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ASCRS 2010
April 9 – 14, 2010, Boston, MA, USA
On the origin of the corneal data
in the Gullstrand eye
Wolfgang Haigis, MS PhD AssProf
Dept. of Ophthalmology, University of Wuerzburg, Germany
The author has no financial interest in the subject matter of this poster
W.Haigis 2010
On the origin of the corneal data in the Gullstrand eye
Background and Purpose
The Gullstrand eye
• is a theoretical (schematic) eye in different modifications (exact,
simplified, without accommodation or under maximum accommodation),
• has been used for decades as an eye model for theoretical-optical
calculations,
• serves as a basis for the calibration of ophthalmic instruments (e.g.
keratometers, topographers, etc.
Question
• To which extent does the cornea of the Gullstrand eye reflect true
biometric measurements ?
• Do applications and instruments based hereupon produce realistic
results ?
W.Haigis 2010
Gullstrand eye
Exact schematical eye in accommodative rest for optical calculations
The Gullstrand theoretical
eye provides exact data for
all components of the optical
system ‘eye‘, i.e. it gives
intraocular distances, indices
of refraction of the ocular
media, and radii of curvature
of the corneal and lenticular
interfaces, together with the
cardinal points (principal,
focal and nodal points) of the
eye.
All this data describe a selfconsistent model eye.
The exact schematical eye in
accommodative rest is axially
1D hyperopic.
From: Siebeck R: Optik des menschlichen Auges. Springer Berlin Göttingen Heidelberg 1960
W.Haigis 2010
Gullstrand eye and corneal power
Keratometer index for instrument calibration
The cornea is a meniscus lens with 2 surfaces.
Both of them determine its refractive power, yet
only the anterior radius is measured by keratometry. The posterior radius and thus the posterior
surface power is not known.
Deriving the corneal power from a measurement of the anterior corneal radius alone makes only sense if there is a fixed
ratio between posterior and anterior radii of curvature. If this
ratio is given by the
Gullstrand ratio R2/R1 = 6.8/7.7
then it is possible to calculate the corneal power K according
to
K  KI 1
R
1
+
where KI : keratometer index and R1: anterior corneal radius.
☺
Surface refractive
powers:
n n
C
1
R
1
?
n n
2
C
R
2
It can be shown that with
• KI =1.3315 (Zeiss index) the total (equivalent) power
• KI =1.3375 (Javal index) the (back) vertex power
of the cornea is obtained.
(In eyes not following the Gullstrand ratio (e.g. after refractive
surgery), neither index produces any reasonable results.)
Haigis W: Corneal power after refractive surgery with myopia: the contact lens method. J Cataract Refract Surg 29(7):1397-1411,2003
W.Haigis 2010
On the origin of the corneal data in the Gullstrand eye
Methods
Methods
Literature research to trace down the numerical properties
of the cornea of the Gullstrand eye to their individual
origins, especially for the anterior and posterior corneal
radii of curvature, the corneal thickness and the refractive
indices of cornea and aqueous.
Gullstrand published his
schematic eye(s) in a
separate chapter entitled
‘Die Dioptrik des Auges‘
(Dioptrics of the eye) in
Hermann von Helmholtz‘s
famous ‚Handbuch der
physiologischen Optik‘
(Handbook of physiologic
optics).
Hermann von Helmholtz
1821-1894
Allvar Gullstrand
1862-1930
H. von Helmholtz: Handbuch der Physiologischen Optik. 3.Auflage, ergänzt und herausgegeben in Gemeinschaft mit Prof.Dr. A.Gullstrand und Prof.Dr.J.von Kries von Prof.Dr.W. Nagel. 1.Band, Hamburg und Leipzig, Verlag
von Leopold Voss, 1909
W.Haigis 2010
Gullstrand eye: refractive index of aqueous/vitreous
Measurement values from contemporary literature
•
•
•
•
•
1.3365 : own measurements by Gullstrand [1].
1.338 : measurements by Chossat, cited in [1].
1.3366 : measurements by Brewster, cited in [1].
1.3420 : measurements by Krause, cited in [1] .
Comparative results from a compilation of various refractometric studies are
cited in [2].
