## Abstract

This study aims to develop a new optical design method of soft multifocal contact lens (CLs) to obtain uniform optical power in large center-distance zone with optimized Non-Uniform Rational B-spline (NURBS). For the anterior surface profiles of CLs, the NURBS design curves are optimized to match given optical power distributions. Then, the NURBS in the center-distance zones are fitted in the corresponding spherical/aspheric curves for both data points and their centers of curvature to achieve the uniform power. Four cases of soft CLs have been manufactured by casting in shell molds by injection molding and then measured to verify the design specifications. Results of power profiles of these CLs are concord with the given clinical requirements of uniform powers in larger center-distance zone. The developed optical design method has been verified for multifocal CLs design and can be further applied for production of soft multifocal CLs.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Multifocal contact lenses (CLs) have been become more popular to correct presbyopia for people after age 40 [1]. Among many types of multifocal CLs, simultaneous multifocal CLs is the most common design to correct presbyopia. It allows the wearers to observe both far objects and close objects in one set of CLs with several concentric optical zones on the optical area [1, 2]. There are many available types of the simultaneous multifocal CLs with different optical power profiles such as bi-concentric, aspheric, multi-concentric and zonal-aspheric multifocal CLs that have been proposed [3, 4]. Normally, multifocal contact lenses need to have a uniform optical power in the central zone to provide the clear near/distance visions for clinical demands. Theoretically, the optical power in the central zone can be constant, but it is very difficult to achieve such target in optical design. Usually the uniform power is only obtained at a relative value by spherical or aspheric curves with small Add powers to generate the optical lens profiles in central zones. Moreover, the lens profiles can be designed with NURBS to approach the central zones as similar to the spherical or aspheric curves for uniform optical power. However, many researches for multifocal CLs or progressive additional lenses using B-spline or NURBS curve have not yet achieved a feasible optimization method for uniform optical powers in central zones [2, 5–9]. So far, there is no feasible design method revealed for continuous distributions in aspheric and zonal multifocal CLs. This study is to develop a new design method of soft multifocal CLs to obtain the uniform power in the large center-distance zone by optimizing NURBS curve.

Since the optical power is correlated to the radius of curvature and the center of curvature, the fitting problem between the NURBS design curve and the corresponding spherical/aspheric curve need to consider not only the fitting error but also the center of curvature error. Thus, the NURBS design curves need to be optimized both to satisfy the given power distribution in the whole optical area and to fit for the given spherical/aspheric curve in center-distance zone. To satisfy those requirements, a multi-objective optimization problem is developed. The first objective function is to minimize power errors between the required optical power and the power of the multifocal CLs. Another two objective functions are related to minimizing the geometric errors between the NURBS curve and its given spherical/aspheric curves in center-distance zone. Such geometrical errors are fitting error and center of curvature error. To solve this problem, a weighted sum method [10,11] is used to optimize three parameters including control points, weights and knots of NURBS. A Simulated Annealing algorithm [12,13] is also adopted to optimize the cubic NURBS design curves to be the anterior profiles of desired soft multifocal CLs. After optimization, the designs of the multifocal CLs with center-distance are presented in two-dimensional (2D) and three-dimensional (3D) computer-aided design (CAD) models for verification of related optical power specification by commercial optical software, Zemax. Finally, four cases of Oculcufil D 55% lenses with different diameters of the center-distance zones have been manufactured from these designs by casting in shell molds made by injection molding process. Measuring these power profiles of four set of multifocal CLs are used to evaluate the proposed design method. Results show that the optical powers in the center-distance zones of the four set of soft multifocal CLs are uniform and closed to given power profiles as well. In addition, based on the feedback of the clinical test results, the power profile of the modified lens sample obtained by this design method is satisfied with the new clinical requirements. Thus, the feasibility of the proposed method can be proven.

