Grading Sphero-Cylinder Spectacle Similarity

Figures & data

Table 1 Variables to Derive Spectacle Comparison GRADE Match

Variable	Description	Formula
SPHtarget	Sphere component of gold-standard, target refraction
CYLtarget	Cylinder component of target refraction
AXIStarget	Cylinder axis of target refraction
SPHcandidate	Sphere component of comparison spectacle
CYLcandidate	Cylinder of comparison spectacle
AXIScandidate	Axis of the cylinder of comparison spectacle
AXIS-ADJcandidate	Axis of candidate refraction rotated the same amount as target would be shifted to 45°	Axiscandidate–(45°-axistarget)
Mtarget	Spherical equivalent	M=SPHtarget+CYLtarget/2
(J0, J45)	Plot of astigmatism vector components
J0-45target	Vector transform of target rotated horizontally with axis at 45°	J0=-(cyl/2)xcos(2*45°)
J45-45target	Vector transformed oblique component rotated to horizontal with axis at 45°	J45=-(cyl/2)xsin(2*45°)
Mcandidate		=SPH+CYL/2
J0-ADJcandidate		J0=-(cyl/2)xcos(2*axis-adj)
J45-ADJcandidate		J45=-(cyl/2)xsin(2*axis-adj)
“a”	Long oval axis	a=SQRT((0.39*(x–h)^2+(y–k)^2)/0.39)
“b”	Short oval axis	b=short axis=0.63*a
Aquadratic	0=A*(Grade)^2+B*(Grade)+C	A=−0.048 (Cyltarget)
Bquadratic	Grade=(-B+sqrt(B^2–4*A*C))/2*A	B=0.29*(Cyltarget)+0.319
Cquadratic	Component of equation	C=-(diameter)=−2*a
Accom	Degree of accommodation required between refractions	Accom=Mcandidate-Mtarget
Accomexcess	Excess accommodation
PD	Pupillary distance	GradePD=∆PD/(4.165*e^-(−0.1*Mtarget)
GradePD	Degree of pupillary distance match	∆PD=|PDtarget–PDcandidate|
A-Grade	Clinical degradation from best corrected acuity to 1 logMAR worse
B-Grade	Degradation from best corrected to 3 logMAR worse visual acuity
C-Grade	Degradation from best corrected to 6 logMAR worse
M	Spherical equivalent	=SPH+CYL/2
J0	Vector transformed Jackson Cross horizontal	J0=-(cyl/2)xcos (2*axis)
J45	Vector transformed Jackson Cross oblique	J45=-(cyl/2)xsin(2*axis)
(h,k)	x and y coordinates of center of an oval
(x,y)	x and y-coordinate of a point on the circumference of an oval
Gradea	Quality of Accommodation match	GRADEa=accom/(0.02*age–1.5)
Slopea		Slopea=0.02(Gradea)
Intercepta		Intercepta=−1.5(Gradea)
Age	Age in years
OverPlus		Overplus=0.5*Grade
Accomexpect		Accomexp=5.25*e^(−0.05*Age)

Figure 1 J0 vs J45 vector transformed astigmatism A, B, and C matched degraded visual acuity. Example subjects with cyl 0.5, 1.0, 1.5, 2.25 contact lens, 2.50, and 5.25 D. The lower graph shows an example spectacle with cylinder axis converted to 45° such that orientation of the concentric A-Grade, B-Grade, and C-Grade ovals have the long axis “a” oriented horizontally.

Figure 2 J0 vs J45 ovals and grades. Method to rotate target to 45° so horizontal and then generate oval boundaries (green A, blue B, and orange C to determine whether a candidate refraction, also rotated, is within a given visual acuity-derived Grade level comparison.

Figure 3 Linear relationship between A-match (green), B-match (blue), and C-match (orange) visual acuity degrades 1 logMAR, 3 logMAR, and 6 logMAR. Diameter of degrade limits versus cylinder power. The slope and intercept for derived long axis length versus cylinder (upper left graph) is related to Grade (upper right graphs). The slope and intercept (lower left) of over-minus spherical equivalent Grade is related to age (lower right).

Figure 4 Spherical equivalent match related to age and accommodation. Over plus produces similar distance blur at all ages while over minus is better tolerated by young patients. A-match (green), B-match (blue), and C-match (orange) visual acuity degrades 1 logMAR, 3 logMAR, and 6 logMAR.