Evaluation for Refractive Surgery
Transcript of Evaluation for Refractive Surgery
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Evaluation for Refractive Surgery
• Want to screen patients prior to performing a surgery that permanently alters the shape of their corneas to ensure they are suitable for the procedure.
• Pupil size is important, especially under dim conditions. Want to ensure the maximum pupil size is not much larger than the optical zone of the treatment.
• Corneal topography is important to ensure cornea irregularities do not exist that might suggest potential disease.
Infrared Pupillometry
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Locate Purkinje Images & Center
Polar Coordinates
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Mask Purkinjes
Edge Filter
111111111100000-1-1-1-1-1-1-1-1-1-1
Convolution
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Find peaks near pupil
Fit points to sinusoid
• Pupil margin is a shifted sinusoidal.• Fit data to truncated Fourier series.• Fit coefficients give center and dimensions of elliptical
pupil.• Iterate (3x) and eliminate outliers.
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Pupil Margin
Pupil Margin
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Eyelashes
Hair & Blur
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Contact Lens
Issues
• Saturation and/or small LCD screen make focusing difficult
• Refresh rate too slow. Currently 10 fps – Need 30 fps• Need “Click” to provide feedback to user that picture was
taken• JPEG compression too high. Defeats Mpix advantage.• Orient LEDs in vertical rectangle to leave sides open.• Use entire imaging chip.
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Image Compression
KeratometryThe keratometer is a device for measuring the radius of curvature of the anterior corneaalong the flat and steep meridians.
Its primary function today is todetermine corneal curvature for IOL implantation (SRK formula).
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Keratometry
hy
d
Radius R
hdy2R =
d and h are fixed incommercial systems and Ris obtained from a calibratedmeasurement of the heighty.
Placido DiskThe Placido disk extends the keratometryconcept to examine curvature at differentpoints on the cornea.
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Corneal Topography
Variations on Placido Concept
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Variations on Placido Concept
Fringe Projection
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Stereophotogrammetry
d
Θ1 Θ2
Knowledge of two anglesand side tells where the tipof the triangle lies.
Requires a diffuse reflection!
Stereophotogrammetry
Fluorescein dye is instilledinto the eye. When illuminatedin blue light, the fluoresceinfluoresces in the green.
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StereophotogrammetryThis system is being linked to excimer lasers in refractive surgery. The ultraviolet laserlight causes the cornea cellsto fluoresce in the blue.
Eliminates the need for dropsand can be done duringsurgery.
Scanning Slit
d
Θ1 Θ2
Knowledge of two anglesand side tells where the tipof the triangle lies.
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Scanning Slit
Technique uses white lightscatter from cornea and crystalline lens.
By refracting through eachsurface, this technique canmeasure anterior and posterior corneal shape, aswell as anterior crystallinelens shape.
Corneal Shape• Axial Power – (sometimes incorrectly called sagittal
power) gives a map of corneal curvature in terms of dioptric power. Is related to the equivalent sphere with the same slope at a given point.
• Instantaneous Power – (sometimes called tangential power) gives a map of corneal curvature in terms of dioptric power. Is related to the curvature (2nd derivative) of a point on the cornea.
• Sag or Elevation – gives a map of the surface height at a given point. The height can be relative to a reference surface.
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Total Corneal Power
D 42.356D 6.108-D 48.346
Diopters 5.6
3771.13374.18.7
3771.01000
=Φ=Φ
⎥⎦⎤
⎢⎣⎡ −
+=Φ
Find the equivalent power surface based solely on the anteriorradius of curvature.
3304.1'n1000
356.428.71'n
=
=−
Keratometric Index of Refraction
• Historically, a keratometric index of refraction nk was chosen based on available data on the cornea and for mathematical simplicity.
• The index nk = 1.3375, which gives a power of 45 D for a surface with radius 7.5 mm was chosen.
• The purpose of the index is to account for the power of the posterior cornea based solely on the radius of the front surface of the cornea.
• Still used today in keratometry and corneal topography.
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Axial Power
Θ
Θ
r
Ra
z
Cornea
Examine each meridian from the axis outward.