## 2. Methodology

#### 2.1 Power profiles

The multifocal contact lens designed in this research is axial rotational and symmetric lens as shown in Fig. 1. To fit the human eye, the posterior optical surface profile is a base (spherical/aspheric) curve that is determined based on the anterior corneal surface of the human eye and lens materials. For popular commercial soft contact lens, the base curves are the spherical curves with the radii from 7.9 to 8.9 mm [14]. In this research, the base curve is selected as 8.6 mm. The anterior surface curve is a NURBS curve that comprises two concentric optical zones having respectively determined different values of optical power. The proposed optical power of the center-distance zone has a uniform power in the center-distance zone while the optical power of the other zone gradually varies from the center zone to the boundary of the optical area. The anterior and posterior peripheral curves and edge are determined based on the design of the optical area to provide comfort, lens alignment and tear flow. This research focuses on the center-distance designs to obtain the uniform power.

The given power distribution in this kind of multifocal lens presented in Fig. 2 is determined in the following equation [2]:

*P*

_{distance}is a constant optical power of the center-distance zone,

*x*is the radius of the center zone, Add is a total addition power,

_{c}*D*

_{op}is optical region diameter and

*x*is a half chord. Since the average pupil diameter of the presbyopic people is from 2 mm to 6 mm, for soft multifocal CLs, the optical diameter

*D*

_{op}should be from 6.0 to 8.0 mm to cover over the pupil [2]. In addition, the diameter of the center-distance zone is chosen less than 4.0 mm or

*x*is less than 2 mm. The range of the overall diameter of the soft contact lens is 14 mm to 15 mm.

_{c}#### 2.2 NURBS summary

Based on the above given optical power profile, a NURBS curve needs to be optimized to be an anterior optical surface profile. NURBS [2, 15] curve of degree *p* defines *C*(*u*) a curve as a piecewise rational polynomial function of a parameter *u*:

*p*defined on the knot vector:

*CC*(

*u*) of that point is determined by the following equation:Where

*N*(

*u*) is an unit normal vector at the above point:

#### 2.3 Multi-Objective optimization

### 2.3.1 Power objective function

The optical power of the multifocal contact lenses is determined via the back vertex power (BVP) by following equation [16]:

where*k*,

_{b}*k*are the curvatures of the anterior and the posterior surfaces,

_{f}*n*is the refractive index of the lens material, and

*t*is the center thickness of the contact lens.

_{c}For a given material and center thickness, the optical power of the multifocal contact lens is a combination of the curvatures of the anterior and posterior surfaces. In other words, when the posterior curves are spherical/ aspheric curves, the power distributions depend on the curvatures of the anterior curves. Thus, the design problem for these multifocal contact lenses is to determined the curvatures of the front curves. Then, a cubic NURBS curve is selected to provide enough flexibility to satisfy the given power profile. Therefore, the first objective function in this research is to minimize the root mean squares (RMS) error between the power (Pw* _{i}*) of the NURBS lens and the required power $(\overline{{\text{Pw}}_{i}})$ that is determined from Eq. (1) at point

*i*along the radial direction, shown in Eq. (10). In this equation,

*i*= 1:

*m*and

*m*is number of data points from the lens center to the boundary of the optical area.

### 2.3.2 Spherical/aspherical fitting in the center-distance zone

To obtain the uniform power in the center-distance zone, the shape of the NURBS curve in this zone should be similar to a spherical or aspheric curve with small additional optical power (Add_{center}). The radius of the spherical curve is determined by Eq. (9) with a given center-distance power P_{distance} at *x =* 0.0. Meanwhile the aspheric curve has two unknown parameters (curvature radius of peak *R*_{o} and the conic constant *k*_{c}); they need two equations to solve these unknowns. The aspheric formula is:

The curvature radius of peak *R _{o}* and the conic constant

*k*

_{c}of the aspheric curve are also determined by solving two Eq. (9) with center-distance power P

_{distance}at

*x*= 0.0 and P

_{distance}+ Add

_{center}at the boundary of the center-distance zone (

*x*=

*x*

_{c}). After the spherical/aspheric curves are defined, the objective functions are fitting functions between the NURBS curves and these corresponding spherical/aspheric curves in the center-distance zones. Figure 3(a) illustrates the data points ${\overline{D}}_{j}({\overline{x}}_{j},{\overline{y}}_{j})$ on the spherical curve and their centers of curvature. Similarly, the data points ${\overline{D}}_{j}({\overline{x}}_{j},{\overline{y}}_{j})$ on the aspheric curve and their centers of curvature $(\overline{CC{x}_{j}},\overline{CC{y}_{j}})$ are in Fig. 3(b), where

*j*= 1:

*s*(

*s*is the number of data points in the center-distance zones). The NURBS curves have to fit with these spherical/aspheric curves in the center zones to ensure that the optical powers in these zones are uniform.

