Download Model of transient drug diffusion across cornea

Journal of Controlled Release 99 (2004) 241 – 258
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Model of transient drug diffusion across cornea
Wensheng Zhanga,*, Mark R. Prausnitzb, Aurélie Edwardsa,*
a
Department of Chemical and Biological Engineering, Tufts University, 4 Colby Street, Medford, MA 02155, USA
b
School of Chemical and Biomolecular Engineering and Center for Drug Design, Development, and Delivery,
Georgia Institute of Technology, Atlanta, GA 30332-0100, USA
Received 26 February 2004; accepted 1 July 2004
Available online 14 August 2004
Abstract
A mathematical model of solute transient diffusion across the cornea to the anterior chamber of the eye was developed for
topical drug delivery. Solute bioavailability was predicted given solute molecular radius and octanol-to-water distribution
coefficient (U), ocular membrane ultrastructural parameters, tear fluid hydrodynamics, as well as solute distribution volume
(V d) and clearance rate (Cla) in the anterior chamber. The results suggest that drug bioavailability is primarily determined by
solute lipophilicity. In human eyes, bioavailability is predicted to range between 1% and 5% for lipophilic molecules (UN1), and
to be less than 0.5% for hydrophilic molecules (Ub0.01). The simulations indicate that the distribution coefficient that
maximizes bioavailability is on the order of 10. It was also found that the maximum solute concentration in the anterior chamber
(C max) and the time needed to reach C max significantly depend on U, V d, and Cla. Consistent with experimental findings, model
predictions suggest that drug bioavailability can be increased by lowering the conjunctival-to-corneal permeability ratio and
reducing precorneal solute drainage. Because of its mechanistic basis, this model will be useful to predict drug transport kinetics
and bioavailability for new compounds and in diseased eyes.
D 2004 Elsevier B.V. All rights reserved.
Keywords: Eye; Ophthalmic drug delivery; Topical; Anterior chamber; Transport
1. Introduction
Topical drug delivery is the most common treatment for diseases of the anterior segment of the eye,
such as glaucoma, but is limited by the low perme* Corresponding authors. Wensheng Zhang is to be contacted at
Tel.: +1-617-627-3528; fax: +1-617-627-3991. Aurélie Edwards,
Tel.: +1-617-627-3731; fax: +1-617-627-3991.
E-mail addresses: [email protected] (W. Zhang)8
[email protected] (A. Edwards).
0168-3659/$ - see front matter D 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.jconrel.2004.07.001
ability of the multilayered cornea, rapid clearance by
tear drainage, and absorption into the conjunctiva.
Hence, the bioavailability of topically administered
drugs is very low. Drug transport processes and
methods to improve bioavailability have been studied
extensively for the past decades, as reviewed, for
example, by Schoenwald [1], Lee and Robinson [2],
and Jarvinen et al. [3].
Pharmacokinetic models have been widely used to
predict the efficiency of new topical agents. Early
physiological models of the eye were developed by
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W. Zhang et al. / Journal of Controlled Release 99 (2004) 241–258
Himmelstein et al. [4] and Miller et al. [5]. Compartment models, in which each tissue is represented as a
separate compartment and is linked sequentially to the
next one as the drug passes through, are commonly
employed today. Their underlying assumptions as well
as their applications have been previously reviewed
[1,2]. In these pharmacokinetic models, absorption
into the cornea and conjunctiva is usually described as
a first-order process, and corresponding rate constants
are determined by fitting the model to a given set of
experimental data. The models are therefore often
valid only for a specific group of molecules, and their
use for the design of new agents is limited.
Taking a more detailed look at drug transport after
topical instillation, drug is absorbed and diffuses in
the cornea and conjunctiva/sclera. The cornea consists
of three main barriers in series: the epithelium, the
stroma, and the endothelium. The corneal stroma and
sclera are highly hydrated fibrous acellular tissues.
The conjunctiva, corneal epithelium, and endothelium
are cellular layers that contain both transcellular and
paracellular pathways; lipophilic molecules preferentially diffuse within cells, whereas hydrophilic molecules permeate mostly through the openings between
cells. The authors previously developed theoretical
approaches to predict solute diffusivity in the cornea
and sclera based upon the ultrastructure of these
membranes and solute physicochemical properties
[6,7].
Building upon these predictions, in this study, a
mathematical model was developed for topical drug
delivery across the cornea that accounts for the
ultrastructure of ocular membranes, solute molecular
size and lipophilicity, tear fluid hydrodynamics, as
well as drug distribution and clearance in the anterior
chamber. This transient model is built upon the
partial differential equation for diffusion across
ocular membranes (i.e., Fick’s law) rather than the
empirical first-order absorption equation commonly
used. All the parameters of this model have a
physical meaning and can be independently measured or estimated, so that the model can be applied
to a variety of compounds. Since the actual
mechanisms of drug transport and elimination are
explicitly taken into consideration, this approach can
yield insights into the main determinants of drug
delivery and may therefore guide strategies to
improve drug bioavailability.
2. Mathematical model
The model developed in this study addresses the
fate of drug molecules applied topically to the eye.
As a drop of solution is instilled into the eyelid sac,
it is assumed to mix instantly with the tear fluid due
to reflex blinks [8]. As shown in Fig. 1, drug
solution thus delivered to the precorneal area (i.e.,
the conjunctival sac and the tear film, considered as a
single, homogeneous compartment) is diluted by
lacrimal secretion and cleared by four different
mechanisms: drainage with the tear fluid towards
Fig. 1. Schematic representation of topical drug delivery model.
W. Zhang et al. / Journal of Controlled Release 99 (2004) 241–258
the nasal cavity, absorption into the palpebral
conjunctiva, absorption into the bulbar conjunctiva/
sclera, and absorption into the cornea. Since solute
absorbed into the conjunctiva/sclera generally does
not subsequently diffuse into the anterior chamber
[9,10], the contribution of that route to solute
deposition into the anterior chamber is neglected,
and the concentration at the interface between the
sclera and the anterior chamber is assumed to be
zero. Molecules that permeate into the cornea diffuse
through the epithelium, stroma, and endothelium
before reaching the aqueous humor in the anterior
chamber. The anterior chamber is considered as a
homogeneous compartment. In the anterior chamber,
the solute partitions between the aqueous humor and
other tissues, such as iris and ciliary body, and it is
cleared either by aqueous humor drainage or by
systemic absorption [1].
In the approach of this study, solute molecules are
characterized by their size and lipophilicity. The latter
is usually quantified using two parameters: the
partition coefficient, which represents the nonionized
molecule concentration ratio between the organic and
aqueous phases, and the distribution coefficient,
which represents the overall concentration ratio (i.e.,
including the ionized solute concentration in the
aqueous phase). Here, the lipid bilayer-to-water
distribution coefficient (U) is used to express concentration equilibrium at the interface between the tear
film and the adjacent epithelial membranes.
There are two pathways available for solutes
diffusing across the epithelia and endothelia: transcellular routes, favored by lipophilic molecules that
can easily penetrate the lipid bilayer of cells; and
paracellular routes, favored by hydrophilic molecules.
Accounting for simultaneous transport through both
routes requires a full two-dimensional representation
of ocular membranes, which lies beyond the scope of
this study. Therefore, it is assumed that solutes
permeate through only one or the other of these two
pathways, which permits a simpler, one-dimensional
approach.
243
portion in excess of the normal lacrimal volume),
following the approach of Chrai et al. [11], so that:
dVt
¼ kd ðVt VL Þ
dt
ð1Þ
where V t is the tear volume at time t, V L is the normal
lacrimal volume (so that V tV L is the excess tear
volume), and k d is the solution drainage constant.
Integration of Eq. (1) yields the tear volume as:
Vt ¼ Vi expð kd t Þ þ VL
ð2Þ
where Vi is the instilled volume (i.e., the initial excess
tear volume).
Alternatively, conservation of tear fluid can be
expressed as:
dVt
¼SQ
dt
ð3Þ
where S is the (fixed) lacrimal secretion rate and Q is
the solution discharge rate. Combining Eqs. (1)–(3)
yields the tear drainage rate as:
Q ¼ S þ kd Vi expð kd t Þ
ð4Þ
The solute concentration in the tear film C t is
assumed to remain homogeneous. Conservation of
solute in the tear film can be written as:
dðVt Ct Þ
dCt
dVt
¼ Vt
þ Ct
dt
dt
dt
¼ Fep fep Aep Fpj fpj Apj Fbj fbj Abj QCt
ð5Þ
where F i is the Fickian diffusional flux of solute
across membrane i whose surface area is A i . The
fraction of A i that is occupied by the diffusional route
being considered (i.e., transcellular or paracellular) is
denoted by f i . The subscripts bep,Q bpj,Q and bbjQ
represent corneal epithelium, palpebral conjunctiva,
and bulbar conjunctiva, respectively. Substituting Eqs.
