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J Opt om. 2010;3(3):125-133
Journal
of
Optometry
P e e r- r e v i e w e d J o u r n a l o f t h e
Spanish General Council of Optometry
ISSN: 1888-4296
Journal
of
Optometry
July-September 2010
| Vo l . 3 | n . 3
Editorial
123
Innovation in contact lenses: basic research and clinical science
Eric Papas, James S. Wolfsohn, Lyndon Jones
Original Articles
125
134
Designing contact lenses for a wide ield of view via ocular wavefront tomography
143
149
158
164
169
Ability of silver-impregnated contact lenses to control microbial growth and colonisation
Xin Wei, Larry Thibos
Development of a new contact lens multipurpose solution:
Comparative analysis of microbiological, biological and clinical performance
Simon Kilvington, Ling Huang, Eugenia Kao, Charles H. Powell
Mark D.P. Willcox, Emma B.H. Hume, Ajay K. Vijay, Robert Petcavich
Visual and optical performance of silicone hydrogel contact lenses for moderate myopia
Nancy Keir, Treford Simpson, Desmond Fonn
Current applications and eicacy of scleral contact lenses – a retrospective study
Boris Severinsky, Michel Millodot
Biochemical analyses of lipids deposited on silicone hydrogel lenses
Shin Hatou, Masaki Fukui, Keiichi Yatsui, Hiroshi Mochizuki, Yoko Akune, Masakazu Yamada
Comparison of the number of visits and diagnostic lenses required to it RGP,
conventional hydrogel and silicone hydrogel contact lenses
Raul Martín, Elena Alonso
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ORIGINAL ARTICLE
Designing contact lenses for a wide Þeld of view via ocular
wavefront tomography
Xin Wei*, Larry Thibos
School of Opt omet ry, Indiana Universit y, Bloomingt on, IN, USA
Received 28 March 2010; accept ed 9 June 2010
KEYWORDS
Lens design;
Cont act lens;
Off-axis wavefront
aberrat ion;
Peripheral vision
Abstract
Pur pose: Correct ing t he of f -axis wavef ront aberrat ion is pot ent ially import ant f or peripheral
vision, f or diagnost ic imaging of t he ret ina, and f or inàuencing ref ract ive development . A new
t echnique called ocular wavef ront t omography (OWT) was adapt ed t o opt imize t he design of
cont act lenses t o improve t he eye’s peripheral opt ical qualit y.
Met hods: OWT is a t echnique f or cust omizing a mult i-surf ace model eye t o mimic t he of f -axis
wavefront aberrat ions for an individual eye. This t echnique was adapt ed for cont act lens design
by est ablishing clear design goals for t he eye + cont act lens syst em. To demonst rat e t he met hod
we opt imized t he shape of an aspheric and bifocal cont act lens t o correct a wide angle model eye
wit h —2D f oveal myopia. Two st rat egies f or correct ion reàect ed alt ernat ive design goals: 1) t o
fully correct cent ral vision while also improving opt ical qualit y peripherally t o enhance vision and
ret inal imaging, or 2) fully correct cent ral vision while int roducing a degree of peripheral myopia
relat ive t o cent ral vision in order t o slow myopia progression.
Result s: The OWT t echnique successfully produced aspheric and bifocal cont act lens designs over
a wide Þeld of view. In addit ion t o correct ing f oveal vision, t he opt imized cont act lens designs
eit her 1) improved t he ret inal image qualit y across t he visual Þeld (< 45º) signiÞcant ly t o obt ain a
visual performance and ret inal imaging beneÞt or 2) produced t he desired level of myopia in t he
peripheral Þeld t o obt ain a refract ive development beneÞt .
Concl usion: The OWT t echnique is a validat ed t ool t o opt imize cont act lens design over a wide
Þeld.
© 2010 Spanish General Council of Opt omet ry. Published by Elsevier España, S.L. All right s reserved.
*Corresponding aut hor: Xin Wei. School of Opt omet ry, Indiana Universit y, 800 East At wat er Ave. , Bl oomingt on, IN 47405. Fax:
812 855 7045.
E-mail address: weix@indiana.edu (X. Wei).
1888-4296/ $ - see front mat t er © 2010 Spanish General Council of Opt omet ry. Published by Elsevier España, S.L. All right s reserved.
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126
PALABRAS CLAVE
Diseño de lent es;
Lent es de cont act o;
Deformación del
frent e de onda
con desplazamient o
de ej e;
Visión periférica
X. Wei, L. Thibos
Diseño de lentes de contacto para un campo visual amplio mediante tomografía ocular
por frente de onda
Resumen
Obj et ivos: la corrección de la deformación del frent e de onda con desplazamient o de ej e es pot encialment e import ant e para la visión periférica, imágenes diagnóst icas de la ret ina y repercut ir
en la progresión de errores refract ivos. Se adapt ó una nueva t écnica denominada t omografía ocular por frent e de onda (OWT, en inglés) para opt imizar el diseño de lent es de cont act o que mej oren la calidad ópt ica periférica del oj o.
