Particle swarm optimisation strategies for IOL formula constant optimisation

Author:

Langenbucher Achim1ORCID,Szentmáry Nóra23ORCID,Cayless Alan4,Wendelstein Jascha15ORCID,Hoffmann Peter6

Affiliation:

1. Department of Experimental Ophthalmology Saarland University Homburg/Saar Germany

2. Dr. Rolf M. Schwiete Center for Limbal Stem Cell and Aniridia Research Saarland University Homburg/Saar Germany

3. Department of Ophthalmology Semmelweis‐University Budapest Hungary

4. School of Physical Sciences The Open University Milton Keynes UK

5. Department of Ophthalmology Johannes Kepler University Linz Linz Austria

6. Augen‐ und Laserklinik Castrop‐Rauxel Castrop‐Rauxel Germany

Abstract

AbstractPurposeTo investigate particle swarm optimisation (PSO) as a modern purely data driven non‐linear iterative strategy for lens formula constant optimisation in intraocular lens power calculation.MethodsA PSO algorithm was implemented for optimising the root mean squared formula prediction error (rmsPE, defined as achieved refraction minus predicted refraction) for the Castrop formula in a dataset of N = 888 cataractous eyes with implantation of the Hoya Vivinex hydrophobic acrylic aspheric lens. The hyperparameters were set to inertia: 0.8, accelerations c1 = c2 = 0.1. The algorithm was initialised with NP = 100 particles having random positions and velocities within the box constraints of the constant triplet parameter space C = 0.25 to 0.45, H = −0.25 to 0.25 and R = −0.25 to 0.25. The performance of the algorithm was compared to classical gradient‐based Trust‐Region‐Reflective and Interior‐Point algorithms.ResultsThe PSO algorithm showed fast and stable convergence after 37 iterations. The rmsPE reduced systematically to 0.3440 diopters (D). With further iterations the scatter of the particle positions in the swarm decreased but without further reduction of rmsPE. The final constant triplet was C/H/R = 0.2982/0.2497/0.1435. The Trust‐Region‐Reflective/Interior‐Point algorithms showed convergence after 27/17 iterations, respectively, resulting in formula constant triplets C/H/R = 0.2982/0.2496/0.1436 and 0.2982/0.2495/0.1436, both with the same rmsPE as the PSO algorithm (rmsPE = 0.3440 D).ConclusionThe PSO appears to be a powerful adaptive nonlinear iteration algorithm for formula constant optimisation even in formulae with more than 1 constant. It acts independently of an analytical or numerical gradient and is in general able to search for the best solution even with multiple local minima of the target function.

Publisher

Wiley

Subject

Ophthalmology,General Medicine

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