{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,24]],"date-time":"2026-07-24T08:45:43Z","timestamp":1784882743326,"version":"3.55.0"},"reference-count":15,"publisher":"Public Library of Science (PLoS)","issue":"12","license":[{"start":{"date-parts":[[2025,12,2]],"date-time":"2025-12-02T00:00:00Z","timestamp":1764633600000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["www.plosone.org"],"crossmark-restriction":false},"short-container-title":["PLoS One"],"abstract":"\n We present a theoretical framework to estimate the anterior and posterior radii of curvature of a thick intraocular lens (IOL) by measuring its back-vertex power in two orientations. Armed with the lens thickness, refractive index, and a potential axial offset\n d<\/jats:italic>\n from haptic angulation, one can determine the individual surface powers and, thus, the geometry of the implant. Using paraxial optics, we derive the back-vertex power in normal and flipped orientations. We consider two cases:\n d<\/jats:italic>\n \u2009=\u20090 (no haptic-induced offset) and\n \n \n \n \n \n d<\/mml:mi>\n \u2260<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n (finite shift). In the\n d<\/jats:italic>\n \u2009=\u20090 case, using standard paraxial relations (y\u2013\n \n \n \n \n \n \u03bd<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n method), we obtain compact expressions that allow direct recovery of the surface powers from the dual back-vertex powers. For\n \n \n \n \n \n d<\/mml:mi>\n \u2260<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , the measured powers are first mapped back to the lens vertex (Eq 11), after which the same closed-form retrieval as for\n d<\/jats:italic>\n \u2009=\u20090 applies. When\n d<\/jats:italic>\n \u2009=\u20090, a closed-form solution yields the surface powers\n \n \n \n \n \n (<\/mml:mo>\n \n P<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n P<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and radii\n \n \n \n \n \n (<\/mml:mo>\n \n R<\/mml:mi>\n \n ia<\/mml:mtext>\n <\/mml:mrow>\n <\/mml:msub>\n ,<\/mml:mo>\n \n R<\/mml:mi>\n \n ip<\/mml:mtext>\n <\/mml:mrow>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . If the lens is shifted by d, we first correct to the vertex plane (Eq 11) and then apply the same closed-form relations. Though lens nominal power alone does not reveal geometry, our dual-orientation approach recovers how much power resides on each surface, benefiting thick-lens IOL power formulas and refining predictions in cataract surgery planning.\n <\/jats:p>","DOI":"10.1371\/journal.pone.0337041","type":"journal-article","created":{"date-parts":[[2025,12,2]],"date-time":"2025-12-02T18:47:32Z","timestamp":1764701252000},"page":"e0337041","update-policy":"https:\/\/doi.org\/10.1371\/journal.pone.corrections_policy","source":"Crossref","is-referenced-by-count":1,"title":["An optical method for deriving the anterior and posterior curvatures of intraocular lenses using dual back-vertex power measurements"],"prefix":"10.1371","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9603-0500","authenticated-orcid":true,"given":"Damien","family":"Gatinel","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"340","published-online":{"date-parts":[[2025,12,2]]},"reference":[{"issue":"2","key":"pone.0337041.ref001","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1097\/j.jcrs.0000000000001378","article-title":"IOL power formula classifications","volume":"50","author":"G Savini","year":"2024","journal-title":"J Cataract Refract Surg."},{"issue":"4","key":"pone.0337041.ref002","doi-asserted-by":"crossref","first-page":"620","DOI":"10.1016\/j.zemedi.2022.11.007","article-title":"Considerations of a thick lens formula for intraocular lens power calculation","volume":"34","author":"A Langenbucher","year":"2024","journal-title":"Z Med Phys."},{"issue":"4","key":"pone.0337041.ref003","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1167\/tvst.10.4.27","article-title":"Determining the theoretical effective lens position of thick intraocular lenses for machine learning-based IOL power calculation and simulation","volume":"10","author":"D Gatinel","year":"2021","journal-title":"Transl Vis Sci Technol."