First, the distance (fitting error) between ${\overline{D}}_{j}({\overline{x}}_{j},{\overline{y}}_{j})$on the spherical/aspheric curves and the corresponding point ${D}_{j}({x}_{j},{y}_{j})$on the NURBS curve should be minimized or the second objective function is the mean of fitting errors between those data points:

Second, the optical powers of the lenses also relate to the locations of centers of curvature, thus the distance between center of curvature $(\overline{CC{x}_{j}},\overline{CC{y}_{j}})$of the spherical/aspheric curves and the corresponding center of curvature $(CC{x}_{j},CC{y}_{j})$of the NURBS curve should be minimized or the third objective function is the mean of center of curvature errors between those centers of curvature:

The spherical curve has only one center of curvature, so$\overline{CC{x}_{j}}=\overline{CCx}$and$\overline{CC{y}_{j}}=\overline{CCy}$.

Since Eqs. (10), (14), and (15) are three objective functions, a multi-objective optimization problem is needed to be solved in this study. A weighted sum method [10,11] is considered as a weighted objective function as following:

where *α*, *β* and *γ* are weighting factors and $\alpha +\beta +\gamma =1$

Priority of these weighting factors can be determined by following consideration [10,11]. The uniform power in center zone is more important than that in outer zone and the Add power in the outer zone is higher. Or the power error in the outer zone can be larger. Therefore, the weighting factors of two last objective function are higher than that of the first function. We can consider that *α* is set equal to 0.1 as well as *β* and *γ* are 0.45 in this study. Equation (16) can be rewritten as:

To minimize the weighted objective function, a nonlinear optimization algorithm i.e. Simulated Annealing algorithm [12,13] is adopted in this study. This optimization method is a popular search algorithm that can escape local minima and converge at global minima. The starting values of three variables including control points, weights and knots are determined by assuming the initial NURBS curve as a spherical curve. The radius of this spherical curve is calculated by Eq. (9) with the given center-distance power and the base curve radius.

As mentioned in the study [2], the line P_{0}P_{1} connecting two first control points P_{0} and P_{1} must be in perpendicular to center axis to ensure the smooth anterior curve at the peak point. Therefore, this constraint is added to optimization in which the XY- coordinate of P_{0} is (0, 0) and the Y-coordinate of P_{1} is 0.

#### 2.4 Analysis, manufacture and measurement

To verify the proposed design method, four cases of soft multifocal CLs have been manufactured and compared. Manufacturing process of the soft multifocal CLs includes many steps such as mold insert machining, injection molding of shell molds, lens casting and lens hydration. The injection mold inserts are shown in Fig. 4(a) and usually machined by single-point diamond turning (SPDT) [18]. The female and male mold inserts are related to the posterior and anterior surface of soft CLs respectively. The shell molds with soft CLs product are shown in Fig. 4(b). Due to the swelling or scaling factor of wet soft CLs from dry lens, the initial optical design dimension needs to consider the scaling factor.

In the optical regions, the diameters of and the Add powers of the different vision zones of the initial designs are 25% larger than the required diameters and Add values except for the center power and the base curve. For examples, the optical diameter of the soft lens equals to 6.0 mm and this diameter of the initial design is 7.4 mm. Based on the updated design parameters, the initial optical designs can be obtained by the proposed method.

After optimizing to obtain the anterior curve, 3D models of the initial CLs are generated in CAD/ CAM software. Then, these initial models are imported to an optical software, Zemax to verify performance of optical design. The 3D models are simulated to check the optical power distribution. Once the powers in the center-distance zones are nearly uniform, these lens models are scaled by a scale factor to design the injection molds. In this study, the scale factor of the injection molds to the initial lens designs is 1/1.25 based on CLs materials supplier.

