-based tear film. We found that enhancing the shear-thinning of the gel-based tears by decreasing the flow behavior index n contributes to increasing the value of the minimum film thickness, which confirms the positive effect of the shear-thinning properties of the natural tears. Low value for the yield stress tends to delay the film breakup and ensures quasi-uniform film thickness around the center of the cornea, which can prevent blurred vision when using gel-based artificial tears. In the range of parameters considered in this research, the maximum value for the minimum tear film is found for values of the gel parameter n = 0.5, τ

_{0}= 1 Pa and k = 0.6 Pa·s. These values can serve as a starting point for an iterative and optimization process to reach perhaps an optimal design of artificial gel-based tear film. We believe that a modeling approach similar to the one presented here can help laboratories design tears on rational basis to alleviate dry eye symptoms.

## 4. Materials and Methods

#### 4.1. Rheology and Spreading of Tear Substitutes

^{−1}). The temperature of the sample was regulated by an integrated Peltier effect system that heats the plate of the cone-plate measurement configuration. A Pt 100 probe controls its temperature. The regulation system ensured that the temperature of the lower plate was accurate to within ±0.1 °C. The measurement geometry was enclosed in an envelope that acts as a solvent trap, thus considerably reducing evaporation on the unconfined surface of the sample. It also reduced heat losses and ensured a uniform temperature around the sample. A rough geometry is used to avoid wall slip (cone 26 mm in diameter, 4° angle, 450 μm gap) [39]. First, a sample was taken directly from its receptacle and placed on the rheometer plate. The cone was then brought up to the plate until the required gap is obtained. We gave enough time to balance the temperature at 25 °C, before applying a shear rate or stress, and the change in torque (stress) or strain is recorded as a function of time until steady conditions are achieved.

_{SA}) is 40 mN/m at 20 °C. It is hydrophobic. The PMMA/water contact angles approach those epithelium/water.

#### 4.2. Numerical Simulations

_{op}= 10 mm and L

_{cl}= 1mm, respectively. The meniscus is pinned to the eyelids during the simulations at the height of h* = 0.5 mm. L(t) indicates the distance between the eyelids during the opening phase. U(t) is the velocity function of the upper eyelid. The domain is considered two-dimensional.

#### 4.2.1. Formulation of the Problem

#### Governing Equations

- $\alpha $= 0: the cell is empty.
- $\alpha $= 1: the cell is full.
- 0 <$\alpha $< 1: the cell contains the interface between the two fluids.

^{–3}, ${\rho}_{g}$= 1.225 kg·m

^{–3}and ${\mu}_{g}$ = 1.7894 × ${10}^{-5}$ Pa·s, ${\mu}_{l}$ = 1.3 × ${10}^{-3}$ Pa·s.

^{T}is its transpose. Thus, the viscosity of Herschel–Bulkley fluid is defined by the following relation [41].

_{0}plays the role of a discontinuous limit; Herschel–Bulkley fluid only flows when the shear stress exceeds the yield stress. $\tau <{\tau}_{0}$ the material remains rigid ($\left|\dot{\gamma}\right|=0$) and for $\tau >{\tau}_{0}$ the material flows as a power-law fluid ($\left|\dot{\gamma}\right|>0$). A regularization procedure is required to handle the discontinuity of the Herschel–Bulkley model at $\left|\dot{\gamma}\right|=0$. We used the regularization already implemented in the Fluent solver [42].

#### Boundary Conditions

- $u=v=0y=0$: at the corneal surface.
- $u=v=0x=0$: at the lower eyelid.
- $u=U\left(t\right)x=L\left(t\right):$ at the upper eyelid.

_{0}, T, and λ are 0.180 s, 0.0163 m/s, 0.0865 s, and 11.6, respectively, for more details (see [43]). t* indicates the time at which the upper eyelid velocity becomes null.

#### 4.2.2. Numerical Method and Validation

^{−6}. The convergence criterion is chosen so that the residual for all equations is less than 10

^{−7}.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 1.**Flow curves of gel tears at 25 °C: yield stress and very high viscosity at a low shear rate and decreasing viscosity during a blink (high shear rate). The grey lines represent the Herschel–Bulkley model (Equation (1)) applied to the product with the lowest yield stress.

