Transient
Analysis of Ocular Drug Delivery  Zero Volume Effect
by
J.
C. Keister^{x,1}, P.S. Heidmann^{2} and P.J. Missel^{3}
1  3M Pharmaceuticals, 3M Center,
Bldg 2704S02, St. Paul, MN 551441000
(612)7330171; FAX (612)7363974; jckeister@mmm.com
2  Computing Devices
International, BLCS2X, 8800 Queen Ave. S. Bloomington, MN 55431
3  Alcon Labs, Inc., 6201 S.
Freeway, Ft. Worth, TX 761342099
Abstract
Dose volume reduction is one method
for reducing the nonproductive loss of ophthalmic drugs caused by premature
drainage from the precorneal area. A new
mathematical method is presented for calculating the bioavailability
enhancement achieved by the dose volume reduction method. This new model
suggests that the steadystate assumption used in a previous paper
overestimated the bioavailability enhancement, depending upon physical factors
such as distribution and diffusion coefficients of the drug in tissue and
solution. The analysis shows that
transient effects can make a difference up to a complete elimination of the
zero volume effect in those cases where the distribution coefficient is large
and the permeability coefficient is small (i.e., for large, lipophilic
molecules).
Introduction and Background
In
a previous paper[1],
an attempt was made to establish the theoretical dependence of ocular
bioavailability on the volume of the dose from an aqueous ophthalmic solution.
The culdesac volume, tear turnover rate and excess volume removal rate
constant were taken from the literature[2]^{,[3]}. The analysis suggested that for drugs having
a low permeability coefficient, up to a fourfold increase in drug
bioavailability might be possible upon reducing the dose volume from 30
microliters to zero volume at constant dose.
As drug permeability increases, this benefit in bioavailability from
reduced volume diminishes appreciably.
The
original work presumed steady state diffusion loss of drug across the cornea, a
valid assumption provided the corneal lag time is short compared to the time
constants for the drainage of excess instilled precorneal fluid and tear
turnover. However, this treatment was
not appropriate for drugs having high distribution coefficients but low
permeability or diffusion coefficients.
For such drugs, the steady state assumption greatly underestimated the
immediate flux of drug into ocular tissues, and overestimated drug losses due
to the drainage of excess precorneal fluid volume after dosing. Thus, estimates of increases in bioavailability
that would result from reductions of the dose volume were overestimated.
The
purpose of this paper is to provide a more accurate estimate of bioavailability
increases which result from reduced volume dosing by taking into account this
previously ignored transient diffusion phenomenon. Even with the current
refinement, the analysis is still limited to modeling only the influences
of the hydrodynamics of aqueous dosage
forms  simply one of several factors which can be manipulated by the formulator
for ophthalmic dosage forms.
Theoretical Section
As was
noted in the previous paper, the ocular bioavailability (OB) for transcorneal
delivery is defined as follows:
_{} (1)
where A_{c} is the area of the cornea, J(t) is the flux of drug
across the cornea, V_{AH} is
the aqueous humor volume, k_{e}
is the aqueous humor elimination rate, and C_{AH}(t) is the aqueous humor drug
concentration^{3}. Using a
theorem for the total masstransport[4],
eq 1 can be rewritten as follows:
_{} (2)
where U_{T}_{ }= _{}, C_{T }(t) is the drug concentration in the tear
film, and k_{per} is the
corneal permeability coefficient:
_{} (3)
where D_{c} is the distribution coefficient and D is the diffusion coefficient for drug
in tissue, and l is the thickness of
the cornea. We assume a uniform drug concentration throughout the entire
precorneal fluid volume and a lack of drug metabolism. Eq 2 predicts bioavailability inside the eye
to be based on the time course of the tear film concentration outside the
eye.
Thus
it is necessary to calculate C_{T}_{
}(t) and evaluate its integral
over time, calculating an exact transient solution rather than the quasisteady
state solution, as was done previously. The timedependent equation for C_{T}_{ }(t) must take into account the mass
diffusion transport across corneal and conjunctival membranes, the loss of
excess fluid due to dosing volume, and the normal tear turnover rate. As in the previous paper, the tear film volume
(i.e., the combined volumes of the initial together with the excess dose
volume) is given by^{2}:
_{} (4)
where V_{d} is the drop volume in microliters, k' is the rate constant for removing the
excess drop volume of fluid, and V_{0} is the natural volume of the
tears. k' (per Ref 2) is given by k' = 0.25 + .0113V_{d} in units of per minute.