Mét odos: La OWT es una t écnica que permit e crear un modelo mult isuperÞcial del oj o que imit a las
deformaciones de frent e de onda con desplazamient o de ej e para un oj o individual. Est a t écnica se
adapt ó para diseñar lent es de cont act o mediant e el est ablecimient o de met as de diseño claras para
el oj o + sist ema de lent e de cont act o. Para demost rar el mét odo, opt imizamos la forma de una
lent e de cont act o asférica y bifocal para corregir un modelo de oj o de ángulo amplio con miopía
foveal de 2D. Dos est rat egias de corrección reàej aron met as de diseño alt ernat ivas: 1) corregir
plenament e la visión cent ral mient ras se mej oraba la calidad ópt ica periférica a Þn de mej orar la
imagen en ret ina y la visión, ó 2) corregir t ot alment e la visión cent ral mient ras se int roduce un
grado de miopía periférica respect o a la visión cent ral para enlent ecer la progresión de la miopía.
Result ados: la t écnica de OWT produj o con éxit o diseños de lent es de cont act o asféricas y bifocales sobre un campo de visión amplio. Además de corregir la visión foveal, los diseños de lent es de
cont act o opt imizadas 1) mej oraron la calidad de la imagen ret iniana a t ravés del campo visual
(< 45º) de forma signiÞcat iva y se obt uvieron beneÞcios en rendimient o visual y en la imagen ret iniana, o 2) produj eron un grado deseado de miopía en el campo periférico que repercut ía de forma beneÞciosa en la progresión de errores refract ivos.
Conclusiones: La t écnica de OWT es una herramient a validada para opt imizar el diseño de lent es
de cont act o en un campo de visión amplio.
© 2010 Spanish General Council of Opt omet ry. Publicado por Elsevier España, S.L. Todos los derechos
reservados.
Introduction
Peripheral vision pl ays an import ant rol e in dail y visual
t asks such as driving 1, 2 and l ocomot ion. 3 Al t hough visual
acuit y for reading let t ers and ot her spat ial resolut ion t asks
declines rapidly in t he peripheral Þeld, visual acuit y f or
det ect ing spat ial pat t erns and obj ect s declines only slight ly
i n t he peri phery. 4-6 Consequent l y, peri pheral det ect i on
acuit y is nearly as sensit ive as f oveal resolut ion acuit y t o
opt ical blur. 7 Overcoming opt ical limit at ions of t he nat ural
eye across t he ent ire visual Þeld wit h advanced designs of
cont act lenses should t herefore provide a signiÞcant visual
beneÞt .
Recent l y cl i ni cal i nt er est i n per i pher al vi si on has
i ncreased dramat i cal l y because of t he possi bi l i t y t hat
peri pheral opt i cal aberrat i ons (especi al l y def ocus and
ast igmat ism) might be import ant for emmet ropizat ion and
myopia development . Animal st udies have demonst rat ed
t hat eye growt h due t o experiment al def ocus or blurring
by a dif f user is cont rolled by local ret inal mechanisms. 8, 9
Likewise, animals t hat consist ent ly experience near obj ect s
in t heir inferior Þeld and dist ant obj ect in t heir superior Þeld
t end t o have longer axial lengt h for t he superior ret ina t han
f or t he inf erior ret ina. 10 The explanat ion of t hese result s
suggest ed by Wallman & Winawer 9 is t hat myopic eyes are
relat ively hyperopic in t he peripheral f ield compared t o
t he cent ral f ield because t he eye is elongat ed along t he
opt ical axis. The homeost at ic signals from t he cent ral ret ina
t hat direct t he eye t o elongat e less would be count ered by
signals f rom t he peripheral ret ina t hat direct t he eye t o
elongat e more. Because t he t ot al number of neurons from
t he peripheral ret ina is large compared t o cent ral ret ina,
t he peri pheral si gnal f or el ongat i on wi l l domi nat e t he
emmet ropizat ion process and lead t o myopia progression.
Smit h et al 11-13 t est ed t hese ideas experiment ally in primat es
and concluded t hat t he peripheral ret ina can cont ribut e t o
emmet ropizing responses and t o amet ropias produced by an
abnormal visual experience.
In spi t e of t he i mpor t ance of per i pher al vi si on, t he
emphasis in cont act lens opt ical design has cent ered on
correct ing f oveal vision. 14-16 Yet cont act l enses are al so
capable of manipulat ing image qualit y in t he periphery. 17
In a t heoret ical st udy, At chison f it a —4D myopic model
eye wit h a spherical cont act lens and an aspheric cont act
lens (conic const ant —0. 25). He f ound t hat t he spherical
cont act l ens i nt r oduced a r el at i ve myopi c shi f t i n t he
periphery but t he aspheric cont act lens eliminat ed such a
myopia shift . That result demonst rat ed t hat cont act lenses
have t he pot ent ial eit her t o improve image qualit y in t he
peripheral ret ina or t o int roduce myopic refract ion pat t ern
in t he periphery which may in t urn slow t he rat e of myopia
progression. 18 However, lit t le is known about how t o design
cont act lenses f or a wide Þeld of view in order t o realize
t hese pot ent ial beneÞt s. Similarly, t he design of spect acle
lenses t o correct refract ive error over t he ent ire visual Þeld
has not yet been achieved alt hough Smit h and colleagues19
successf ully opt imized an opht halmic lens t o correct one
meridian of a wide-angle schemat ic-eye. 20
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Designing cont act lenses for a wide Þeld of view via ocular wavefront t omography
Classic opt ical design of cont act lenses largely involves
select ing surf ace shape so t hat t he aberrat ions associat ed
wit h f oveal vision are minimized. 14, 15 Designing cont act
lenses t o improve peripheral vision is more complicat ed
because mul t ipl e obj ect ives are possibl e. For exampl e,
apart from correct ing cent ral vision, t he designs could eit her
1) improve peripheral ret inal image qualit y signiÞcant ly t o
obt ain a visual performance and/ or ret inal imaging beneÞt
or 2) produce a desired level of peripheral myopia t o obt ain
a ref ract ive development beneÞt . In t his report we show
how t hese t wo design goals can be achieved using a new
t echnique called ocular wavef ront t omography (OWT). 21
The OWT t echnique was developed t o creat e opt ical models
of t he eye t hat mimic t he of f -axis wavef ront aberrat ions
measur ed i n i ndi vi dual eyes. Gi ven such a model , t he
same t echnique can t hen be used t o opt imize t he design
of a cont act lens (or ot her opht halmic t reat ment s such as
spect acles, int raocular lenses, corneal inlays, et c.) for use
in conj unct ion wit h t he eye t o achieve t he desired opt ical
behavior of t he eye + lens syst em across t he visual Þeld.