( ) a2
22
2
2
a
Rr
dr/dz1
dr/dzsin
drdz1sin
drdz
sin1sin
drdz
drdz tanand
Rrsin
=+
=Θ
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛+Θ=⎟
⎠⎞
⎜⎝⎛
Θ−
Θ=
=Θ=Θ
( ) ( )( )2
k
a
ka
dr/dz1r
dr/dz1nR
1n
+
−=
−=Φ
Axial Power - Example
constantR
1nrR
rdrdz
rRRz
ka
22
22
=−
=Φ
−=
−−=Sphere:
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Instantaneous Power
r
RI
Cornea
Examine each meridian from the axis outward.
( )[ ]( )
( )[ ] 2/32
22k
I
kI
2/32
22
I
dr/dz1
dr/zd1nR
1ndr/dz1
dr/zdR1
+
−=
−=Φ
+=z
From Calculus:For a given curve in the r-z plane, thereexists a circle of radius RI which istangent to the curve at (r,z) and has the same curvature 1 / RI as the curve.
Instantaneous Power - Example
[ ]constant
R1n
rRR
drzd
rRr
drdz
rRRz
ki
2/322
2
2
2
22
22
=−
=Φ
−=
−=
−−=Sphere:
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Axial & Instantaneous Power
( )
( ) ( ) ( )( )
( ) ( ) 'dr'rr1r
drrd
f1df/dr ff
f11
drdf1n
drrd
f1f1nr
drdzf Define
r
0Ia
Ia
2/322ka
2k
a
∫Φ=Φ
Φ=Φ
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛
++
+−=
Φ
+
−=Φ
=
Elevationr
Cornea
z
rCornea
z
Elevation maps tend to obscuresubtle features on the cornea, unlessa reference surface is subtracted.
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Astigmatism
RK Incisions
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Lasik
Central Island
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KeratoconusLocalized thin spot in the cornea that progressively bulges.
Map Comparison
Axial Instantaneous Elevation Elevation - Ref
www.euclidsys.com
Derivation of Axial and Instantaneous Curvature Jim Schwiegerling
March 5, 2004
Corneal topographic data is typically specified in terms of a dioptric power. Dioptric power is not a true power per se, but instead a description of the surface shape. Two different types of topographic maps are common. The first is typically referred to as the axial power map, or more rarely, the sagittal power map. The second type is called the tangential or instantaneous power map. In this paper, the local curvature of a rotationally symmetric surface is derived and its relationship to axial and instantaneous power is shown. Fundamental Forms of Geometry Differential geometry provides the tools to determine the local curvature on a surface. If a continuous surface is defined as z = f(x,y), then the local curvature can be obtained from the First and Second fundamental forms of geometry. At each point on this surface, there exist two principal curvatures κ1 and κ2. These curvatures represent the maximum and minimum curvature through this point and they are always in orthogonal directions. The surface curvature in other directions is at some value between κ1 and κ2. The First Fundamental quantities are given by
E = 1 + (dz/dx)2 F = (dz/dx)(dzdy) G = 1 + (dz/dy)2
The Second Fundamental quantities are given by
[ ] 2/12
22
FEGdx/zdL
−= [ ] 2/12
2
FEGdxdy/zdM
−= [ ] 2/12
22
FEGdy/zdN
−=
From these, the mean curvature is given by
( ) ( )212 21
FEG2FM2GLENH κ+κ=
−++
= ,
and the Gaussian curvature is given by
212
2
FEGMLNK κκ=
−−
=
The principal curvatures can also be extracted from these quantities as
KHH
KHH2
2
21
−−=κ
−+=κ.
From these relationships, the axial and instantaneous curvature of a surface will be derived.
First Derivatives The display of corneal topography dioptric power maps has an underlying assumption that the surface is rotationally symmetric. In this special case, the axial curvature is given by κ2 and the instantaneous curvature is given by κ1. To calculate these curvatures for a rotationally symmetric surface, it is convenient to describe the surface in polar coordinates. Since all the derivatives in the preceding expression are given in Cartesian coordinates, a transformation is in order. Converting the x derivative to polar form is achieve as follows:
θ⎟⎠⎞
⎜⎝⎛ θ
+⎟⎠⎞
⎜⎝⎛=
dd
dxd
drd
dxdr
dxd ,
where the standard definition of polar coordinates hold (i.e. 22 yxr += and . Calculating the derivatives gives
)x/y(tan 1−=θ
θ=+
= cosyx
x221
dxdr
22 and
rsin
xy
x/y11
dxd
222
θ−=⎟
⎠⎞
⎜⎝⎛ −
+=
θ
since 21
x11)x(tan
dxd
+=− . Thus, the final result is
θθ
−θ=dd
rsin
drdcos
dxd
.