Simultaneously, the peripheral curves of the anterior and posterior curves are determined to complete the mold deign step. Since the soft CLs are very thin, the edges of the lenses have sharp shapes. Based on [19], shell molds of posterior surface and anterior surface are produced by injection molding process. Two set of mold inserts need to be manufactured and such shell molds are used to cast soft CLs. The UV light is applied to obtain dry lenses. Then, dry lenses are hydrated to become soft lenses. All processes to manufacture soft multifocal contact lenses are conducted in the Seinoh Optical Co., Ltd at New Taipei City, Taiwan. The power profiles of these samples are measured by a Contest Plus device provided by the same company. Finally, the power profiles of the actual multifocal contact lens samples are compared with these of the simulation lenses and the original requirements to evaluate the applicability of the developed design method.

## 3. Result and discussion

The design method has been analyzed on four cases of Oculfilcon D 55% multifocal CLs with center-distance. Firstly, four cases with the same the center power but different center-distance zone diameters and total Add power values. The first two cases have the same power distributions and the other two cases have different power distributions. The optical power in the center-distance zones of four cases are all required uniform. The main required parameters of these multifocal CLs are listed in Table 1 in which the parameters in Case 4 are set up based on the clinical requirement of the largest radius of the center-distance zone, *x _{c}* as 1.2 mm from the collaborated company. All power distributions of these center-distance multifocal lenses are determined by Eq. (1) and presented in Fig. 5(a).

As mentioned in the methodology section, the first step in this process is to design the initial lenses. The setting parameters of the initial designs listed in Table 2 are different from the required parameters of the soft lenses in Table 1. To obtain the uniform power in the center zone, the shapes of the NURBS front curves in the center-distance zone should be fitted with spherical or aspheric curves with very small Add power values. The power distribution of the initial design are illustrated in Fig. 5(b). In the Case 1, the central curve is spherical curve while they are aspheric curves with different Add power values in three other cases. The Add_{center} values in the center-distance zones are 0.0, 0.05, 0.10 and 0.15 respectively. From those Add_{center} values, the given base curve, center thickness and refractive index of the lens material, the radius of curvature at vertex of the aspheric curve, the conic constant of the aspheric curves are determined.

After that, NURBS curves are optimized to fit with the given power profiles in Fig. 5(b) and the above spherical/aspheric curves in the center-distance zones. From the given power distributions, three parameters i.e. control points, knots and weights of the cubic NURBS curves are optimized to be the anterior lens curve in each case by Simulated Annealing algorithm. This study uses a step size 0.05 mm in radial direction, thus the number of points using to compute the multi-objective function is 71 points (the radius of the optical region is 3.5 mm). The number of control points are determined after some optimization tests with a range of number of control points from 7 to 13. After analyzing the objective function results, smoothness of the anterior curve, time computing, the number of control points should be selected from 9 to 12. This research uses 9 control points for all cases. The optimization results of the three objective function are small enough and listed in Table 3.

From the optimized results, the NURBS curves are drawn in CAD/CAM softwares. After revolving, 3D models of the CLs are obtained in the standard for the exchange of product model data (STEP), stereolithography (STL), or initial graphics exchange specification (IGS) format for analysis of optical performance. Those lens models are imported to ZEMAX software to verify the power distributions of the proposed models by simulating power profiles as shown in Figs. 6 and 7. All these power distributions are closed to those of the initial designs shown in Fig. 5(b). However, the optical power in the center-distance zone of Case 1 reduces slightly from the center before increases at the peripheral area of this zone. In reality, the power of the spherical lens in this design has the power reducing gradually from the center. Thus, when Case 1 uses this spherical curve for the center-distance zone, the power trend is moving down. Since these multifocal CLs in this study have center-distance, the optical power should increase from the lens center. To solve this problem, aspheric curves are used in the center zones instead of the spherical curves. Therefore, the power distributions of Case 2, 3 and 4 gradually increase from center to the peripheral areas of the outer zones. By observation, the additional powers in the center zones of these cases are less than 0.3 D and the highest value in Case 4 with the largest center zone diameter.

After simulation, the initial lens designs are scaled with the above scale factor to obtain mold insert surfaces. Then, the shell molds are manufactured by injection molding process. Those shell molds are used for casting the contact lenses. Under UV light, the dry lenses are obtained and they are hydrated to be soft lenses as shown in Fig. 8 noted as Case 1 to Case 4.