The cont ent of t he paper i s as f ol l ows. The Met hods
sect i on descri bes how t o adapt t he OWT t echni que f or
cont act l ens design. The Resul t s sect ion il l ust rat es t he
met hod by designing t wo t ypes of cont act lenses, one f or
each of t he t wo goals list ed above (opt imizing peripheral
image qualit y, or deliberat ely int roducing Þeld curvat ure
for myopia cont rol). The Discussion sect ion brieày discusses
t he t radeof f bet ween cent ral and peripheral correct ions,
t he formulat ion of design goals, and t he modeling of human
eyes across a wide Þeld of view.
Methods
Summary of ocular wavefront tomography
Ocular wavef ront t omography (OWT) is a comput at ional
pr ocess f or cust omi zi ng a wi de angl e schemat i c eye
t o achi eve t he t wi n goal s of anat omi cal accur acy and
f unct ional equivalence. 21 Anat omical accuracy is achieved
by const raining t he paramet ers of t he model t o lie wit hin
accept able limit s. Funct ional equivalence is achieved by
adj ust i ng t he model ’ s par amet er s unt i l t he wavef r ont
aber r at i on f unct i on of t he model al ong t he f oveal
line-of -sight , and along mult iple peripheral lines-of -sight ,
mat ch t he aberrat ions measured in an individual eye or
some represent at i ve eye. The OWT procedure consi st s
of f our st eps. St ep 1 conf igures a generic model eye as
an init ial t emplat e t hat serves as a st art ing point f or t he
opt imizat ion process. This init ial model should include all
of t he anat omical f eat ures deemed import ant . Depending
on t he int ended applicat ion, t his t emplat e model eye might
include a single ref ract ing surf ace, or mult iple surf aces,
or a gradient index lens. St ep 2 det ermines t he wavef ront
aber r at i ons of t he eye al ong mul t i pl e l i nes- of - si ght
t hat adequat el y sampl e t he range of vi sual f i el d t o be
correct ed. For example, a modiÞed clinical Shack-Hart mann
aberromet er 22,23 or a scanning Shack-Hart mann Aberromet er 24
could be used t o obt ain such aberrat ion measurement s in an
individual eye. Alt ernat ively, t he eye might be charact erized
by aberrat ions of a t ypical eye across t he visual f ield f or
a t arget populat ion. St ep 3 f ormulat es a good measure of
127
t he dual cust omizat ion goal s (anat omical simil arit y and
funct ional equivalence) in t he form of a merit funct ion (eqn.
(1)) t hat quant iÞes t he degree t o which t he current st at e
of t he model sat isÞes t he design obj ect ive. The Þrst part
of t he merit f unct ion represent s t he anat omical similarit y
bet ween t he cust omi zed model and t he anat omi cal
dimensions common t o all eyes or, if available, t he speciÞc
dimensions measured for an individual eye. The second part
of t he merit f unct ion measures t he dif f erence bet ween
t he wavef r ont measur ement s of i ndi vi dual eyes al ong
mult iple lines-of-sight and t he t heoret ical values obt ained
by ray t racing t hrough t he cust omized model. The relat ive
weight ing of t hese t wo part s, and of t he various f act ors
wit hin each part , is àexible and applicat ion-dependent . 21
St ep 4 f ormul at es t he t omography probl em of adj ust ing
t he t emplat e model eye t o become anat omically similar
and f unct ional l y equival ent t o t he subj ect ’ s eye int o an
opt imizat ion problem of Þnding a cust omized model eye t hat
achieves a global minimum of t he merit funct ion. A variet y
of opt imizat ion t echniques can be used f or t his purpose,
including damped-least squares, simulat ed annealing, neural
net works, case-based reasoning, and expert -syst em. These
comput at ional-int ensive t echniques solve t he opt imizat ion
problem it erat ively.
→
→
→
Merit (eye) = Merit funct ional_equivalence (eye) + Merit anat omical_similarit y (eye) (1)
Applying ocular wavefront tomography
to the design of contact lenses
Given a wide-angle model of an eye, t he OWT t echnique
can be used t o opt imize t he design of a cont act lens used in
conj unct ion wit h t he eye. We do t his by Þxing t he paramet ers
of t he eye model while opt imizing t he paramet ers of t he
cont act lens t o achieve t he desired opt ical behavior of t he
eye + lens syst em across t he visual Þeld. Again t he met hod
involves four st eps: t he const ruct ion of a design t emplat e,
t he speciÞcat ion of design goals, t he formulat ion of a merit
f unct ion, and t he opt imizat ion of t he design. St eps 3 and
4 are t he same as described above in sect ion 2. 1, but t he
Þrst t wo st eps require modiÞcat ion as described below.