(2) and (3) into Eq. (5) yields:
2.1. Precorneal area
After a drop is instilled, it is assumed that the
change in precorneal tear volume over time is
proportional to the excess tear volume (i.e., the
Fep fep Aep Fpj fpj Apj Fbj fbj Abj SCt
dCt
¼
dt
VL þ Vi expðkd t Þ
ð6Þ
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W. Zhang et al. / Journal of Controlled Release 99 (2004) 241–258
where W is the tissue-to-water distribution coefficient.
Fickian diffusional flux F i is expressed as:
2.2. Anterior chamber
To determine solute concentration in the aqueous
humor as a function of time, conservation of solute
in the anterior chamber is expressed as:
Fi ¼ Di
dðVd Ca Þ
¼ Fen fen Aen Cla Ca
dt
2.4. Anatomical and physiological parameters of the
eye
ð7Þ
where V d is the distribution volume of solute in the
anterior chamber (which accounts for partitioning
between the aqueous humor and the iris/ciliary
body), C a is the solute concentration in the anterior
chamber (which is a weighted average that accounts
for solute bound to tissue and free in solution), and
Cla is the drug clearance rate in the anterior
chamber. As described by Schoenwald [1], the
clearance rate of solute can be several times higher
than that of aqueous humor, as metabolism and
systemic uptake by the vascular tissues of the
anterior uvea constitute alternate routes of elimination. The subscript benQ denotes the endothelium at
the endothelium–anterior chamber interface. Rearranging Eq. (7) yields:
dCa
Fen fen Aen Cla Ca
¼
dt
Vd
ð8Þ
@Ci
@x
ð12Þ
Anatomical and physiological parameters were
determined based on experimental measurements
found in the literature, or, when data were not
available, using theoretical estimates as described in
previous publications [6,7]. Baseline values for these
parameters are summarized in Table 1. Note that the
distribution volume (V d) and clearance (Cla) in the
anterior chamber vary with each solute. For the
baseline case, midrange values of 600 Al and 13.5
Al/min, respectively, were therefore assumed. The
baseline instillation volume was taken as 30 Al. It was
also assumed that the diffusional length, diffusivity of
the solute, and its lipid bilayer-water distribution
coefficient (U) were identical in the conjunctival
epithelium and in the corneal epithelium and that the
fractional surface area occupied by the paracellular
route (i.e., the porosity f ) was 15 times greater in the
2.3. General approach to diffusion in the tissues
Table 1
Anatomical and physiological parameters in human eyes
The general solute conservation equation in tissue i
is expressed as:
Parameters
Value
Reference
Normal tear volume (V L) [Al]
Tear secretion rate (S) [Al/min]
Solution drainage rate constant
(k d) [min1]
Corneal surface area (A c) [cm2]
Conjunctival surface area (A j ) [cm2]
Palpebral conjunctiva surface area
(A pj) [cm2]
Bulbar conjunctiva surface area
(A bj) [cm2]
Corneal thickness [mm]
Volume of aqueous humor in
anterior chamber (Va) [Al]
Aqueous humor secretion rate,
as a percentage of Va [%/min]
Distribution volume in anterior
chamber (V d) [Al]
Clearance in anterior chamber
(Cla) [Al/min]
7.0
1.2
1.45
[2]
[2]
[11]
1.04
17.65
8.82
[38]
[38]
See text
7.78
See Text
0.52
261–310
[38]
[2,39,40]
1–2%/min
[2]
150–3000
[1]
1–30
[1]
@Ci
@ 2 Ci
¼ Di
@t
@x2
ð9Þ
where D i is the solute diffusivity across the corresponding layer and x is the axial coordinate as shown
in Fig. 1.
At the interface between two adjacent layers,
interfacial boundary conditions can be expressed as:
Cj
Ci
¼
Wi
Wj
ð10Þ
Fi fi Ai ¼ Fj fj Aj
ð11Þ
W. Zhang et al. / Journal of Controlled Release 99 (2004) 241–258
conjunctival epithelium than in the corneal epithelium. These assumptions are discussed later.
2.5. Diffusion across cornea
Diffusion across the cornea was modeled following the approach of Edwards and Prausnitz [6,7]. The
detailed model can be found in those papers; the
predicted parameters needed for this study are
reported in Table 2. Briefly, transport across the three
primary layers that make up the cornea was considered. The cellular tissues of corneal epithelium and
endothelium were modeled based on two transport
pathways: one following a tortuous route through
tight junctions and around cells, utilized primarily by
hydrophilic compounds, and the other, utilized
primarily by lipophilic compounds, involving partitioning into, moving along, and partitioning out of
cell membranes. The largely acellular tissue of stroma
was modeled as a fiber matrix, where molecules
diffuse around the lamellar collagen fibrils embedded
in a fibrous extracellular matrix. The model was
developed using independently determined parameters that represent physical properties of drugs and
tissues.
In this analysis, the corneal epithelium constitutes
the main barrier to the transport of hydrophilic
molecules across the cornea; for hydrophilic solutes
with a radius between 0.35 and 0.55 nm (i.e.,
approximately 150–450 Da), the relative resistances
to diffusion through the corneal epithelium, stroma,
and endothelium are predicted to be approximately
92%, 1%, and 7%, respectively (Table 2), in
reasonable agreement with experimentally measured
relative resistances of 80:1:2 for mannitol [12].
For lipophilic molecules, the resistance of the
cellular layers increases as U decreases. For highly
lipophilic compounds (e.g., U=100), the relative
resistance of the corneal epithelium is found to
decrease from 41% to 20% as solute radius
increases from 0.35 to 0.55 nm. For less lipophilic
compounds (U=10), the relative resistance is 81–
67%. These predictions agree with the results of
Huang et al. [13], who experimentally found that the
resistance of the epithelium layer to h-blockers is 1–
20% that of cornea for highly lipophilic molecules
(UN40), and 45–70% for less lipophilic compounds
(Uc10).
245
Table 2
Diffusional parameters across cornea and scleraa
Lipophilicity
Hydrophilic
Solute radius (nm)
Epithelium L b (Am)
f c (%)
D (106
cm2/s)
Wd
P e (106
cm/s)
k f (%)
Stroma
L (Am)
f (%)
D (106
cm2/s)
W
P (106
cm/s)
k (%)
Endothelium L (Am)
f (%)
D (106
cm2/s)
W
P (106
cm/s)
k (%)
Sclera
D (106
cm2/s)
W
Corneal P (106 cm/s)
0.35
151.7
0.18
10.9
a
b
c
0.45
151.7
0.18
6.54
Lipophilic
0.55
151.7
0.18
4.33
0.35
151.7
1
0.02
0.45
151.7
1
0.02
0.55
151.7
1
0.02
0.86 0.83 0.79 10
1.1 0.64 0.40 13.2
10
10
13.2 13.2
92
450
1
6.1
75
450
1
3.5
92
450
1
3.5
92
450
1
2.2
81
450
1
6.1
67
450
1
2.2
0.73 0.71 0.69 0.73
98.9 55.5 33.9 98.9
0.71 0.69
55.5 33.9
1
12.2
0.18
11.1
11
15.5
1
0.02
18
15.5
1
0.02
26
15.5
1
0.02
0.88 0.84 0.81 10
14.4 8.37 5.33 129
10
129
10
129
7
7
7
8
6.96 4.16 2.72 6.96
7
7
4.16 2.72
1
12.2
0.18
6.71
1
12.2
0.18
4.46
0.762 0.752 0.741 0.762 0.752 0.741
1.03 0.59 0.37 10.7 9.84 8.84
Parameter evaluation is described in Refs. [6,7].
