MACROMOLECULAR
FROM
DRUG RELEASE
POROUS
IMPLANTS:
SOURCES OF MATRIX TORTUOSITY
by
Ronald
B.S.
M.S.
Alan
University of
Submitted
in
Q Ronald A.
grants to
copies of
by:
1984
M.I.T. permission to
this thesis document
of
reproduce
in whole or
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44
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'CK
by:
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Ala
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Author:
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1984
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of the
DOCTOR
The author hereby
and to distribute
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(1975)
ENGINEERING AND
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-i-
MACROMOLECULAR DRUG RELEASE
SOURCES
FROM POROUS.IMPLANTS:
OF MATRIX TORTUOSITY
by
RONALD ALAN
SIEGEL
Submitted to the Department of Electrical Engineering
and Computer Science on February 20, 1984
in partial fulfillment of the requirements for the
degree of Doctor of Science
ABSTRACT
Release of the highly water-soluble proteins Bovine
Serum Albumin (BSA),
-lactoglobulin and lysozyme chloride
from ethylene-vinyl acetate copolymer (EVAc) matrices was
studied at three drug powder particle sizes and ten drug
loadings.
It was found that at small particles and low
loadings, much of the BSA is virtually trapped inside the
matrix.
Even at higher loadings and particle sizes, where
all drug was released, the release rate was orders of
magnitude slower than would be predicted for diffusion
through waterfilled channels, which have previously been
shown to be the conduits for drug release.
For the two smaller particle
sizes it
was found that
as
loading increases, a sharp transition occurs in the total
fraction of releasable drug.
This transition is similar to
other transitions that are described by percolation theory.
A percolation-type
model was applied to the data, with
qualitative agreement but quantitative disagreement.
It was
conjectured that the differences are due to inhomogeneities
in the drug particle
distribution
in the matrices.
Three possible mechanisms of retardation of drug release
were studied theoretically.
It was shown that neither the
concentration dependence of the diffusion coefficient of
proteins nor the random pore topology are sufficient to
account for the slowness of release.
However, it was shown
that constricted throats connecting pores, which have
previously been identified by scanning electron microscopy,
can account for the retardation of release.
The constricted
throats are much larger in diameter than the protein
molecules,
so the retardation
is not due to sieving or
hydrodynamic effects.
Rather, the constricted throats make
it more difficult for diffusing molecules to find their way
out of pores.
-ii-
Thesis
Supervisors:
Dr. Robert S. Langer
Associate Professor of Biochemical
Engineering
Dr. Alan J. Grodzinsky
Associate Professor of Electrical
Engineering and Computer Science
-iii-
ACKNOWLEDGEMENTS
I would
supervising
first
like
to
thank Professor Robert Langer
my doctoral work, and
for
financial support he has given me.
would have had
Very
personal and
few biochemists
(i.e.
patience)
to work with a
theoretically inclined electrical
engineer.
I appreciate
training and
the forsight
the
for
the opportunities
that have opened
the
up as a
result of working with Bob.
I
thank my other co-supervisor,
Grodzinsky,
for the
part played
the course he gives on
flows
was
to physiology.
extremely helpful
in
Professor Alan
the
thesis,
the application of
the
committee members, both made
thesis.
committee
enjoyed his
Drs.
always
suggestions
that
my graduate
the
other
strengthened
has
been on
program, and
every
I've
presence.
Larry Brown, Yosi
been around
experimental,
I
faced during
forces and
from that course
Deen,
Incidentally, Professor Weiss
I've
for
present work.
Professors Thomas Weiss and William
this
fields,
The knowledge I gained
in pursuing
and also
to discuss
of drug
thank Rajan
Kost, and
Elazer
any aspect,
Edelman have
especially
release.
Bawa
for "taking
me under
his wing"
when
I
-iv-
started
on
project.
this
opened my mind
ignored.
(By
acknowldege
Late
Rajan
in my
Larry Brown
thesis.
thank Brian Marasca
I
that I otherwise might have
to many aspects
the way, Dr.
says
Please do
"I
for assisting me
in
polymer matrices, and Arthur Grant, Howard
Dr.
Margaret Wheatley for
Redner,
hundreds of
fabricating
Bernstein,
and
boring readings
Nikalaos Peppas,
Alan Waxman,
helpful in theoretical discussions.
Cris
Fitch did
like
for me.
last-minute
some very valuable
to thank Genia Kost
Also,
equation
There are
working here
I thank
Sid
Anna Balazs
programming work.
for
the
figure drafting
Phyllis Vandermolen
for some
stencilling.
countless other
both at MIT
and
Richard Baker,and
very
did
labs,
Marcus Karel,
and Drs.
I would
she
to
I was away.
Professors
were
taking
forgot
it for me.")
the
while
with Rajan
arguments
night
people
and at Children's
I have met at Langer
Hospital, who have made
a pleasure.
Finally, I thank my wife,
patience
and
end with
the nitty
Carol Fishman,
encouragement, and also
gritty
of
pasting
for her
for helping me at
the
thesis
the
together.
I also
thank my
supportive
in my
parents, who have
endeavors.
This work was
by
also been completely
supported
in part by NIH grant GM26698 and
a grant from the Paint Research Institute.
-vi-
TABLE OF
CONTENTS
Abstract......................
.
Acknowledgements.............. ..
Table
of
.
.
..
..
.
.
..
.
.
.
.
..
.
..
.
.
..
.
.
i
i
Contents.........................................vi
List of
figures..........................................ix
List
tables.........................................xiii
I.
of
Introduction...........................................I
I
I
I
I
I
I
II.
1
2
3
4
5
6
Overview of pharmaceutics .
ue
Overview of drug delivery i ss ue
Implants.
e
..
Reservoir and ma trix
devi c e
S..
EVAc matrices...
Scope of thesis.
References......
Background
on EVAc
sustained
.1
.3
.6
.7
10
17
20
release matrices........23
11.1
11.2
II.
II.
Introduction.....................
.23
Factors influencing release kinet ics....
.23
2.1 Effects of loading and partic
le size
.23
2.2 Effects of release media (pH, ionic s t re n g h
.
and temperature)................
.25
II. 2.3 Effect of coating............
.26
II.3 Microstructural studies..........
.26
II. 3.1 Optical microscopy...........
.26
II. 3.2 Scanning electron microscopy. .... *
.27
11.4 Conceptual model of the release proce S..
.27
References...........................
.33
III.
Review
III.
III.
III.
III.
IV.
1
2
3
4
of
Introduction.......
Maxwell-type models
Percolation models.
Models of influence
References.........
Release
IV.1
IV. 2
IV.
IV.
IV. 3
IV.
IV.
transport in heterogeneous media.. ...
kinetics
of
pore
structure
Introduction..........
2.2 Methods...........
Results...............
3.1 Introduction.......
Physical
34
.
34
35
37
46
52
from EVAc matrices.................55
Materials and methods.
2.1 Materials.........
3.2
........
.....
........
........
...
parameter
5
of
5
labs
.55
.55
.55
.57
.60
.60
.63
...
-vii-
IV.3.3 Porosity............................
IV.3.4 Kinetics............................
IV.3.4.1 Axes of graphs..................
IV.3.4.2 Bovine Serum Albumin............
IV.3.4.3 a-lactoglobulin and lysozyme chl 0 r i de
IV.4 Discussion..............................
IV.4.1 Tortuosities
and retardation
factors
IV.4.2 "Fast" and "slow" phases of release.
References..............................
V.
Modelling the
.
.
.
.
.
.
.
.
.
incompleteness of release.......... .
65
67
67
68
76
80
80
84
89
.90
V.1 Introduction....... ................................
90
V.2 Computer representa tion of the matrix..............92
V.3 Calculation of total amount o f drug released........95
V. 3.1 Descrip tion of algorithm.
.....0 ...
...........
95
V. 3.2 Results and discussion...
.98
V.3.3 Comparison to percolation theory...
.101
V.3.4 Effect of release from matrix rim
.106
V.4 Simulation of desorption (rel ease) from
lattices.....................
.110
V.4.1 Introduction.............
.110
V.4.2 Description of the algori thm .......
.110
V.4.3 Deterministic algorithms for compar
results..................
.114
V.4.4 Results and discussion...
.116
V.5 Comparison with data.........
.117
V.5.1 Total fraction
of drug re lease....
.117
V.5.2 Rate of release....... ...
. 131
References................ ...
.133
VI.
Modelling
the
slowness of release..................134
VI.1 Overview of chapter......................... 0 n .. 134
VI.2 Effect of concentration dependence on diffus i0 n
coefficients................................
..135
VI.2.1 Introduction............................
.135
VI.2.2 Tracer and mutual diffusion coefficients
.. 137
VI.2.3 Experimental determinations of diffusion
coefficients for proteins............... .....
140
VI.2.4 Effect of concentration dependence of
diffusion
coefficient on desorption from
slabs................. ..
..
..
..
..
..
..
. 145
VI. 2.4.1 Introduction.....
. . . . . . . . . 145
VI.2.4.2 Effect of concent ration dependence on
VI.2.4.3
steady state diff usion..................145
Calculation
of tr ansient concentration
dependent diffusi on:
theory............148
VI.2.4.4 Application
VI.2.4.5
VI
VI.3
VI
2.5
...
to
sp ecific
models........... ....
Discussion.......
Summary...............
of po re
Models of the effect
3.1 Introduction..........
diffusivity
.............
1.....
152
. . . . . . . . . 156
. . . . . . . . . 158
geometry.............160
. . . . . . . . . 160
...
...
...
-viii-
Development of first passage time equations..162
VI.3.
2.1 Introduction.............................162
VI.
VI.
2.2 First passage time distribution..........165
VI.
2.3 Mean first passage times.................170
VI.3.
First passage times in model geometries.......171
VI.
3.1 Introduction.............................171
VI.
3.2 Straight pore (figure 10a)................173
VI.
3.3 Cylindrically conical pore (figure 10b)..174
VI.
3.4 Spherically conical pore (figure 10c)....175
VI. 3.3 .5 Discussion............ ................... 176
VI.3. 4 F irst passage times in square pores with
c onstricted inlets and outlets............... 176
V 1.3.4 .1 Introduction.... ......................... 178
V 1.3.4 .2 The geometric setting.................... 179
V 1.3.4 .3 First passage out of a square............ 184
V 1.3.4 .4 Description of the Monte Carlo Brownian
motion simulation........................ 193
VI. 3.4 .5 Results of the Brownian motion procedure. 198
VI.3.4.6 Discussion....
207
VI.4 Combination of factors affecting release.
214
References............
216
VII.
Suggestions
for
further
work.......... ........... 219
Appendix
I.
Zero Order Release from Monolithic
Devices..................................221
Appendix
II.
Programs for Chapter V.................. .235
Appendix
III.
A Second Set of Data.................... .237
Appendix
IV.
Numerical Procedure and Computer Program
for Concentration Dependent Diffusion....252
Appendix V.
Biographical
The RSPASS
program.......................258
Note.......................................267
-ix-
LIST OF
FIGURES
Figure
I.1.
Time course of drug plasma
Figure
1.2.
Reservoir device and monolithic (matrix)
device.......................................8
Figure
1.3.
Typical in vitro release kinetics of BSA
from EVAc matrices..........................11
Figure
1.4.
Fabrication procedure for' EVAc sustained
release matrices............................13
Figure
1.5.
10pm section of an EVAc
Figure
II.l.
Scanning electron micrograph (SEM) of an
empty pore..................................28
Figure
11.2.
Schematic of EVAc
Figure
11.3.
Two-dimensional representation of
conceptual model of pore structure...........32
levels............2
matrix..............14
sustained matrix...........30
Figure III.l.
Stages of formation
network in a porous
Figure 111.2.
A typical percolation lattice of lattice
sites containing conductors or insulators...40
Figure 111.3.
Plots of percolation probability and
conductivity of a random simple cubic
lattice.....................................43
Figure
111.4.
Conductivity of NAFION
perfluorinated
ionomer membranes...........................45
Figure
111.5.
Electrical
analog
of a percolating
pore
rock....................38
of constrictions...........48
Figure 111.6.
Model of steady state diffusion through
pores with varying widths....................50
Figure IV.l.
Assembly used
Figure IV.2.
Release kinetics of
PS=90-106pim.......
Figure
IV.3.
Release kinetics of Bovine Serum Albumin,
PS=150-180pm.......................................72
Figure
IV.4.
Release kinetics of Bovine Serum Albumin,
PS=300-425pm
................................ 74
Figure
IV.5.
Release kine tics of Bovine Serum Albumin
from the THIN slab.............................77
in
release
experiments.........61
Bovine Serum Albumin,
.......................... 69
s-lactoglobulin
....
78
lysozyme chloride..
79
Figure
IV.6.
Release kinetics
Figure
IV. 7.
Release kinetics of
Figure
IV. 8.
Total fraction
F6 of originally
incorporated dr ug r eleased during
"fast phase"
....................
Figure V.1.
Figure V.2.
Figure V.3.
Figure V.4.
Figure V.5.
of
Illustration of lat tice
polymer/drug ma trix
0 . ...
88
representation 0 f
.94
Result of iterative process determining
releasable and trap ped drug............ .....
97
Results of computer simulation of total
fraction releasable drug............... .....
99
Distribution of tra pped and releasable
drug for three volu me loadings..........
.. 102
Results of computer simulation of
percolation probabi lity................
104
Figure V.6.
Comparison of total fraction released a nd
percolation p robabi lity................ .. 0 .105
Figure V.7.
Comparison of
Figure V.8.
Cumulative fr action
RRWALK.....
Figure V.9.
Figure V.10
Figure V.11.
Figure
Figure
Figure
VI.1
VI.2
VI.3
total
Simulated k in e tics
desorption.
fraction released.. b
.109
released simulated b y
....................
.118
of random walk
. 119
Comparison of resul ts of the
simulation with exp erimental
chapter IV.
USEFUL2
results of
.. 124
Comparison of data and predictions
of t he
total fract ion rele ased for two models. ..
.129
Tracer and mutual d iffusion coefficient
of BSA, for pH=4.7.
s
.141
Mutual diffusion
co efficient of BSA,
pH=7.4.
............
.144
fo r
Schematic of test
c ell for measuring
steady state diffusion........ .
147
*
Figure
VI.4
Values of D(C)/D 0 , R
SS
Keller
and
Anderson
(C)
and RT (C
for
.. 153
models...
*
Figure
VI.5
Values of D(C)/DO, RS(C)
and RT(C
for
-xi-
Butterworth models....................
...
155
Figure
VI.6
Concentration profile for slab with hi gh
value of C ........................... .. 0 ... 159
Figure
VI.7
Schematic of Brownian motion in a
constricted pore...................... ..... 161
Figure VI.8
Schematic of
passage time
Figure VI
9
Local
Figure
10
Geometries of model pores for which me an
..... 172
first passage times can be solved..... o.
VI
Figure VI. 11
situation for which first
distributions are solved. ..... 163
reflecting pore wa 11...168
geometry near
Comparison of first passage times for
pores with geometries of figure 10.... .......
177
..
180
two adjacent pores......... .....
Figure
VI. 12
Example of
Figure
VI. 13
Parameters for simulation of Brownian
motion throu gh constricted pores........... 182
Figure
VI. 14
Pore outlet
Figure
VI. 15
Imbedding ap proach for determining hitting
times....... ............................................ 185
Figure
VI. 16
c.d.f.
configurations................. 183
and p .d.f. of hitting
Figure VI. 17
Probability
Figure VI. 18
Illustration
time..........
188
density for hitting spot....... 191
of Brownian motion procedure.. 195
results... 199
Figure
VI. 19
Comparison o f RSPASS
Figure
VI. 20
Diffusing pa rticle
Figure
VI. 21
Retardation
Figure
VI. 22
Constriction
Figure
AI. 1
Geometric parameters and concentration
profile for slab in which release is
governed by diffusion.................. .... 222
Figure
AI.2
Concentration profile for slab in which
release is dissolution controlled...... ....
224
Concentration
diffusion and
profile of slab with both
dissolution.............. ...