To design a cont act lens using OWT, t he f irst st ep is t o
creat e a design t emplat e consist ing of a generic cont act
lens in apposit ion t o a Þxed model eye. The paramet ers of
t his generic cont act lens will be it erat ively opt imized while
t he Þxed model eye remains unchanged as a mast er syst em
in t he t emplat e. The choice of model eye depends on t he
speciÞc applicat ion. For example, t o cust omize a cont act
lens for an individual eye, t he eye model could be obt ained
f rom wavef ront aberrat i on measurement s by t he OWT
t echnique, using t he corneal t opography dat a in t he merit
funct ion t o ensure accurat e geomet ry of t he corneal front
surface. Alt ernat ively t o design a more generic cont act lens
f or improving peripheral vision f or a populat ion of eyes, a
st at ist ical model of t he eye based upon populat ion dat a may
be pref erred. In t he examples report ed below, we used a
published model eye 20 t o enable a comparison wit h known
result s from t he lit erat ure.
The second st ep of OWT l ens design is t o specif y t he
desi red opt i cal perf ormances of t he l ens + eye syst em
in cent ral and peripheral visual f iel ds. In t his st udy, we
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128
X. Wei, L. Thibos
developed t wo different design goals. Bot h goals aimed t o
opt imize image qualit y (e.g. RMS of wavefront error) along
t he cent ral line-of-sight , but t hey differed for t he peripheral
f ield. Test case 1 aimed t o improve image qualit y in t he
periphery whereas case 2 aimed t o manipulat e t he variat ion
of peripheral refract ive error across t he visual Þeld for t he
purpose of myopia cont rol.
Design goals f or wide Þeld-of -view require deÞning t he
opt ical performance (e.g. wavefront error) along peripheral
lines-of-sight , where oblique viewing of t he iris causes t he
ent rance pupil t o appear ellipt ical. Several met hods are
available t o def ine wavef ront aberrat ions over ellipt ical
pupi l s, whi ch we compare and cont rast el sewhere. 25, 26
Par t i cul ar l y f or t est case 2, i n or der t o cal cul at e t he
spherical refract ive error along t he oblique line-of-sight it
was convenient t o use t he scaling met hod 22,26 which st ret ches
t he ellipt ical pupil int o a circle of const ant diamet er for all
lines-of-sight . Zernike aberrat ion coefÞcient s obt ained from
t he st ret ched wavefront map can be convert ed t o spherical
ref ract ive error using t he f ormula derived by At chison et
al (eqn. B24 in At chison’s paper 27). By ignoring t he higher
order t erms in t he original equat ion, t he formula becomes
M=
—[(2√3C20 —6√5C40) (1 + cos2w) + (√6C22 —3√10C42)sin2w]
R2 cos2w
z(x) =
(2)
where M is t he means spherical refract ive error in diopt ers,
Cnm are t he Zernike coefÞcient s calculat ed from t he scaling
met hod, R i s t he radi us of t he ci rcul ar ent rance pupi l ,
and w i s t he angl e bet ween t he per i pher al and f oveal
lines-of -sight . Since t he eye models in our examples have
rot at ional symmet ry, t his eqn. (2) applies t o any meridian.
The desired variat ion of M wit h Þeld angle w speciÞes t he
design goal for peripheral refract ive errors in t est case 2.
Wit h t he st art ing t emplat e set up in st ep 1 and rigorously
f ormulat ed design goals in st ep 2, t he merit f unct ion can
be readi l y f ormat t ed i n st ep 3. Si mi l arl y t o t he cl assi c
OWT approach, t he f irst part of t he merit f unct ion was
formulat ed t o measure t he difference bet ween t he speciÞed
design goals and t he ray t racing predict ion of t he design
t emplat e (cont act lens + model eye). The second part of
merit f unct ion incorporat es t he mechanical const raint s
( e. g. edge t hi ckness) of t he cont act l ens f r om t he
f abricat ion or peripheral zone design. This merit f unct ion
(Eq. (3)) is analogous t o t he merit f unct ion t hat measures
t he f unct ional equivalence and anat omical similarit y (Eq.
(1)) of model eyes in sect ion 2. 1. The weight ing inside
and bet ween each part of t he merit f unct ion are àexible
and can be it erat ively adj ust ed during t he design st age t o
achieve t he balance of t he design. 28-30 Aft er formulat ing t he
merit f unct ion, t he opt imizat ion engine can be applied t o
Þnd t he opt imal design t hat achieves t he global minimum of
t he merit funct ion in t he Þnal st ep.
→
→
→
Merit (CL) = Merit design_goal (CL) + Merit mechanical_const raint (CL)
Thi s wi del y ci t ed model capt ures t he mai n anat omi cal
f eat ur es of t he human eye wi t h mi ni mum compl exi t y.
Besides a spherically curved ret ina, t his model eye consist s
of f our ref ract ive surf aces: ant erior & post erior cornea
and ant erior & post erior lens. The model exhibit s realist ic
off-axis aberrat ion performance at moderat e Þeld degrees
(10º-45º). 20,22 To simulat e axial myopia (—2D, 550 mm), t he
lengt h of t he schemat ic eye was increased appropriat ely
(post erior chamber lengt h = 17. 1005 mm). Ent rance pupil
diamet er was set at 5 mm, which is large enough t o include
signiÞcant amount s of higher-order aberrat ions.