L=diffusional length (i.e., actual thickness times tortuosity).
f=Fraction of the surface area occupied by the paracellular
route.
d
W=Tissue-to-water distribution coefficient. This coefficient
was taken as the octanol-to-water distribution coefficient (i.e., U)
for partitioning of lipophilic molecules into epithelium or endothelium, which is determined by solubility. In contrast, W is governed
by steric exclusion for lipophilic and hydrophilic molecules in the
aqueous stroma, and for hydrophilic molecules in the water-filled
paracellular spaces of cell layers. In these latter cases, W was
calculated as described in Edwards and Prausnitz [6].
e
P=permeability, calculated as DfW/L. For cornea, P c=1/(1/
P ep+1/P st+1/P en). For lipophilic compounds, P was based on W=10.
f
k=Ratio of the membrane diffusional resistance to that of the
cornea. Resistance is the inverse of permeability.
2.6. Diffusion across sclera and conjunctiva
Solute diffusion in the sclera was modeled as in
stroma, following the approach of Edwards and
Prausnitz [6]; the diffusional parameters are shown
in Table 2.
246
W. Zhang et al. / Journal of Controlled Release 99 (2004) 241–258
The conjunctiva consists of ultrastructurally heterogeneous epithelia, with 10–15 cell layers towards the
cornea and five to six layers at the eyelids [14].
Lacking more specific ultrastructural data, it was
assumed that the surface area of the palpebral
conjunctiva, which covers the inner surface of the
eyelid, is the sum of the surface area of the bulbar
conjunctiva (which covers the visible part of the sclera)
and that of the cornea. The effective diffusional length
across conjunctiva was also assumed to be the same as
that across corneal epithelium, based on the transmission electron microscopy (TEM) pictures published by Breitbach and Spitznas [15], which suggest
that the thickness of bulbar conjunctiva is close to 60
Am, comparable to that of the corneal epithelium.
The porosity of the conjunctiva was estimated as
2.7%, based on experimental measurements by Hamalainen et al. [16] showing that the porosity of the
conjunctiva is 15 times greater than those of the
cornea. The porosity estimate in this study is several
orders of magnitude larger than that of those investigators since theirs characterizes the superficial tight
junction layer of the conjunctival (as well as corneal)
epithelium, whereas the value in this study is an
average for the entire epithelial layer. These porosity
calculations are supported by the experimental measurements of pore size distribution by Doughty [17],
which yield a calculated porosity between 0.25% and
44% (7.6% on average) for palpebral conjunctiva, and
at least 0.8% for bulbar conjunctiva; these ranges
comprise the estimate of 2.7% in this study. This
porosity value corresponds to a conjunctival-to-corneal permeability ratio (c) for hydrophilic molecules of
17, based on the assumption that the diffusivities in
both conjunctival and corneal epithelia are the same
and the observation that the corneal epithelium
accounts for 90% of the resistance of the entire cornea.
This prediction is in good agreement with experimental data by Wang et al. [18], who measured the
conjunctival-to-corneal permeability ratio of hydrophilic h-blockers as 20, and by other investigators who
reported values in the range of 14–25 [12,16,19].
It also appears reasonable to assume that lipophilic
molecules diffuse within the cell membranes of
corneal and conjunctival epithelia at the same rate;
thus, the permeabilities of these epithelia were taken
to be equal, which yields predictions of c between 2
and 5 for solutes with a molecular radius between 0.35
and 0.55 nm and a distribution coefficient between 10
and 100. These estimates are comparable to the
experimental data of Sasaki et al. [20] and Wang et
al. [18], who reported c values between 2 and 10 for
lipophilic molecules.
It is possible that the uncertainty in conjunctival
parameter values might influence the accuracy of
predictions of corneal absorption and bioavailability.
Indeed, this model predicts that the amount of solute
absorbed by the conjunctiva is much greater than that
diffusing across the cornea, due to the large value of c.
In the baseline case, the fractions absorbed into the
anterior chamber vs. the bulbar and palpebral conjunctiva were found to be equal to 0.13% vs. 16% and
22%, respectively, for hydrophilic molecules, and
3.2% vs. 31% and 43%, respectively, for lipophilic
molecules. However, a sensitivity analysis showed
that even a twofold increase in epithelial permeability
of either conjunctiva alone, or both conjunctiva and
cornea yielded changes in these fractional absorption
values of less than 40%.
2.7. Overall fractions of solute clearance
The overall amount of solute that has penetrated
into a given ocular membrane (i.e., cornea, palpebral
conjunctiva, or bulbar conjunctiva) at time t can be
obtained from an interfacial balance at the boundary
between the tear film and the epithelium (i.e., at x=0):
Z t
dCi M i ðt Þ ¼ A i f i D i
ð13Þ
dt:
0 dx x¼0
Mass conservation in the tear film at time t also
yields:
Mc ðt Þ þ Mpj ðt Þ þ Mbj ðt Þ þ
R
t
0
QCt dt þ Ct ðt ÞVt ðt Þ
ðCt Vt Þjt¼0
¼1
ð14Þ
This equation can be used to verify the accuracy of
the numerical results, as described later.
Similarly, the overall amount of drug that has been
absorbed by the aqueous humor at time t, M a(t), can
be obtained either from an interfacial balance at the
boundary between the corneal endothelium and the
W. Zhang et al. / Journal of Controlled Release 99 (2004) 241–258
aqueous humor, or from mass conservation in the
aqueous humor:
Z t
dCen
Ma ðt Þ ¼ Aen fen Den
0 dx
Z t
þ Cla Ca dt
j
dt ¼ Vd Ca
eneq
ð15Þ
247
(the time needed to reach C max), and M a (the
fractional amount of solute diffusing into the
chamber) are important to clinical applications and
therefore are often measured in experimental studies,
the effects of various parameters on these three
measurements were predicted.
0
3.1. Effect of molecular size on drug transport
where "en–aq" denotes the endothelium–aqueous
humor interface. By comparing the right-hand side
of Eq. (15) with the center term, the accuracy of the
numerical method can be estimated. As time goes to
infinity, the value of M a(t) should approach that of
M c(t).
2.8. Numerical methods
Eqs. (6), (8), and (9) constitute a system of
differential equations that must be integrated to yield
solute concentration in the precorneal area, the ocular
membranes considered here, and the anterior chamber.
At t=0, C t=C 0, and all other concentrations are equal
to zero; interface (boundary) conditions are expressed
by Eqs. (10) and (11). These differential equations
were solved using second-order implicit finite difference approximations. The overall magnitude of solute
clearance or delivery was calculated based upon Eqs.
(13)–(15). The program was written using MatLab 6.1
(The Mathworks, Natick, MA). Simulations were run
on a personal computer with an AMD Athlon 1800+
processor (Advanced Micro Devices, Sunnyvale, CA)
and lasted typically between 10 and 20 min. The
difference between the right-hand side and the center
term of Eq. (15) was always less than 103 at each
time step, and so was the difference between the righthand and left-hand side of Eq. (14), which shows
good numerical accuracy.
3. Results
To address the bioavailability of topical eye
drops, the objective of this study was to model the
transient absorption of a solute from the precorneal
area into the anterior chamber, based upon the
morphology and physiology of the eye, and upon
the solute properties. Because C max (the maximum
solute concentration in the anterior chamber), t max
Increasing molecular size is known to decrease
drug bioavailability and limit topical delivery applications [1–3]. Shown in Table 3 are predicted t max,
C max, and M a values based on the diffusional
parameters in Table 2. Increasing molecular radius
of hydrophilic molecules from 0.35 to 0.95 nm
decreases absorption into the ocular tissues. This has
the effect of reducing drug transport into the anterior
chamber (M a), palpebral conjunctiva (M pj), and bulbar
conjunctiva (M bj), as well as C max. Likewise, it
increases the time needed to diffuse into the anterior
chamber (t max and t 90) and the fraction of solute
cleared by tear fluid drainage (M t). For lipophilic
compounds, the same trends are generally seen,
except that more drugs are absorbed into the ocular
tissues over longer periods of time and less are
Table 3
Effect of molecular size on solute deliverya
Molecular U
radius
(nm)
b
b
c
C max
M ac M pj
M bjc M tc t 90
t max
5
(min) (10 ) (%) (%) (%) (%) (min)
0.35
0.45
0.55
0.95
0.35
0.45
0.55
0.95
42
55
68
119
99
105
112
145
a
Hydrophilic
Hydrophilic
Hydrophilic
Hydrophilic
10
10
10
10
4.3
2.7
1.8
0.37
50
45
40
18
0.17
0.13
0.11
0.06
3.3
3.2
3.0
1.9
26
22
18
9
41
43
44
53
19
16
13
7
33
31
29
20
55
62
69
84
23
23
24
25
100
160
240
759
275
305
340
590
Parameter values are that of the baseline case (see text), unless
otherwise specified.
b
C max is the maximum concentration in the anterior chamber,
normalized by the initial concentration in the tear fluid, and t max is
the time needed to reach this peak concentration.
c
M a, M pj, M bj, and M t represent the fraction of solute that has
reached the anterior chamber through the corneal route, been
absorbed into the palpebral conjunctiva, been absorbed into the
bulbar conjunctiva, and has been eliminated by tear fluid drainage,
respectively, at t 90 (i.e., when 90% of molecules that have been
absorbed into the cornea have diffused into the anterior chamber).
