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F. Thorn, J. Gwiazda; Mathematical Modeling of Myopia Progression in Children. Invest. Ophthalmol. Vis. Sci. 2009;50(13):3955.
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Modeling of myopia progression requires closely fitting both the available data and the known characteristics of myopia progression. A least squares fit alone does not determine the best model. Our purpose is to (1) compare the Gompertz function to other functions with similar least squares fits for individual children, and (2) show how curve fitting to group data relates to models for individuals.
Linear, exponential, sinusoidal, parabolic, and Gompertz functions were fit to individual longitudinal refractive data of both eyes of 36 children in the NECO Children’s Vision Laboratory and to the combined data for the whole group after adjustment for the time of myopia onset. The functions for individual eyes had to satisfy a number of logical constraints.
All functions fit the data closely when the data were limited to a 4 - 6 year period of myopia progression, but most failed to approximate the entire myopization process. The sinusoidal and parabolic models deviated abruptly from the process before and after myopia progression. The exponential function started in high hyperopia and usually continued progressing into middle or old age. The Gompertz function was better able to render realistic pre- and post-progression refraction levels with limited data. Only the Gompertz function improved its fit when the full length of the longitudinal data extending several years before and after the period of myopia progression was used, and only it rendered realistic projections beyond the data set. In aim two, the data for the whole group adjusted for age of myopia onset are highly scattered creating poor fits. An exponential function fit best even if the Gompertz function was the best fit for individual children. Mathematically, a simple exponential function approximates the average of a set of best fit Gompertz functions or linear functions.
Only the Gompertz function improved its fit as the individual data sets were extended before and after myopia progression, thus rendering a realistic model for the full course of myopia progression. Functions derived from fitting group data often suggest a simple exponential function when the actual function for individual eyes is something else.
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