Test case 1 was a monof ocal aspheri c l ens opt i mi zed
t o improve peripheral opt ical qualit y (RMS of wavef ront
error). The t emplat e cont act lens used rigid mat erial wit h
a refract ive index of 1.492. The back surface of t he cont act
lens was spherical wit h t he same radius of curvat ure as t he
ant erior corneal surface. A t hin t ear Þlm layer of refract ive
index 1.336 was placed bet ween t he ant erior cornea and t he
post erior surf ace of t he cont act lens. The f ront surf ace of
t he cont act lens for t est case 1 was aspheric wit h a surface
sag z given in Eq. (4),
(3)
Test cases
A rot at ionally symmet ric, wide-angle model-eye20 was chosen
as t he f ixed mast er syst em in t he OWT design t emplat e.
x2 / r 2
1 + √1 —(k + 1)x2 / r 2
, | x| ≤ h
(4)
where r is t he radius of curvat ure, × is t he radial coordinat e
in lens unit s, k is t he conic const ant and h is t he radius of
t he opt ical zone.
Test case 2 was a t wo-zone bif ocal designed t o correct
t he cent ral vi si on and an out er annul ar zone desi gned
t o i nt roduce rel at i ve myopi c ref ract i on pat t ern i n t he
peripheral Þeld. It s surface had sag proÞles given in Eq. (5)
z(x) =
{
x2 / r inner
2
1 + √1 —(kinner + 1)x2 / r inner
x2 / r out er
2
1 + √1 —(kout er + 1)x2 / r out
er
, | x| ≤ h1
(5)
+ C, h1 < | x| < h2
where r inner and r out er are t he radii of curvat ure and k inner
& k out er are t he conic const ant s of t he t wo opt ical zones
respect ively, × is t he radial coordinat e in lens unit s, h1 and
h2 are t he radii of t he inner and out er opt ical zones. The
const ant C ensures t he same sag val ue at t he boundary
bet ween t he inner zone and out er zones.
Results
Test case 1: Contact lens to improve the peripheral
retinal image quality
The goal of t est case 1 was t o correct f oveal wavef ront
aberrat i ons whi l e si mul t aneousl y i mprovi ng peri pheral
image qualit y of t he Navarro myopic eye (—2D) Since foveal
vision correct ion is of high priorit y, 80 %of t he weight was
assigned t o minimize t he RMS of cent ral wavef ront . The
remaining 20 %of t he weight was equally assigned t o reduce
t he RMS of t he wavefront s along ot her oblique line-of-sight
wit hin 50 degree visual Þeld. To prepare t he st art ing design
t emplat e, we f it t he myopic model eye wit h an aspheric
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Designing cont act lenses for a wide Þeld of view via ocular wavefront t omography
design (CV in t able 1) t hat solely gives t he diffract ion-limit ed
performance for t he cent ral vision. Aft er OWT opt imizat ion,
we found a design (PV in t able 1) t hat correct s t he cent ral
vi si on and meanwhi l e i mpr oves t he per i pher al i mage
qualit y.
Figure 1 (a) shows t he root -mean-square (RMS) of f -axis
wavefront error of lens + eye as a funct ion of Þeld angle of
t he myopic Navarro model eye. Design CV is a t radit ional
design t hat only aims t o correct refract ive error in cent ral
vi si on (CV). Thi s desi gn pr ovi ded di f f r act i on-l i mi t ed
performance cent rally and served as t he st art ing t emplat e
for opt imizat ion wit h OWT. Design CV also improved image
qualit y in t he near periphery (Þeld angle < 30º) but image
qualit y in t he far periphery was act ually worse wit h t he lens
t han wit hout for reasons described lat er. By comparison, t he
design PV gives priorit y t o correct ing foveal refract ive error
while simult aneously aiming t o improve image qualit y f or
peripheral vision (PV). This design reduced RMS wavefront
error along t he f oveal line-of -sight t o 0.14 .m (1/ 4 wave)
over a 5 mm pupil and improved peripheral image qualit y
out t o 45º visual Þeld relat ive t o t he uncorrect ed eye. Design
PV provided superior image qualit y compared t o design CV
f or all peripheral f ield angles. The slight penalt y f or t his
improved performance in t he periphery was a small residual
wavef ront aberrat ion (1/ 4 wave RMS) al ong t he cent ral
l i ne-of -si ght (compar ed t o t he desi gn CV’ s di f f r act i on
limit ed cent ral correct ion).
The primary effect of t he cont act lens is t o change mean
spherical ref ract ive error M as comput ed by eqn. (2). As
shown in Figure 1b, M is slight ly larger for design CV t han for
design PV at t he f ovea, but M increases more rapidly wit h
Þeld angle for design CV t han for design PV. The relat ively
small amount of refract ive error in t he periphery for design
PV is t he primary fact or t hat account s for t he superior image
qualit y of t his design relat ive t o t he ot her t wo condit ions
shown in Figure 1a. Ot her fact ors (oblique ast igmat ism and
higher-order aberrat ions) play a smaller, but signif icant ,
role also. Peripheral ast igmat ism becomes larger for design
PV t han design CV. The ast igmat ism of PV and CV at 50º Þeld
angle are 4.49 .m and 3.65 .m respect ively. On t he ot her
A
Design
Myopic eye
Design CV
Design PV
4
2
0
r (mm)
k
h (mm)
8.0156
8.0536
—0.4223
—0.1518
4.97
4.96
CV
PV
hand, t he coma t erm of design PV is about 25 % smal l er
t han design CV. The net effect of t hese aberrat ion change is
superior peripheral opt ical qualit y for design PV compared
t o design CV as shown in Figure 1a.