226
Higuchi model
for slab and
Fig ure
Figure
AI. 3
AI.4
factors
and RSPASSD
"cutting corners"....... 206
for values
factor for cubic
of w and
z.. 208
pore..... ....
coated
213
-xii-
hemisphere.................................229
Figure
AI.5
Figure AIII.1
Coated hemisphere with all drug in
solution...................................231
Total fraction released for preliminary
set of BSA disks...........................251
-xiii-
LIST OF TABLES
Table
I.1
Macromolecular
Table
IV.l.
Physical
Table
IV.2.
Average porosities, tortuosities, and
retardation factors..........................83
Table
IV.3.
Average fraction of incorporated drug that
is connected by pores to the surface..........87
Table
VI.1
Frequency distribution of stepsizes for
simulation of Brownian motion through
constricted pores............................201
Table
VI.2
First passage times, standard deviations,
and retardation factors.....................202
drugs and
their
half-lives....7a
parameters for EVAc disks............62
-1-
I.
of Pharmaceutics
I.1 Overview
has
Pharmaceutics
The
is
first goal
to
disorders
the
body processes
subfields of
to
absorbed
into
the various
concentrate
other
toxic and
Furthermore,
the
drug at
tissues.
"targeting.")
to
the
system and
rate
range"
by which drug
the
(This
For example,
is
to
site,
localization
is
Many
the body.
1) above
the drug
is often desirable
target
in
of
(figure
below which
it
delivery
the administration of
compartments
a "therapeutic
becomes
ineffective.
from
the
second concern is
drugs possess
the drug
the
drug
of drug
field
the
the concerns of
of
A
delivery of
Often
site.
successful
circumvent physiological barriers
drugs.
the
to
barrier.
blood-brain
One
proper
the
gastrointestinal
such as the
administration,
is
third goal
the
provides barriers
human body
one
pharmacodynamics, metabolism,
The
to
is understanding
leads
This goal
pharmacokinetics,
bioactive agents
or
associated with
second goal
drugs.
and excretory physiology.
drugs
concerned with
interactions
The
treated.
be
Those
design of new drugs.
the rational
biochemical
biophysical and
goals.
broad and complementary
three
in drug design are primarily
involved
how
INTRODUCTION
and
to
which
is
to
keep
of drug is
cancer chemotherapy
it
away
called
there
-2-
Drug
Plasma
Toxic level
.
Minimum effective level
Level
a
Therapeutic range
Thrpuirag
b
Time
Figure 1.1.
Time courses of drug plasma levels in response
to conventional drug delivery (a and b),
and sustained drug delivery (c).
-3-
exist many drugs which could
but are not useful because
cells.
nonmalignant
will be
It is
effectively kill
they are also
lethal to
that one
hoped
day
that only malignant
targeted such
cance'r cells,
cancer drugs
will be
tissues
destroyed.
The
The
three goals
chemical
described above are
structures of
the drug molecule
determine
the pharmacodynamics of
metabolic
pathways by which the
Furthermore, rates
rate
the
1.2
at
which
should
drug is
be
and
the receptor
drug action and
the
removed.
and removal often
of drug uptake
drug
interconnected.
determine
administered.
Overview of drug delivery issues
The
delivery
present work concerns methods
of various drugs and
at
therapeutic
of
the
issues
levels.
involved
of prolonging
maintaining drug concentrations
section provides
This
in and
the
strategies
a brief review
this
towards
evolved
end.
A common method of sustained
tablet.
reaches
gut).
to
When
the
ingested,
appropriate
the
drug delivery
remains
tablet
intact
site for dissolution
The chemical composition
of
the
is
until
it
(stomach or
tablet can
provide desired rates of dissolution.
the
be altered
-4Tablets
problems
is
labile
provide a simple
in drug delivery.
through
tablet of
To be absorbed
the
passed
is alkaline.
also
release
The
the
bioavailability.
bioavailability
of
considerations
in
bolus
the
of
or complexing
the
tract are
the
is in
the
the gut, which
agent should
of problems of
Drugs
excretory
of importance
This can be
directly into
pharmacokinetic
are
drug may never reach the
and
site
removed from
processes.
is
the
the
A
circulatory
determined by
injecting a
the circulation and measuring
circulating drug activity as a function
Often a drug's half-life
the
drug.
the circulation,
parameter
drug
bioavailability
having
stability of drugs in
examples
become important.
of a drug.
the
Once
presence of alkali.
incorporated
circulation via metabolic and
pharmacokinetic
Thus a
A drug delivery system should maximize
Once a drug
half-life
tablet enters
the
problems associated with
the gastrointestinal
swallowing,
strongly acidic.
from acid attack.
the coating
drug when
that
contain a coating or a complexing
the stomach,
Thus
two
in an alkaline
the drug must, after
stomach, which is
this drug should
through
solution of
Consider for example a drug
that protects the drug
agent
the
in an acid environment but stable
environment.
pass
example of
is very
of
time.
short, and
of action.
pharmacokinetics are
much of the
Thus
intertwined.
A
-5-
the
circulation over a
that is
hand,
a drug
(long
half-life),
does not require
absorption of drug
drug and a suitable
tablets,
Besides
preparations whose
the
of
the drug
insulin are
often
the
free drug
excretion.
Finally,
drug
transport of
physiological
into
some
drug
into
processes
the
[1].
sense
is
the presence
blood and
and
such
Drugs
of agents
prevent
(in
the
immediate
intramuscular
the body,
circulation
most
the
levels, but is
patients.
in an effective depot in
the
slow
example is
protamine) which slow
subdermal
prolongs
skin or mucus membrane on
plasma drug
the
the
for drugs
application, and
Another
injected in
insulin, zinc or
of
the
a substrate
only for bedridden
of
place
into
pharmaceutical
localize and/or prolong
site of
the
for controlling
feasible
generally
release
to
administration, which in
direct method
the case
is
purpose
ointment is placed.
the
intravenous
as
Thus,
the circulation.
several other
there are
them at
localizes
release
which
rate of
the
rate of drug delivery.
Ointments provide
action of drugs.
which
to alter
gut into
the
circulation
sustained administration.
been designed
from
the other
design of a tablet can ensure both bioavailability of
proper
the
slowly removed from the
forms have
Tablet
On
time.
period of
long
drug into
the
such a drug should release
system for
delivery
injections
and
the
is regulated
by
-61.3
Implants
Unfortunately,
tablets,
sustain the release
generally
relatively
short time
many drugs
it
weeks,
are
focussed on
delivery
systems.
Recently,
developed
for
long
control)
in several
In such cases,
useful.
ruled
the
past 25
(for
it is
by an
implant.
enzymatically
can be absorbed.
Furthermore, many
out.
sustained
essential
Almost
all
in
stomach or
the
Thus enteric administration
macromolecules
For
example,
do not provide a good constant basal
that many of
the
long
retinopathy, nephropathy)
levels,
leading
manner.
must be
dosage
insulin
birth
that sustained
in a
felt
(which
[4,5].
administered
is
years much
systems have been
dental applications
be mediated
they
forms
of
implanted
and progesterone
[21
macromolecules are degraded
is
In
polymeric
For macromolecular drugs,
gut before
For
Sustained release polymers have also proved
[3].
drug delivery
hours or days.
term release of pilocarpine
ocular pressure)
reduces
of drugs over
the development of implantable drug
commercial
the
injections
sustain release over periods
to
generally more
work has
i.e.
intervals,
is desirable
and
and/or absorption
months, or even years.
devices
useful
ointments,
to
are
term symptoms of
due
to poor
pathological
insulin
level.
It
diabetes (e.g.
control of basal
blood glucose
levels
-7-
an
Thus
[6].
provides
system which
implantable
improved
insulin delivery is desirable.
that macromolecules are
fact
the
Despite
sustained release implants,
candidates
for
clinically
used implants
for
this will be discussed
thus
dalton) drugs.
The
far
reason
the next section.
some macromolecular
drugs
delivery would be efficated by a sustained release
Many of
implant.
in small
drugs
these
so
quantities,
(and others)
their use
in
exist at
therapy
present
has been
However, with the advent of biotechnological
limited.
techniques
may
in
I provides a list of
Table
whose
(<600
low molecular weight
for
best
the only
developed
that have been
are
the
among
such as genetic engineering, many
soon be
such quantities as
producible in
of
these drugs
them
to make
clinically useful.
and matrix devices
1.4 Reservoir
There are
two classes
normally considered in
The
first class
devices
(see
figure
low
2a),
the
a saturated
permeability
is much
that are
applications
In
reservoir devices.
polymeric membrane.
coefficient of drugs
implants
polymeric
sustained release
consists of
encapsulated by a
solely as a
of
barrier.
lower
in
solution of drug
The membrane
Since
the
[7].
these
is
functions
the diffusion
polymer
than in
-7a-
Macromolecular Drug
Molecular Weight
Half-life
4700
<5 min.
1200
15
sec.
Bradykinin
1060
30
sec.
Calcitonin
3600
ACTH
Angiotensin
I
Enkephalins
Gonadotropic
<40 MIN.
2
600
min.
hr.
30000
.5-3
22600
<25 min.
Hormones
Growth Hormone
Heparin
3-37,000
Insulin
6000
Oxytocin
1007
Parathyroid
9500
<2 hr.
<25
min.
2 min.
<15
min.
Hormone
Vasopressin
Table
I.1
Macromolecular
4 min.
1200
drugs and
their
half
lives
[131.
-8-
,
poiymer
Tim
00
Tine T >0
druq dispsd
b.
-
.0.
-- .-**-- 0
in paymer
Time 0
Figure 1.2.
.
'
.
- -.
-
'ime'T
a)
b)
-in
0
disrsed
pO@yim
>0
Reservoir device.
Monolithic (matrix) device.
-9-
water,
drug
the
for a
reservoir device can
long period of
consists of
devices
inside
the
(see
figure 2b),
implant
that as
is
of
the
achieved
devices
the
the membrane
reservoir
to the
However,
to make,
implants
these
both as
the drug
into
if
reservoir
It is
type
easy to show
is held at
"zero
order"
as one must surround a
the
the
release
First, reservoir
Second,
and more
somehow ruptured,
patient,
the other
they are ruptured.
popular as reservoir
described
devices,
with
the
toxic
hand, will
However,
liquid
important,
device will
effects.
not disgorge
they have not
because it was
believed
they could not release drugs at a constant rate.
true
for slabs
controlling.
devices
a
there are several
polymer.
polymer membrane is
their contents
is
The
type.
reservoir system.
Monolithic devices, on
that
functions
interior drug solution
of dr.ug by the
all
been as
In
of sustained release polymers
[7].
are not easy
solution
release
of
the
is dissolved or dispersed
concentration, a constant, or
disadvantages
the
The second class
possesses one clear advantage.
long as
saturation
if
Thus,
the examples
in section 3 are
rate
drug
release
and as a storage device.
All
of
time.
monolithic, or matrix devices.
the polymer.
barrier
retain and slowly
can
releasing
In appendix
provide
This
drugs for which diffusion is
I other cases
zero order
where monolithic
delivery are
discussed.
rate
-10-
For macromolecules,
monolithic
devices are
macromolecules
daltons.
there is
preferred,
larger
distance between polymer strands
is
if not essential.
have a molecular weight greater
Their radii are
membrane
another reason why
impermeable
to the
than the
in
Most
than 1000
characteristic
the membrane.
macromolecule,
Thus
so a
the
reservoir
device cannot work.
1.5
EVAc matrices
Langer and Folkman
devices
comprised of ethylene-vinyl
when loaded
release
Since
[8] demonstrated that monolithic
with various macromolecules
their contents over
then,
monolithic
others have reported
implants
of
the
for bioassays
macromolecule
in powdered
periods greater
the use
form,
than one month.
of such EVAc
in which
(9-11].
macromolecule was of interest
prevent
acetate copolymer (EVAc),
The
the
response
polymer
from diffusing away
to a
serves to
from the
site
interest.
EVAc
is
biocompatible
[121,
Food and Drug Administration
Further,
it is
delivery of macromolecular
Figure
3 shows
typical
has been approved by
for certain medical
hydrophobic and
features make EVAc matrices
and
uses.
swells negligibly.
potential
candidates
These
for
the
drugs.
in vitro
release
curves
for a
the
-11-
S10
2Z i
30
SORT TIME (HOURS 2
Fig. 4 Kinetics of release for bovine serum albumin (BSA) from
EVAc matrices at various drug loadings and particle sizes. Abscissa is
square root of time. Ordinate is cumulative fraction of incorporated
BSA that is released.
A Loading =0.10. particle size range 150-18O0s
A Loading 0.10. particle size range =3OO-425s
* Loading - 0.30. particle size range 150-80s&
O Loading - 0.30. particle size range - 3O0-42S5s
* Loading - 0.50. particle size range - lSO-180s.
Figure 1.3.
Typical in vitro release kinetics of BSA from
EVAc matrices.
Reproduced from [131.
-12-
protein, Bovine Serum Albumin
abscissa
is
corresponds
the
the
square
square root of
to 400 hours.
for months.
(BSA)
Also,
root of
the
time.
Clearly,
fact
time
suggests
like other polymers,
macromolecules.
possible
of
it is necessary
procedure
figure 4)
[14].
is
is
suspension
congeals, and
suspension
poured into a cooled
is
5a
separate.
solution
polymer and
Figure
is
rate
it
to answer
fabrication
in which
(-80
from the
the
resulting
C)
the macromolecular
mold.
are
The
to evaporate.
mold,
over four
this manner
the
drugs
0
the
an organic
the
and
the
days.
shows a photomicrograph of a
section of a matrix cast in
the
the
solvent begins
then removed
thin
[15].
(104)
It is clear
powder phases
5b shows a photomicrograph of a similar
matrix after release.
It is
to
Lyophilized or crystalline
After mixing,
the
the
how
is a poor solvent for
solvent is evaporated
Figure
that
is
In order
to describe
of interest, since
suspension
remaining
be released
dichloromethane (methylene chloride),
typically water-soluble.
2
to
raised:
drug?
is
macromolecular drugs
1/
proportional
impermeable
solvent
Dichloromethane
the
[7].
powder is mixed into an EVAc
The
hours
protein can
protein
solvent.
20
that
that diffusion
the
this question,
(see
Thus,
Thus the question was
to obtain release
Note
that release is
determining mechanism of release
EVAc,
[13].
seen
that
the
powder
is
-13-
PREPARATION of SUSTAINED
RELEASE POLYMERS
F'
WASH POLYMER in ALCOHOL
4
DRY
DISSOLVE POLYMER
P
in METHYLENE CHLORIDE
4
ADD DRY PROTEIN
POWDER
CAST
at-80C
DRY of-200C
4
DRY at 200C
ACTIVATE with PHYSIOLOGICAL SALINE
I
FIg. 1 Preparation of ethylene-vinyi acetate copolymer (EVAc)
matrices by solvent casting.
Figure 1.4.
Fabrication procedure for EVAc sustained release
matrices. Reproduced from [13].
-14-
a)
a. Before reinst.
b)
b. After relen.
Fig. 3 5S& thick sections of EVAc polymers.
Figure 1.5.
a) 10Qm section of an EVAc matrix loaded with myoglobin
before release.
b) l1 pm section of an EVAc matrix, after release is complete.
Reproduced from [13].
-15the pores
dissolved away, but
release
a pore
through
occurs
network formed
through
macromolecule powder, and not
pores, causing
the
permeates
leaving
behind a
porous
out,
other
drug molecules must pass
of
An overall picture
[15].
developed
It is
formed due
particles
from the
If
pores
of
should
the drug
A
device.
by
the
been
is mediated by
through water-filled pores
the drug
polymer.
in
the
EVAc
by
in water and by the
characteristic
implant,
is
through
then
the
rate
of
D
coefficient
the diffusion
smallest dimension L of
time of
release
tc
can be
the
computed
equation
L 2/D
t
This
characteristic
its
the matrix.
phase separation of
the
be determined
(1)
of how
through which
that release of
the diffusion of macromolecules
water-filled
release
to
Water
itself.
release process has
postulated
the macromolecules
diffusion of
the
to dissolve and
leaving
from EVAc matrices
macromolecular drugs
that are
the
"carcass"
before
by
the EVAc
the drug powder
leach
that drug
It appears
remain.
c
=
long
it will
time gives an
take
incorporated drug.
coefficient
for
order-of-magnitude estimate
the device
For example,
to
release most of
for BSA,
in water is approximately 7X10
-7
the
diffusion
2
cm /sec
[16].
