Richard Y. Hwang 1, PhD; Dan Gauthier 2, PhD; Dana Wallace 1, MD; Natalie A. Afshari 1, MD 1 Department of Ophthalmology, 2 Department of Physics Duke University Durham, NC The authors have no financial interest. Research supported by Research to Prevent Blindness.

 DSEK Posterior lamellar transplant Indicated for patients with endothelial dysfunction Typically induces unpredictable hyperopic shift  Purpose To develop a mathematical model to predict refractive changes after DSEK How: Evaluate effect of DSEK on Gullstrand eye model Light cornea Anterior cornea Posterior cornea lens Eye refractive power has 2 components: 1) Corneal power 2) Lens power Corneal refractive power has 2 components: 1) Anterior corneal power 2) Posterior corneal power Gullstrand eye model

Posterior corneal graft changes the posterior radius of curvature. Posterior corneal power has 3 components: n 3, index of refraction of aqueous humor n 2, index of refraction of cornea r pc, radius of curvature of posterior cornea (meters) (Rao, Leung et al. 2008) (Scorcia, Matteoni et al. 2009) DSEK surgery affects the posterior corneal radius of curvature

Recipient posterior corneal surface r pc =recipient radius of curvature r pc ’= r pc -t thickness Ideal donor corneal shape (even width) t thickness Recipient posterior corneal radius of curvature represented as a circle post-DSEK posterior corneal radius of curvature represented as a circle - Ideal shape of corneal graft would be parallel to the host cornea (even width) - New posterior curvature of even width = host posterior curvature – transplant thickness Visual axis - Radius of curvature can be approximated as a circle.

Recipient posterior corneal surface t transplant = Central thickness of donor cornea (C) Peripheral thickness of donor cornea (P) h’’=1.5 cm C P ? Radius of curvature w t transplant = transplant thickness w = difference in peripheral width between ideal and non-ideal corneal transplant = t transplant * (1/CP – 1) h‘’ = height at which CP ratio is measured -Quantify un-even graft with central to peripheral graft thickness ratio, CP ratio (C/P) -How do we estimate the new posterior radius of curvature? (Yoo, Kymionis et al. 2008)

y r xx X 2 +Y 2 =R 2 (sag equation) Assume s << r y= ½ chord length s=  x = saggital depth r= radius of curvature Chord s=  x=Saggital depth (sag) y=0.5 x Chord length r =radius of curvature To estimate the new posterior radius of curvature, we can relate 3 measurements 1) posterior radius of curvature (r) 2) saggital depth (s) 3) ½ chord length (y) with the sag equation: r = y 2 /(2s)

 w Recipient posterior corneal surface Central thickness of donor cornea t transplant si’si’ r pc ’ h’ w Recipient posterior corneal surface Central thickness of donor cornea t transplant si’’si’’ r pc ’’ h’’ Uniform width graft Non-uniform width graft Sag equation for uniform width graft Sag equation for non- uniform width graft Note: r pc ‘ is the posterior radius of curvature of a uniform width graft Note: r pc ‘’ is the posterior radius of curvature of a non-uniform width graft

 wsin  w Recipient posterior corneal surface Central thickness of donor cornea t transplant si’si’ s i ’’ r pc ’ r pc ’’ h’ h’’ r pc ’’ r pc ’ wcos  Equations h’ = h’’ + w sin  s i ’’ = s i ’ + w cos  s i ’’ - s i ‘= w Make assumptions Assume  = 0 (very small) h’ = h’’ s i ’’ = s i ‘+ w Combine the sag equations… The magic of arithmetic Post-DSEK radius of curvature

Anterior corneal powerModified Posterior corneal power Modified total corneal power Modified distance between cornea and lens principal planes Modified component of Gullstrand eye model Refractive shift = F eye+DSEK -F eye- DSEK Modified total eye power

 4 variables required to calculate change in power of the eye Transplant thickness (obtained via transplant bank) CP ratio (obtained via transplant bank) Host corneal thickness (preop-pachymetry) Host posterior radius of curvature  Steps to estimate refractive change after DSEK surgery Obtain 4 pre-surgical variables Calculate pre-surgical and post-surgical eye power in diopters Subtract pre from post surgical eye power = (-1) * refractive change  Model applied to 4 patients PatientPre-op thickness (micom) Pre-op graft thickness (microm) Pre-op graft CP ratio Posterior radius of curvature (cm) Post op month Observed shift refractive (D) Predicted shift in corneal power Predicted shift in power of eye Average mean refractive change: 0.89 D Average predicted mean corneal power change: 0.84 D (94%) Average predicted mean eye power change : 0.79 D (88%)

Corneal power Eye power Transplant thickness (10 -4 m) Host corneal thickness (10 -4 m) Host PRC (10 -4 m) CP Ratio Corneal powerEye power Host PRC (10 -4 m) Graphical representation of Gullstrand eye model equations.

 DSEK math model approximately estimates refractive change This model provides a suitable starting point for building a more sophisticated math model  Implications To correlate refractive change with 1 variable, it would be ideal to hold the other 3 variables constant Graft tissue thickness and CP ratios have significant impact on refractive change Both hyperopic and myopic shifts possible In theory, tailoring the shape of donor tissue can be targeted toward a refractive goal  Future refinements Account for transplant and recipient corneal deturgescence Account for corneal changes (e.g. change in recipient radius of curvature without graft) after surgery Prospective studies are required to refine validity of model Account for estimation errors