Richard Y. Hwang1, PhD; Dan Gauthier2, PhD; Dana Wallace1, MD; Natalie A. Afshari1, MD

1 Department of Ophthalmology, 2Department of PhysicsDuke University

Durham, NCThe 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

corneaAnterior 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 power2) Posterior corneal power

Gullstrand eye model

pcr

nn 23pcF

Posterior corneal graft changes the posterior radius of curvature.

Posterior corneal power has 3 components:n3, index of refraction of aqueous humorn2, index of refraction of cornearpc, 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

rpc=recipient radius of curvature

rpc’= rpc -tthickness

Ideal donor corneal shape (even width)

tthickness

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

ttransplant = Central thickness of donor cornea (C)

Peripheral thickness of donor cornea (P)

h’’=1.5 cm

C

P

? Radius of curvature

w

ttransplant = transplant thicknessw = difference in peripheral width between ideal and non-ideal corneal transplant = ttransplant * (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)

yr

x

X2+Y2=R2

(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

s

yr

s

ysr

ysrs

yrsrsr

yrsr

yrrs

2

2

2

2

2

22

22

2222

22

22

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 = y2/(2s)

w

Recipient posterior corneal surface

Central thickness of donor cornea

ttransplant

si’

rpc’

h’

w

Recipient posterior corneal surface

Central thickness of donor cornea

ttransplant

si’’

rpc’’h’’

Uniform width graft

Non-uniform width graft

'2

)'('

2

pci r

hs Sag equation for

uniform width graft

''2

)''(''

2

pci r

hs Sag equation for non-

uniform width graft

Note: rpc ‘ is the posterior radius of curvature of a uniform width graft

Note: rpc ‘’ is the posterior radius of curvature of a non-uniform width graft

wsin

w

Recipient posterior corneal surface

Central thickness of donor cornea

ttransplant

si’

si’’

rpc’

rpc’’

h’ h’’

rpc’’

rpc’

wcos

Equationsh’ = h’’ + w sin si’’ = si’ + w cos si’’ - si‘= w

'2

)'('

2

pci r

hs

''2

)'(

''2

)''(''

22

pcpci r

h

r

hs

'2'

'2'''

'2'''

'2'''

'''2'

'

2

''

2

1))11

((2

1''

1))11

((21

1)(21

1)'''(21

11)'''(2

2

)'(

2

)'('''

pc

transplant

pc

pc

transplant

pc

pcpc

pc

ii

pc

pcpc

ii

pcpc

ii

rhCP

tr

rhCP

t

r

rh

w

r

rh

ss

r

rrh

ss

r

h

r

hss

Make assumptionsAssume = 0 (very small)h’ = h’’si’’ = si‘+ w

Combine the sag equations…

The magic of arithmetic

Post-DSEK radius of curvature

))((F3

eye lenscornealenscornea FFn

dFF

))()((2

pcactransplant

pcaccornea FFn

ttFFF

pcr

nn 23pcF

acr

nn 12acF

))(*

)(**1(005678.0

2

3transplant

cornea

actransplant ttFn

Fttnd

Anterior corneal power Modified Posterior corneal power

Modified total corneal power

Modified distance between cornea and lens principal planes

Modified component of Gullstrand eye model

Refractive shift = Feye+DSEK-Feye-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

Patient Pre-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

1 650 142 0.88 6.33 23 1.0 0.73 0.69

2 816 129 0.85 7.08 22 0.75 0.84 0.79

3 573 147 0.86 7.10 5 1.0 0.86 0.81

4 764 95 0.79 6.98 3 0.82 0.91 0.85

Average mean refractive change: 0.89 DAverage 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) Transplant thickness (10-4 m)

Host corneal thickness (10-4 m)Host corneal thickness (10-4 m) Host PRC (10-4 m)

40

41

42

43

56

57

58

59

CP Ratio CP Ratio

Corneal power Eye power

24

68

24

68

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