Similarly,
θθ
+θ=dd
rcos
drdsin
dyd
.
Second Derivatives The second derivatives in polar form are given by the product of the first derivates in polar form. Care must be taken when distributing the differentials as they can act on the components they multiply. For the second order derivatives,
⎟⎠⎞
⎜⎝⎛
θθ
−θ⎟⎠⎞
⎜⎝⎛
θθ
−θ=dd
rsin
drdcos
dd
rsin
drdcos
dxd
2
2
, which gives
2
2
2
2
2
2
2
22
2
2
dd
rsin
dd
rcossin2
drd
rsin
drdcos
dxd
θθ
+θ
θθ+
θ+θ= .
Similarly,
2
2
22
22
2
22
dd
rcossin
dd
rsincos
drd
rcossin
drdcossin
dxdyd
θθθ
−θ
θ−θ−
θθ−θθ=
and
2
2
2
2
2
2
2
22
2
2
dd
rcos
dd
rcossin2
drd
rcos
drdsin
dyd
θθ
+θ
θθ−
θ+θ= .
Calculating the First and Second Fundamental Forms For a rotationally symmetric object, the θ derivatives in the preceding expression are all zero. Taking this into account and plugging the derivatives into the expression for the First and Second Fundamental forms gives
2
2
drdzcos1E ⎟
⎠⎞
⎜⎝⎛θ+=
2
drdzcossinF ⎟
⎠⎞
⎜⎝⎛θθ=
22
drdzsin1G ⎟
⎠⎞
⎜⎝⎛θ+=
2/122
2
22
drdz1
drdz
rsin
drzdcosL
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛+⎥
⎦
⎤⎢⎣
⎡ θ+θ=
2/12
2
2
drdz1
drdz
rcossin
drzdcossinM
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛+⎥
⎦
⎤⎢⎣
⎡ θθ−θθ=
2/122
2
22
drdz1
drdz
rcos
drzdsinN
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛+⎥
⎦
⎤⎢⎣
⎡ θ+θ=
Calculating the Gaussian and Mean Curvature The Gaussian and mean curvatures can now be determined for the rotationally symmetric surface.
22
2
2
drdz1r
drdz
drzdK
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛+⎥
⎦
⎤⎢⎣
⎡= (Gaussian Curvature)
2/322
2
2
drdz12
drdz1
drdz
r1
drzdH
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛+
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛++= (Mean Curvature)
Calculating the Principal Curvatures Finally, the Principal curvatures can be determined from the Gaussian and mean curvatures such that
2/32
2
2
1 drdz1
drzd
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛+=κ (Instantaneous Curvature)
2/12
2 drdz1r
drdz
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛+=κ (Axial Curvature)
The curvatures κ1 and κ2 are typically converted to a dioptric power by multiplying the respective curvatures by (nk - 1), where nk is the keratometric index of refraction. Principal Directions The orientation of the Principal Curvatures are described by a unit vector xix + yiy, where i = 1 for κ1 and i = 2 for κ2. The elements of this vector are given by1
( ) ( )2i
2i
ii
ELFM
FMxκ−+κ−
κ−= and
( ) ( )2i
2i
ii
ELFM
ELyκ−+κ−
κ−−=
For i = 1, and , where θθ=κ− cossinAFM 1 θ−=κ− 2
1 sinAEL
2/32
2
2
2
drdz1
drdz1
drdz
r1
drzd
A
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛+
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛+−
=
Plugging these expressions in, the orientation of the Instantaneous Curvature is given by cosθ x + sinθ y, or in other words, the Instantaneous Curvature is in the radial direction. Similarly, for i = 2, θθ=κ− cossinBFM 2 and , where θ=κ− 2
2 cosBEL
2/12
2
2
2
drdz1
drdz1
drdz
r1
drzd
B
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛+
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛+−
=
Plugging these expressions in, the orientation of the Axial Curvature is given by sinθ x + cosθ y, or in other words, the Axial Curvature is in the azimuthal direction.
References 1. Cohen E, Riesenfeld RF, Elber G. Geometric Modeling with Splines. AK Peters: Natick, MA: 2001. pg. 367.