A
-16implant we might consider
typical
For
the
this case,
characteristic
is a
time,
slab of
depth 0.1
given by
cm.
equation
(1)
is
t
(0.1 cm)
=
1.4X10 4sec
=
4 hrs.
Considering
months,
2
=
this
/(7X10
-7
cm
2
that figure 3 shows
/sec)
release over periods of
result is paradoxical.
Classically, the retardation of diffusion
media
is
attributed
The channels
to the
and pores
"tortuosity"
through which
pass are
irregular, and
distance
that a molecule must
equation
(1)
should
travel
Introducing a dimensionless
to
the medium
porous
[17].
the macromolecules must
hence sinuous.
be modified
effective depth of the
of
through
is
take
tortuosity
slab becomes TL,
Thus,
the
effective
increased,
and
into account.
this
factor T,
the
and equation
(1)
is
replaced by
(2)
A
2
tc =(tL) 2/D
tortuosity factor of 2 would
factor of 4,
the
factors
while
characteristic
of at
least
a tortuosity
release
10
are
increase
the
factor of
10 would
time hundredfold.
required
to
release time by a
predict
increase
Tortuosity
drug
release
-17-
that
will
continue
values of
Typical
sands
To
[18]
obtain
unusual
and
(and
the
biological
"intelligent")
crossed each other
diffusing molecules.
II.)
than
lie
between
such as
/T and /T.
10 would require
organization of
a highly
the pore
the channels
each other.
(If
and
channels
they would provide short cuts
Channels and
pores
the
pores,
are
for
will be defined
are cast such that
situated at random,
unlikely that such an organization could exist.
Therefore,
10
[191
Given that EVAc matrices
drug particles, and hence
is
tissues
so much and also avoid
in chapter
it
in porous media
tortuosities
in a polymer matrix such that
pores wind
pores
months.
tortuosities greater
structure
or
for
could
it
is unlikely
be due
solely to
that
tortuosity
the sinuousness
factors as
of
the
large as
pore
structure.
A
primary goal
understanding of
EVAc matrices.
topological
the basis
1.6
Scope
This
release of
of this
is
to achieve an
the retardation of diffusion and release
It will be shown
properties of
for
thesis
the
the
retardation
that geometrical and
water
of
in
filled pores
can
provide
diffusion.
of Thesis
thesis
is devoted
to
macromolecular drugs
the
understanding of
from
EVAc matrices.
the slow
Chapter
-18-
II
provides a brief
It
will
called
materials
previous work on
In chapter
inhomogeneous media.
problem.
this
to a
belong
the EVAc matrices
the field of
of
review
that
seen
be
review of
of
class
III a brief
inhomogeneous media
transport through
is presented.
Chapter
set
IV comprises
the
release slower than might be expected, but also that
is drug
certain conditions release becomes
under
presented
In chapter V a model is
amount
amount
of drug released
of drug
the kinetic
against
tested
predictions and
the model
VI an explanation for the
release
further
its
research are
figure,
the
data
and
of chapter
in
the
i.e.
that can be
is made
IV.
between
In
chapter
of macromolecular
is provided.
Suggestions
for
chapter VII.
in a self-contained
set of
section numbers.
the
model also
This
Comparison
sustained nature
provided
are written
trapped.
data.
own bibliography and
reference,
is
the drug
predicts
that
the kinetic curves
from EVAc matrices
Chapters
with
of
some
the matrices,
from
that will not be
certain aspects of
predicts
that for
slow
the matrix.
in
trapped
so
assumed that
purposes it can be
practical
total
large
not only
that
seen
be
will
It
determinations.
of kinetic
drug
experimental effort--a
If
figure,
referral
fashion,
each
equation,
is
made
equation or section from another chapter,
the
to a
-19-
referral
contains
the roman numeral for
the
latter chapter.
Referrals within a chapter do not contain roman numerals.
-20-
REFERENCES
[1]
Robinson, J.
(ed.
), Sustained and Controlled Release
Drug Delivery Sys tems,
[2]
Friedrichs, R.L.,
system,"
delivery
[3]
Zador, G.,
"Clini cal
progestrone syste m
559
[41
[51
experience
Haffajee, A.,
"Periodontal
therapy by
tetracylcline
,"
J.
local
Cahill,
G.F.,
Kopec,
D.R.
N.
Washin gton,
Engl.
J.
Agents,
vol.
Medicine,
135
and
Med.
F.W.
43 of Advances
15
Harris,
eds.,
(1976).
1004
"Control
(1976).
"Controlled release
in Experimental
of Bioactive
Biology and
eds .
Plenum,
(1974).
R.
and Folkman, J.,
release
of
proteins and
( 1976).
Release
Freinkel, N.,
294,
E.,
flouride ion
in Controlled Release
Langer,
797
inorganic
p.
(1979).
Korostoff,
A.C. Tanquary and R.E. Lacey,
New York, p.
83
in Controlled
Baker, R.W. and Lonsdale, H.K.,
mechanism and rates,"
L.,
Paul and
Etzwiler, D.D.,
diabetes,"
S.S.
delivery of
"Re lease of
Formulations,
Amer. Chem. Soc.,
263,
Contraception 13,
Clin. Periodontology 6,
Solomon, 0.,
Halpern, B.D.,
Polymeric
[81
),"
uterine
and Socransky,
from rigid polymer matri ces,"
[7]
the
a nd
(1976).
Gordon, J.M.,
and
with
(Progestasert
and Ackermann, J.L.,
[61
Westrom, L.
Sjoberg, N.D,
drug
(1974).
1279
6,
Ophthal.
Nilsso n, B.,
J.,
Wiese,
A new
"The pilocarpine Ocusert:
Ann.
(1978).
York
New
Marcel Dekker,
"Polymers
other
for
the
macromolecules,"
sustained
Nature
-21-
[9]
Langer, R.
their
and Murray, J.,
delivery
systems,"
"Angiogenesis inhibitors and
Appl.
Bioch.
Biotech.
8,
9
(1983).
[10]
Gospodarowicz, D.,
Bialecki, H.,
and Thakral, T.K.,
"The angiogenic activity of
the
epidermal growth
Exp. Eye Res.
factors,"
fibroblast and
28,
501
(1979).
[111
Langer, R.,
K.,
"A
simple method
sustained
Can.
[12]
J.
Microbiol.
[13]
attractants
26,
Brem, H.,
polymeric delivery
Biomed. Mater. Res.
Gryska, P.V.,
and Bergman,
for studying chemotaxis
release of
Langer, R.,
of
Fefferman, M.,
274
from inert polymers,"
(1980).
and Tapper, D.,
systems
15,
using
267
"Biocompatibility
for macromolecules," J.
(1981).
Siegel, R.A. and Langer, R.,
"Controlled release of
polypeptides and other macromolecules,"
Pharm. Res.
1,
2 (1984).
[14]
Rhine, W.,
Hsieh, D.S.T.,
the sustained
systems
kinetics,"
J. Pharm. Sci.
Bawa,
Controlled Release
R.S.,
Langer, R.,
release of macromolecules:
fabricate reproducible
[15]
and
Ethylene-Vinyl
Acetate
69,
[16]
methods
265
(1980).
of Macromolecules
Copolymer Matrices:
Thesis,
(1981).
Kozinski, A.A.
and Lightfoot, E.N.,
ultrafiltration: A
filtration,"
general
A.I.Ch.E. J.
"Protein
example of boundary
18,
to
and constant release
Microstructure and Kinetic Analyses, M.S.
M.I.T.
"Polymers for
1030
(1972).
from
-22[17]
[18]
Greenkorn, R.A.,
"Steady flow
A.I.Ch.E. J.
529
27,
Klinkenberg, L.J.,
(1981).
"Analogy between diffusion and
electrical conductivity
Soc. Amer.
[19]
Safford,
62,
R.E..
diffusion in
muscle,"
559
through porous media,"
in
porous rocks,"
Geol.
(1951).
and Bassingthwaighte,
transient and
Biophys.
Bull.
J.
20,
steady
113
and
(1977).
J.B.,
"Calcium
states
in
-23-
II.
Introduction
II.1
This chapter reviews
EVAc matrices.
factors
In
influencing
3, microstructural
information
developed
the
much of
The
drug release are
studies
are
influencing
2 and
This model
From
The
release
the
provides
the basis
for
kinetics
size
the
diameters.
particle
size
the
the
size
The drug
Since
is
the
powder
is
range
is
sieved
suspended in
to enhance
the
the powder
solvent dichloromethane,
drug
the
release
the drug
total weight of
is done
release kinetics.
loading and
defined as
ranges before it
This
drug
the
Drug loading is
drug particle
particle
solution.
In section
3, a conceptual model
loading and particle
size.
divided by
polymer.
polymer
identified.
two most important variables affecting
particle
of drug
from
studies are reviewed and
reviewed.
sections
of macromolecular drugs are
powder
of macromolecules
thesis work.
Effects of
11.2.1
2 kinetic
in section 4.
Factors
11.2
section
provided by
towards
previous work geared
the mechanism of release
understanding
is
SUSTAINED RELEASE MATRICES
BACKGROUND ON EVAc
the weight
plus
the
of (dry)
to narrow
the
polymer
reproducibility
does
drug
not dissolve
of
in
the
the
powder granules maintain
-24-
their
integrity as
the
Thus
the
size
range when the
drug particle
Figure
size and
that
is
1.3
size
range
polymer solvent is
shows
examples
the abscissa
is
the cumulative fraction of
dividing
time
point by
into
the matrix.
It
is
of mass
more
drug
total mass
obvious
that as
inside the
loading
drug molecule.
in some
the
the
sense
drug
It
particle
is
observation
drug
the
of BSA
[1].
time and
so
originally
loading
does
rate
increasing
resistance
to a
incorporated
simply because
the
is
increases,
figure
of
the
the rate
there
1.3
of
Note
the ordinate
This
However,
In other words,
is
indicates
release per
drug
the matrix
loading
to
release
molecules.
also seen
in figure 1.3
sizes, not all
Apparently
root of
of drug
pore
of particle
of drug released up
matrix.
lessens
the
completely evaporated.
the effects
increase,
increases,
into
drug released.
the mass
transfer should
that as
of
the square
(section 1.5).
translates
loading on the release kinetics
obtained by
of
solvent evaporates
in
drug
these cases
will
release mechanism
be
that at
is released
much of
important
(section 4).
in
low
from
the drug
is
formulating
loadings and
the matrix.
trapped.
a model
for
This
the
-25-
11.2.2
Effects of release
media
(pH,
ionic
strength,and
temperature)
The effect of
studied
The
[2].
very close
to
of BSA has been
release occurs at pH=5.0, which is
slowest
of BSA
the isoionic point
rate increases with pH,
the release
becomes
pH on release kinetics
Above pH=5.0
[3].
i.e.
the BSA
as
molecule
more negatively ionized.
One
experiment has been
carried out
strength of
the
release media was varied
experiment,
the
ionic
for BSA.
However,
in which the
[2].
pH was not controlled,and
close
of BSA
be acidic
(unbuffered water
diffusion coefficient
twofold with
ionic
physiological pH
diffusion
strength.
effect of
tends
It has
of air).
the
due
Near the
Therefore, more
strength on
the
solution
isoionic
is much
is
to
that
release
the
than
buffered at
point, however,
less affected
complete experiments
rates need
the
isoionic point
been shown elsewhere
if
coefficient of BSA
ionic
to
to
rates
the pH's of
of BSA in water can vary more
strength
[4].
that
strength did not affect release
release media were probably quite
dissolution
In
ionic
the
by ionic
testing
the
to be
performed.
As
A plot
inverse
temperature
of
the
log of
temperature
increases,
the
slope
so does
of
the
(Arrhenius plot)
the release
rate
[5].
release curves against
yields
an "activation
-26-
energy"
of
identical
plot
approximately
to
the activation
for diffusion
that the
of BSA
release of BSA
11.2.3.
4.5
is
[7].
[6].
dipped several
times
is allowed
decreased
that EVAc
is
the original
dipping
to
fabricated as
into
a 20%
dry,
the
is
evidence
in
figure 1.4,
EVAc solution,
and
and
if
resulting release rates
to uncoated EVAc matrices
This might seem surprising considering
impermeable to macromolecules.
Perhaps
polymer in the matrix dissolves
during
process, allowing
some
of
the drug
coating and
become exposed again.
process may
be imperfect, with small
could
flow
Alternatively,
defects
the
into
the
the
coating
through which
studies
Optical microscopy
Because
visualize
to
some of
escape.
Microstructural
11.3.1
Thin
This
However, the dipping process does not completely
release.
11.3
from an Arrhenius
diffusion controlled.
considerably compared
eliminate
drug
is almost
Effect of coating
the coating
are
This
energy derived
in water
When EVAc matrices are
then
kcal/mole.
the
pore
the matrix
(10pm)
sizes are
large
ultrastructure
sections have
(-100pm),
using
been obtained
one
can
light microscopy.
using a cryomicrotome
-27-
[8].
Figures I.5a,b
release,
show
release
maintenance
the
figure
of
I.5b.
without
Phase
respectively.
polymer before
is
polymers before
clear from figure
pore structure after
and
release is
seen in
Similar photomicrographs of EVAc matrices
to
the
microscopes, a
detectable pores
small depth of focus
[8].
information on
an SEM
picture of a pore.
connected
allowed by
scanning electron microscope
obtain
the pore structure
pores
to other
The
pore
is
the diameter of the
bulging pores.
to
11.4
Conceptual
the release
presented
in
matrices are
model of
several
this
salient
chapter.
the
points
First,
granules are randomly incorporated
network.
Since
itself,
they leave
used
Figure
small
to
1 shows
(By narrow
relative
to
channels are still
process
that
the
into
have
leach out
been
EVAc sustained
The
protein
and when
porous
do not permeate
through
release
powder
the matrix,
behind a highly random
the macromolecules
they must
was
size of protein molecules.)
heterogeneous materials.
leach out
[8].
are
The
large when compared
are
(SEM)
through narrow channels.
quite
There
light
a bulging body,
the channel diameters
meant that
polymer
I.5a,
Scanning electron microscopy
Due
they
and after
between drug and
separation
drug show no optically
11.3.2
it is
loaded
the
the pore
network.
-28-
-r
S.
Pore Body
Connecting Channel
Figure II.l.
Scanning electron micrograph (SEM) of an empty pore,
illustrating a bulging pore body and narrow connecting
channel. Reproduced from [1].
-29-
Second,
at
drug
forever trapped
as
is
low drug loadings
follows:
at low
the
surface of
completely
the
This
matrix.
the
is small.
the polymer matrix,
the
so
that an
in chapter
third
trapped.
important point
from
slabs
is
provide
For example,
the
drug's
governed
the
by solubility
diffusion
Since
network,
for
the
brought
the
out
low particle
can be
on
the
simple model will be
2.2 and
by
t1/2 kinetics.
is
by
the
the fact
This
does not
the only governing
gradients
that
for diffusion may be determined by
However,
only
the release
through
the
through
the water
rates
effect of
are
solubility
process.
diffusion
rate of
in
At
suggested
the concentration
is
pore geometric and
slow
is
section
follows
driving force
solubility.
2.
that drug release appears
This
necessarily mean that diffusion
process.
drug cannot
VII.
be mediated by diffusion.
that release
likely be
fraction of
This
explained
two drug
drug particles
even greater
drug will be
that
it will
fraction of
surface,
be
the
a drug particle is
only a small
activation energy argument of
on
Unless
in figure
quantitated
the
likelihood
illustrated
The
sizes, much of
This can
is
incorporated
to
particle
surrounded by polymer, which the
penetrate.
sizes,
the
loadings,
particles make contact
at
in
and
this
topological
diffusion.
chapter is
The
that
filled
factors must account
fourth salient
the
pore
pores
seem
point
to be
-30-
Polymer matrix
Releasable
Trapped
drug
b.t
Figure 11.2.
