# Pupil Function in Pseudophakia: Proximal Miosis Behavior and Optical Influence

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Participants

#### 2.2. Pupil Size Measurement

^{®}(R2016b, MathWorks, MA, USA). The participants looked at a white circle (with angular size of ${5.5}^{\circ}$ and luminance of $65.3\phantom{\rule{0.277778em}{0ex}}\mathrm{cd}\phantom{\rule{4.pt}{0ex}}{\mathrm{m}}^{-2}$) displayed on an LCD monitor (LG model 23MP65HQ), with a Maltese cross (angular size: ${1.0}^{\circ}$) on its center used as a fixation point. The object distances tested, given by the monitor position, were $3.0$, $1.0$, $0.66$, $0.50$, $0.40$ and $0.33$ m. The stimulus angular size was kept constant for the given distances. The amount of light reaching the corneal plane was $0.45$ lux (luxmeter Minolta, model T-10A, Konica Minolta, Europe) and was also kept constant for the different object positions. The pupil diameters were measured from the furthest (3.0 m) to the closest (0.33 m) object position. The participants sat in a dark room with the head on a headrest. Initially, the participants had 2 m of light adaptation period staring at the white circle positioned at 3.0 m, and then, for each object position, there was a 5 s light adaptation period followed by 10 s of pupil diameter measurement. Prior to the measurements, each participant was instructed to focus the Maltese cross and blink normally. For each object position, the pupil diameter was calculated by removing the blinks from the sampled data, using a Hampel filter, followed by averaging of the sampled data. An example of pupil size acquisition during 15 s (5 s light adaptation plus 10 s test) is presented in Figure 1, where both the raw data and the processed data for blink removal are displayed.

#### 2.3. Pupil Size Statistical Modeling

^{®}(R2016b) using the function nlmefit from the Statistics and Machine Learning Toolbox, with the option LME (method of maximum likelihood). The fitting routine also provides an estimate of the variance–covariance matrix D.

#### 2.4. Optical Performance Modeling

^{®}(R2016b). Before performing the through focus computation of the VSMTF and the aMTF values, it was necessary to optimize the eye model, using the ray tracing software implementation of the LBME, to resemble a distance corrected pseudophakic eye. First, the object was set to the distant position (infinity) and the pupil size was selected as the mean far distance pupil obtained from the experimental measurements. After retrieving the set of Zernike coefficients, using the Zernike Analysis of Wavefront feature of OSLO Premium, and convert it to the OSA standard version [39], the $\mathrm{MTF}\left(\right)open="("\; close=")">{f}_{x},{f}_{y}$ and ${\mathrm{MTF}}_{\mathrm{DL}}\left(\right)open="("\; close=")">{f}_{x},{f}_{y}$ functions were generated for a set of defocus values. Then, using Equation (8), the VSMTF values were computed from these functions. The defocus value that maximizes the VSMTF was retrieved and the corresponding vitreous length was computed.

## 3. Results

#### 3.1. Pupillary near Response

#### 3.2. Optical Performance

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Pupil diameter recordings during 15 s and for the different observing distances.

**Upper**set: The raw data for the RE and LE.

**Lower**set: The pupil sizes after blink removal for the RE and LE using the Hampel filter.

**Figure 2.**(

**a**) Far distance ($3.0$ m) pupil diameter versus subjective spherical equivalent. (

**b**) Far distance ($3.0$ m) pupil diameter versus age.

**Figure 3.**(

**a**) Fixed effect fit and $95\%$ confidence limits for the mean (dashed green line) and for a single prediction (dashed red line) for a given value of pupil diameter. (

**b**) Scatter plot of standardized residuals versus fitted values for the first mixed effects model.

**Figure 4.**Area under MTF as a function of object vergence in diopters, from near ($OV=-3\phantom{\rule{0.277778em}{0ex}}\mathrm{D}$) to far ($OV=0\phantom{\rule{0.277778em}{0ex}}\mathrm{D}$) object distances: (