248
W. Zhang et al. / Journal of Controlled Release 99 (2004) 241–258
drained away by tear fluid. As shown in Fig. 2, M a
decreases by 65% and 43% for hydrophilic and
lipophilic solutes, respectively, as molecular radius
increases from 0.35 to 0.95 nm (i.e., from 125 to 2500
Da); for a given solute size, M a is about 20 times
larger for lipophilic molecules relative to hydrophilic
molecules. In the remainder of this study, results are
given by assuming a molecular radius of 0.45 nm, an
average representative of current therapeutic compounds, unless otherwise specified.
3.2. Effect of lipophilic vs. hydrophilic transport
pathways on drug transport
Notwithstanding the variations due to molecular
size, bioavailability is predicted to be small for
lipophilic molecules that follow transcellular pathways (on the order of 3%) and still smaller for
hydrophilic compounds that follow paracellular pathways (on the order of 0.1%; see Table 3 and Fig. 2).
This difference can be explained by the very small
fraction of the epithelial surface area that is occupied
by intercellular openings ( f ) available for hydrophilic
solute transport, and the fact that the conjunctival-tocorneal permeability ratio (c) is significantly higher
for hydrophilic molecules.
The peak time t max is predicted to be close to 1
h for hydrophilic molecules and 2 h for lipophilic
compounds in the range of solute radii examined
(Table 3). These predictions agree with the exper-
imental data of Fukuda and Sasaki [21], who
measured t max as 2.25 h for the lipophilic molecule
erythromycin lactobionate, and as ~1 h for the
hydrophilic compounds chloramphenicol and levofloxacin.
It is interesting to note that the corneal layer that
controls steady-state permeability values is not necessarily the same as that which controls the kinetics of
transient diffusion. Permeability scales with diffusivity (D) and the fractional cross-sectional area available for transport ( f ), as well as other parameters.
Because D for lipophilic molecules within epithelial
layers is low and f for hydrophilic molecules is also
small, the epithelium controls corneal permeability for
both lipophilic and hydrophilic molecules. In contrast,
transport time lag (and therefore t max) depends on
diffusivity (D) and tissue thickness (L), where the
characteristic time is on the order of L 2/D. Given their
low diffusivity in epithelial layers, t max for lipophilic
molecules is also controlled by the epithelium.
However, the epithelial diffusivity of hydrophilic
molecules is much larger and it is the larger tissue
thickness contributed by stroma that predominantly
determines t max. The overall result is that the steadystate corneal permeability is greater for lipophilic
molecules, whereas transient transport lag time is
shorter for hydrophilic molecules.
As expected, M a significantly depends on permeability. The results indicate that a twofold increase in
the permeability of epithelial layers significantly
increased bioavailability in the anterior chamber: for
hydrophilic molecules, M a increased by 85%, and for
lipophilic molecules, M a increased by 50%. In
contrast, when the permeability of the layer that
controls t max (i.e., epithelium for lipophilic molecules
and stroma for hydrophilic molecules) was held
constant, simultaneous twofold changes in diffusivity
and membrane thickness (i.e., the parameters that
affect transient transport lag time) yielded 30–40%
changes in t max, but no significant variations in M a
were observed.
3.3. Effect of distribution coefficient on drug transport
Fig. 2. Effect of solute molecular radius on bioavailabiliy in the
anterior chamber (M a).
To more closely examine the effects of lipophilicity, the effects of distribution coefficient on solute
transport are illustrated in Fig. 3, assuming that drug is
transported exclusively through the transcellular path-
W. Zhang et al. / Journal of Controlled Release 99 (2004) 241–258
Fig. 3. Effect of lipid bilayer-to-water distribution coefficient M a
and t max, assuming that solutes are exclusively transported across
cellular membranes through transcellular pathways (i.e., neglecting
delivery through paracellular pathways).
ways across cellular layers even for U less than 1 (i.e.,
neglecting the possible delivery through paracellular
pathways). As U rises from 0.01 to 1000, bioavailability initially increases until it reaches a maximum at
U=15, and then it decreases. This functionality is due
in part to the observation that increases in U raise the
permeability of the conjunctiva to a greater extent
than that of cornea, since the corneal permeability
includes the acellular stroma (the intrinsic permeability of which is not affected by U). As U increases
up to 15, the conjunctiva-to-cornea permeability ratio
(c) remains sufficiently small so that corneal absorption experiences a net increase. Thereafter, the
increases in permeability of cornea are outweighed
by still greater increases in permeability of palpebral
conjunctiva (which is composed of epithelium only),
such that absorption into palpebral conjunctiva
dominates at the expense of absorption into the
cornea, as well as bulbar conjunctival absorption
and precorneal drainage.
Distribution coefficient also affects transport lag
time. In these simulations in which U was varied, the
clearance rate (Cla) and the distribution volume (V d) in
the anterior chamber were fixed, whereas these two
parameters are in fact dependent upon solute lipophilicity, among other factors. As described later,
although Cla and V d have a small effect on solute
bioavailability, they are significant determinants of
C max and t max. The effect of U on C max and t max
249
should therefore be examined in conjunction with that
of U on Cla and V d. However, it is not possible to
predict the effect of U on clearance rate and
distribution volume. Assuming that the latter two
parameters do not vary with the distribution coefficient, the simulations indicate that as U increases from
0.01 to 1000, t max decreases initially, reaches a
minimum for Uc5, and increases thereafter. Indeed,
for molecules with UV1, increasing U raises the
epithelial concentration at the epithelium–tear interface, thereby reducing the residence time in the
precorneal area and decreasing t max. As U increases
beyond 5, partitioning between the cell layers and the
stroma is significantly reduced, so that diffusion across
the stroma is considerably slower and t max increases.
In agreement with these predictions, Huang et al.
[22] found that timolol (U=2.2) diffuses more rapidly
across the cornea than both acebutolol, which is
hydrophilic (U=1.6), and bufuralol, which is highly
lipophilic (U=200). The review of Lee and Robinson
[2] also suggests that the optimum partition coefficient
for corneal drug absorption is between 10 and 100.
To facilitate analysis, as discussed in the previous
section, this model assumes that any given solute
permeates through either the transcellular or paracellular route, but not both. The model, therefore, does
not apply to the entire range of distribution coefficients and is only valid when transport via one of
these two types of routes is predominant. To identify
the appropriate range of applicability, the predicted
bioavailablity by each route at different distribution
coefficients was compared. If U is varied from 1 to
0.1 to 0.01, M a is calculated as 1.2%, 0.14%, and
0.014%, respectively, assuming that all molecules
follow the transcellular routes across the epithelium
and endothelium. Assuming that solutes all follow the
paracellular pathways, M a=0.13%, independently of
U. Thus, at Uc0.1, the predicted M a values are
comparable for both routes and the model is not
rigorously correct. However, for much more lipophilic
compounds (Uz1) or much more hydrophilic compounds (UV0.01), the model’s assumptions are valid.
3.4. Effect of anterior chamber distribution volume
and clearance rate on drug transport
Unlike solute properties, such as molecular size
and distribution coefficient, which can be easily
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W. Zhang et al. / Journal of Controlled Release 99 (2004) 241–258
controlled and varied experimentally, study of ocular
physiological properties, such as anterior chamber
distribution volume and clearance rate, is experimentally difficult and therefore especially benefits from
theoretical analysis. Understanding the effects of
anterior chamber fluid mechanics is especially important in cases of glaucoma, ocular inflammation, and
pediatric patients. The anterior chamber distribution
volume accounts for solute dissolved in the aqueous
humor as well as partitioned into the iris/ciliary body,
and the clearance rate takes into consideration not
only aqueous humor turnover but also metabolic
degradation and systemic uptake by vascular tissues.