As observed in measured results of Fig. 8, the power distributions of the real CLs are close to the required power profiles. The optical power profiles in the center-distance zones are similar to the simulation distributions. Especially, the powers at the centers of the fours CLs are near to −4.0 D that is the original specifications for optical designs. The small difference can be improved by a little tuning value of the setting parameters of the initial designs with scaling factor. A close look on the measurement results also show that the power in the center-distance zone of Case 1 is bent in the center-distance zone, while they gradually increase from the lens centers on the other cases. These results match very well with the simulation results. Thus, the simulation can be used to determine the results of the final samples.

The additional powers in the center-distance zones of the real lenses are listed in Table 4. The maximum value is 0.264 D in the case fitting the center-distance zone by the spherical curve. Meanwhile, these values are less than 0.1 D in the other cases. Therefore, for the center-distance multifocal contact lenses, the anterior lens curves in the center zone of should be similar to the aspheric curves. For the center-near multifocal contact lenses, the spherical curves can be considered to present the lens curves in the center-near zones because the power distributions reduces gradually from the lens center in these designs. Since the Add_{center} are small values in three last cases, the powers can be considered as uniform. In addition, the power error curves in the center-distance zones between the original requirements as in Fig. 5(a) and the measurement results of the lens samples as in Fig. 8 are presented in Fig. 9. According to the radii of center-distance zones in Table 1, the maximum power errors are small and less than 0.1 D. Compared with the power profiles of the commercial contact lenses in [4], these values can be accepted. For Case 2 to Case 4, the power error curves are nearly uniform. Especially, in Case 4 with the largest center-distance zone, the power errors are small and uniform values.

However, the original requirements of the optical powers are only relative values as shown in Fig. 5(a), thus the power errors are not yet the final factors to evaluate the lens samples in the center-distance zones. The final step to verify these lenses are clinical tests. There are some feedback based on the clinical test results provided by the collaborated company. The uniform levels of the optical powers in the center-distance zones are accepted. However, the additional power at diameter of 4.0 mm has to be higher than 4.0 D and the diameter of the total optical region has to be increased to 6.0 mm. Based on the new requirements, two parameters of Case 4 in Table 2 are changed i.e. the D_{op} = 7.6 mm, the total Add power = 14.0 D. After using the proposed method with new parameters, the optical power profile of the soft CLs sample in Case 4-V2 after modifying is shown in Fig. 10. In this modified lens, the Add power at the diameter of 4.0 mm is 4.05 D, D_{op} is 6.0 mm, and Add_{center} is 0.2 D. These values satisfy the new requirements based on the previous clinical test.

By analyzing all above cases and comparison, the power distributions of the fabricated soft CLs agree closely with these of the clinical requirements. The multi-objective optimization problem has been solved to achieve uniform power in the large center-distance zones. The efficient applicability of the proposed method has been proven for designing multifocal CLs by optimized NURBS.

## 4. Conclusion

This study has developed a new design method of soft multifocal contact lens with uniform optical power in large center–distance zone. In this method, a multi-objective function problem is defined by considering not only the optical power but also the center of curvature and fitting distance. The NURBS design curves are optimized to be the anterior curves in the optical region of the multifocal CLs and to satisfy three objective functions simultaneously by Simulated Annealing method. Four cases of Oculfilcon D 55% soft multifocal CLs have been manufactured and measured for verification. Results of power profiles of the measurement results of the soft multifocal CLs agree with that of the original requirements for these optical designs. The uniform levels of the optical powers in the center-distance zones have been achieved. Based on clinical test feedback, the power profile of the modified lens sample is consequently satisfied with the new clinical requirements by applying the proposed method. Experimental results have shown the feasibility of optimized NURBS. Therefore, the developed design method of multifocal contact lenses with uniform powers in the larger center-distance zones has been verified. Future study of this developed method can be further investigated for multifocal CLs with center-near applications.

## Funding

The ministry of science and technology, Taiwan: MOST 104B0223-2.

## Acknowledgement

The authors would appreciate for the kind assistance of Seinoh Optical Co., Ltd for manufacturing and measurement the CLs samples.

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