**a**) Case 1 models with constant pupil miosis and varying distance pupil diameter (PD) values; and (

**b**) Case 2 models with constant distance pupil and varying pupil miosis.

**Figure 5.**Predicted visual acuity in logMAR (MAR: Minimum Angle of resolution) variation as function of object vergence in diopters, using the simulated area under MTF curve and Equation (10), from near ($OV=-3\phantom{\rule{0.277778em}{0ex}}\mathrm{D}$) to far ($OV=0\phantom{\rule{0.277778em}{0ex}}\mathrm{D}$) object distances: (

**a**) Case 1 models with constant pupil miosis and varying distance pupil diameter (PD) values; and (

**b**) Case 2 models with constant distance pupil and varying pupil miosis.

Surface | Radius (mm) | Thickness (mm) | Conic Constant | $\mathit{n}\left(\right)open="("\; close=")">\mathit{\lambda}=555\phantom{\rule{0.277778em}{0ex}}\mathbf{nm}$ |
---|---|---|---|---|

Anterior cornea | $7.77$ | $0.500$ | $-0.18$ | $1.376$ |

Posterior Cornea | $6.40$ | $3.160$ | $-0.60$ | $1.336$ |

Iris | ∞ | $1.512$ | − | $1.336$ |

Anterior IOL | $13.00$ | $1.057$ | − | $1.461$ |

Posterior IOL | $-10.00$ | $17.721$ | − | $1.336$ |

Retina | $-12.00$ | $0.000$ | − | − |

Object Vergence (D) | |||||||
---|---|---|---|---|---|---|---|

0.33 | 1.00 | 1.50 | 2.00 | 2.50 | 3.00 | ||

Pupil Size (mm) | Mean | 4.44 | 4.24 | 4.10 | 3.99 | 3.91 | 3.84 |

SD | 0.87 | 0.90 | 0.88 | 0.87 | 0.86 | 0.86 |

**Table 3.**Estimated parameters and fit statistics for the proposed mixed effects models. The fixed effects parameter estimates are given by mean (standard error).

Parameter | First Model | Second Model | |
---|---|---|---|

Fixed parameters | ${\beta}_{1}$ ${\beta}_{2}$ ${\beta}_{3}$ ${\beta}_{4}$ | $\phantom{\rule{1.em}{0ex}}4.451\phantom{\rule{0.277778em}{0ex}}\left(\right)open="("\; close=")">0.117$ $-0.227\phantom{\rule{0.277778em}{0ex}}\left(\right)open="("\; close=")">0.022$ $\phantom{\rule{2.em}{0ex}}-$ $\phantom{\rule{2.em}{0ex}}-$ | $\phantom{\rule{1.em}{0ex}}5.971\phantom{\rule{0.277778em}{0ex}}\left(\right)open="("\; close=")">0.725$ $-0.496\phantom{\rule{0.277778em}{0ex}}\left(\right)open="("\; close=")">0.136$ $-0.022\phantom{\rule{0.277778em}{0ex}}\left(\right)open="("\; close=")">0.010$ $\phantom{\rule{1.em}{0ex}}0.004\phantom{\rule{0.277778em}{0ex}}\left(\right)open="("\; close=")">0.002$ |

Variance components | ${\sigma}_{{b}_{1}}^{2}$ ${\sigma}_{{b}_{2}}^{2}$ ${\sigma}_{{b}_{1}{b}_{2}}$ | $\phantom{\rule{1.em}{0ex}}0.789$ $\phantom{\rule{1.em}{0ex}}0.025$ $-0.052$ | $\phantom{\rule{1.em}{0ex}}0.732$ $\phantom{\rule{1.em}{0ex}}0.023$ $-0.042$ |