**Figure 3.**Effect of flow index, n, on variation of film thickness for τ

_{0}= 1 and 4.5 Pa, k = 0.07 and 0.6 Pa·s at t = 0.18 s. ρ = 10

^{3}kg·m

^{−1}, μ = 1.3 × 10

^{−3}Pa.s (Newtonian case), σ = 0.045 N/m, h* = 0.0005 m, L

_{cl}= 0.001 m, L

_{op}= 0.01 m, U

_{0}= 0.0163 m/s. (

**a**–

**d**) indicate the same simulation with different values for the parameters indicated within the figure.

**Figure 5.**Variation of film thickness for different τ

_{0}, n = 0.5 at t = 0.18 s. (

**a**) k = 0.07, (

**b**) k =0.6, (

**c**) k =2.5 Pa·s.

**Figure 6.**Depth average velocity for different consistency k, at time = 0.18 s. (

**a**) over the whole cornea surface (

**b**,

**c**) a closeup near the lower and upper eyelid.

**Figure 7.**Depth average velocity for different. yield stress τ

_{0}at time = 0.18 s. (

**a**) over the whole cornea surface, (

**b**,

**c**) a closeup near the lower and upper eyelid.

**Figure 8.**Variation of wall shear stress: (

**a**,

**b**) consistency k effect; (

**c**,

**d**) yield stress τ

_{0}effect.

**Figure 10.**Evolution of film thickness for different mesh grids in the case of non-Newtonian film n = 0.5, k = 0.6 Pa·s and τ

_{0}= 1 Pa at t = 0.18 s.

**Figure 11.**Model validation against the model by Aydemir et al.’s work [24]. (

**a**,

**b**) film thickness distribution along the planar subtract at 0.04 and 0.18 sec, respectively. (

**c**) Evolution of the minimum tear film thickness with time. The parameters of the simulations are: ρ = 10

^{3}kg·m

^{−1}, μ = 1.3 × 10

^{−3}Pa.s, σ = 0.045 N/m, h* = 0.001 m, L

_{cl}= 0.002 m, L

_{op}= 0.01 m, U

_{0}= 0.0163 m/s.

Composition | Concentration (%) | Yield Stress (Pa) | k (Pa.s^{n}) | n |
---|---|---|---|---|

Carbopol^{®} 974 P | 0.25 | 4.7 | 2.5 | 0.5 |

0.3 | 23.8 | 18.1 | 0.4 | |

0.3 | 26 | 14.4 | 0.4 | |

0.3 | 27.7 | 20.5 | 0.4 | |

0.3 | 28.1 | 16.9 | 0.4 | |

Carbopol^{®} 980 NF | 0.13 | 7.3 | 3.4 | 0.5 |

0.2 | 15.9 | 7.9 | 0.4 | |

0.2 | 31.3 | 12.4 | 0.4 | |

0.2 | 32.1 | 15 | 0.4 | |

0.2 | 33.8 | 15.5 | 0.4 |

h_{min} at Upper-Lid (10^{−7} m) | |||||
---|---|---|---|---|---|

τ_{0} = 0 Pa | τ_{0} = 0.2 Pa | τ_{0} = 1 Pa | τ_{0} = 4.5 Pa | ||

n = 0.5 | Newtonian | 0.574 | |||

k = 0.07 Pa·s | 0.603 | 0.628 | 0.660 | 0.664 | |

k = 0.6 Pa·s | 0.599 | 0.591 | 0.671 | 0.625 | |

k = 2.5 Pa·s | 0.589 | 0.627 | 0.641 | 0.631 |

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**MDPI and ACS Style**

Mehdaoui, H.; Ait Abderrahmane, H.; de Loubens, C.; Nait Bouda, F.; Hamani, S.
Dynamics of a Gel-Based Artificial Tear Film with an Emphasis on Dry Disease Treatment Applications. *Gels* **2021**, *7*, 215.
https://doi.org/10.3390/gels7040215

**AMA Style**

Mehdaoui H, Ait Abderrahmane H, de Loubens C, Nait Bouda F, Hamani S.
Dynamics of a Gel-Based Artificial Tear Film with an Emphasis on Dry Disease Treatment Applications. *Gels*. 2021; 7(4):215.
https://doi.org/10.3390/gels7040215

**Chicago/Turabian Style**

Mehdaoui, Hamza, Hamid Ait Abderrahmane, Clement de Loubens, Faïçal Nait Bouda, and Sofiane Hamani.
2021. "Dynamics of a Gel-Based Artificial Tear Film with an Emphasis on Dry Disease Treatment Applications" *Gels* 7, no. 4: 215.
https://doi.org/10.3390/gels7040215