In the simulation results presented previously, effective distribution volume (V d) and clearance rate (Cla)
were taken as 600 Al and 13.5 Al/min, respectively,
which are the most frequently reported values;
however, measurements of V d and Cla vary from
150–3000 Al and 1–30 Al/min, respectively [1].
The effects of V d and Cla were assessed by varying
one of the two parameters, while keeping the other
fixed at its baseline value. As shown in Fig. 4A and B,
variations in Cla or V d have almost no effect on solute
bioavailability in the anterior chamber; for lipophilic
molecules, M a is slightly reduced as Cla decreases to
~1 Al/min. Indeed, at lower clearance, the accumulation of solute produces a higher concentration in the
anterior chamber, which slows diffusion due to a
reduced concentration gradient across the cornea; the
concentration gradient is even reversed after a long
period of time, so that backdiffusion of solute from
the aqueous humor to the corneal endothelium occurs.
However, as described previously, Cla is generally
larger than the aqueous humor secretion rate, which is
at least 2.6 Al/min, as shown in Table 1. The effects of
Cla and V d on M a are therefore negligible in most
cases.
Not surprisingly, increases in Cla or V d significantly
lower C max, as shown in Fig. 5A and B. Of greater
impact on drug delivery, the model also predicts that
t max decreases as the clearance rate increases, as
illustrated in Fig. 6A. As indicated by Eq. (8), solute
concentration in the anterior chamber (C a) is determined by the balance between the diffusional influx
from the cornea ( F enA en f en) and the efflux by
clearance mechanisms (ClaC a). Initially, the influx is
larger than the efflux; however, the clearance rate
continues to increase over time while the influx
Fig. 4. Effect of (A) clearance rate (Cla) and (B) distribution volume
(V d) in the anterior chamber on M a.
reaches steady state and then decreases. When the
clearance rate equals the diffusional influx, the
concentration reaches its maximum value. Therefore,
increasing clearance rate reduces the time needed for
influx and clearance rate to become equal (i.e., t max).
In contrast, t max increases with increasing distribution volume (Fig. 6B) because this slows the rate at
which the anterior chamber concentration increases
(i.e., dC a/dt; see Eq. (8)) and thereby increases the
time needed for the clearance rate to catch up to the
diffusional influx. Altogether, if Cla and V d are
simultaneously varied over the entire interval (1–30
Al/min and 150–3000 Al, respectively), t max is
calculated to range between 17 and 155 min for
W. Zhang et al. / Journal of Controlled Release 99 (2004) 241–258
251
rapidly partition into the cellular epithelia, t 1/2 is
calculated to be as short as 5 s. Another consequence
of partitioning is that the amount of solute cleared by
precorneal drainage (M t) is much greater for hydrophilic than lipophilic molecules (62% vs. 23%).
Precorneal drainage rate ( Q) can be altered by
changes in instilled volume (Vi), drainage rate constant (k d), and tear turnover rate (S), as indicated in
Eq. (4). As shown in Table 4, order of magnitude
increases in k d or S affect hydrophilic and lipophilic
molecules by reducing M a by ~40% and 10%,
respectively, and increasing M t by ~20% and 140%,
respectively. The different magnitudes of these effects
depending on lipophilicity stem from the fact that
lipophilic solutes are much more rapidly absorbed into
Fig. 5. Effect of (A) clearance rate and (B) distribution volume in
the anterior chamber on C max.
hydrophilic molecules and between 63 and 340 min
for lipophilic molecules.
3.5. Effect of precorneal solution drainage on drug
transport
Bioavailability can be dramatically affected by
precorneal drainage, which can be altered by dry eye,
tearing due to irritation by an eye drop, and drug
formulation [2]. For hydrophilic molecules delivered
in a conventional aqueous formulation, the half-life of
solute concentration in the precorneal area t 1/2 is
predicted as 70 s, which is close to that measured by
Lee and Robinson [23] for pilocarpine in the rabbit
eye (1–2 min). For lipophilic molecules, which
Fig. 6. Effect of (A) clearance rate and (B) distribution volume in
the anterior chamber on t max.
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W. Zhang et al. / Journal of Controlled Release 99 (2004) 241–258
Table 4
Effect of precorneal drainage
U
Drainage rate
Ma
Tear turnover t max
constant, k d
(min) (%)
rate, S
(1.45 min1) (1.2 Al/min)
Mt
(%)
Hydrophilic
Hydrophilica
Hydrophilic
Hydrophilic
Hydrophilic
10
10a
10
10
10
0
1
1
10
10
0
1
1
10
10
0
62
76
77
87
0
23
53
51
70
a
0
1
10
1
10
0
1
10
1
10
58
55
55
55
54
108
105
93
105
93
0.35
0.13
0.08
0.08
0.05
4.1
3.2
1.9
2.0
1.2
Baseline case.
the corneal and conjunctival epithelium, which
reduces their drainage by the tear fluid and makes
the corneal epithelium a solute reservoir. As concentration in the precorneal area decreases over time,
solute absorbed in this reservoir diffuses back to the
precorneal area and is cleared by tear drainage. In
contrast, changing precorneal drainage parameters has
very little effect on t max, since the precorneal
concentration does not affect the time needed to
diffuse across the cornea.
If solution drainage is totally eliminated due to
occlusion of the puncta, which are the two nasal ducts
through which fluid leaves the precorneal area, M a
increases by 170% and ~30% for hydrophilic and
lipophilic molecules, respectively, and M t is, of
course, zero (Table 4). This is consistent with the
observation of Patton and Robinson [24] that occlusion of the puncta should increase the amount of a
mildly hydrophilic molecule, pilocarpine, delivered to
the aqueous humor by onefold to threefold. It also
agrees with Linden and Alm [25], who suggested that
occlusion may increase fluorescein concentration in
aqueous humor by a factor of four.
Administration of eye drops, which increases
precorneal fluid volume, causes increases in lacrimal
drainage as well as tear turnover, which further dilute
solutes in the tear film [11]. As shown in Fig. 7, an
order-of-magnitude decrease in Vi from 50 to 5 Al
increases M a by 150% and ~30% for hydrophilic and
lipophilic molecules, respectively, because Q is
proportional to Vi. However, since the overall amount
of instilled drug is concomitantly reduced by a factor
of 10, the actual amount of drug delivered to the
anterior chamber (calculated as M aC i Vi, where C i Vi is
the amount of solute being instilled) decreases by 75%
and 90% for hydrophilic and lipophilic molecules,
respectively. Therefore, a large instilled volume is
preferred (at the maximum possible concentration).
3.6. Additional comparison of predictions with
experimental measurements
To assess the validity of the developed model,
predictions were compared with experimental measurements. Predicting bioavailability is of paramount
interest to applications and can be estimated by
determining M a. However, opportunities for direct
comparison with experimental data are limited, since
experimental measurements of M a are difficult or
costly, and therefore other parameters, such as C max
and t max, are more often reported. Therefore, the
predicted values of M a, C max, and t max were compared
with reported measurements, despite intrinsic limitations in predicting C max and t max when the clearance
rate and the distribution volume in the anterior
chamber have not been measured. Since both calculations and measurements of AUC (i.e., the area under
the concentration–time curve in the anterior chamber)
vary with the duration of simulations and experiments, this parameter was not included in the later
analysis. Results are shown in Table 5.
As described later, the overall difference between
the predicted estimates of M a, C max, and t max and
experimental measurements for compounds whose V d
and Cla have been measured is always less than a
Fig. 7. Effect of instilled volume on M a.
