Se6ling Dynamics of a Contact Lens Kara L. Makia, David S. Rossa, and Emily K. Holzb, aSchool of Mathema-cal Sciences and bDepartment of Chemical and Biomedical Engineering, Rochester Ins-tute of Technology, Rochester, NY Introduc-on Methods: Modeling Bending The U.S. contact lens industry is a mul--­‐billion dollar industry; 34 million people wear contact lenses na-onwide. Studying the se\ling dynamics of a contact lens is of great importance to manufacturers, as it allows them to op-mize pa-ent comfort and ocular fit. Contact lens is designed Clinical trial assesses fit and comfort The empirical design process of the industry could be accelerated with the aid of mathema-cal modeling. Figure 1: Contact lens design process Design is modified based on results §  We couple fluid and solid mechanics to create a mathema-cal model of the contact lens on the eye. §  First, we consider the effects of bending forces in the lens alone on the pressure distribu-on. §  The tear film is modeled as a Newtonian fluid with a viscosity near that of water. Lubrica-on theory governs its mechanics. §  To simplify the mechanics, the contact lens is modeled as a solid beam. §  Gravity and momentum are both neglected. §  The shape of the contact lens is approximated as that of a spherical cap sa-sfying a height h above the x-­‐axis and a radius r at the y-­‐intercept. Only the mo-on of the contact lens center line is considered. §  The cornea is modeled as a flat, solid surface. §  The equa-on for the pressure underneath the lens results from a force balance and the applica-on of boundary condi-ons. poutside(r) } z } r §  We consider so^ silicon hydrogel contact lenses in our model. Silicon hydrogels allow for oxygen transmission and fluid flow through the lens. §  A^er the ini-al se\ling that occurs following its inser-on, the lens sits over the cornea immersed in tear fluid. Contact Lens Post Lens Tear Film Cornea Sclera Figure 2: Model of the contact lens on the cornea, separated by a layer of tear fluid §  A contact lens is subjected to forces from both the tear film as well as those of an eye blink, and in response, the lens bends and stretches. §  The ver-cal force applied to the lens from a blink is es-mated to be about 40-­‐50 dynes §  The horizontal force of the blink is es-mated at 20,000 – 25,000 dynes. §  These forces cause rota-onal, up and down, in and out, and side to side lens movement. Srr =
p(r) 3
Eτ
1 ∂
∂
p(r) = poutside (r) +
r
2
12(1 − σ ) r ∂r
∂r
�
h(r) 1 ∂
r ∂r
�
dR
dr
�
�
�
��
�
2
�
2
1 + h (R) − 1 + g (r)
R−r
�
+σ
r
1 + g � (r)2
∂
r
(h(r) − g(r))
∂r
���
§  Maple was used to find the solu-ons to the pressure equa-on for different parameters based on the input given. §  The shape of the contact lens that sa-sfies a certain pressure curve, h(r), was computed based on an es-mated pressure distribu-on underneath the lens. §  The es-mated p(r) was a cubic func-on that sa-sfied the following condi-ons: ∂p
|r=0 = 0
∂r
§  The maximum pressure value was determined based on experimentally measured values in the literature. Sθθ = σSrr + Eτ (r)
1 dh
R dR
h� (R)
RSrr �
1 + h� (R)2
RSrr
1 + h� (r)2
�
+ p(R) = 0
− p(R) −
d
dR
�
Sθθ
�
h� (r)2
1+
h� (R)
Results of ver-cal blink forces: §  The lens moves downward on the cornea with the ver-cal force of the eyelid. §  Viscous drag slows its mo-on, and ul-mately it reaches the intersec-on of the sclera and cornea, where the radius of curvature changes. §  The lens then acquires energy from resul-ng elas-c (hoop) stresses, which drives its return to the eye’s center. Lens ProperEes • Radius, r = .7 cm • Thickness , τ = 1 * 10-­‐2 cm • Young’s modulus, E = 1 * 107 g/cm*sec2 Tear Film ProperEes • Viscosity, μ = 7 * 10-­‐3 g/
cm*sec • Post lens tear film thickness, h = 5*10-­‐4 cm Eye ProperEes • Corneal radius of curvature: .
78 cm • Scleral radius of curvature: .
115 cm Table 1: Material and physical proper-es of the system’s components §  The equa-on for the -me rate of change of the tear film thickness was derived based on lubrica-on theory. Scaling the equa-on revealed that if bending was the only force involved, the contact lens would take more than a year to relax to its rest shape a^er it was subjected to the forces of the eyeblink. §  This scaling suggests that bending plays a minor role in the centra-on and suc-on of the contact lens. §  This was confirmed by solving for h(r) with the given pressure distribu-on: Lens center Lens center p(r) Lens edge Pressure (dynes/cm2) h(r) §  For the results shown, the rest shape of the contacts lens g(r) is given by a parabola shown in blue. Lens center Broader Lens Shape Distance to the Cornea (cm) p(r) Lens edge Pressure (dynes/cm2) Height (cm) Lens edge Radius (cm) Figure 4: Es-mated pressure distribu-on Radius (cm) Lens center Distance to the Cornea (cm) p(r) Lens edge Pressure (dynes/cm2) Radius (cm) Radius (cm) §  As shown, the rela-onship between the suc-on pressure and deforma-on of the contact lens is complicated and not straighoorward. Conclusions Radius (cm) The goal of our work is to explore the roles of the different forces in influencing centering and lens fit. This informa-on is then used to determine the pressure and fluid flow beneath the lens. =0
Stretching Results Narrower Lens Shape Bending Results �
§  These two equa-ons can be combined to form a third-­‐order ordinary differen-al equa-on (ODE) for the radial displacement R(r). §  We solve the ODE in MATLAB with the following boundary condi-ons: zero displacement at the origin, and specify zero radial tension as well as zero pressure at the lens edge. Radius (cm) Results of horizontal blink forces: §  The lens bends and stretches, pushing tear fluid out at the edges. §  Fluid is drawn back in as the lens relaxes. §  This acts to create a nega-ve (suc-on) pressure under the lens. R−r
r
§  We balance forces in the normal and tangen-al direc-ons along the center of the contact lens to derive two equa-ons for the two unknowns, R(r) and the pressure p(r). §  Normal force balance: �
�
dR h� (R) dR
Bending Force �
Eτ (r)
1 − σ2
�
§  Tangen-al force � balance: d
1
d
1
�
Figure 3: Balance of forces on the lens §  To determine the effects of lens stretching on the tear film and pressure distribu-on, we first determine the radial and angular tensions along the center line of the contact lens. §  We assume post-­‐lens tear film takes a known shape h(r) and that the contact lens must conform to this known shape. §  The deforma-on (or stain) of the contact lens from its known rest shape g(r) is then given by a single radial displacement func-on R(r). §  By applying Hooke’s law, we find the tensions to be τ Background Pre Lens Tear Film Methods: Modeling Stretching Figure 5: Resul-ng lens shape – lens is flipped inside out §  The bending force plays a minor role in keeping the contact lens on the eye. To achieve the es-mated pressure distribu-on under the lens through bending forces alone, results suggest that the en-re lens would need to be flipped outward. §  Consequently, the force due to bending can be neglected when modeling lens dynamics, shi^ing the focus to an analysis of the elas-c forces in the lens. §  Further work is in progress to explore the role of stretching in the lens’s centra-on and the underlying pressure distribu-on. §  Future research involves applying the results of these models to an in vitro study of the detrimental effects of contact lens wear on the cornea.