The mass balance equation
governing the drug content of the tear film after dosing is as follows:
_{} (5)
where A is the total tissue area available for exposure to tears in the
precorneal area and
where j(t), (the drug flux entering
the corneal/conjunctival membranes when generated by a unit step in the drug
concentration in the tear film), is given by:
_{} (6)
That is, we account for the mass
balance of drug in the tear film as it is depleted by the three factors of 1)
drainage of excess tear fluid volume (from eq 4), as
represented by the first term on the right hand side of (5)), 2) the natural
turnover of the tear reservoir due to production of tears (the term v' C_{T
}(t), where n‘
is the tear turnover rate in units of volume/time, as represented by the second
term on the right hand side of (5)), and 3) the loss of drug by transient
diffusion through ocular tissue membranes (the last two terms in eq 5). It is, of course, assumed that the ocular
tissues are uniform as far as drug diffusion is concerned. The term _{} cancels out on both
sides. The secondtolast term on the right hand side of equation (5) is equivalent to the
solution to the problem of a drugfree membrane having diffusion coefficient D, thickness l and area A placed
against a donor solution of constant drug concentration
C_{0} at time zero[5]. The final term on
the right hadn side of equation (5) arises from applying Duhamel’s theorem[6] to the previously
mentioned secondtolast term to account for the changes in the drug
concentration of the tear film with time. The value of the area A is taken to be one third of the total corneal and conjunctival
area as deduced from the fit to eq (10) of reference 1.
Thus,
the differential equation we wish to solve is:
_{} (7)
The solution to this differential
equation would need to be integrated over time to obtain the U_{T} term noted in eq 2
above. However, if use is made of a
standard theorem for Laplace transforms[7],
there is no need to solve the differential equation. The Laplace transform
of C_{T}(t), evaluated at the
zero value of the transform variable, yields the integral of interest:
Lim_{ s }_{®}_{ 0} C (s) = _{}_{} (8)
Thus, the basic procedure is to
take the Laplace transform of eq 7, and then take the limit of the Laplace
transform of C_{T}(t) as s approaches zero. The Laplace transform of eq
7 is as shown below:
_{}_{} (9)
where s is the transformed
variable, C (s) is the Laplace transform of C(t), and k is equal
to Ak_{per}. The complicated coth terms on the r.h.s. of
eq 9 arise from the Laplace transformed solution to the timedependent membrane
diffusion problem, with a unit step concentration applied to the donor surface
(see Appendix 1).
The
value of C (s) is therefore given by:
_{} (10)
The above formula can be iterated
to obtain an infinite series for C (s). Per the definition of U_{T} in
equation (3) together with equation (8), a series expression for U_{T}
can be obtained by setting s = 0 in this infinite series for C (s)
:
_{}
(11)
Because C (s) is a decreasing function of s when s is real, the series will be convergent when V_{d}/V_{0}
Ł 1. When V_{d}/V_{0} > 1, the
series is divergent; that is, each successive term alternates between a lower
and higher value and where the magnitude of successive
terms eventually becomes unbounded.
Nevertheless, the actual value of U_{T} can be estimated in
these cases, since it is alternately bracketed by succeeding partial
sums. This latter condition is of prime
interest because with a typical V_{0} value of 7.5  8 microliters and
typical dose volumes greater than 15 microliters, the ratio V_{d}/V_{0}
is almost always significantly greater than 1.
The variable U_{T} (V_{d}/V_{0})
can be normalized by dividing it by U_{T} (0). In this way, a comparison can be made between
the bioavailability of a given dose volume to the bioavailability of a zero
volume dose. This ratio is readily
obtained from the previous series as follows:
_{}
(12)
A somewhat more useful
expression is simply the inverse of the ratio on the left hand side of equation
(12) which will be defined as R. R is
called the bioavailability improvement ratio of a zero volume dose to the dose
volume in question, V_{d}. It
simply means the total dose required for a given dose volume, V_{d}, to
supply a specified amount of drug to the target tissue divided by the total
dose required for a zero volume dose to supply the same amount of drug to the
target tissue. To determine the
estimates of the value of R, the U_{T}(V_{d}/V_{0}) to
U_{T}(0) ratio of equation (12) was equated to the series on the right
and the resulting value of this ratio was inverted to obtain R.