W. Zhang et al. / Journal of Controlled Release 99 (2004) 241–258
253
Table 5
Comparison between model predictions and experimental observationsa
Compound
logU
V d (Al) Cla (Al /min) t max (min)
Clonidine
Phenylephrine
Thiamphenicol
Timololc
Pilocarpined
Flubiprofen
Fluorescein
Chloramphenicole
2.2
1.7b
0.27b
0.34
0.460.22
0.75
4.3
2.5
Levofloxacine
1.3b
530
423
2200
446
580
620
600
600
200
600
150
600
300
600
150
Erythromycin lactobionatee 1.57
Bufuralolc
2.31
14.9
14.6
28
25.4
13.0
14.4
13.5
13.5
7
13.5
4.5
13.5
27
13.5
30
C max (105)
M a (%)
Reference
O
P
O
P
O
P
15–30
40–60
45
25–60
20–30
60–120
120
61
40
35
57
80
[9–111]
100
62
75
63
72
65
155
110
158
160
200
5.0
2.8
352
9.3
7.7
2.9
141
1.63
0.36
190
0.1
50
110
1.9
18
40
18
58
170
110
78
50
65
135
10
58
9.9
260
[27]
[41]
[28]
[22]
0.62– 0.8 [0.41– 4.1] [24]
7.0
5.5
[42]
[25]
[21]
[21]
[21]
[22]
a
Experimental measurements of V d and Cla are from Ref. [1] except for timolol [43]. For compounds with unknown V d and Cla values,
either baseline values (600 Al and 13.5 Al/min, respectively) or fitted values were used in order to match experimental measurements of t max and
C max. LogU values are from the listed references or from Ref. [26]; O=experimental observation; P=model prediction.
b
Although the model can only be rigorously applied to logUN0 or logUb2, predictions are given here for intermediate U values,
assuming that all molecules were carried through paracellular routes.
c
Instilled three times at 2-min intervals.
d
Predictions for pilocarpine are given as a range (see text); the lower and upper limits are obtained by considering only the hydrophilic
pathways across cell layers, and by adding the contributions of the hydrophilic and lipophilic pathways, respectively.
e
Instilled three times at 15-min intervals.
factor of two, which represents good agreement for a
mechanistic model that does not rely heavily on
empirical fitting. The errors in M a were less than 50%
in two of three cases; those in C max were less than
50% in four of five cases examined. Similarly, the
errors in t max were less than 25% in three of six cases,
and less than 50% in five cases. Since V d and Cla have
small effects on M a, this model should give reasonable
estimates of M a even for drugs with unknown V d and
Cla values.
Comparisons between predicted and measured M a
values were possible for only three compounds:
clonidine, pilocarpine and flubiprofen. There is good
agreement in the case of flubiprofen, with a relative
difference between the two values of M a equal to ~
20% (t max and C max values are also in good agreement). The logU of pilocarpine has been reported to
lie between 0.22 and to 0.46 [26]; therefore, the
lower and upper limits of M a were calculated, based
upon the diffusion across hydrophilic pathways only,
or the contributions of both lipophilic and hydrophilic
pathways, respectively. The value of M a reported by
Patton and Robinson [24] falls within the predicted
range, and is close to the lower limit, thereby
suggesting that pilocarpine may be weakly hydrophilic. The measured M a of clonidine is 1.6% [27]
(i.e., more than four times higher than the prediction).
This large discrepancy may be due to the fact that
clonidine is highly ionized [26], which is not
accounted for in this model. The predicted value of
t max for clonidine is, however, much closer to the
experimental one.
The values of V d and Cla were available for only 6
of 11 compounds examined, including the three
solutes examined previously. Among the three other
solutes phenylephrine, thiamphenicol, and timolol, the
agreements between predicted and measured t max and
C max values are good for the first two. Since the
distribution coefficient of thiamphenicol is 0.54, the
model does not strictly apply in that case, and
predictions were based upon the assumption that all
thiamphenicol molecules follow paracellular pathways within cellular layers. Aldana et al. [28]
observed that the anterior chamber concentration of
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W. Zhang et al. / Journal of Controlled Release 99 (2004) 241–258
thiamphenicol remained elevated several hours after
t max (which they measured as 45 min), most likely
because of the non-negligible fraction of molecules
following transcellular pathways. The logU of timolol
has been reported as ranging from 0.39 to 0.34
[22,26]. Assuming that timolol is lipophilic (i.e.,
assuming the upper limit value for U), the predicted
t max (80 min) is close to that reported by Ellis et al.
[29], but significantly greater than that measured by
Huang et al. [22].
For the other five solutes in Table 5 for which V d
and Cla values have not been reported, either the
baseline values (600 Al and 13.5 Al/min, respectively)
or fitted values of V d and Cla were used in order to
match experimental measurements of t max and C max.
Since V d and Cla are strong functions of solute
lipophilicity, and since these two parameters significantly affect C max and t max values, the predictions
should be viewed with caution. Using baseline values,
the agreement between predictions and experimental
data was generally poor. With fitted V d and Cla values,
good agreement was obtained for chloramphenicol
and levofloxacin. For fluorescein, however, t max is
significantly underestimated and C max is greatly
overestimated independently of V d and Cla values. A
previous study suggested that the solute may adhere to
the sclera [30]; since the composition of stroma and
sclera is similar, it is possible that fluorescein binds to
the corneal stroma as well, which would explain the
discrepancy between calculated and measured values.
For erythromycin lactobionate, the predicted t max is
close to experimental data, but there is a factor of 10
difference in calculated and measured C max values; it
is possible that erythromycin lactobionate is degraded
within the cornea, as observed for levobunolol by
Tang-Liu et al. [31]. Lastly, the discrepancy between
calculated and measured t max values in the case of
bufuralol (the latter is as low as 10 min) may stem
from the fact that for such a lipophilic molecule
(logUN 2.3), the amount of solute delivered to the
anterior chamber through the conjunctiva/sclera may
not be negligible.
4. Discussion
The theoretical model developed in this study
should be considered in the context of previous work.
Drug delivery through ocular membranes has been
modeled extensively [1–3]. A semiempirical approach
involving compartment models has often been used
due to the complicated morphology of the eye, the
lack of specific ultrastructural data related to membrane composition, and the difficulty in accounting for
certain transport phenomena such as binding, blinking, and tear drainage [4,5,23,32,33]. In those
approaches, clearance and transport processes for
solutes are described by first-order rate equations,
and several rate constants are typically obtained by
fitting the model to experimental data. Although such
compartment models yield adequate predictions, they
are generally limited to the specific type of molecules
studied by the investigators; and because they do not
explicitly account for the ultrastructure and the
hydrodynamics of the eye, they must be validated
for every case being considered.
4.1. Model strengths and limitations
In this study, a mechanism-based transient diffusion
model was developed in order to examine the effects of
ultrastructure and hydrodynamics on topical drug
delivery to the anterior chamber through the corneal
route, and to provide a broadly predictive tool for
estimating drug bioavailability. In contrast to compartment models, transport rates across the ocular tissues
were calculated based upon the physicochemical
properties of the solutes as well as the ultrastructural
parameters of the cornea, sclera, and conjunctiva,
without relying heavily on fitted empirical parameters.
Its modeling ability should extend to any drugs within
the broad range of physicochemical properties considered and requires knowledge of molecular radius,
octanol–water distribution coefficient, as well as
anterior chamber distribution volume and clearance.
This model may also help predict the effect of
transiently modifying the integrity of the corneal
epithelium in order to enhance corneal drug permeability. For example, Grass et al. [34] have proposed
using calcium chelators to dissolve extracellular
matrix material to widen intercellular spaces and
other compounds to perturb lipid bilayers. Chung et
al. [35] have proposed the use of peptides to open
hydrophilic pathways across cornea. Such changes in
the ultrastructure of the cornea could be directly
reflected in the parameters of the model.
W. Zhang et al. / Journal of Controlled Release 99 (2004) 241–258
Despite this model’s general ability to capture the
important physical processes that govern topical drug
delivery to the eye, it has limitations. Although it was
assumed that solutes enter the anterior chamber
through the corneal route exclusively, drugs can also
be absorbed into the eye through conjunctiva and
sclera. However, the assumption appears to be
generally valid because studies indicated that the
noncorneal route bypasses the anterior chamber and
results primarily in delivery to the iris, ciliary body,
lens, and possibly vitreous humor by mechanisms that
may involve uptake and redistribution via the bloodstream [10,14,36].
To avoid the complexity of a full two-dimensional
representation of ocular membranes, it was assumed
that solutes permeate through either the paracellular or
transcellular route, but not both. This assumption is
valid for strongly hydrophilic molecules (Ub0.01) and
strongly lipophilic molecules (UN1) (i.e., cases when
one of the two routes carries at least 10 times more
solute than the other one). Especially at intermediate
levels of lipophilicity (0.01bUb1), model predictions
are therefore underestimates.
In this study, the octanol-to-water distribution
coefficient was used, instead of the partition coefficient employed in many other studies, to characterize solute lipophilicity, which is viewed as a strength
because the distribution coefficient accounts for the
ratio between the organic and aqueous phase concentrations of both ionized and nonionized molecules.