Schematic of EVAc sustained matrix.
a)
Low drug loading.
b) High drug loading.
drug
-31-
by narroww"channels"
connected
it will be
slowness
of
The porous
which is
a
two
of figure
of throats
network might be envisioned as
dimensional conceptualization.
3.
The
random walks
field of
transport
media has received much attention
in
figure 3,
We
imagine
through
the random
through random
in recent years,
next chapter a brief review of work
presented.
can explain the
diffusion process.
that drug molecules execute
maze
In chapter VIII
"throats."
that the presence
shown
the
or
in
that
field
and
is
in
the
-32-
Figure 11.3.
Two-dimensional representation of conceptual
model of pore structure. Pore topology is
random. Pore geometry is characterized by bulging
pore bodies connected by narrow throats (channels).
-33-
REFERENCES
[1]
Siegel, R.A. and Langer,
polypeptides and
2
[2]
R.,
"Controlled release of
Pharm. Res.
2,
and Rhine, W.,
in
other macromolecules,"
(1984).
Langer, R.,
Hsieh, D.S.T.,
Brown, L.,
Better Therapy with Existing Drugs:
Delivery Systems,
p.
179,
New Uses and
Information Corp.,
Biomedical
New York (1981).
[3]
Phillies, G.D.J.,
Benedek, G.B.,
"Diffusion in protein solutions
a
J.
[41
[51
study by quasielastic
Chem. Phys.
Doherty, P.
and Benedek, G.B.,
"The
effec t of electric
Phys.
5426
12,
Rhine, W.D.,
Sustained Release
Matrices,
R.
Thesis
M.S.
"Temperature
and Folkman,
J.
Phys.
J.,
Chem.
J.
from polymers,"
E.
A.
Nakajima,
of Macromolecules
from
(1979).
dependence of diffusion
Chem.
58,
770
(1954).
"Controlled release
macromolecules
Golberg and
,"
(1974).
Longsworth, L.G.,
Langer,
spectroscopy,"
(1976).
the diffusion of macromolecules
York
[8]
light scattering
1883
in aqueous solutions,"
[71
at high concentrations:
charge on
Polymeric
[6]
65,
and Mazer, N.A.,
of
in Biomedical Polymers,
eds.,
p.
113,
Academic, New
(1979).
Bawa,
R.S.,
Controlled
Ethylene-Vinyl
Acetate
Release
of Macromolecules
Copolymer Matrices:
Microstructure and Kinetic Analyses, M.S.
M.I.T.
(1981).
Thesis,
from
-34-
CHAPTER
III.1
III.
REVIEW OF TRANSPORT IN HETEROGENEOUS MEDIA
Introduction
In chapters I and
macromolecular drugs
is
II
it was
caused
indicated
that release of
by diffusion of
the
drug
molecules
through a randomly
connected, water filled
network.
Because
is random,
for whatever
Clearly one
is
to
cannot
and
to model
can be
the approach of
work
Rather,
examining
the
for important statistical
these features
all possible
the
release
approach one
features of
in such a way
of
transport
in porous
received much attention.
that has been done on
that
necessary
to
make
takes
the medium
the
phases,
is
various
one
which
transport
permeable.
impermeable
sustained release
to
polymers
is a review of
Since
to
Specifically,
the mobile
of
review
two
species and one
applicability of
the problem of diffusion
will
it is
this
media consisting
The relationship and
reviewed works
heterogeneous
range of materials,
to heterogeneous
is
other
This chapter
some restrictions.
be restricted
and
this problem.
heterogeneous media cover a broad
which
also random.
a porous medium and calculating
problem
media has
will
boundary conditions
predicted.
The
the
take
for each realization.
search
the
transport equations apply are
realizations of
process
the milieu
pore
be assessed.
in EVAc
the
-35-
Maxwell-type
111.2
The
models
first attempt at modelling
transport
heterogeneous media was made by Maxwell
considered
dilute
the
problem of
to
steady state
problem by assuming
others'
presence,
the
current
subject to
at the
the
of
the
(1)
the medium
conductivity of
conductive
the
phase,
dilute media
insulators
is varied
such as
is
near
insulators
insulator
the
composite, and
then Maxwell's
enough
that their
solved Laplace's
that
If
the normal
a0
is
spheres are
the
imbedded,
e the volume fraction
result is
1.
have been derived for
in which the
(i.e.
cylinders
In general,
s
He
zero.
into which
More complicated expressions
shapes
is
affected by each
dilute
interact.
the
[2e/(3-e)]a0
a
similar
spheres are not
a sphere
through a
(Conduction is
boundary condition
surface of
conductivity of
a
the
Maxwell
He approached
spheres are
induced dipole fields do not
equation
spheres.
diffusion.)
that the
i.e.
[1].
electrical conduction
suspension of insulating
analogous
through
spheres
[2],
shape of
are
spheroids
is
because
suspended
replaced by
[3],
these Maxwell-type models
This
the
fraction
(1-)
are valid only when
the assumption
gets
large.
convex
etc.).
that
do not affect each other breaks down as
volume
other
After
the
the
the point
-36-
where a Maxwell-type model
adequate
to
describe
function of e.
statistical
between
solely
the behavior of
When
factors
the insulators
besides e
positions of
regularly spaced
than a dispersion of
placed
and Glandt
[4]
values
that
the
elaborates
system,
In might be
order
of
the
the
the
EVAc
rates
polymer) does not
the
the
been pursued
For
in this
than 1.
be
present
are not
a
EVAc matrices,
realistic
pointed
out
the direct
for which Maxwell-type
it is
space)
insulating
these
point.
introduce
lower,
the conductive
that
is
suspended
fraction (i.e.
consist of convex paricles, as
models.
this
less
could
it should
polymers,
the water filled pore
The
at
by Chiew
simplest form,
one
polymers
the heterogeneous media
insulating medium.
have not
that
However,
In EVAc
insulating
e is the
the operant parameter
sustained release
(i.e.
Maxwell-type
a different overall
detail on
their
an
conductive
A recent review
in much more
transport
porosity e.
models are designed.
fraction
in a
randomly placed
medium.
argued
Correlations
For example,
spheres
possess
in
a
interact, other
statistical characterization of
predict
opposite of
in an
same
of e,
the Maxwell-type models,
then
as
the matrix, which is considerably
appropriate.
and
in
the EVAc
porosity of
higher
conductivity a
(which are not determined
insulating
conductivity
Thus,
is
start to
the insulators
at lower values
In
the
single model
become important.
medium will,
spheres
no
e) can have a significant effect.
by
array of
breaks down,
the
is assumed
reasons, Maxwell-type models
work.
by
-37-
111.3
Percolation models
Percolation
been
first
theory
is a relatively young
introduced in
1957
[5,6].
a problem in
the
a
surrounded by water,
large
wets
rock
the rock.
rock, at
low,
Small
in
the
rock,
surface
to
critical porosity,
that pervades all
"percolate"
porosity, or
however,
,
regions of
"percolation
these
percolating
Figure
When
rock
(figure
lb).
that
the
the
a network of
rock,
so
the water
into
connected
porosity is
As
but those
they are
la).
As
surface of
the
the
pores
is formed
that water may
there
to
the
At a certain
1c).
At the
critical
are still
many
the
pervading
(or
the porosity
increases
even
isolated pore clusters are
recruited
into
the
ld).
1 is reminiscent of figure 11.8.
the
percolation model will be
the
amount of drug
Although percolation
Consider
(figure
to
threshold,"
that are not
that is
media.
be wetted,
connect
rock (figure
network (figure
name from
introduced
rock will
the
the
"percolating") pore network.
further,
slowly
of
denoted e
the
and assume
its
rock will not, because
are wetted
throughout
through porous
of the
the surface
pores
It derives
the rock.
increases, more pores
so more
pores,
in
interior of the
connected
porosity
flow
pores are
random positions
pores at the
pores
not
study of
field, having
used
in
releasable
A variation of
chapter V
to analyze
from EVAc matrices.
theory is named
for
the wetting of
-38-
0
0
0
0
0
b.
00
0
O
0
o
C.
0 00
0
0
d.
e
Figure III.l.
connected to surface
-
pore
-
isolated
pore
Stages of formation of a percolating pore
network in a porous rock.
a)
E<<E C,
b)
E< c,
c)
E=c
, d)
E >Ea
-39-
porous rocks,
areas of
it has
physics
found widespread
that deal with phase
sense,
the onset of a percolating
viewed
as
a phase 'transition.
can be modelled using
metal/insulator
the
metal
used as
is
increased
the vehicle
also been
study of
two phase
Unlike
inherently
or
lattice,
literature.
[61.)
conductor,
this
A
insulator
to
[8-11].
be used
of which are
(This
is called
For a review
With
and
then
a
belonging
the
If
threshold
the
to
site
lattice
effect can be seen
for
and
[151),
"conductive"
with
insulator
(see
random
such as
the
a
and
latter
figure
lattice
using
the
the
percolates
clear
a
of percolation,
with
large,
are
the
"filled"
that
the
or
is
the
in
a regular
percolation in
cluster
is
the
[5,6,13,141.
idealized as
statistics
a
[6,121,
percolation models
of conductive clusters
of conductors
lattice.
rocks
as
conduction was
theory has
either
is filled
tabulates
distribution of sizes
a conductor,
Percolation
percolation program generates
rule,
transition in a
(but not always
"site"
E
it
that
the wetted
of different types
probability
otherwise
In a
in place of
through porous
Typically
sites
Since
theory of epidemics
heterogeneous medium is
the
transition
the
the Maxwell-type models,
"insulating."
2a).
flow
stochastic.
porous
see
the
phase
theory is
for discussion.
applied in
[6,7].
discussing Maxwell-type models,
metal/insulator mixture will
rock as
Another
from an
framework for
transitions
several
channel in a rock can be
percolation
composite
fraction
the
application in
fraction
throughout
percolation
statistic
(see
-40-
a.
- insulator
- conductor
1.0
face
centered
simple
cubic
cubic
tetrahedral
0
b.
a0.5-
0.
CL2 0.3o 0.4 05 -0,
CONDUCTIVE FRACTION
Figure III.2.
a)
A typical percolation lattice of lattice sites containing
conductors (filled squares) or insulators (empty squares).
Conductor fraction E-0.56.
b)
Probability P that a conductive site belongs to a percolating cluster, for various 3-dimensional lattices. Redrawn
from [7 ].
-41-
figure 2b).
Just above the
of conductors
lattice
value
lattice
the
top
have
that
universal
probability
cluster,
That
is,
P(
Most
quantities
More
3-D
lattice
filled
) ~
(E-E )
,
C
E>E
the
is
threshold
If P(Es)
in the
is
[7]
is
the
percolating
associated
C
with
percolation
power
law exponents are
structure, given a
shown
encountered
to
the
conductive
far
networks
to e-e c.
independent of
the
fixed dimensionality of
be
in phase
related
to
transitions
critical value
organization, and
So
precise calculations
lattice site
law dependence with respect
However,
the
in figure 2b can be
structures.
power
been
3 dimensional
the asymptotic behavior near
that a
recruited.
then
(2)
has
are
very
virtually independent of
the curves
of each other.
for all
fraction
cluster rises
for regular
is
threshold itself,
shifted on
the
the percolation behavior, except for
structure.
shown
percolating
the
that,
2b also shows
structures,
of
to
threshold,
soon virtually all conductors
quickly, and
Figure
belonging
percolation
to various
have
a
Remarkably, all
actual
the
lattice
lattice.
(This
"universalities"
and critical
phenomena
[16].)
E c does depend on the lattice
also higher order
properties of
the
sites.
in
this
section,
the
only concern has
been whether
-42-
a mixture
Nothing
of conductors and
has
been said about
composite above
the
conductivity a on
the
threshold
e c,
the
relation holds
This
is
types
~
(E-
Below
the
for 3-D lattices
)
The dependence of
sites
e has
the percolation
isolated
composite
from each other,
is
zero.
the following
Above
asymptotic
[71:
1.5
in figure 3.
3-D lattices,
give rise
of
(e>e C),
threshold
plotted
of
threshold.
conductive clusters are
percolation
a
conductivity a of the
actual
[8-11].
overall conductivity
(3)
the
fraction of conductive
studied, however
the
can conduct current.
the percolation
been well
so
insulators
This
relationship holds
although once again, different
to different
values
of
E
for all
lattices
.
C
Note
as
the
the
that the
rise
rise
in the
in conductivity
fraction of
percolating cluster.
cluster
This
contains many regions
of
current flow,
and
of
the
These
cluster.
Further,
the
tortuously
connected
Equation
membranes
closer
(3)
in at
thus
E
conductive
sites
is
the percolating
because
that are not on
do not contribute
regions are
is
is not nearly as
to
often
to
fast
belonging
the main
the
to
pathway
conductivity
termed "dead
ends."
E , the more tenuously and
C
the percolating cluster will be.
has been
least one
tested
case:
for conduction
in porous
that of perflourinated
ionomer
-43-
I.-
P
0.8-
0.6-
0.4-
02-
0.0 0.2
0.3
0.4
0.5
0.6
~
0.7'
-
0.8
0.9
10
Conductive volume fraction E
Figure 111.3.
Plots of percolation probability P and conductivity a
of a random simple cubic lattice of conductors and
insulators, as a function of 6, the volume fraction
of conductors. Reproduced from [9].
-44membranes
[17].
water-filled
angstroms,
In
these membranes,
pores whose radii are
so that
the pores
modelled as belonging
membrane.
As
shown
quite well near
The
near e
c
power
statistical
determine
law
the
developed
with
to
finite
explain
be
The
the
there
the
to
is
equation (3)
fits
the
the
data
threshold.
holds only for e
of
the
and higher
order
conductive sites
the percolation probability
medium"
theories
[81
the dependence of a on e
is no
of
threshold
that
lattice.
a
finite depth.
ec
,
In general,
conductivity
since
and
have been
for 6 away from
of
the
diffusion
the conduction
that
percolation models
[19,20].
of macromolecules
as
the
lattice
[18]
Thus,
at
the
e there is a
will
span
depth gets
the
smaller,
rises.
through percolating
problem
on
For such
for any
chain of conductors
relation
diffusion
that
the positions
Einstein
the
of
regions can be
lattice structure
3-D lattices
problem of
related
tens
lattice
results have been reported recently
the expected
-
order of
success.
probability
depth of
,
"Effective
conductivity of
lattices
c
e-dependence of
some
Some
the
dependence discussed above
features of
conductivity.
C
the percolation
conductive sites are
insulating
to an extensive
in figure 4,
Away from E
.
and
on
the
clusters
can
through a suitable
first glance
could explain
the
it might
seem
slowness of
through EVAc matrices.
After all,
-45-
10
10-
10-2
7 10-2
C:
b
/
z/
10-2
Figure III.4.
0
C/
c/
10~1
Conductivity of NAFIONO perfluorinated innomer membranes
as a function of volume fraction e of membrane that is
saline., Dashed line is prediction of percolation theory
[eq. (3)]. Reproduced from [17].
-46-
as E+E,9
the
Two
a+O.
lattice sites
release
large
can be
depth.
near
equation
small,
so
depth
porous network to
would
the form of
predicted
included
shapes
species
by
in
of
pores
can
move
discussed
111.4
in
Models
So
far
composite
medium
this
Since
of
the
the EVAc
pores
matrix,
depth
included.
lattices
In
are of
in pores
do not have
be released.
to
fact,
it
predict
that
high conductance
(3)
is a
that are
transport
it can be
scaled by
might also affect
the
from pore
to pore.
chapter,
the
translates
rate
Thus,
might be
factors not
For example,
by which
These
scaling
the
mobile
aspects
will
section.
influence of
primary
(i.e.
proportionality.
the model.
is
finite
contend with a
Second,
lattice aspect of
this
the
sustained
the behavior as a function of e
the next
of
in
assume
seem to
equation (3),
the
First,
slow even at high porosities, where
But notice
diffusivity).