When V_{d}/V_{0} is > 1 and in those
cases where the distribution coefficient is high and the diffusion coefficient
is low, the first few terms alternate closely around the actual value for the ratio of equation (12). When the reverse is true, (i.e., when the
distribution coefficient is low and the diffusion coefficient is high), the series
is more widely divergent, and a different mathematical technique is required to
evaluate this series. In Appendix 2, eq
(12) is transformed into a more suitable form for the method described there. It is important to note that although the new
series brackets the ratio of equation (12) more closely when recast, it still
is divergent whenever V_{d}/V_{0} > 1. Nevertheless, the transformed series yields
values for R which are accurate to at least 3 significant figures, (even when V_{d}
= 30 microliters). The reason this is so
is that the successive partial sums differ from one another in the first few
half dozen (or so) terms only in the third significant figure, at most.
Permeability
coefficient and distribution coefficient data are available for a number of
drugs[8].
The permeability coefficient data are steady state across excised rabbit
cornea, and the distribution coefficient is from octanol/buffer solution at pH
of 7.65. Calculations were made for the
bioavailability improvement ratio for 30 microliters as compared to zero microliters
for each of these drugs, using both the steady state expressions of the
previous paper as well as the transient calculations of this paper. A V_{0} of 7.5 microliters and a tear flow of 0.66
microliters/min. were used for these calculations (2). The results are shown in table I.
The
largest discrepancies between the transient and quasisteady state results
occur for drugs which have high distribution coefficients and average or low
diffusion coefficients. Under these
circumstances, the drug flux is considerably higher than it would be under
quasisteady state assumptions, especially for large dose volumes. Consequently, the steady state calculations
will yield lower values for U_{T}
than the transient calculations, especially for large volumes. This means that the actual dose required for
a given specified amount of drug to the target tissue will be more for the
quasisteady state calculations. Thus, improvement ratios calculated using
quasisteady state assumptions would be artificially high. On the other hand,
for those cases where the distribution coefficient is low and the diffusion
coefficient relatively high, the lag time will be comparatively short, thus
producing transient calculational results which are considerably closer to the
steady state calculations. Under these
circumstances, the improvement ratio remains virtually unchanged from the
previous steadystate prediction.
Figure
1 depicts a family of curves which show the improvement ratio as a function of
the distribution coefficient. Each of
the six curves represents different (but constant) permeability coefficients. When the distribution coefficient is small
and when the permeability coefficient is held constant, this corresponds to
large diffusion coefficients and short lag times, a situation in which the
membrane diffusion very quickly comes to “quasisteady state” equilibrium. Alternatively, when the distribution
coefficient is very large this corresponds to small diffusion coefficients and
very long lag times. Under these
circumstances, the diffusion effects are equivalent to those sustained by a
semiinfinite medium. Finally, there is a third transition region between the
first two regions described, which corresponds to conditions where transient
phenomena are important, but where the membrane thickness is still of some
importance as well. Each of these three
regions is noted on the figure. The
steady state region corresponds to a ratio of permeability coefficient to
distribution coefficient < 7.8 x 10^{6} cm/sec; the transient
region corresponds to the same ratio being > 780 x 10^{6} cm/sec,
and the transition region lies between these two values. These two “break points” are somewhat
arbitrary, and were determined by an examination of Figure 2, as discussed
below. An examination of Table 1 shows
the various drugs listed in order of increasing permeability coefficient to
distribution coefficient ratio, ranging from 0.355 to 52 x 10^{6}
cm/sec. The transient  transition
“break point” falls between metoprolol and nadolol such that of the drugs
listed, 9 are in the transient region, 6 are in the transition region. None of these drugs are actually in the
steady state region, although atenolol and tobramycin are very close.