However, this model did not account for effects of
ionization state on diffusion rates, which may be
especially important for diffusion through the corneal
epithelium, which is negatively charged at normal pH
[37].
This model did not consider drug metabolism or
binding within the cornea in the absence of specific
data for each solute of interest. The corneal epithelium
possesses drug-metabolizing enzymes [2], and possible binding of solute to the stroma was also reported
[30]. Drug metabolism should lower M a and C max,
whereas binding is expected to elongate t max and
decrease C max.
The effects of the mucin layer on solute penetration, not considered in this study, are likely to be
significant for charged molecules as well as macromolecules. As charged molecules are sequestered by
the mucin glycocalyx membrane, which forms the
255
exterior surface of the tear film, t max is expected to
increase. Other factors likely to affect transport rates
that are not considered in this analysis include
molecular shape, viscosity of the instilled solution,
as well as the effects of reflex blinks.
4.2. Improving the efficiency of drug delivery
Topical drug delivery remains generally inefficient,
as tear drainage and absorption into the conjunctiva
are the dominant clearance mechanisms, in good
agreement with experimental data. This model predicted that only a small fraction of solute present in
the instilled drop reaches the anterior chamber: less
than 5% for lipophilic molecules and less than 0.5%
for hydrophilic molecules. Lacrimal drainage continuously depletes solute in the tear film, and the solute
that remains is preferentially absorbed by the conjunctiva. Indeed, the latter membrane has a surface
area that is 17 times greater than that of cornea [38],
and the permeability of the conjunctiva is also
manyfold greater than that of the thicker, three-layer
cornea.
Strategies to optimize ocular drug delivery
usually involve one of the following approaches:
increasing drug residence time in the tear film by
using suspensions, gels, or inserts for instance;
reducing lacrimal secretion and tear fluid discharge
by eye closure or punctum occlusion; or enhancing
corneal permeability. These strategies are consistent
with the model predictions. Increasing the instilled
volume may not necessarily improve corneal
absorption because of spilling and more frequent
blinking; moreover, the resulting higher systemic
drug load may be accompanied by undesirable side
effects.
Of note, this study suggests that since simultaneous
enhancement of the permeability of cornea and
conjunctiva does not necessarily increase drug bioavailability, strategies should focus on reducing the
conjunctival-to-corneal permeability ratio (c), such as
designing drugs, and their formulations, to have lipidto-water distribution coefficients between 10 and 20.
As U becomes very large, conjunctival absorption is
increasingly favored to the detriment of corneal
absorption, since increases in U reduce the diffusional
resistance of epithelia but not that of stroma, thereby
raising c. Hence, maximum corneal permeability does
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W. Zhang et al. / Journal of Controlled Release 99 (2004) 241–258
not necessarily mean enhanced absorption by the
cornea relative to conjunctiva, and there may not
always be a proportional correlation between corneal
permeability and bioavailability. Experimental observations by Huang et al. [22] support the findings:
although the lipophilicity and corneal permeability of
acebutolol, timolol, and bufuralol increase in that
order, the mildly lipophilic timolol has the greatest
C max of all three compounds.
Experimental determination of the quantitative
effects of tear secretion, lacrimal discharge, blinking
frequency, and punctum occlusion on topical drug
administration remains uncertain. This model indicated that a factor of 10 change in the rate of lacrimal
secretion and solution drainage only yields a ~40%
variation in the amount of solute delivered to the
anterior chamber. That is because an increase or
decrease in solute removal by tear drainage is
primarily compensated for by changes in solute
absorption into the conjunctiva, and vice versa.
Punctum occlusion is predicted to increase the amount
of solute that reaches the anterior chamber by a factor
of two for hydrophilic molecules, but only by ~30%
for lipophilic molecules.
5. Conclusion
The developed model of transient transport across
the cornea explicitly considers the ultrastructure and
hydrodynamics of the eye as well as the physicochemical properties of diffusing solutes. Predictions in
this study are in good agreement with experimental
data on topical drug delivery. The model suggests
promising strategies to increase the bioavailability of
drugs applied topically to the eye, involving (1)
increasing corneal permeability without a corresponding increase in conjunctival permeability, and (2)
increasing drug–cornea contact time by reducing drug
loss via tear fluid drainage and/or providing drug for
an extended period of time (e.g., using controlledrelease approaches). The model’s greatest strength,
however, is its expected ability to predict bioavailability and drug delivery kinetics for new drugs and
abnormal ocular anatomical and physiological states
due to the model’s mechanistic basis, use of parameters with true physical meaning, and avoidance of
fitted empirical parameters.
Nomenclature
Ai
Surface area of ocular membrane i
Ci
Solute concentration in compartment i
C0
Initial solute concentration in tear fluid
C max Maximum solute concentration in aqueous
humor normalized by C 0
Cla Solute clearance rate in the anterior chamber
Di
Diffusivity in membrane i
Fi
Diffusional flux across membrane i
f
Fraction of surface area occupied by diffusional
pathway
kd
Lacrimal discharge rate constant
L
Length of diffusional pathway across membrane
Mi
Fraction of solute clearance or absorption by
compartment i
P
Permeability
Q
Lacrimal discharge rate
rs
Solute molecular radius
S
Lacrimal secretion rate
V
Volume
Vd
Distribution volume in the anterior chamber
Vi
Instilled drug solution volume
VL
Volume of tear film under normal physiological
conditions
Vt
Tear volume or volume in precorneal area
t
Time
t 90
Time when 90% of solute absorbed into the
cornea has reached the anterior chamber
t max Time when solute concentration in anterior
chamber reaches its maximum
Greek
U
W
c
k
symbols
Octanol-to-water distribution coefficient
Tissue-to-water distribution coefficient
Conjunctival-to-corneal permeability ratio
Membrane-to-cornea diffusional resistance
ratio
Subscripts
a
Aqueous humor or anterior chamber
c
Cornea
en
Corneal endothelium
en–aq Endothelium–aqueous humor interface
ep
Corneal epithelium
bj
Bulbar conjunctiva
pj
Palpebral conjunctiva
st
Corneal stroma
t
Tear film or tear flow
W. Zhang et al. / Journal of Controlled Release 99 (2004) 241–258
Acknowledgements
The authors thank Henry Edelhauser for helpful
discussions. This work was supported, in part, by the
National Institutes of Health.
References
[1] R.D. Schoenwald, Ocular pharmacokinetics and pharmacodynamics, in: A.K. Mitra (Ed.), Ophthalmic Drug Delivery
Systems, Marcel Dekker, New York, 2003, pp. 135 – 179.
[2] V.H. Lee, J.R. Robinson, Topical ocular drug delivery: recent
developments and future challenges, J. Ocul. Pharmacol. 2 (1)
(1986) 67 – 108.
[3] K. Jarvinen, T. Jarvinen, A. Urtti, Ocular absorption following
topical delivery, Adv. Drug Deliv. Rev. 16 (1995) 3 – 19.
[4] K.J. Himmelstein, I. Guvenir, T.F. Patton, Preliminary
pharmacokinetic model of pilocarpine uptake and distribution
in the eye, J. Pharm. Sci. 67 (5) (1978) 603 – 606.
[5] S.C. Miller, K.J. Himmelstein, T.F. Patton, A physiologically
based pharmacokinetic model for the intraocular distribution
of pilocarpine in rabbits, J. Pharmacokinet. Biopharm. 9 (6)
(1981) 653 – 677.
[6] A. Edwards, M.R. Prausnitz, A fiber matrix model for the
permeability of sclera and corneal stroma, AIChE J. 44 (1998)
214 – 225.
[7] A. Edwards, M.R. Prausnitz, Predicted permeability of the
cornea to topical drugs, Pharm. Res. 18 (11) (2001)
1497 – 1508.
[8] G.M. Grass, V.H. Lee, A model to predict aqueous humor and
plasma pharmacokinetics of ocularly applied drugs, Investig.
Ophthalmol. Vis. Sci. 34 (7) (1993) 2251 – 2259.
[9] I. Ahmed, T.F. Patton, Importance of the noncorneal absorption route in topical ophthalmic drug delivery, Investig.
Ophthalmol. Vis. Sci. 26 (4) (1985) 584 – 587.
[10] R.D. Schoenwald, G.S. Deshpande, D.G. Rethwisch, C.F.
Barfknecht, Penetration into the anterior chamber via the
conjunctival/scleral pathway, J. Ocular Pharmacol. Ther. 13 (1)
(1997) 41 – 59.