EVAc
the finite
is not so clear when the
(3)
out here.
the mapping between conductivity and
in EVAc matrices is
media,
to the
surface of the membrane
complicated
that
compared
In particular, diffusing molecules
the
while
pointed
just described should be
shown that
diffusivity
be
dimensions of pores.
lattice depth must be
considerations
be
should
in a representation of an
polymer have the
are relatively
the
things
pore
focus
factor
has
been
determining
conducting
into
structure
the
volume
porosity.
placed
the
conductance
fraction.
In
on models
this
of
a
For porous
section
-47-
models which deal with the
As
are
to
was
in
beds and
rocks, and
are
porous
of
the field
convex.
media
of pores
in section 2,
be regarded as convex.
models
pores
stressed
shape
of
these
Archie
the pores
There are
petrology
in EVAc matrices
several
that handle
models
[211
are reviewed.
published
compacted
similarly assume
showed
porous
that the
that many compacted
that are saturated with water have a conductance
form
(3)
a = E-m
where m
the
is
called
porous
the
"cementation
media being
studied.
factor,"
structural
the conductance
Probably
structure
to
He
can be
quite
then
measured
electrodes,
he simulated
This
inlets and
cube
is
was
(see
outlets.
the
the
due
similar
is
to
large,
it
then
porosities.
postulated pore
to Owen
[22].
that shown
Owen
in figure
conductance of a Bakelite cube
By
figure 5).
the effect of
Owen
current
If m
to predict
even at large
that used a
found
almost proportional
because
proposed
medium.
small,
structure quite
saline
the
the
predict conductivity was
filled with
pore
in
the first model
proposed a pore
11.9.
factors
on
The constant m must be
determined empirically, and no model was
based on
which depends
tends
to
to
varying
size
constrictions at
that
the
the
the
the
the
the
the conductance
area of
take
of
of
electrodes.
path of
least
-48-
I:
z z
- saline
\insulator
a.
TV
0
.11
(
b.
Figure 111.5.
Electrode
Electrical analog of constrictions due to Owen [22].
width determines resistance. As electrode becomes wider, more
saline will conduct current, so resistance decreases. Current
is indicated by arrows.
a)
Narrow electrode (high resistance)
b)
Wide electrode
(low resistance)
-49resistance.
irrelevant
With small electrodes, much of
for
Michaels
steady
state
the passage
[231
state
as
Michaels'
identical
Figure 6a
model.
is
the wide
to
the
the narrow
to
those
a schematic
This is analogous
inverse
of
to
2r,
the
network is
network is
which is
(see
shown
2Rr/(R+r).
considerably
two
of
pores
of
resistor network
solute)
pore widths.)
For R>>r,
previous pore network
for steady
the essence
the
shown
in figure
of
two straight, parallel
same
figure 6c).
For R>>r
than
(R/r is equal
The network
this
the
total
length as
The analagous
in figure 6d.
less
r.
and
equals approximately R/2.
one wide and one narrow, with the
electrical
this
ratio of the
(R+r)/2.
is not
having equivalent resistance R
consider an alternate network
pores,
the
is
This
in
for electrical
pores having equivalent resistance
resistance
Now
pores
model,
boundary conditions
(voltage-concentration, current-flow of
with
is
result for
a mathematical
Each pore column consists of
different widths.
6b,
same
electrical analog.
the equation and
diffusion are
conduction.
virtually the
but by using
contradistinction to Owen's
surprising,
cube
of current.
arrived at
diffusion,
the
The
resistance of
approximately equals
resistance of
the
first
network.
The
difference
between the
explained.
In figure
the path
least resistance.
of
6d,
two overall
current
In
resistances
can choose
to
figure 6b,
all
flow
can be
through
current must
-50-
V
C=C*
R
a.O
r
r
R
c=o
-7-
C.
- -- - -
V
c=c*
d.
R
r
R
r
C=O
Figure 111.6.
Model of steady state diffusion through pores with varying
width, after Michaels [23].
Solute flows from region of
concentration C* to region of concentration 0.
a)
Wide pores meeting narrow pores.
b)
Electrical analog of a.
c)
No longitudinal variation in pore width.
d)
Electrical analog of c.
-51-
flow
through a high resistance element.
The
state
previous examples
dealt with quasistatic or
transport through porous
EVAc matrices
Michaels
is a
transient process,
do not apply directly.
general model
specialized
and VI
domains.
for
the model will
so only
a brief
Transport
through
the porous medium
top
could
level
is
randomly
by
the
that
width
from pore
shape
of
this rate
to
pore
to
pore.
the pores.
of hopping
width.
the
lattice), while
individual pores.
The
A
decreases
in chapters
regarded at
lattice
the
is
the
two
of pores
bottom
shown
with
A
V
provided here.
level
thought
of hopping
be
presented a
porous medium.
particle
rate
It will
has
be utilized
is
in
the models of Owen and
[24]
description is
that of motion on
be a percolation
motion within
so
Pismen
,
The
However, diffusion
transient diffusion in a
case of
steady
is
levels.
(which
is
to
the
"hop"
determined
in chapter
VI
ratio of channel
-52-
REFERENCES
[1]
Maxwell, J.C.,
Trea tise
Clarendon, London
of Electricity
Lord Rayleigh, Phil. Mag.
[3]
Fricke, H.,
Rev. 24,
[4]
Chiew, Y.C.
on
94,
and Glandt,
629
Frisch,
11,
894
E.D.,
J.
structure
Col 1.
Interf.
Camb .
Phil.
Soc.
J.
"Percolation
Soc.
Ind.
Appl.
Math.
(1963).
45,
Adler, D.,
9
Physiol.
"Perco lation
mazes," Proc.
topics,"
and Kirkpatrick, S.,
theory, "
574
Adv. Phys.
20,
"An
introduction
325
Rev. Mod.
(1973).
Flora, P.,
and
Senturia, S.D.
systems,"
Sol
,
"Electrical
State Comm.
(1973).
Straley, J.P.,
"Critical
phenomena
in
to
(1971).
"Percolation and conduc tion,"
conductivity in disordered
[101
"The effect of
H.L. and Hammersley, J.M.
Kirkpatrick, S.,
12,
electrical
systems,"
dispersion,"
crystals and
Shante, V.K.S.
Phys.
the
(1957).
percolation
[9]
of
90 (1983).
processes and related
[8]
treatment
Broadbent, S.R. and Hammersley, J .M.,
53,
[7]
(1892).
(1924).
processes I.
[61
481
capacity of disperse
the conductivity of a
Sci.
[5]
575
34,
mathematical
conductivity and
Magnetism,
( 1873).
[2]
"A
and
resistor
-53networks,"
[11]
J.
Phys.
Winterfeld, P.H.,
C:Sol.
Sciven,
State
L.E.,
Phys.
J.
(1976).
H.T.,
random two-dimensional
Phys.
C:
Grassberger, P.,
"On
the
general
process with dynamical
Sol.
783
and Davis,
"Percolation and conductivity of
composites,"
9,
State
Phys.
14,
2361
(1981).
[12]
epidemic
Math. Biosc. 63,
[13]
[14]
Larson,
Scriven,
Chem. Eng.
(1981)..
Chandler, R.,
J.
Fluid
Vics-ek, T.
36,
Mech.
to
Stanley, H.E.,
[17]
Hsu,
V.,
Phys.
A:
Barkley, J.R.,
Math.
198,
Phys.
Straley, J.P.,
random networks
14,
L31
of
(1981).
to Phase Transitions and
New York
(1971).
P.,
"Ion
transition
in
(1980).
conductivity
J.
Gen.
test
ionomer membranes,"
Allain, M.,
,"
continuum:
insulator-to-conductor
Mitescu, C.D.,
networks
and Willemsen,
and Meakin,
13,
cal
H.T.,
(1982).
in a
Macromolecules
"Electri
percolation,"
"Monte Carlo renormaliztion
percolation
Nafion perfluorinated
[19]
249
Phenomena, Oxford,
W.Y.,
the
flow in porous media,"
Lerman, K.,
Introduction
percolation and
[18]
phase
119,
and Kertesz,
universality," J.
Critical
57
Koplik, J.,
group approach
[16]
and Davis,
L.E.,
two
Sci.
behavior of
(1983).
"Percolation theory of
J.F.,"
[15]
R.G.,
157
critical
A:
Guyon,
of
Math.
"The ant
near the
in
E.,
finite-si
Gen.
the
and Clerc,
ze
15,
percolation
2523
labyrinth:
critical
J.P.,
(1982).
diffusion in
percolation
-54-
threshold,"
J.
Phys.
C:
Sol.
State Phys.
13,
2991
(1980).
[20]
Mohanty, K.K.,
and
transport
Ottino, J.M.,
and Davi s,
in disordered composite media:
introduction of percolation concepts,
37,
[21]
146,
in core
54
[23]
Sci.
"Electrical
analysis
resistivity analys is
interpretation,"
Trans.
as
an
A.I.M.E.
(1942).
Owen, J.E.,
body,"
Chem. Eng.
(1982).
Archie, G.E.,
aid
[22]
905
"Reaction
H.T.,
"The
resistivity of
Trans. A.I.M.E.
Michaels, A.S.,
195,
169
a fluid-filled
(1952).
"Diffusion on a pore
section--a simplified
porous
treatment,"
of
irregular cross
A.I.Ch.E. 5,
230
(1959).
[241
Pismen, L.M.,
structure,"
"Diffusion in
Chem.
Eng.
Sci.
porous media with
29,
1227
(1974).
random
-55-
CHAPTER IV.
IV.l
RELEASE KINETICS
FROM EVAc MATRICES
Introduction
Studies
of release
kinetics
of various
drugs
from EVAc matrices have been presented
works
[1-4].
In none of
drug particle
In
this
size and
chapter,
these
studies
loading been
the results
have
presented,
along with
The studies
focus
loading.
Their purpose
which
models
IV.2
the
Materials
results
lysozyme
effects
of
on
(BSA)
study
to date are
the release kinetics of
chloride.
on the
is
the
of the most comprehensive
kinetics of Bovine Serum Albumin
P-lactoglobulin and
in previous
studied systematically.
of release
some
macromolecular
effects of particle
size
to provide
a data
base
of chapters V and VI
can be
compared.
and
against
and Methods
IV.2.1 Materials
The
polymer ethylene-vinyl acetate copolymer
(EVAc, 40%
w/w vinyl acetate) was obtained under the product name ELVAX
40P
from DuPont Chemical Co.,
All
Wilmington, Delaware.
proteins were obtained
Louis, Mo.
Proteins
used were
from Sigma Chemical
bovine
serum albumin
Co.,
St.
(cat. no.
-56A4503,
m.w.
18000),
68000),
and
The
lysozyme chloride
from Sigma
4912);
no.
sulfate
(cat. no.
following chemicals were
sodium chloride
and
L2506, m.w.
L2879, m.w.
14,500).
(crystalline form) was
G3632).
used
(Mallinckrodt, Inc.,
sodium phosphate dibasic
7917-1);
(cat. no.
(cat. no.
antibiotic gentamycin
also obtained
The
beta-lactoglobulin
in
the
Paris,
anhydrous
release medium:
Ky.,
cat. no.
(Mallinckrodt, cat.
sodium phosphate monobasic monohydrate
(MCB
Manufacturing Chemicals, Cincinatti, Ohio).
The
solvents dichloromethane
(methylene chloride) and
xylene were obtained from Mallinckrodt, St.
Protein
powder was
sieved using
U.S.A.
Louis, Mo.
standard
sieves, with no. 40,50,80,100,140
and
manufactured by W.S. Tyler,
Inc.,
Mentor, Ohio.
agitated
also
Inc.
in a
sieve
shaker,
vortexed
tubes
on a Vortex-Genie
Bohemia, N.Y.,
molds
170 gauge meshes,
obtained
Drug-polymer suspensions were mixed
polypropylene centrifuge
model
(VWR cat.
mixer
12-812).
Sieves were
from W.S.
no.
21008-940),
level
Matrices were
(Fisher
Solutions were
Molds
Scientific, cat.
cut using a #3
and
(Scientific Industries,
(Dow Corning, Midland, MI).
a circular
Tyler,
in Falcon
cast
prepared with ordinary window glass cemented
silastic
testing
or a #6
were
no.
into glass
with
leveled with
12-000).
cork borer
(A.H. Thomas,
-57-
Phil.,
a
PA).
Physical dimensions of
micrometer
(Starret
Wax
Co.,
Athol,
slabs were measured with
MA).
was obtained from Lancer,
Sherwood Medical, St. Louis,
Paraplast
,
in pellet
from Becton,
All
form.
MO, under
Repipet Jr.,
the
liquid
transfers were
performed with a
Berkeley, CA, or
Pipetman.
Spectrophotometric determinations were made
Perkin-Elmer 553 fast scan
super
obtained
Rutherford, NJ.
obtained from Labindustries,
with a Gilson
trade name
Syringe needles were
Dickenson and Co.,
quantitative
a division of
on a
spectrophotometer equipped with a
sipper.
IV.2.2 Methods
EVAc
slabs containing
manner similar
acetate
to
copolymer
beads were washed
a clay
butylated
EVAc was
or 8%
that shown
in
figure 1.4.
(EVAc) was obtained
several
coating added
then washed
protein drugs were
times
in
then
dissolved
(w/v) solutions.
(BHT),
form.
distilled water
exhaustively in ethanol at
37
in a
Ethylene vinyl
in bead
during manufacturing.
hydroxytoluene
fabricated
The
to
The beads
remove
were
0C to remove
an antioxidant.
in dichloromethane
The clean
to give
Protein drug powder was
either
10%
sieved at 4 0 C
-58-
for
30 minutes
containing
using a mechanical
particles of specified
For each casting,
(weight/weight) drug
solution was placed
determined
by
of
to
10%
used
and
poured
be called
had
been
for
5 minutes.
in
those
into a leveled,
precooled
into a
transferred
left for
The
low
slab.
the THIN
used.
dessicator
for
two
The
drying
days.
days.
stages.
The
drug were
the
total
(There is one
1.5
slab.)
grams
In most
loadings
so
the
the 8%
flat 7cmX7cmXlcm glass mold which
pried
it on a flat slab
so
of dry
the mixture
slab was
room
of approximately 4cmX4cmXlmm
0
C
freezer, where
temperature)
shrinkage
and
and
it was
to a vacuum
left
there
occured during the
a rubbery slab
(see
to
Ten minutes after
then transferred
Considerable
result was
-20
ice
fast that no
loose with a cold spatula
screen- in a
The
drug
mixture was vortexed
(The cooling was
slab was
EVAc
such that
temperature caused
(600 millitorr,
more
and
At higher
cases.
and
the
became very viscous,
by placing
to a wire
two
and
and
3 grams.
convection currents were observed.)
pouring, the
tube,
this--one slab was cast with only
polymer-solvent-drug mixture
congeal
(PS)
prescribed.
loading,
EVAc solution was
solution was
yield fractions
size
The amounts of EVAc
prescribed
mass--this will
cases,
(L) were
to
ranges.
particle
polymer and protein equalled
exception
total
size
in a polypropylene
then added.
the
the drug
loading
powder was
mass
shaker,
with dimensions
section 3.2).
-59-
The
pouring step caused
protein, which stayed behind
to
the
viscosity of
true as
the
drug
polypropylene
in all
suspension
left behind
fraction of
in the
the mass
(diameter=1.18 cm)
from each slab.
The weight
that was
was
(mg)
used
and
Each disk was
through
circular faces
center
of one of
syringe needle.
was
then glued
cap
using RTV
the needle
placed
on
to
the
far enough
the
needle
rubber.
that a
tip and
then be
The drop of RTV at
rear end
inside of a
silicone
assembly could
off
The
the
the
tip
punch out
impaled
by a 5/8",
25
assembly
scintillation vial
pushed back on
drop of RTV
touch
poured.
thickness
then
The matrix was
not
to
the needle
20 ml
small
screwed
of
cases
polypropylene
(mm) of each disk.was measured.
gauge
due
However,
cork borer
the
tube,
loading increased.
only a small
five disks
polymer and
increasingly
tube was
#6
the
of
This was
of
A
in
loss
the suspension.
the mass
the
some
could also be
the matrix.