Goodness -of-fit | $Bias$ $RMSE$ ${R}^{2}$ | $\phantom{\rule{1.em}{0ex}}0.000$ $\phantom{\rule{1.em}{0ex}}0.129$ $\phantom{\rule{1.em}{0ex}}0.985$ | $\phantom{\rule{1.em}{0ex}}0.000$ $\phantom{\rule{1.em}{0ex}}0.129$ $\phantom{\rule{1.em}{0ex}}0.985$ |

Model | DF | AIC | BIC | Loglikelihood | LRT | p-Value |
---|---|---|---|---|---|---|

1 | 6 | $16.723$ | $39.837$ | $-2.362$ | ||

2 | 8 | $14.702$ | $45.519$ | $0.649$ | $6.0217$ | $0.0493$ |

**Table 5.**Previously published results for (mean and $95\%$ CIs) VA estimation (logMAR) for different viewing distances. The results for the present study are added, with simulated bounds computed from the regression model single prediction $95\%$ CIs as explained in the text.

Study | Distance $6.0$ m | Intermediate $1.0$ m | Near $0.33$ m | |
---|---|---|---|---|

Shentu et al., 2001 [52] | $+0.04\phantom{\rule{0.277778em}{0ex}}\left(\right)open="("\; close=")">-0.08,0.10$ | − | $+0.61\phantom{\rule{0.277778em}{0ex}}\left(\right)open="("\; close=")">0.34,0.74$ | |

Pieh et al., 2002 [53] | $+0.10\phantom{\rule{0.277778em}{0ex}}\left(\right)open="("\; close=")">-0.06,0.18$ | $+0.16\phantom{\rule{0.277778em}{0ex}}\left(\right)open="("\; close=")">-0.11,0.29$ | $+0.58\phantom{\rule{0.277778em}{0ex}}\left(\right)open="("\; close=")">0.34,0.70$ | |

Rocha et al., 2007 [54] | $+0.03\phantom{\rule{0.277778em}{0ex}}\left(\right)open="("\; close=")">-0.05,0.07$ | $+0.33\phantom{\rule{0.277778em}{0ex}}\left(\right)open="("\; close=")">0.04,0.47$ | $+0.39\phantom{\rule{0.277778em}{0ex}}\left(\right)open="("\; close=")">0.06,0.55$ | |

Navavaty et al., 2009 [55] | $-0.02\phantom{\rule{0.277778em}{0ex}}\left(\right)open="("\; close=")">-0.22,0.08$ | − | $+0.57\phantom{\rule{0.277778em}{0ex}}\left(\right)open="("\; close=")">0.28,0.71$ | |

Present study | $-0.03\phantom{\rule{0.277778em}{0ex}}\left(\right)open="("\; close=")">-0.10,0.05$ | $+0.07\phantom{\rule{0.277778em}{0ex}}\left(\right)open="("\; close=")">0.00,0.11$ | $+0.34\phantom{\rule{0.277778em}{0ex}}\left(\right)open="("\; close=")">0.27,0.42$ |

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**MDPI and ACS Style**

Fonseca, E.; Fiadeiro, P.; Gomes, R.; Sanchez Trancon, A.; Baptista, A.; Serra, P.
Pupil Function in Pseudophakia: Proximal Miosis Behavior and Optical Influence. *Photonics* **2019**, *6*, 114.
https://doi.org/10.3390/photonics6040114

**AMA Style**

Fonseca E, Fiadeiro P, Gomes R, Sanchez Trancon A, Baptista A, Serra P.
Pupil Function in Pseudophakia: Proximal Miosis Behavior and Optical Influence. *Photonics*. 2019; 6(4):114.
https://doi.org/10.3390/photonics6040114

**Chicago/Turabian Style**

Fonseca, Elsa, Paulo Fiadeiro, Renato Gomes, Angel Sanchez Trancon, António Baptista, and Pedro Serra.
2019. "Pupil Function in Pseudophakia: Proximal Miosis Behavior and Optical Influence" *Photonics* 6, no. 4: 114.
https://doi.org/10.3390/photonics6040114