Test case 2: Contact lens to introduce myopic
refraction pattern
The goal of t est case 2 was t o correct f oveal wavef ront
error whil e simul t aneousl y changing rel at ive peripheral
ref ract ive errors f rom hyperopic (in t he Navarro model )
t o myopic (in t he correct ed eye). This opt imized design
(Design BF) is a concent ric, t wo-zone bifocal design (Eq. 5)
based upon t he 5mm on-axi s ent rance pupi l of myopi c
model eye. Similar t o t est case 1, a merit funct ion was set
up wit h 60 % of t he weight assigned t o minimize t he RMS
of cent ral wavef ront error and t he remaining 40 % of t he
weight was equally dist ribut ed over t he peripheral lines of
sight t o manipulat e t he peripheral refract ive pat t ern. The
st art ing design t emplat e f or t his t est case is summarized
in t he f irst row of Tabl e 2. Af t er OWT opt imizat ion, t he
bif ocal (BF) design in Tabl e 2 achieves t he dual goal s of
correct ing t he cent ral ref ract ive error and changing t he
peripheral ref ract ive pat t ern f rom relat ive hyperopia t o
relat ive myopia.
The BF design achieved dif f ract ion-limit ed perf ormance
(5 mm on-axis ent rance pupil) along t he foveal line-of-sight
by using t he inner zone t o correct t he myopic eye’s on-axis
wavefront error wit h an asphericit y t hat produces negat ive
spherical aberrat ion t o compensat e f or t he eye’s posit ive
6
Refractive error, M (D)
RMS (mm)
Table 1 Summary of asphercial cont act lens designs CV
and PV. Paramet ers refer t o equat ion (4)
B
6
129
Myopic eye
Design CV
Design PV
4
2
0
–2
0
10
20
30
Line of sight (degree)
40
50
0
10
20
30
Line of sight (degree)
40
50
Figure 1 Perf ormance of cust omized cont act lens t hat improves t he peripheral opt ical qualit y (5 mm on-axis ent rance pupil,
550nm). (a) Root -mean-square (RMS) of wavef ront errors (lower- and higher- order aberrat ions) as a f unct ion of Þeld angle of
peripheral lines-of-sight for Navarro myopic eye (empt y t riangle), t he design CV (solid circle), and t he design PV (solid diamond);
(b) The spherical refract ive errors along lines-of-sight for Navarro myopic eye (empt y t riangle), t he design CV (solid circle), and t he
design PV (solid diamond).
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130
X. Wei, L. Thibos
Table 2
Summary of bifocal cont act lens designs (BF). Paramet ers refer t o equat ion (5)
Design
Init ial
BF
r inner (mm)
kinner
r outer (mm)
kouter
h1 (mm)
h2 (mm)
8
8.0156
0
—0.4223
8
7.9499
0
0.0497
—
2.48
—
4.97
Table 3 Summary of bifocal cont act lens designs based on different ent rance pupil sizes (4 mm, 5 mm, 6 mm).
Paramet ers refer t o equat ion (5)
EP diameter (mm)
r inner (mm)
kinner
r outer (mm)
kouter
h1 (mm)
h2 (mm)
4
5
6
8.0156
8.0156
8.0156
—0.4223
—0.4223
—0.4223
8.0431
7.9499
7.8178
0.3739
0.0497
—0.2247
1.99
2.48
2.95
4.74
4.97
5.20
from t he original bifocal design (BF). Smoot hing increased
RMS wavef ront error slight ly along t he f oveal line-of -sight
from 0 (design BF) t o 0.07 .m (0.13 wave, design BF smoot h)
over a 5 mm ent rance pupil. Nevert heless, t he peripheral
opt ical qualit y of t his design is bet t er t han t he classic design
CV wit hin 45 degree Þeld of view (Figure 2b).
For bifocal lenses, t he assumed size of t he ent rance pupil
(EP) along t he f oveal line-of -sight is an import ant design
paramet er f or det ermining t he relat ive sizes of inner and
out er zones of t he lens. Moreover, t he relat ive dimensions
of t he t wo zones affect t he balance achieved bet ween t he
cent ral vision and peripheral vision correct ions. The bifocal
design BF report ed in Table 2 was designed for t he 5 mm EP
of t he myopic model eye. Applying t he same OWT procedure
t o ot her pupil sizes, t he bif ocal lens opt imized f or a 4mm
EP or a 6 mm EP differ signiÞcant ly from t he 5mm design as
report ed in Table 3. Af t er smoot hing t hese bif ocal designs
by pol ynomi al f i t t i ng, we cal cul at ed t hei r per i pher al
refract ive errors. In each design, t he peripheral refract ive
error varies onl y sl ight l y wit h pupil size. Theref ore t he
per i pher al r ef r act i ve er r or of each desi gn r epor t ed i n
A
B
6
6
Myopic eye
Design CV
Design BF
Design BF smooth
4
Myopic eye
Design CV
Design BF smooth
5
RMS (mm)
Refractive error, M (D)
spherical aberrat ion. The sign of peripheral refract ive error
of design BF remains hyperopic in t he near-peripheral visual
f ield (< 35º), but beyond 35 degree visual f ield, t he sign
changes t o myopic. This pat t ern of peripheral ref ract ive
errors is markedly different from t he CV design, for which
peripheral refract ive errors are always hyperopic. This result
reveals t he design àexibilit y provided by bifocal lenses for
manipulat ing peripheral refract ive errors.