Figure
2 is depicts a family of curves of distribution coefficient versus diffusion
coefficient, each of which represents a different value for the bioavailability
improvement ratio. These curves asymptotically approach the quasisteady state
results, which are shown as vertical lines on the graph. The straight line asymptotes shown on the
upper left part portion of the Figure correspond to conditions where the tear
film drug concentration decay due to partitioning is so rapid that the
diffusion membranes themselves act like semiinfinite media of absorption. Each of the curves, therefore, approaches
either the quasisteady state result (when the distribution coefficient is
small) or the semiinfinite slab (when the distribution coefficient is very
large). The drugs noted in Table 1 are
represented as single points on the graph.
The 30 microliter to zero volume improvement ratio is shown in
parentheses underneath the points for each drug. A curve fitting formula which yields bioavailability
improvement results accurate to within 5% of the exact calculational results of
the method described in the Appendix 2 was developed empirically. It is as shown below:
_{} (13)
_{}
It should be noted once again that
Dc is the distribution coefficient (obtained as noted in Reference 8), Vd is
the dose volume in microliters and k_{per} is the permeability
coefficient in cm/hr. This formula can
be used to determine the improvement ratio, R, of any one given dose volume to
another by simply taking the ratio of the ratios. For example, if the 30 microliter ratio (to
zero volume) is 3.5 and the 5 microliter ratio (to zero volume) is 1.4, then
the improvement ratio of 30 microliters to 5 microliters would be 3.5/1.4 =
2.5.
Application section
A study conducted with timolol[9]
shows that with a 25 microliter drop, the presence of a thickening agent (which
(in effect) might produce close to a zero dose volume condition by essentially
eliminating excess fluid drainage) resulted in an improvement ratio ranging
between 1.4 and 2.8, depending on the specific tissue being examined. The calculated improvement ratio for timolol,
(based on a 25 microliter to zero volume reduction), is 1.84 which is well
within the range of the reported experimental error.
It should be emphasized that this
calculation does not take into account the generation of any additional liquid
volume due to side effects of the drug (e.g., stinging for example). It is possible, therefore, that changing a
formulation from a drop to an insert or a viscous gel might increase the
improvement ratio more than these calculations would predict, especially if the
presence of the insert or gel were to reduce or eliminate the stinging effects
of the drug as compared to when the drug is in a solution type
formulation.
Conclusions
A more exact solution to transient
effects of drug transport through the eye has been developed which shows that
the quasisteady state approach in effect yields an upper bound to the
improvement in ocular bioavailability from reducing precorneal loss factors
while maintaining constant dose of drug.
Both distribution coefficient and permeability coefficient in general
play key roles in determining the volume improvement ratio. Bioavailability
improvements calculated using quasisteady state assumptions are artificially
high when the drug distribution coefficient is high and the diffusion
coefficient is low to average, but improve in accuracy as the diffusion
coefficient increases or the distribution coefficient decreases. As is apparent from an examination of Figure
1, the transient analysis can make a difference up to a complete elimination of
the zero volume effect in the case where the distribution coefficient is large
and the permeability coefficient is small (i.e., for large lipophilic
molecules). For the molecules shown in
Table 1, the difference is not as much.
Acknowledgements
The authors gratefully acknowledge
discussions and helpful suggestions obtained from Drs. E. Cooper, D. Hager, J.
Lang, and G. Kasting.
Appendix 1
Laplace
transform to the solution of the Unit Step Concentration applied to the donor
side of a membrane
One starts with the standard
diffusion equation, as follows:
_{} (A11)
In addition, we apply the spatial
boundary conditions of C = 0 at x = l,
C = 1 at x = 0, and the initial condition and C = 0 for 0 < x Ł l at t = 0. Taking the Laplace
transform of the above differential equation yields the following:
_{}_{} (A12)
The general solution to the above
equation can be expressed in terms of sinh and cosh functions as follows:
_{} (A13)
The Laplace transform of the
solution at x = 0 (i.e., 1) is simply 1/s, while the Laplace transform of the
solution at x = l is 0. A and B are
thereby determined and the final solution for C(s) is:
_{} (A14)
The quantity of interest is the
flux of drug into the membrane, f(s), defined below:
_{}
_{} (A15)
By making use of the distribution
coefficient (D_{c}) and the definition of the permeability coefficient
from eq (3), and also noting that the total diffusion loss of drug into the
membrane must be the product of the area times the flux, we see that this loss
term must be given by:
_{}
(A16)
which is equivalent to the second
loss term in equation 9 of the text. The
final loss term is obtained by making use of the convolution property of the
Laplace Transform.