 1.336 adopted by Gullstrand
[1]:
[2]:
H. von Helmholtz: Handbuch der Physiologischen Optik. 3.Auflage, ergänzt und herausgegeben in Gemeinschaft mit
Prof.Dr. A.Gullstrand und Prof.Dr.J.von Kries von Prof.Dr.W. Nagel. 1.Band, Verlag von Leopold Voss, 1909, S.87
Freytag G. Vergleichende Untersuchungen über die Brechungsindizes der Linse und der flüssigen Augenmedien des
Menschen und höherer Tiere in verschiedenen Lebensaltern. Wiesbaden 1907
H. von Helmholtz: Handbuch der Physiologischen Optik. 3.Auflage, ergänzt und herausgegeben in Gemeinschaft mit Prof.Dr. A.Gullstrand und Prof.Dr.J.von Kries von Prof.Dr.W. Nagel. 1.Band, Hamburg und Leipzig, Verlag
von Leopold Voss, 1909, S.285
W.Haigis 2010
Gullstrand eye: refractive index of the cornea
Measurement values from contemporary literature
•
1.3770 and 1.3721 :
•
1.3746
:
•
1.3763
:
refractometrically measured on a 50 year old male and
a baby aged 2 days by Aubert and Matthiesen [1].
calculated from measurement results for the individual
corneal components [2].
last compilation by Matthiesen [3].
 1.376 adopted by Gullstrand
[1]: Aubert H: Grundzüge der physiologischen Optik, Leipzig, 1876
[2]: Lohnstein Th: Über den Brechungsindex der menschlichen Hornhaut. Arch. f. d. ges. Physiologie. LXVI, 1897
[3]: Matthiessen L. Die neueren Fortschritte in unserer Kenntnis von dem optischen Baue des Auges der Wirbeltiere.
Hamburg 1891
H. von Helmholtz: Handbuch der Physiologischen Optik. 3.Auflage, ergänzt und herausgegeben in Gemeinschaft mit Prof.Dr. A.Gullstrand und Prof.Dr.J.von Kries von Prof.Dr.W. Nagel. 1.Band, Hamburg und Leipzig, Verlag
von Leopold Voss, 1909, S.279
W.Haigis 2010
Gullstrand eye: corneal center thickness
Measurement values from contemporary literature
•
•
•
0.482 – 0.668 mm: determined ophthalmometrically by Blix [1] for 10 eyes
in vivo.
0.506 – 0.576 mm: results of Blix if 1 eye with extreme values is excluded
(i.e.n=9).
0.46 und 0.51 mm: own measurements by A.Gullstrand [2] in vivo on 2
eyes of different subjects.
 0.5 mm adopted by Gullstrand
[1]: Blix M: Oftalmometriska studier. Upsala Läkareförenings Förhandlingar. XV. 1880
[2]: H. von Helmholtz: Handbuch der Physiologischen Optik. 3.Auflage, ergänzt und herausgegeben in Gemeinschaft
mit Prof.Dr. A.Gullstrand und Prof.Dr.J.von Kries von Prof.Dr.W. Nagel. 1.Band, Verlag von Leopold Voss, 1909
H. von Helmholtz: Handbuch der Physiologischen Optik. 3.Auflage, ergänzt und herausgegeben in Gemeinschaft mit Prof.Dr. A.Gullstrand und Prof.Dr.J.von Kries von Prof.Dr.W. Nagel. 1.Band, Hamburg und Leipzig, Verlag
von Leopold Voss, 1909, S.280-282
W.Haigis 2010
Gullstrand eye: anterior corneal radius Ra
Measurement values from contemporary literature
•
7.858: (8.396 – 7.280) mm: average for 110 males (m) [1].
•
7.799: (8.487 – 7.115) mm: average for 46 females (f) [1].
 7.829 mm: mean (m,f), adopted by Helmholtz [2] henceforth.
•
•
7.843
7.723

7.80 mm (8.50 - 7.00) mm adopted by Gullstrand
[1]:
[2]:
[3]:
[4]:
mm: average of 1916 Augen [3].
mm: average [4] with n smaller than in [3].