One disadvant age of t he bif ocal design is t hat t he slope
of t he surf ace is discont inuous at t he boundary bet ween
inner and out er zones. This discont inuit y is a disadvant age
for fabricat ion of t he lens and for achieving robust opt ical
performance across different pupil sizes and different Þeld
angles. To avoid t hese problems, a t ransit ion zone is usually
incorporat ed bet ween t he inner zone and out er zone f or
t his purpose. We implement ed a smoot h t ransit ion zone
by l east -square f it t ing of t he l ens surf ace wit h a set of
polynomials up t o t he 30t h order. The RMS of t he residual
Þt t ing error was 0.08 .m, which is negligible compared t o
t ypical fabricat ion t olerances. Residual refract ive errors for
t his smoot hed design (BF Smoot h) were indist inguishable
2
0
4
3
2
–2
1
0
–4
0
10
20
30
Line of sight (degree)
40
50
0
10
20
30
Line of sight (degree)
40
50
Figure 2 Performance of cust omized cont act lens t hat int roduces a relat ive myopic pat t ern ont o myopic eye’s peripheral visual
f iel d. (a) Spherical ref ract ion pat t ern of t he Navarro myopic eye (sol id square), design CV (sol id circl e), design BF (empt y
upper-t riangle), and design BF Smoot h (empt y lower-t riangle). (b) Root -mean-square (RMS) of wavefront errors (lower- and higher
order aberrat ions) of t he Navarro myopic eye (solid square), design CV (solid circle), and design BF Smoot h (empt y lower-t riangle).
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Designing cont act lenses for a wide Þeld of view via ocular wavefront t omography
A
B
131
1
2
0
0.6
MTF
Refractive error, M (D)
0.8
–2
0.4
4 mm design
5 mm design
6 mm design
–4
4 mm EP/4 mm design
5 mm EP/4 mm design
6 mm EP/4 mm design
6 mm EP/5 mm design
6 mm EP/6 mm design
0.2
–6
0
0
10
20
30
Line of sight (degree)
40
50
0
5
10
15
20
Frequency (cyc/deg)
25
30
Figure 3 Perf ormance of cust omized bif ocal cont act lenses designed t o int roduce relat ive myopic pat t ern ont o t he peripheral
visual Þeld (a) Relat ive peripheral refract ive error of t he 4 mm, 5 mm, and 6 mm designs when t he ent rance pupil size mat ches t he
design pupil size. (b) Modulat ion t ransfer funct ions (MTF) of foveal correct ions for different combinat ions of ent rance pupil sizes
and design pupil sizes.
Figure 3a, which was comput ed at t he design pupil size, is
represent at ive of all pupil sizes. Figure 3a reveals t hat t he
peripheral ref ract ive error pat t erns of t he “ 4 mm design”
is more ef f ect ive t han t he “ 5 mm design” or t he “ 6 mm
design” at int roducing peripheral relat ive myopia. This is
because t he ‘ 4 mm design’ has a larger out er zone, which
manipulat es peripheral ref ract ive error more ef f ect ively.
The penalt y of a small inner zone is reduced ret inal image
qual i t y al ong t he f oveal l i ne-of -si ght when t he act ual
EP becomes l arger t han t he design size. This penal t y is
quant iÞed by t he modulat ion t ransf er f unct ions (MTF) in
Figure 3b. The MTF f or t he 4 mm design is signif icant l y
depressed f or a 5 mm EP and even more depressed f or a
6 mm EP. By comparison, t he MTF provided by t he 5 mm
desi gn i s superi or when t he EP i s 5 mm but once agai n
becomes depressed if t he EP exceeds t he design size (e.g.
6 mm EP, 5 mm design pupil). Theref ore, in pract ice, t he
relat ive dimensions of inner and out er zones of t he bifocal
should be select ed carefully t o achieve t he desired balance
bet ween f oveal image qual it y and peripheral ref ract ive
errors for habit ual pupil sizes.
Discussion
In t his st udy, we successf ully applied t he OWT t echnique
t o design cont act lenses t o correct a 4-surf ace schemat ic
eye wit h axial myopia over a wide f ield of view. Besides
correct i ng cent ral vi si on, t he t wo desi gns (PV and BF)
reduce t he myopic model eye’s peripheral refract ive errors
reasonably well from different perspect ives. The design PV
improves t he peripheral image qualit y over a ±45 degree
visual Þeld (Figure 1a) whereas t he design BF ef f ect ively
int roduces a myopic pat t ern of refract ive error in t he model
eye’s f ar periphery (> 35º, Figure 2a). Through t hese t wo
exampl es, we have demonst rat ed t he ef f ect i veness of
opt imize cont act lens design via OWT.