Appendix 2
If we let
the function Z(x) be defined as follows:
_{} (A21)
and the
related subfunctions Z_{n}(x):
_{} (A22)
Using
this approach, eq 12 of the text is transformed into eq A3 below:
_{} (A23)
Series such as eq A23 often do not
produce the desired limit with sufficient precision. Techniques must be applied to extend the
useful region of these expansions in cases where the
series is known to be divergent. The _{} transform of Brezinski[10]
was found to extract the antilimit* from the
series with sufficient accuracy.
The _{} transform is defined[11]
as follows for any nonconstant sequence^{a} _{}
_{}
_{}
_{}
Where the subscript represents the
order of the transform, the superscript the term of the transformed sequence,
and the difference operator _{}.
This recursive transform^{b}
was originally derived from the _{} transform^{c}
(p. 727 of reference 10). Both of these
transforms have been shown to have accelerative properties on large classes of
convergent sequences[12]^{,[13],[14],[15]}
and antilimit extraction capabilities on
certain classes of divergent sequences[16]^{,[17]}.
The results presented in this paper
were obtained by applying the _{} transform to the
sequence of partial sums of the divergent series for
U_{T}(V_{d}/V_{0})/ U_{T}(0). When performing these calculations, the
following practical considerations were taken into account:
80 bit floating point objects were used.^{d}
For many values of _{}, _{} was found to be
sufficient. For larger values of _{}, however, _{} was required.
The series for U_{T}(V_{d}/V_{0})/ U_{T}(0). is a divergent series whenever V_{d}/V_{0}
> 1. The transform of this series is also a divergent series. _{} will have a larger
region of usefulness than _{} which in turn will
have a larger region of usefulness than the direct series for U_{T}(V_{d}/V_{0})/
U_{T}(0)**.
____________________
^{a}In this paper, the sequences are successive partial sums of the asymptotic series (that is, _{}).^{}
^{b}This transform is more accurately described as a hierarchy of transforms. The even ordered transforms provide the desired properties, the odd ordered transforms diverge. The higher order transforms will typically accentuate the desired characteristics.
^{c}For information on the e transform, consult [13], [14], and [15]
^{d}The q transform, because it involves successive divisions by progressively smaller differences, typically
requires this much precision
*When the
successive partial sums of a divergent series are known to oscillate above and
below the value which the partial series approximates, when these sums contain
convergent as well as divergent components and these convergent components
dominate in the first few terms, the limit of these convergent components is
called the “antilimit” of the divergent series.
By analogy, the antilimit of a divergent, oscillating series is the
counterpart to the limit of a convergent, oscillating series in cases where the
first few partial sums of the divergent series display successively smaller
differences from one another.
**By a
larger region of usefulness, we mean that Q_{4} produces a more
accurate and (more useful) estimate of the antilimit than eqn. (12) itself
produces in the region where the series diverges. In this regard, the eqn. (12) estimates of
the antilimits for several of the drugs noted in this paper were no better than
10  50% accurate for large V_{d}, in contrast to the 3 significant
figure accuracy of the Q_{4} series for these
drugs.
References
1. Keister, J. C., Cooper, E. R., Missel, P. J., Lang, J. C., and Hager, D. F. J. Pharm Sci. 1991, 80, 5053
2. Chrai,
S.S., Patton, T.F., Mehta, A., Robinson, J.R.
J. Pharm Sci. 1973, 62, 11121121.
3. Patton,
T.F. in Ophthalmic Drug Delivery Systems.; Robinson, J.R., Ed.; A.Ph.A.:
Washington, 1980; Chapter 2
4. Keister,
J. C., J. Membr. Sci. 1986, 29, 333344
5. Crank,
J., The
Mathematics of Diffusion, Clarendon Press 1975, pg. 50, eq 4.22, where C_{2} is zero and C_{1}
is C_{0} of eq 4.