},{"issue":"3","key":"pone.0337041.ref004","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1097\/j.jcrs.0000000000001584","article-title":"Accuracy of thick and thin intraocular lens power formulas using paraxial vergence calculation","volume":"51","author":"K N\u00e6 ser","year":"2025","journal-title":"J Cataract Refract Surg."},{"issue":"6","key":"pone.0337041.ref005","doi-asserted-by":"crossref","first-page":"556","DOI":"10.1097\/j.jcrs.0000000000001159","article-title":"Surgeons need to know more about intraocular lens design for accurate power calculation","volume":"49","author":"T Olsen","year":"2023","journal-title":"J Cataract Refract Surg."},{"issue":"10","key":"pone.0337041.ref006","doi-asserted-by":"crossref","first-page":"3404","DOI":"10.3390\/jcm12103404","article-title":"Theoretical impact of intraocular lens design variations on the accuracy of IOL power calculations","volume":"12","author":"D Gatinel","year":"2023","journal-title":"J Clin Med."},{"issue":"31","key":"pone.0337041.ref007","doi-asserted-by":"crossref","first-page":"7327","DOI":"10.1364\/AO.34.007327","article-title":"Standardized methods for assessing the imaging quality of intraocular lenses","volume":"34","author":"NE Norrby","year":"1995","journal-title":"Appl Opt."},{"issue":"3","key":"pone.0337041.ref008","doi-asserted-by":"crossref","DOI":"10.1111\/aos.15726","article-title":"Impact of uncertainties in biometric parameters on intraocular lens power formula predicted refraction using a Monte-Carlo simulation","volume":"102","author":"A Langenbucher","year":"2024","journal-title":"Acta Ophthalmol."},{"issue":"11","key":"pone.0337041.ref009","doi-asserted-by":"crossref","first-page":"1995","DOI":"10.1016\/j.jcrs.2009.06.021","article-title":"Testing the dioptric power accuracy of exact-power-labeled intraocular lenses","volume":"35","author":"KJ Hoffer","year":"2009","journal-title":"J Cataract Refract Surg."},{"issue":"9","key":"pone.0337041.ref010","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1167\/tvst.11.9.5","article-title":"Theoretical relationship among effective lens position, predicted refraction, and corneal and intraocular lens power in a pseudophakic eye model","volume":"11","author":"D Gatinel","year":"2022","journal-title":"Transl Vis Sci Technol."},{"key":"pone.0337041.ref011","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.ajo.2023.05.005","article-title":"A simplified method to minimize systematic bias of single-optimized intraocular lens power calculation formulas","volume":"253","author":"D Gatinel","year":"2023","journal-title":"Am J Ophthalmol."},{"issue":"9","key":"pone.0337041.ref012","doi-asserted-by":"crossref","DOI":"10.3928\/1081597X-20250707-03","article-title":"Influence of manufacturing tolerance and formula thickness type in the prediction error of multifocal intraocular lens power calculation","volume":"41","author":"J Fern\u00e1ndez","year":"2025","journal-title":"J Refract Surg."},{"issue":"1","key":"pone.0337041.ref013","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1016\/S0886-3350(03)00556-X","article-title":"Effect of optic edge design and haptic angulation on postoperative intraocular lens position change","volume":"30","author":"V Petternel","year":"2004","journal-title":"J Cataract Refract Surg."},{"issue":"1","key":"pone.0337041.ref014","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1097\/ICU.0000000000001093","article-title":"Update on intraocular lens formulas","volume":"36","author":"M Raimundo","year":"2025","journal-title":"Curr Opin Ophthalmol."},{"issue":"8","key":"pone.0337041.ref015","doi-asserted-by":"crossref","first-page":"1640","DOI":"10.1038\/s41433-022-02211-2","article-title":"Influence of decentration and tilt of Tecnis ZCB00 on visual acuity and higher order aberrations","volume":"37","author":"E Mart\u00ednez-Plaza","year":"2023","journal-title":"Eye (Lond)."}],"container-title":["PLOS 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