One import ant aspect of applying OWT t o design cont act
l enses i s t o achi eve a bal ance among var i ous desi gn
goal s, especi al l y bet ween t he goal s rel at ed t o cent ral
and peripheral vision. Since t hese dual design goal s are
compet it ive, achieving bal ance bet ween t hem requires
adj ust ing t he weight ing of t he corresponding operands
in t he merit f unct ion. For example in t est case 1, if t he
operands relat ed t o cent ral vision are weight ed 100 %, t he
OWT opt imized design will be t he classic aspheric design CV
(Figure 1 in sect ion 3. 1) t hat achieves dif f ract ion-limit ed
correct ion cent rally but only improves peripheral image
qualit y t o a limit ed ext ent . If 80 %of t he weight is assigned
t o t he operands for cent ral vision and 20 %of t he weight is
assigned t o t he operands f or peripheral vision, t hen OWT
achieved a balanced design PV, which has slight ly worse
foveal correct ion (1/ 4 wave RMS) but effect ively improves
peripheral image qual it y up t o 45 degrees. Usual l y t his
weight ing adj ust ment procedure is it erat ive, t he essence of
which is similar t o lens design. 28,30
Anot her i mport ant aspect of appl yi ng OWT t o desi gn
cont act lenses is t he generalit y of specifying design goals.
Alt hough t he on-axis and off-axis wavefront aberrat ions in
obj ect space were adopt ed t o formulat e t he merit funct ion
in t his paper, ot her indicat ors of image qualit y can also be
adopt ed. For example, t he MTF can be used t o represent t he
correct ion along t he f oveal lines-of -sight . The peripheral
spheri cal correct i on i n i mage space 25, 31, 32 or peri pheral
t hrough-f ocus33 can be used t o indicat e peripheral opt ical
qualit y t oo.
We demonst r at ed ef f i cacy of t he OWT met hod by
designing cont act lenses t o correct a schemat ic eye t hat
i s r epr esent at i ve of t ypi cal human eyes. Al t hough t he
emmet ropic version of t he Navarro wide-angle model-eye
overest imat es t he on-axis spherical aberrat ion and achieves
l it t l e agreement at smal l angl es, it nevert hel ess agrees
wit h t he experiment al dat a reasonably well at moderat e
Þeld angles (10-40 degrees). 20 The myopic version of t his
model also predict s t he relat ively hyperopic shif t in t he
periphery (Figure 2). Theref ore it is a reasonable model
t o be used as a mast er syst em in t he design t emplat e t o
demonst rat e cont act l ens design f or wide Þ el d of view.
Mor e sophi st i cat ed model eyes ( e. g. myopi c model
eyes17) could also be used. However individual variabilit y
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132
i n wavef r ont aber r at i ons acr oss t he vi sual f i el d may
require t he use of cust omized wide angl e model eyes 21
as t he mast er syst em. These cust omi zed model eyes in
gener al ar e r ot at i onal l y asymmet r i c, whi ch suggest s
t hat opt i mi zat i on based on mul t i pl e semi -mer i di ans
is required. Nevert hel ess since t he OWT procedure is a
general f ramework 21, it is rel at ivel y st raight f orward t o
generalize t he applicat ion of OWT t o design cont act lens
based on one or more semi-meridians.
Besides applying OWT t o opt imize t he design of cont act
l ens t o achi eve desi red opt i cal perf ormance, i t i s al so
import ant t o budget appropriat e t olerance for how t he lens
int eract s wit h t he eye. In general, t he misalignment of a
cont act lens on t he eye affect s bot h cent ral and peripheral
correct ions. The sensit ivit y of t he peripheral correct ions
t o t he mi sal i gnment i s compar abl e t o t hat of cent r al
correct ion. Furt hermore, due t o t he high priorit y assigned
t o correct ing cent ral vision, t he primary goal of t olerance
analysis f or wide angle designs should also aim t o ensure
good opt ical correct ion along t he cent ral LoS. From t his
perspect ive, t he t olerance analysis f or wide angle designs
is similar t o t he analysis for t he classic cont act lens designs
t hat f ocus sol el y on cent ral vision correct ion. However,
since t he wide angle designs usually realize t heir beneÞt
for peripheral vision by compromising slight ly in correct ing
cent ral vision, t hey usually have t ight er t olerance t han t he
classic designs. For example, t he smoot hed bif ocal design
(‘ BF Smoot h’ ) reduced t he RMS wavefront error t o 0.07 .m
(0. 13 wave) over 5 mm ent rance pupi l , whi ch i s worse
t han dif f ract ion-l imit ed correct ion of cl assic CV design.
To ensure RMS wavefront error wit hin 0.13 .m (1/ 4 wave),
t he classic design CV t olerat es t he decent rat ion of up t o
0.4 mm, while t he smoot hed bifocal design t olerat es much
less decent rat ion (0.14 mm).
In t hi s st udy, we i nt r oduced t he OWT t echni que f or
designing cont act lens and demonst rat ed it s ef f ect iveness
t hr ough t he t wo exampl es i n t he r esul t sect i on. Yet a
successf ul cont act l ens design needs t o achieve opt imal
per f or mances among mul t i pl e desi gn goal s. Some of
t hese goal s may i ncl ude t he opt i cal per f or mances of
t he correct ed eye f or pol ychromat ic l ight , various pupil
si zes, t hr ough- f ocus per f or mance, and mechani cal
st abilizat ion on t he cornea. Modern cont act lens design is
a f ramework f or Þnding pract ical and balanced solut ions
t o t his mul t i-dimensional probl em. We regard t he OWT
t echni que as a subset of t hi s l ar ger f r amewor k t hat
st r esses r eal i zi ng t he benef i t of cor r ect i ng per i pher al
vision of human eyes.
Acknowledgement
Funded in part by NIH grant NEI R01 EY 05109 awarded t o
Larry Thibos.
Conàict of interest
The aut hors have no propriet ary, Þnancial or commercial
i nt er est i n any mat er i al or met hod ment i oned i n t hi s
st udy.
X. Wei, L. Thibos
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