6. Carslaw,
H.S. and Jaeger, J.C., Conduction of Heat in Solids, 2nd
edition, Clarendon Press 1959,
30&31
7. Nilsson,
J.W., Electric Circuits, Addison Wesley 1986, 586589
8.
Schoenwald, R. D., The Control of
Drug Bioavailability from Ophthalmic Dosage Forms, Ch. 6 from Controlled
Drug Bioavailability, Vol 3, Ed. Smolen & L. Ball, Wiley, 1985,
264266
9. Thermes,
Florence, et. al, Bioadhesion: The Effect
of Polyacrylic Acid on the Ocular Bioavailability of Timolol, Int. J.
Pharm., 81, 1992, 5965
10.
Brezinski, C., Acceleration de suites a convergence
logarithmique, C. R. Acad. Sc. Paris, 18 October 1971, 727730
11. Smith, D.
A., Ford, W. F., Acceleration of Linear and Logarithmic Convergence, J. of Numerical
Analysis, Fol 10, No. 2, April 1979,
223240; see especially pg. 225
12. Ibid.,
Pgs. 225226; pg 229; pgs 232238
13. Smith, D.
A., Ford, W. F., Numerical Comparisons of Nonlinear Convergence Accelerators,
Mathematics of Computation, , Vol 38, No. 158,
April 1982, pgs. 481499; see
especially pgs. 482486
14. Wynn.
P., On
the Convergence and Stability of the Epsilon Algorithm, Siam J. of Numerical Analysis, Vol 3, No. 1, 1966, 91122
15. Wynn.
P., On
a Device for Computing the e_{m}(S_{n}) Transformation, Mathematical Aids to Computation, Vol 10, 1956, 9196
16. Shanks,
D., Nonlinear
Transformations of Divergent and Slowly Convergent Sequences, J. of Mathematics and Physics, Vol 34, 1955, pgs. 142; see especially pgs.
1621
17. Reference
13, pgs. 486488
Table
1. 30 microliter to zero volume improvement ratio
Drug 
k_{per} x10^{6},cm/sec 
D_{C} 
(k_{per}/Dc
= D/l)
x 10^{6} cm/sec 
Bioavailability Enhancement Ratio 





Steady State 
Transient 
Transient/St State 
Bufuralol 
72.4 
204 
0.355 
1.51 
1.07 
0.709 
Bevantolol 
67.6 
155 
0.436 
1.54 
1.08 
0.701 
Acebutolol 
0.85 
1.58 
0.538 
3.86 
3.05 
0.790 
Propranolol 
47.6 
41.7 
1.141 
1.68 
1.17 
0.696 
Levobunolol 
17.4 
5.24 
3.321 
2.26 
1.62 
0.717 
Cyclophosph. 
11.3 
2.4 
4.708 
2.56 
1.96 
0.766 
Timolol 
11.7 
2.2 
5.318 
2.53 
1.97 
0.779 
Oxyprenolol 
27.5 
4.9 
5.612 
1.97 
1.53 
0.777 
Nadolol 
1.0 
0.151 
6.623 
3.83 
3.65 
0.953 
Phenylephrine 
0.94 
0.100 
9.400 
3.85 
3.73 
0.969 
Pilocarpine 
17.4 
1.7 
10.235 
2.26 
1.91 
0.845 
Sotalol 
1.6 
0.056 
10.256 
3.69 
3.66 
0.992 
Metoprolol 
24 
1.9 
12.632 
2.06 
1.77 
0.859 
Atenolol 
0.68 
0.0302 
22.517 
3.91 
3.91 
1.000 
Tobramycin 
0.52 
0.0100 
52.000 
3.98 
3.98 
1.000 
Figure 1. 
Family of curves for the therapeutic improvement ratio versus distribution
coefficient. Representative of the
improvement in bioavailability which occurs upon reducing the dose volume from
30 mliters to
zero at a constant dose.
Figure 2.  Family of curves of
distribution coefficient versus permeability coefficient, each
representing a different therapeutic improvement ratio. The therapeutic ratio represents the
improvement in bioavailability which occurs upon reducing the dose volume from
30 mliters to
zero at a constant dose.