[11] S.S. Chrai, T.F. Patton, A. Mehta, J.R. Robinson, Lacrimal and
instilled fluid dynamics in rabbit eyes, J. Pharm. Sci. 62 (7)
(1973) 1112 – 1121.
[12] A.J. Huang, S.C. Tseng, K.R. Kenyon, Paracellular permeability of corneal and conjunctival epithelia, Investig. Ophthalmol. Vis. Sci. 30 (4) (1989) 684 – 689.
[13] H.S. Huang, R.D. Schoenwald, J.L. Lach, Corneal penetration
behavior of beta-blocking agents: II. Assessment of barrier
contributions, J. Pharm. Sci. 72 (11) (1983) 1272 – 1279.
[14] I. Ahmed, The noncorneal route in ocular drug delivery, in:
A.K. Mitra (Ed.), Ophthalmic Drug Delivery Systems, Marcel
Dekker, New York, 2003, pp. 235 – 363.
[15] R. Breitbach, M. Spitznas, Ultrastructure of the paralimbal and
juxtacaruncular human conjunctiva, Graefe Arch. Clin. Exp.
Ophthalmol. 226 (6) (1988) 567 – 575.
257
[16] K. Hamalainen, K. Kananen, S. Auriola, K. Kontturi, A. Urtti,
Characterization of paracellular and aqueous penetration
routes in cornea, conjunctiva, and sclera, Investig. Ophthalmol. Vis. Sci. 38 (3) (1997) 627 – 634.
[17] M.J. Doughty, Scanning electron microscopy study of the
tarsal and orbital conjunctival surfaces compared to peripheral
corneal epithelium in pigmented rabbits, Doc. Ophthalmol. 93
(4) (1997) 345 – 371.
[18] W. Wang, H. Sasaki, D.S. Chien, V.H. Lee, Lipophilicity
influence on conjunctival drug penetration in the pigmented
rabbit: a comparison with corneal penetration, Curr. Eye Res.
10 (6) (1991) 571 – 579.
[19] J.L. Ubels, E.M. Woo, W.J. Watts, L.K. Smith, U. Zylstra, J.
Beaird, M.D. McCartney, Conjunctival permeability and
ultrastructure. Effects of benzalkonium chloride and artificial
tears, Adv. Exp. Med. Biol. 438 (1998) 723 – 730.
[20] H. Sasaki, Y. Igarashi, T. Nagano, H. Yamahara, K. Nishida, J.
Nakamura, Penetration of beta-blockers through ocular membrane in albino rabbits, J. Pharm. Pharmacol. 47 (1995) 17 – 21.
[21] M. Fukuda, K. Sasaki, General purpose antimicrobial ophthalmic solutions evaluated using new pharmacokinetic
parameter of maximum drug concentration in aqueous, Jpn.
J. Ophthalmol. 46 (4) (2002) 384 – 390.
[22] H.S. Huang, R.D. Schoenwald, J.L. Lach, Corneal penetration
behavior of beta-blocking agents: III. In vitro–in vivo
correlations, J. Pharm. Sci. 72 (11) (1983) 1279 – 1281.
[23] V.H. Lee, J.R. Robinson, Mechanistic and quantitative
evaluation of precorneal pilocarpine disposition in albino
rabbits, J. Pharm. Sci. 68 (6) (1979) 673 – 684.
[24] T.F. Patton, J.R. Robinson, Quantitative precorneal disposition
of topically applied pilocarpine nitrate in rabbit eyes, J. Pharm.
Sci. 65 (9) (1976) 1295 – 1301.
[25] C. Linden, A. Alm, The effect of reduced tear drainage on
corneal and aqueous concentrations of topically applied
fluorescein, Acta Ophthalmol. 68 (6) (1990) 633 – 638.
[26] M.R. Prausnitz, J.S. Noonan, Permeability of cornea, sclera,
and conjunctiva: a literature analysis for drug delivery to the
eye, J. Pharm. Sci. 87 (12) (1998) 1479 – 1488.
[27] C.H. Chiang, R.D. Schoenwald, Ocular pharmacokinetic
models of clonidine-3H hydrochloride, J. Pharmacokinet.
Biopharm. 14 (2) (1986) 175 – 211.
[28] I. Aldana, D. Fos, E. Gonzalez Penas, A. Gazzaniga, V.
Gianesello, N. Ceppi Monti, P.G. Figini, M.A. Zato, L.
Bruseghini, A. Esteras, Ocular pharmacokinetics of thiamphenicol in rabbits, Arzneim.-Forsch. 42 (10) (1992)
1236 – 1239.
[29] P.P. Ellis, P.Y. Wu, D.S. Pfoff, D.C. Bloedow, M.R. Riegel,
Effect of nasolacrimal occlusion on timolol concentrations in
the aqueous humor of the human eye, J. Pharm. Sci. 81 (3)
(1992) 219 – 220.
[30] M.R. Prausnitz, A. Edwards, J.S. Noonan, D.E. Rudnick, H.F.
Edelhauser, D.H. Geroski, Measurement and prediction of
transient transport across sclera for drug delivery to the eye,
Ind. Eng. Chem. Res. 37 (1998) 2903 – 2907.
[31] D.D. Tang-Liu, S. Liu, J. Neff, R. Sandri, Disposition of
levobunolol after an ophthalmic dose to rabbits, J. Pharm. Sci.
76 (10) (1987) 780 – 783.
258
W. Zhang et al. / Journal of Controlled Release 99 (2004) 241–258
[32] H. Sasaki, M. Ichikawa, S. Kawakami, K. Yamamura, K.
Nishida, J. Nakamura, In situ ocular absorption of tilisolol
through ocular membranes in albino rabbits, J. Pharm. Sci. 85
(9) (1996) 940 – 943.
[33] H. Pospisil, H.G. Holzhutter, A compartment model to
calculate time-dependent concentration profiles of topically
applied chemical compounds in the anterior compartments of
the rabbit eye, Altern. Lab. Anim. 29 (3) (2001) 347 – 365.
[34] G.M. Grass, R.W. Wood, J.R. Robinson, Effects of calcium
chelating agents on corneal permeability, Investig. Ophthalmol. Vis. Sci. 26 (1) (1985) 110 – 113.
[35] Y.B. Chung, K. Han, A. Nishiura, V.H. Lee, Ocular absorption
of Pz-peptide and its effect on the ocular and systemic
pharmacokinetics of topically applied drugs in the rabbit,
Pharm. Res. 15 (12) (1998) 1882 – 1887.
[36] I. Ahmed, R.D. Gokhale, M.V. Shah, T.F. Patton, Physicochemical determinants of drug diffusion across the conjunctiva, sclera, and cornea, J. Pharm. Sci. 76 (8) (1987) 583 – 586.
[37] Y. Rojanasakul, J.R. Robinson, Transport mechanisms of the
cornea: characterization of barrier permselectivity, Int. J.
Pharm. 55 (2–3) (1989) 237 – 246.
[38] M.A. Watsky, M.M. Jablonski, H.F. Edelhauser, Comparison
of conjunctival and corneal surface areas in rabbit and human,
Curr. Eye Res. 7 (5) (1988) 483 – 486.
[39] S.T. Fontana, R.F. Brubaker, Volume and depth of the anterior
chamber in the normal aging human eye, Arch. Ophthalmol.
98 (10) (1980) 1803 – 1808.
[40] R.D. Schoenwald, Pharmacokinetics in ocular drug delivery,
in: P. Edman (Ed.), Biopharmaceutics of Ocular Drug
Delivery, CRC Press, Boca Raton, FL, 1993, pp. 159 – 191.
[41] D.S. Chien, R.D. Schoenwald, Improving the ocular absorption of phenylephrine, Biopharm. Drug Dispos. 7 (5) (1986)
453 – 462.
[42] D.D. Tang-Liu, S.S. Liu, R.J. Weinkam, Ocular and systemic
bioavailability of ophthalmic flurbiprofen, J. Pharmacokinet.
Biopharm. 12 (6) (1984) 611 – 626.
[43] H. Sasaki, K. Yamamura, T. Mukai, K. Nishida, J. Nakamura,
M. Nakashima, M. Ogasawara, M. Ichikawa, In vivo ocular
pharmacokinetic model for designing dosage schedules and
formulations of ophthalmic drugs in human, Acta Med.
Nagasaki. 42 (3–4) (1997) 45 – 50.