The whole
into a scintillation vial.
prevented
the matrix
from falling
the needle.
A diagram of the whole assembly
Drug
buffer
liter
release was
(pH=7.4:
H20)
contained
antibiotic
concentration of
actuated by placing
11.499 g Na HPO4 and
into 20 ml
the
is given
gentamicin
0.2 mg/ml.
sulfate,
figure
16 ml of
2.622
scintillation vials.
in
phosphate
g NaH 2P04
The buffer
at
1.
in
1
also
a
The cap-needle-matrix assemblies
-60were
then
screwed onto
matrix in
placed
the
on a
the
release medium
shaker (100
At each
vials,
rpm)
totally
(see
figure
the old vials
was
1).
The
incubator
the cap assemblies were
timepoint,
using UV
then assayed
vials were
at 28.5+1 0 C.
transferred
the assay was
concentrations where
at 280nm.
performed
the
in
spectrophotometry.
At high protein concentrations, which occured early
release,
the
The amount of protein
containing fresh buffer.
to vials
an
inside
immersing
At
280nm reading was
in
the
low
lower
than
.100,
made at 220nm [1].
readings were
IV.3 Results
IV.3.1
Introduction
Table
1 lists all
combinations
loading
sizes
the weight/weight
to
.50
in
received
powder
studies
that
tested.
that were
loadings
increments
from
the
of particle
were
case.
90-106 pm and
of
of
.05.
supplier did
size
90-106pm and
.10,
For
The
batches
not
of
two drug
150-180pm),
slabs were varied
BSA
from
for a
full
.05
powder
enough
permit harvesting
only done with loadings
loading
most extensive
the
300-425 pm
The THIN slab was
The
Serum Albumin.
diameter
(particle
size-(weight/weight)
drug-particle
performed with Bovine
study was
particle
the
study,
so
.05,.l0,.15,.20,.25 for
cast with
but with half
particle
the
size
thickness
of
the
-61-
vial cap
polymer/dru
5/8" syringe needle
polymer/drug
matrix
RTV drop
release medium
-
Figure
IV.l.
-
Assembly used
20 ml scintillation vial
in release
experiments.
-62-
LOADING
L
BOVINE SERUM
ALBUMIN
90-106 pm
BOVINE SERUM
ALBUMIN
150-180pm
0.05
0. 10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
WEIGHT
W
141.
136.
139.
128.
126.
121.
115.
109.
116.
112.
(mg)
DEPTH
d
1. 316
1. 276
1. 270
1. 166
1. 152
1. 110
1. 072
1. 014
1. 088
1. 102
4.
5.
+ 5.
+ 8.
4.
+T 5.
+T5.
3.
+T 2.
2.
+
+T
0.05
0. 10
0. 15
0. 20
0. 25
0. 30
0. 35
0.40
0.45
0. 50
136.
132.
127.
122.
116.
114.
114.
90.
87.
86.
4.7
+T 3.1
6.5
+T3.1
T+_2.6
2.9
+T 3.2
0.9
+T 3.4
3.3
1. 244
1. 170
1. 140
1. 100
1. 060
1. 046
1. 074
1. 068
1. 052
1. 086
BOVINE SERUM
ALBUMIN
300-425-pm
0.05
0. 10
0. 15
0. 20
0.25
146
137
134
124
87
6
7
5
2
3
1. 266
1. 226
1. 246
1. 174
0. 886
BOVINE SERUM
ALBUMIN-THIN
90-106 pm
0.10
49.1
+
0
1
2
4
2
(mm)
ESTIMATED
POROSITY
£
+ .017
+
+
.064
.041
.072
.073
.034
.067
.055
.048
.015
T
+T
+T .042
.07 +
.12
.14 +
.19 +
.25
.30 +T
+T
.36
.40
.46
+T
.53
.023
.047
.038
.029
.045
.032
.011
.075
.042
.04
.07
.13
.19
.25
30
.37
.53
.58
.64
.051
.102
.070
.038
.018
.00
.07
.16
.22
.32
02
03
01
01
03
02
02
02
02
01
.00
+T.01
+T .02
+T .03
.00
.02
.03
.01
.02
.00
.01
.04
.03
.03
.01
2.6
0.476
+
.043
.15
+
.04
-LACTOGLOBULIN
106 - 150 pm
0.10
100.2 +
7.7
0.920
+
.076
.10
+
.03
-LACTOGLOBULIN
300-425-pm
0.10
0.30
142.7 +
98.5 +
5.6
4.2
1.330 +
1.020 +
.037
.046
.12 +
.38 +
.01
.02
LYSOZYME CHLOR.
106- 150 pm
0.10
141.9
+
4.6
1.264 +
.046
.07
+
.02
LYSOZYME CHLOR.
300-425 pm
0. 10
0.30
135.2
128.6
+
+
9.3
5.4
1.262 +
1.184 +
.099
.061
.12
.30
+ .01
+ .02
Table
IV.1
Physical parameters
Values are mean + 1
for EVAc di sks.
std. dev.
-63-
corresponding slab
in the
cast with P-lactoglobulin
limitations
these
on
the
row
were punched
size
and
in table
the
the
chloride.
available
in
loadings
of
the
same mother
of
extensive
(L=.10,.30) and
tested.
data
the
for five
specified
disks
the physical properties
slab show
that
particle
standard deviations of
(disk weight W and depth d)
Typically,
batches
slabs
1 summarizes
Means and
Again,
supplier precluded
from a mother slab at
loading.
table.
from
from
parameters
physical parameters
the
sizes
Slabs were also
(PS=106-150pm,300-425im) were
Physical
Each
sequence.
lysozyme
so only illustrative
particle sizes
IV.3.2
and
particle
proteins received
studies,
"main"
the
are
listed
of disks
a relative variation on
in
taken
the
order of 5-10%.
Some
slabs.
of
that
the
As
and
drug are
the
polymer
carries a
is
less
(except
3 grams.
given
increases,
decrease.
slabs
weight of
trends in weight and
loading
slabs
all
can be
clear
This is
the
the weights
somewhat
THIN slab)
However, a simple
for this
phenomenon.
in suspension
outweighs the
large
both
depth exist for
volume of
polymer and hence
before
less
the depths
surprising, given
were
cast
at
a
total
physical explanation
Recall
casting.
drug powder,
solvent.
and
the BSA
so
that
the
polymer
At
low
loadings,
the
At higher
solvent,
so
the
suspension
loadings,
there
total volume
-64-
of
the
suspension is
into molds of
Therefore
a
occur
the height of
slab, it
from
the
The
thicker
will
occur
the
top,
the
After
from
bottom, and
slab,
the more
the
Thus
the
high
loading
Since
the disks,
(7cmx7cm).
the
this
the mold must
horizontal
low
support
is
thus
the slab.
leading
to
slabs will dry
than will
cork borer provides
to
can
that evaporation
slabs
translates
of
slab
this explanation.
For
BSA PS=90-106 pm,
.05-.40.
the
10%
EVAc
dimensions
the
a constant
the observed
example,
after drying
for slabs
solution was
The horizontal dimensions
of the
measured
after drying, and were
found
(3.7cm)2
to
(4.1cm)2 as
increased.
and
8%
EVAc was used.
.50,
increasing
a
the
slight
Notice
depth of
rise
in
also that
has weight and
the
sides of
loading
thicker
evaporation
in disk weight.
Some measurements
L=.45
poured
suspension "freezes"
sides,
more horizontally compact,
slabs.
the
likely it
into
trends
suspensions are
the mold, and
horizontal
shrinkage.
diameter for
the
suspension in
loading.
is removed
through
horizontal
Now
fixed horizontal dimensions
decrease with drug
into
smaller.
loading
the
to
Thus, more
slabs were
increase from
For
frozen suspensions.
the THIN BSA
sequence"
depth
slab
thickness considerably
for loadings
loadings
.45
solvent was added,
disk weight and
corresponding "main
used
containing
pm, L=.10)
than half
For
that at
occurs.
(PS=90-106
less
slab.
Note
the THIN
that of
slab,
-65-
most evaporation
the
slab,
is probably
through the
so most shrinkage will be
in
top
and
bottom of
the vertical
direction.
There
and L=.25
in
the
is a sharp
for BSA,
drop in depth and weight between L=.20
PS=300-425im.
fabrication for
the glass
L=.25.
mold after pouring,
This
is
due
to an anomaly
That slab would not
pry from
so
the
early stage
of
evaporation was carried out with
the
slab
the
instead
mainly
of on a screen.
through the
than would
There
is
Particle
there
IV.3.3
slab,
thinner slab
leading
in weight,
occur
is no clear
in depth
to a
occur
size
seems
explanation for
and weight with
to have
There
decrease slightly as
but not depth,
PS=150-180pim.
for P-lactoglobulin
physical parameters.
weight
to
and L=.40 for BSA,
decreases
to
caused evaporation
also a sharp drop
Unfortunately,
appear
the
mold
have resulted otherwise.
between L=.35
The
top of
This
inside
is an
a
and
loading also
lysozyme
relatively
chloride.
minor
indication
particle size
this.
effect
that depth and
increases.
Porosity
An estimate
for
the porosity of each EVAc
disk was
on
-66-
obtained
By
using
porosity
is meant
not polymer.
of
the
be
the weight
e =
Then
V =
Equations
E =
of
drug,
of depth d and radius
tr d
.
(1)
and
toge ther
(2)
p =965 mg/cm
p
3
[2].
the region
)
(cm
of
the
is
the
Let W
disk,
p
p
the
(w/w) drug
given by
r,
yield
of 1.18
Since
cm,
contains
average
the
respective
the cork borer
r=.59 cm.
from section 3.2.
to
to be
1.
that is
or release medium.
porosity
measurements
equal
disk volume
taken
the polymer, and L
the estimated
inner diameter
For
is
table
2
1-(l-L)W/p nr d
p
For EVAc,
an
pore space
the
in
1-(l-L)W/p V
For a disk
(3)
the
(mg) and V the volume
(mg/cm 3)
loading.
(2)
Thus
depth measurements
the fraction of
disk containing air,
density
(1)
the weight and
The
L,
W,
used has
and
d are
last column of
table
estimated porosity values,
plus
1
their
standard deviations.
(BSA,90-106pm),
the
(w/w)
(BSA,150-180pim),
the
loadings.
average porosities are almost
This
for L=.05-.35.
is also
There was
true
a
for
large
increase
-67in
porosity for L=.40-.50.
drop
in weight at
of air
those
This
corresponds
loadings.
incorporated into
Apparently
those slabs
For BSA,(PS=300-425pm,L=.05),
0.00.
This may be due
of
For L=.10,.15,.20,
d.
closely.
IV.3.4
At L=.25,
to
to
the
sharp
there was a
during casting.
the
computed
porosity was
a faulty measurement, most
the
lot
porosities
porosity increases
follow
to
the
likely
loadings
.32.
Kinetics
IV.3.4.1 Axes of graphs
Kinetic
curves are presented
section.
These
release.
Each
a
curves represent
point represents
and discussed
1197
the
loading-particle size condition,
standard
deviation.
dimensionless
root of
the
water,
the
ratio
(reduced)
reduced
(cm 2/hr)
coefficient
and d
is
cumulative
of
the
The kinetic
the
form.
of
the
depth
fraction of
cumulative
drug originally incorporated
Using
the
drug
mass
the dimensionless
average of
and
the
2
, where D
at
are +1
in a
is
the
square
is
the
diffusion
dilution in
(cm).
released F ,
of
drug released
in
the disk.
parameters
of
five disks
plotted
infinite
disk
days)
e rror bars
curves are
drug at
of
(50
The abscissa
e=D0 t/d
time
hours
in this
0 and
The
ordinate
defined as
is
the
to
the mass
of
F0
facilitates
-68-
comparison across experimental
experiment and
e
molecule.
theory.
F
normalizes
therefore
matrix
allows
the
to be
drug.
at
the
results
loading was
to enlarge
(figure
4),
shown
versus
structure of
in
the
9
the
figure 2-4.
for particle
tested.
(figs.
the F
loading curves in more detail
300-425 m
depth and
Plotting F
only the
diffusion coefficient of BSA was
the
curves are
3a),
separate
axis and
show
the
2b and
3b).
to be
low
For
loadings were
taken
m
full
and
(figs.
low
the
Because
2a
90-106
sizes
In both cases,
lower loadings
graphs were made
per drug
studied.
150-180pm, respectively.
bunched
between
Serum Albumin
Figures 2 and 3 display
spectrum of drug
release rate
the pore
Release kinetics for BSA are
and
the
effects of matrix
the effect of
on diffusion
IV.3.4.2 Bovine
reflect
the
diffusion coefficient of
conditions, as well as
tested.
The
2.52x10-3 cm 2/hr
[4].
First consider
marked
similarities.
extremely
As
slow
loading
however,
release
almost
rate
stops.
for
loadings
the
3a.
to
is a
The
two
plots
show
drug escapes at an
less
fractional
that for
progresses
There
2a and
In both cases,
increases,
It appears,
.40,
figures
than or
release
loadings
less
a certain point,
tendency for
equal
rate
to
.20.
increases.
than or equal
to
after which it
the kinetic
curves to
-69-
Figure
IV.2 Release kinetics of-Bovine Serum Albumin
(PS=90-106pm), plotted in reduced form.
D =2. 52X10
a)
b)
3 cm2 /hr.
All loadings.
Low loadings.
-70-
BOVINE SERUM ALBUMIN
90-108um
1.0(ft(
WEIGHT
LOADING
0.8-C-A-
0.50
0.45
-0-
0.35
0.30
z
0
0.4-
*1
U
'a.
-0-4-
0.4- r
3c'
v1
0.2-
0.06
i
$
20
a
1/2
*
SQRT(D 0t)
Figure
1/2
IV.2a
/d
0.40
0.25
0.20
0.15
0.10
0.05
NO MATRIX
-71-
BOVINE SERUM ALBUMIN
90-10um
0.220.200.18-
0.16WEIGHT
0.14-
LOAD ING
0.12-
S0.25
0.20
0.15
0.10-
0.10
0.05
-C
as
cu
Da
4
:C.
0.080.06.
0.04
0 .02
0 ^^
I
II
1
10
1/2 -
SQRT(D t) 1/2/d
Figure
IV.2b
20
-72-
Figure
IV.3. Release kinetics of Bovine Serum Albumin
(PS=150-180im), plotted in reduced form.
D 0 =2.52X10
a)
b)
-3
2
cm /hr
All loadings.
Low loadings.
a
-72...
BOVINE SERUM ALBUMIN
150-180um
1.0-
-
-- --
--
>
CD
WE IGHT
LOADING
0.8.
oa
to
0.50
0.45
ac
0.8-
-0-
0.4-
-+-
0.2-
0
10
9 1/2
-
15
SQRT(D 0 t) 1/2 /d
Figure
IV.3a
20
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
NO MATRIX
-73-
BOVINE SERUM ALBUMIN
150-180um
0.35-
'5.
0 30-
-l
A
'53
ui
4
'a'
-3
hi
0 25WEIGHT
LOADING
z
0
-4
I-.
(.3
4
0. 20-0
'5.
hi
'a.
-4
0. 15-
S0.05
4
-.3
K
(.3
0. 10-
0. 05-
0. mn0
0.25
0.20
0.15
S0.10
5
1o
a1/2 =
15
SQRT(D 0 S1/2/d
Figure
IV.3b
20
-74-
BOVINE SERUM ALBUMIN
300-425um
1.00.9-
-l
0.80
0.7-2
WEIGHT
LOADING
0.25
IC
0.5-
-0-_
0.4-
0.20
0.15
0.10
0.05
NO MATRIX
0.3- L
0.2-L
0.1
0.0
10
5
e
Figure
IV.4.
1/2
-
15
SQRT(D0 t0
20
1/2
/d
Release kinetics of Bovine
(PS=300-425m),
plotted
in
D -2. 52X10
0
-3
2
cm /hr.
Serum Albumin
reduced form.