Donders FC: On the anomalies of accommodation and refraction. London, 1864
Helmholtz H. Handbuch der Physiologischen Optik. 2. Auflage. Hamburg und Leipzig, 1885
Steiger A. Beiträge zur Physiologie und Pathologie der Hornhautrefraktion. Wiesbaden, 1895
Sulzer, zit, in: H. von Helmholtz: Handbuch der Physiologischen Optik. 3.Auflage, ergänzt und herausgegeben in
Gemeinschaft mit Prof.Dr. A.Gullstrand und Prof.Dr.J.von Kries von Prof.Dr.W. Nagel. 1.Band, Verlag von Leopold
Voss, 1909
H. von Helmholtz: Handbuch der Physiologischen Optik. 3.Auflage, ergänzt und herausgegeben in Gemeinschaft mit Prof.Dr. A.Gullstrand und Prof.Dr.J.von Kries von Prof.Dr.W. Nagel. 1.Band, Hamburg und Leipzig, Verlag
von Leopold Voss, 1909, S.272
W.Haigis 2010
Gullstrand eye: posterior corneal radius Rp
Gullstrand‘s own measurements (n=6)
Gullstrand‘s measurements of the ratio Ra/Rp with a keratometer according to Blix:
*):
#
Ra/Rp
 Rp/Ra
 Rp [mm] *)
1
1.1822
0.846
6.60
2
1.1811
0.847
6.60
3
1.1864
0.843
6.57
4
1.1734
0.852
6.65
5
1.1486
0.871
6.79
6
1.1427
0.875
6.83
mean ± sd
1.169 ± 0.019
0.856 ± 0.014
6.67 ± 0.11
calculated for Ra = 7.8 mm (selected by Gullstrand as ‘ophthalmometric average‘ of contemporary
measurements)
Tscherning, Lagrange et Valude. Encyclopédie d’ Ophthalmologie. III S.109, Paris 1904: Rp=6.5 mm
H. von Helmholtz: Handbuch der Physiologischen Optik. 3.Auflage, ergänzt und herausgegeben in Gemeinschaft mit Prof.Dr. A.Gullstrand und Prof.Dr.J.von Kries von Prof.Dr.W. Nagel. 1.Band, Hamburg und Leipzig, Verlag
von Leopold Voss, 1909, S.283-284
W.Haigis 2010
Gullstrand eye: corneal radii Ra and Rp
Justification for Gullstrand‘s choice of Ra and Rp
Ra = 7.80 mm: choosen as average from contemporary measurements.
Rp = 6.67 mm: obtained from own measurements (n=6).

Rp / Ra = 6.67 / 7.80 ≈ 6.70 / 7.80 = 0.859.
But:
Rp / Ra = 6.80 / 7.70 = 0.883 adopted in final Gullstrand eye.
Gullstrand‘s reasoning: Measurements are performed in the optical zone (optical axis), not at the
intersection of the cornea with the fixation axis. Due to the measurement, Rp/Ra is smaller than
real, hence Rp must be larger and Ra must be smaller than measured.
Thus: Rp = measured value (6.7) + 0.1 mm = 6.8 mm.
Ra = measured value (7.8) – 0.1 mm = 7.7 mm.

Rp = 6.8 mm, Ra = 7.7 mm adopted by Gullstrand
H. von Helmholtz: Handbuch der Physiologischen Optik. 3.Auflage, ergänzt und herausgegeben in Gemeinschaft mit Prof.Dr. A.Gullstrand und Prof.Dr.J.von Kries von Prof.Dr.W. Nagel. 1.Band, Hamburg und Leipzig, Verlag
von Leopold Voss, 1909, S.285
W.Haigis 2010
On the origin of the corneal data in the Gullstrand eye
Summary and Conclusion
Summary
•
Values for the optical properties (refractive indices, thickness, radii of
curvatures) of the corneal system of the Gullstrand eye were mainly
derived from the scientific literature of the late 19th century.
•
Only few measurements were performed by Gullstrand himself.
•
Particularly the determination of the posterior corneal curvature is based on
a very small number of measurements.
Conclusion
•
Although the Gullstrand eye plays an important role as a self-consistent
theoretial model eye in visual optics, it does not seem to be apt to serve as
a realistic biometrical model for the cornea.
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