-75-
"hunch over"
deviations
as
from
loading decreases.
t1/2 kinetics
That
is,
concave downward
become more apparent with
lower
loadings.
Figures
2b and 3b show
loadings
in more detail.
there
an
is
release.
initial
a small
release never
2 and
short burst,
3,
release
is
smoother
however,
differences
for L<.20,
particle
size
released
in
the
there
(i.e.
no
is an extremely
the drug.
trend
i.e.
in
slab
but dependent
increases,
the
is
so does
burst phase.
is
for
.03,.06,
is a
slow but
It
the
should be
(section 3.2).
the
to be
short bursts
independent of
the
As
fraction of drug
(The values
that a certain
there
on particle size.
the
in
anomalously thin
the casting procedure
and 4 appears
As
For L=.25,
discrete burst).
that the L=.25
burst are approximately
ths
loadings,
timepoints,
and
of F
.20
150-180pim, and 300-425pm, respectively.)
of
lowest
that for L=.05-.20,
release of
in figures 2b,3b,
loading
the
(BSA,300-425pm).
The cumulative fraction released
seen
that
by extremely slow
that even at
results for
after which
( 1/2)
to
followed
it appears
t1/2
due
burst,
it appears
stops completely.
steady
recalled,
behavior at low
amount released at all
Figure 4 shows
figures
release
For L=.05-.20,
Notice, however,
there is
the
fraction of
A
at
the
end of
for PS=90-106p±m,
rough explanation
the
incorporated
-76-
drug
lies on
when
the
is
more
the surface of
polymer
likely
is
the
immersed.
a given
that
disk.
For
The model
the
quantifies
this explanation and
released quickly
will
lie
sizes,
on
part
it
the
in chapter V
presented
does
in
predict
the
observed
predicts higher release
that model
although
is
larger particle
particle
surface of
trend,
disk and
levels
overall.
A comparison between
disks
at
(90-106im,L=.10,.15)
in figure
abscissa
longer
5.
is
for
the
THIN disks
thinness
drug
of
shown
in figures
coefficients
taken
to
be
respectively
For
curves for
sequence is
the
is
the
burst appears
higher for
because
the
the
Lysozyme
chloride
respectively.
lysozyme
chloride
The diffusion
P-lactoglobulin and lysozyme chloride were
for
2.82x10
-3
2
cm /hr
and 3.74x10
-3
2
cm /hr,
[5,61.
$-lactoglobulin,
consistent with
of
surface.
and
7,
small,
and
shown
that a greater fraction
P-lactoglobulin and
6 and
is
the
sequence disks
implies
-lactoglobulin
Kinetic
explains why
the main
touch
the main
The value F
the THIN slab
particles can
of
the THIN disks
This
THIN disk.
than for
IV.3.4.4
are
Because d for
expanded.
(90-106pm,L=.10)
the THIN disks
the
loading
the observations
effect
for BSA.
is
qualitatively
However,
the
-77-
THIN VS. THICK
0.220.206.
0.18-
hi
0.16-
LI
4
hi
hi
0.14-
z
0
s-I
I..
C.,
4
0.12-
6.
0.10-
WEIGHT
LOADING
0.15
.. 0.10
-
0.10 THIN
hi
I-I
0.08-
4
s-a
E
0.060.040.020.00
0
10
20
9
Figure
IV.5.
-
30
SQRT(D,
Ot)
1/2/
40
d
Release kinetics of Bovine Serum Albumin
from THIN slab (PS=90-1064m,Lm.10), plotted
in reduced form.
D =2. 82X10
-3
2
cm /hr.
-78-
BETA LACTOGLOBULIN
0.407
0.35
hi
0.30-
C,,
hi
hi
WEIGHT
0.25-
LOADING
z
PARTICLE
SIZE
0
I-
0. 30 300-425um
0.10 300-425um
0.20-
0. 10 108-150um
I..
hi
0.15I..
'-3
U
0.100.05A. n
6
10
01
Figure
IV.6
15
SQRT(D 00
Release kinetics
in reduced form.
D
0
=2.82X10
20
/d
of
2
cm /hr.
P-lactoglobulin, plotted
-79-
LYSOZYME CHLORIDE
0.35,-
0
hi
U)
0 .304
0.25-
hi
-3
hi
z
WEIGHT
LOADING
PARTICLE
SIZE
0 .204
0
I..
h.
hi
0.30 300-425um
0. 10 300-425um
0.10 106-isoum
0. 154
-3
E
0.10-
U
0.05-
0.00
10
t
91/2
Figure
-
15
I
20
SQRT(D 0 t) /2/d
IV.7 Release kinetics
in reduced form.
-3
D3. 74XlO
of
2
cm /hr.
lysozyme chloride,
plotted
-80-
particle size
the
larger
comparison at L=.10 gives
particle size,
Lysozyme
conditions
chloride
displayed,
less.drug escapes.
shows
a burst of
followed
this
observed
for BSA and P-lactoglobulin.
rate-of-release
PS-300-425pim,
L=.30
release for all
by a definite
slope of
slow phase
reversed results--at
is considerably
behavior for
slow
phase.
larger than
There
is a
three
The
that
reversal
in
lysozyme chloride--for
the L=.10 disks
release much
faster
than
the
disk.
IV.4 Discussion
IV.4.1
If
Tortuosities
and retardation
there were no matrix
diffusing drug molecules,
reduced
form, would
(1)
=
F
1
-
(This assumes
During
the
(2)
the
F9
=
providing
then
the
that
form [7]
(4//T)9 1/2
to
the
in
[71
/(2n+l)
the diffusion
stages of
resistance
release kinetics,
follow the equation
(8/n2) 1 e
n=0
early
simpler
factors
.
coefficient is constant.)
release,
(F
<.6),
equation
(1)
has
-81-
(1)
Equation
the highest
in
1/2,
the
matrices would be
has
the
(3)
Fe
real
to
of
the
reach a fixed
R =
2)
(eq.
of Fe
to
to
reach
the
time
that
required
level
for
is
16/ra2
the
of
ratio
reduced
times
is equal
to
the
ratio
of
times.)
path
further
the
length
tortuosity factor T
factor
retardation
(5)
level
from a matrix
for release
time required
given NO MATRIX
Assume
the
curve
a kinetic
ae 1 /2
=
3)
(Note:
to
the retardation due
Suppose
obtainable.
linear
form
release
(4)
release, even
2-7 were
in figures
then a simple assessment of
The ratio R
(eq.
the
loadings.
the actual kinetic curves
If
the EVAc
clear that
is quite
curve in
"NO MATRIX"
provide significant retardation of
matrices
at
It
4.
2a,3a, and
figures
the dashed
plotted as
is
-
=
that retardation
drug molecules
is equal
R,
so
aR1/2 .7
= 4/ a/
that
to
the
is
due
to an
must negotiate.
square
root of
increase
in
Then the
the
-82-
linear
1/2,
in
This
Therefore no
kinetic curves
procedure
However, an arbitrary
curve.
factors and
to compute retardation
was devised
the
exists for comparing
SLAB"
"NO
are generally not
downwards.
but are concave
consistent method
with the
2-7
figures
kinetic curves in
The
tortuosities.
results
procedure allows comparison with the modelling
of chapter V.
kinetic
a =
(F
8
For
-
8
)/(
2
1/24
corresponding
respectively.
(5)
and
As
other
the
to
method
values
the
However,
consistently,
it can be
if a
similar procedure
This
will
be
Computed
done
used
is
was
"a"
timepoints,
then
plugged into
values
of x
(4)
and T.
completely
is
as
long as
arbitrary.
Choice
lead
the method
to
the
to quite
is applied
to make comparisons with
applied
in chapter
the
of 8
timepoints for determining slopes will
different results.
the
slope
fourth
second and
This value of
this
of
)1/2
2
to yield values of R
stated,
slopes
release, but after
the
each curve,
computed, where 82 and 84 are
was
of
F
4
of
an early stage
curves at
original burst.
(6)
the
consists of estimating
The method
theory,
theoretical curves.
V.
and R
are given
in
table
2.
Of
-83LOADING
L
POROSITY
E
TORT. RETARDATION
T
R
BOVINE SERUM
ALBUMIN
90-106pm
0 .05
0 .10
0 .15
0 .20
0 .25
0 .30
0 .35
0 .40
0 .45
0 .50
0
0
0
0
0
0
0
0
0
0
07
12
14
19
25
30
36
40
46
53
698
676
463
268
127
77
45
25
15
11
486804
456472
214301
71612
16211
5924
1983
642
221
127
BOVINE SERUM
ALBUMIN
150-180pm
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0
0
0
0
0
0
0
0
0
0
04
07
13
19
25
30
37
53
58
64
333
297
201
168
84
51
45
24
15
13
111158
88423
40341
28358
7111
2617
2038
567
211
172
BOVINE SERUM
ALBUMIN
300-425pm
0. 05
0. 10
0. 15
0. 20
0. 25
0. 00
0. 07
0. 16
0. 22
0. 32
144
95
79
64
57
20836
8994
6242
4063
3188
BOVINE SERUM
ALBUMIN THIN
90-106pm
0.10
0. 15
897
804524
S-LACTOGLOBULIN
106-1501im
0.10
0. 10
405
164168
-LACTOGLOBULIN
300-425pm
0.10
0.30
0.12
0.38
1101
78
1211967
6128
LYSOZYME CHLORIDE
106-150-pm
0. 10
0.07
209
43472
LYSOZYME CHLORIDE
300-425pm
0.10
0.30
0.12
0.30
134
173
17824
29838
Table
IV.2
Average porosities, tortuosities,
factors.
and retardation
-84-
that when comparing BSA (90-106im)
interest is
the values of
(150-180pm),
the
largest particle
similar
loadings
tortuosities
presented
size
there
trapped
halt
in
for
"Fast" and
slow
appears
slopes
that the
slope of
approximately
after
release curves.
the
the original
model
low loading and
fraction of
particle
is
drug that
one would expect a premature
a certain time.
t1/2
seen in
that,
at
followed by
(1/2
the kinetic
same
As
It appears
the case.
that follows
release
anomalous,
the conceptual
of
not
is
the
loadings of BSA.
of release
that at
is
and
release begins with a burst,
loadings,
burst are
phases
release after
this
The
sizes.
chloride are
lysozyme
"slow"
at
P-lactoglobulin are close to those
polymer matrix,
the drug
extremely
also
for
should be a significant
in the
the
to
tortuosities measured
the smaller particle
in chapter II
figures 2b,3b,4-7,
low
than the
consequences
the
One of
tortuosity for
tortuosities
the
(300-425ptm),
size
anomalous behavior of
the
reflecting
the
quite similar
corresponding particle sizes
measured at
IV.4.2
for
assessed
tortuosities
The
lower
are
for BSA disks
similar for
the main sequence BSA (PS=90-10Lm,L=.10) slab.
for
tortuosity
At
(PS=90-106pm,L=.l0) is
slab
the THIN
quite
Also,
(especially L>0.25).
loadings
equal
T and R are
to BSA
) kinetics.
curves after
for L=.05,.l0,.15,.20.
burst cluster around
the value
It
the
The
-85-
a =
/d91/2
dF
Plugging
retardation
~
.006
this
value of "a"
factor of about
The persistent,
burst in disks with
extremely slow release of
low
that release
first,
The
phases'.
interconnecting
The
the matrix.
of vortexing
It will
any disk
allows
drug
through
to
is
is
in two
the
of
surface
the
release of
the
during
defects in
the
slow
phase
of air as a
or by
slabs,
result
the
process.
fast phase
in
the
the
polymer matrix.
the
trapping
fabrication of
be assumed now
estimation of
is
mechanism of
could be caused by
Specifically,
the
that
phase
It
however.
evacuation of
the
is
that extends
the
that when
drops below a certain
completed.
above),
phase
or "fast"
second, "slow"
evaporation
solvent
This will
from EVAc matrices proceeds
porous network
diffusion
is
defects
These
the conceptual
that
theme,
central
the
drug after
II must be modified.
the
strong candidate for
release
a
yields
drug by an unidentified mechanism.
"trapped"
A
implies
loadings
done without abandoning
now assumed
(6)
into equation
160000.
release from chapter
model of
be
.
if dF
point,
/d61/2
is assumed
the release
then
the
falls
total
fraction F
pore
space
connected
of
to
fast
below
terminated.
the
rate from
This
phase
.006
is
(see
criterion
incorporated
the matrix
-86surface.
for
chapter.
Figure 8
is
seen that at low
size.
and
is a plot of
At high
150-180ptm.
Because
to
than
the
data
latter particle
If
loadings
the
in table
the
would
lie on
It
only for 90-106im
size yielded
results
plot were
3.
particle
increases with
the former,
the right.
the points at higher
F
there exists
loadings,
higher porosities
are shifted
loadings,
the BSA data
the above
of this
in the experiments
used
the disks
criterion
derived using
the values of F
3 lists
Table
for
of F
top
the
latter
versus L,
of each other.
-87-
LOADING
L
POROSITY
F 00~
E
BOVINE SERUM
ALBUMIN
90-106 ur
0. 05
0. 10
0. 15
0. 20
0. 25
0. 30
0. 35
0. 40
0. 45
0. 50
0.07
0.12
0.14
0.19
0.25
0.30
0.36
0.40
0.46
0.53
0 .031
0 .034
0 .042
0 .072
0 .195
0 .296
0 .521
0 .832
0 .965
1 .007
BOVINE SERUM
ALBUMIN
150- 180pm
0. 05
0. 10
0. 15
0. 20
0. 25
0. 30
0. 35
0. 40
0. 45
0. 50
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
04
07
13
19
25
30
37
53
58
64
0. 066
0. 074
0. 103
0. 189
0. 306
0. 467
0. 521
0. 851
0. 959
0. 983
BOVINE SERUM
ALBUMIN
300-425-pm
0. 05
0. 10
0. 15
0. 20
0. 25
0
0
0
0
0
00
07
16
22
32
0
0
0
0
0
BOVINE SERUM
ALBUMIN-THIN
90- 106pm
0.10
0.15
0.068
-LACTOGLOBULIN
106- 150pm
0.10
0.10
0.089
-LACTOGLOBULIN
300-425im
0.10
0.30
0.12
0.38
0.016
0.323
LYSOZYME CHLORIDE
106-150pm
0. 10
0.07
0.101
LYSOZYME CHLORIDE
300-425 pm
0.10
0.30
0.12
0.30
0.289
0.200
Table
IV.3.
197
224
270
327
456
Average fraction F, of incorporated
drug that is connected by pores to
the surface.
-88-
a
1.04U
A
A
U
0.8-
A
0.6a
F
A
IL
BSA
BSA
BSA,
90-106 pm
150-180 pm
300-425 pIm
A*
0.4A
0.21
U
A*
A
a
0.0
0.0
a
OR
i
i
0.2
0.4
0.6
0.8
POROSITY
Figure
IV.8
of originally incorporated drug
Total fraction F
that is released during "fast" phase, as a
function of volume loading (porosity) and particle
Porosities obtained from table 1.
size.
-89-
REFERENCES
[1]
Rhine, W.,
of
the sustained release
fabricate reproducible
kinetics,"
[21
Miller,
J.
E.S.,
water-soluble
for
to
systems and constant release
69,
and
N.A.
Peppas,
copolymers,"
acetate
methods
macromolecules:
Pharm. Sci.
bioactive
"Polymers
and Langer, R.,
Hsieh, D.S.T.,
agents
Chem.
Eng
265
(1980).
"Diffusional release of.
from ethylene-vinyl
Comm.
22,
303-315
(1983).
[3]
Kozinski, A.A
and Lightfoot, E.N.,
ultrafiltration:
filtration,"
[4]
Lehninger,
York
[5]
[6]
A.I.Ch.E.
A.L. ,
J.
18,
of boundary
1030-1040
layer
(1972).
Biochemistry, Worth Publishers,
New
(1977).
Edsal,
K.,
a general example
"Protein
J.T.,
eds.),
Crank,
(1975).
J.,
in The
Proteins,
Academic Press,
Mathematics
New
(Neurath, H.
York,
and
Bailey,
(1953).
of Diffusion, Clarendon, Oxford
-90-
V.1
Introduction
that for
loadings,
low drug
incorporated drug remains
chapter
(see
When drug particles are
particles
to
loadings,
only particles near
contents
into
IV it was
In chapter
After
the original
loadings,
there
is
of drug.
The mechanism for
picture
maintain
results
of
the surface
mechanism,
the
some of
the
At low
connect to and
this
picture
the
data of chapter
is almost
release at
low
slow but persistent release
the
of chapter IV
with
the matrix.
behind a porous
can
"burst" of drug
an extremely
keeping
that
the release medium.
release, however.
In
of
assumed
release medium.
latter
phase is
do not alter
It
is
still
that almost all release occurs
network.
it was
they leave
shown that
correct.
The
the
communicate with
is
explain this
between particles allow
Connections
their
the end
At
to
Basically,
released,
carcass.
release
the release
situated randomly within
particles are
drug
trapped drug.
11.3).
figure
indicate
the originally
the matrix after
a heuristic model was given
II,
phenomenon
chapter II,
a fraction of
in
there exists
i.e.
complete,
in
discussed
Previous results,
the
RELEASE
INCOMPLETENESS OF
MODELLING THE
CHAPTER V.
basic
unknown.
the fundamental
to
possible
through the
hypothesis
on
pore
the
release
IV was analyzed with the
slow
-91-
phase of
release factored
In chapter
describes
it was
the
depth of
would
EVAc
of
pores)
in
percolation
of
the
matrix.
the
total
1V can
shape of
Section
theory
(and
[1,21.
In
the
bordering
the
release, and
these
Thus,
there
be made.
Both the
the
simulations of
that are modelled as
total
fraction of drug
release kinetic
the
3 describes
functions of particle
the
particles
clusters
to
present case,
percolation
drug
computer
the
program for
size and
representation of
presents
loading.
simulating release
corresponding results.
curves are
algorithm for computing
fraction of drug released and
the
the
literature with which a direct comparison
drug from matrices
the
the
the whole matrix
Section 2 describes
the
presents
cluster
However,
present case,
typical percolation studies.
lattices.
released and
(in
Moreover,
chapter consists of computer
desorption of
describes
a
the
smaller than
[1,2].
that spans
in
(in
the whole system
size of
data of chapter
studied.
much
size
the matrix also contribute
theory
the
is
fields.
theory
theory deals with systems
release system, small
not counted
This
as
size of
sustained
is no
size)
the
surface of
percolation
the matrix)
predict
therefore
that
that percolation
from other
characteristic grain
drug particle
the
indicated
similar results
characteristic
are
III it was
pointed out
in which
the
some
out.
In
the
results
Section 4
kinetics
section 5,
and
-92-
between the
experimental results
of chapter IV.
that the
topology in
random pore
This motivates
the work
for
the
of a
The dispersion
simulations.
lattice with dimensions
there are N
layers
long.)
64x64=4096
is
Each cube
The
drug.
The
sites
cubes.
(It was
the volume
fillings
The
drug or
polymer does
filling of
the
subroutine RANDM,
lattice
which is
two
That
not
(cubes),
found
is,
each layer
and
convenient
is
that
to
64 bits
either with polymer or with
loading,
of any
That
.
on a VAX word
a given
that
statistically independent.
with
bits
as
is modelled as a
64x64xN
lattice
is "filled"
probability
specified by
matrix.
of
lattice sites
represent
to be
be considered
random dispersion was
simple cubic
consists of
drug
the
thick.
A simple representation
chosen
sustained
the matrix and
The matrix can
particle diameter d .
particles
VI.
the matrix
depth L of
the
release matrix are
N =L/d
experimentally.
characterizing a
lengths
two important
The
section 5
in
shown
the
simulation does not
the
chapter
of
Computer representation of
V.2
It is
the slow release observed
to explain
suffice
and
theoretical results
comparison's are made
cube
is
listed in
the
e, of the
lattice are
whether cube
influence
sites
in
with drug
filled
or porosity
cubes
is,
is
how cube
executed
A
B
by the
appendix ITI.
is
filled
is filled.
VAX MACRO
-93-
The
and
two
drug
Drug
important fabrication variables,
particle size,
loading is
loading.
can be represented
represented via
It should be noted
normally
the
that
loading and
the drug
The
second
parameter for the
fabrication
enters
the
matrix
depth L
An example
example,
depth
=6,
N
of a
the
lattice
the
with drug particle
(porosity)
for
lattice
is
im),
the
cubes are
layers of
In
the
this
facilitate
use
either
the
drug
scheme.
particle diameter,
depth NP,
shown in
once
the
1.
In
"D".
drug
N
included
a lattice
explanations.
clarity,
is 0.25.
simply changes
lattice are
For
of depth
this
Each
1 shows
lattice.
"-"
The volume
Note
that
particle
to
3,
in
the
coordinate
lattice
1.5mm,
figure
represented by dashes
are marked by
chapter
figure
150-180im.
size
full 64x64x6
this example
one
to
represent a matrix
Imm deep matrix with a larger
300-425
in order
size ~250pm.
lOxlOx6 core of
cubes
translation
lattice might represent a matrix of
Alternatively, it could
drug-filled
a
is
specified.
so
Polymer-filled
loading
present simulation
1 mm, with drug particle
only a
scheme.
Therefore
parameter, drug
simulation via
is
this
porosity must be derived,
theoretically or experimentally,
loading as a
loading
porosity, or volume
taken as a weight fraction.
between drug
in
drug
and
loading
to model a
diameter
so
(say
that only
three
simulation.
system
site
is
used
to
has coordinates
-94-
D
-D
D
D
0
D
D
-D
D
D
D
8
D-
0
LAYER 1
D D -
-0
D
(TOP)
D
LAYER 4
D
DD
DD-
D
D
D
-
D
D
-0
D
LAYER
2
D D-
D
0-
--
-
- D
D
LAYER 5
0---
D
D
D D
-0
D
D0
D
D
D
-0
-
-0
D5-
D0
LAYER 3
-
- D-
-8D
-0--
-
-0
-D-
FIGURE V.1.
-0
-0
D
-
LAYER 6
(BTrToM)
0
D
-
Illustration of lattice representation of polymer/drug
matrix. "-" indicates polymer, "D" indicates drug.
Drug particle on lattice site (4,5,3) is circled, and
its neighbors are circumscribed by squares. N =6, £=0.25
-95-
(r,c,t),
corresponding
coordinates
N
.
r and
The
(row,columnlayer).
c vary from
For example,
coordinates
to
the
on
such
real drug
described
the drug
the matrix,
subject
cannot overlap.
artificial,
particles
(see
chapter
qualitative
is
not cubic,
a rough
but are
total
they are
that
distributed
in
two particles
of
layers
parameter.
irregularly
studies
in model
their centers
can appear anywhere
very existence
these differences
differences
of
Rather,
constraint
evidenced by many
A very simple
The
to the
so that N
III),
line up with
is
Second,
drug
shaped.
in percolation
should not
lead
theory
to
predictions.
amount *of drug
released
Description of algorithm
V.3.1
total
do not
particle centers
the
Calculation
V.3
1 has
can only provide crude
lattice.
Thus
are
However, as
in figure
controlled release matrices.
particles
of a regular
that
from 1 to
(4,5,3).
lattices
sites
varies
encircled drug particle
representations of actual
First,
and I
1 to 64,
The
algorithm was
fraction of drug released
main driver
program
chosen
from the
for the algorithm is
USEFUL2, which
is
listed
to calculate
the
simulated matrices.
coded as
the
in appendix II.
FORTRAN
-95-
Before describing
required.
one
If
and
Two
lattice sites
only one of
one views
considered
scheme,
the
their
lattice
neighbors
if
lattice
lattice
sites as
particles are considered
situated
on neighboring
lattice.
cubes,
two cubes are
then
a square
face.
sites are
In
1.
this
not neighbors.
encircled
drug
particle
"neighbors"
sites
are
a chain of neighboring
on
said
if
they are
the simple
to be
cubic
"connected"
drug particles
between
of all drug particles connected
if
them.
to a
drug particle.
a "$",
drug particle
is
if
to a cluster
it belongs
eventually find
that
belongs
is
to a
considered
their way
formed
cluster
around
then
polymer and
cannot release
polymer
is
particle
is
With
drug
impermeable to
marked
of
by a
in
that
not extend
particle
its
is
to
particle
the matrix
the cluster.
that does
the matrix,
that
out
releasable, and marked
that extends
The macromolecules
the matrix.
space
lattice
drug particles
"cluster" consists
A
of
Two
is
single
differs by
the
Two drug
A
coordinates
they share
sites of
"neighbors" if
1 are indicated by circumscription with a square.
figure
there
few definitions are
are considered
diagonally adjacent lattice
The neighboring
of
the algorithm, a
macromolecules.)
surface
will
through the pore
If the
particle
to
surface of
the
"trapped"
molecules.
the
within
the
(Recall that
the
In
this case,
"T".
these definitions,
by
the algorithm for determining
the
-96-
the
fraction of drug can be
releasable
sites are
lattice
.
of altering this
consequences
lattice
Each iteration
After
to drug particles
several
iterations,
are marked with a "$".
When
the
iterative
markings,
particle
All
the
"$"
in which drug particles are marked with a
are neighbors
"$".
iterative routine
through
consists of a pass
in
discussed
assumption are
initialization, an
After the
section 3.4.)
(The
of a matrix.
faces
and
1=1
that drug release
assumption
the
two broad
the
through
ensues.
to
if
releasable drug particles
is no
there
they
by a
that are already marked
all
the
surface
The
i.e.
bottom layers,
top and
corresponds
This
occurs
the
so
drug are releasable,
containing
particles are marked with "$"'s.
surface drug
I=N
sites
lattice
surface
First, all
described.
change
is
procedure
drug particles not marked with a "$"
in
the
terminated.
then marked with
are
a "T".
The
marked with a "$"
appendix II as
The
example
lattice
via
the algorithm,
marked
This
1,
is
when applied
process,
shown
in figure
first consider
by a circle.
the chain of particles
(3,1,1).
ESCAP2.
iterative
of figure
listed in
in VAX MACRO and
the subroutine
result of the
illustrate
(4,2,3),
is coded
are
which drug particles
algorithm for determining
It
is connected
(4,1,3),
chain is marked by
the
(4,1,2),
diamonds
in
to
the
To
2.
particle at site
to
the
(3,1,2)
the
surface
and
figure.
-97-
--
--
e~-
-
TT
-
LAYERI
T
$-
-
-
$-
(TOP)
(20
- -
--
-
- - $- -
$ -
- - - - -
- T
T --
-9$
T T-T - - - - - - T
$--------
T------T--
* ---------T - -T$
- - - - --- -- -
~~T
-
-
T-
- ---
- - -- - -
T--T- --.-
T-------LAYER 5
- - - - - - -- - - - - - - -T
- - -- - - - - - -
- - T- - -T - -- T
T - T - - -- - - -
-----------------
-- - -
- - -
-
-
- - .-
-
- - - ---------
9---
---
T----
---
-
p-p--p
$
- -
-
- - -
p
--
- -
- -
-
--------
T - - 7--s--7
- T -$
- --
FIGURE V.2.
LAYER 6
(BOTTOM)
---- ----
T --
LAYER 4
- T - -T - - - - T
2-----------9--T
LAYER 3
T
.T
-
- - - - - - - T- -
- $--T
- -0
LAYER
T--
-
- -
-- T T ----
T
T T
--- - -- -- -----$ - - 0
- - $ - - - - - - - - -
-
---
Result of iterative process determining releasable and
trapped drug, applied to figure 1. "-"
indicates polymer,
"$" indicates releasable drug, and "T" indicates trapped
drug. Chain of drug particles connecting circled drug
particle at (4,2,3) is indicated by circumscription by
diamonds.
Trapped drug particle at (4,6,2) is circumscribed by a square.
-98-
Thus
the
with
a
particle at
"$".
(4,6,2)
On
so
After
the
the other
(marked by a
(polymer),
(see
(4,2,3)
the
appendix
it
is
square),
of
releasable
V.3.2
counts the
these
particle
is surrounded
marked
at
site
by dashes
"-"
thus marked with a "T".
the MACRO subroutine
number of
"$"'s,
is
to
taken
COUNTD
and also counts
(i.e.
two numbers
it is
"D"'s).
be
The
fraction of
the
drug.
Results and Discussion
results are
average
and volume
the
and all
shown in
figure 3.
Each point
Figure
increases
e.
loading
drug particles
3 shows
that
with volume
layers.
the
The
those
Clearly, as
volume
for a particle
its
of being
case Np=2
top and
layers, a
surface via
smaller clusters
trivial,
bottom
layer,
releasable.
decreases with
increases,
neighbors
in a cluster extending
With fewer
is
layers are
loading
to have
increases.
an
total fraction released
loading, and
opportunity
chances
in
represents
of random lattices with
lattice consists only of a
the
The
for NY=2,3,4,7,10,12,15.
taken over 10 realizations
depth N
of
drug
filled sites
Simulations were run
since
the
trapped and
total number of drug
quotient
hand,
iterative routine,
II)
so
is releasable,
particle
the number
the
increases,
to
the surface
can connect
which can exist at
so
to
the
lower volume
-99-
1.0+
V
0.8+
z 0 . 6-0
-A-
H
rT4
0 04-4-
-c0-
2
3
4
7
10
12
15
LAYERS
LAYERS
LAYERS
LAYERS
LAYERS
LAYERS
LAYERS
0
0. 2-
0.0
0.1
0.2
0.3
VOLUME LOADING
Figure V.3.
0.4
0.5
E
Results of computer simulation of total fraction drug
released as a function volume loading E and number of
lattice layers N .
-100loadings.
The
second
prediction
the
If F
loading,
(1)
with
that
loading,
result, of course,
larger
translates into
particle
sizes
at
fraction of releasable drug will
is
the
total
then
the
following
lim F
= 2/N
lim F
=
=2d
fraction released and
the
a fixed volume
increase.
E is the volume
limits must hold:
/L
and
(2)
Both
1
cc
E+*1
limits are
consequence of
evidenced
the
fact
in
figure 3.
that two
layers,
and always will
chances
of making contact
release
to
of
deeper
to e.
lattice models
of a porous matrix.
approximately for any model
distribution, especially for
It
is
at
E=0.5.
interesting, however,
An
interesting
feature
(1)
the
N
(1)
is a
layers are
their contents.
proportionally
Limit
Limit
As
surface
e-0, the
layers vanishes
holds
for all
It will
homogeneous
also hold
that assumes homogeneous drug
d <<L.
that
Limit
limit (2)
in figure
3
is
(2)
is
obvious.
is nearly
that
the
reached
-101-
transition between
released
to
sharper as
size
the region of a
is
large,
come only from
the volume
figure
for
to
the
the majority of
the
large
simple cubic
results of
e=0.10,
larger.
(see
Shown are
drug lies
but not all
of the
clusters
be
e=0.30).
For
practically all
indicates
that
This will
be
V.3.3
As
small,
on
particles
the
+o,
the
in
section 3.3).
for understanding
the
lattices generated
for
For E=0.,10,
surface
layers.
(If
N
the
so
true no matter
practically
For
e=0.30,
releasable,
is
the amount of
clusters are
be
curves
until
probability curve
interior drug becomes
the
will
large and
the releasable
made
larger,
releasable
drug
lattice at
large
drug particles are releasable.
this
particle
fraction;
proceeds deeper into
e=0.50,
the
releasable drug
can now be formed.
seen as one
is
percolation
then a very clear gradient in
can
increases (or
and 0.50, with N =6.
releasable
because
fraction released becomes
As Np
is a heuristic aid
0.30.,
fraction
clusters which do not appear
lattice
figure 3.
total
because when N
3 should approach the
the
some
is
loading gets
Figure 4
all
This
clusters contribute
when N
large
number of layers N
the
decreases).
small
the region of a small
how
that
Figure
large
N
3
is.
discussed below.
Comparison
indicated
to percolation
in
section
1,
theory
the
present simulations
are
-102-
-6--
Layer
(t p
6-----
.....
---
----
1
--
S----g-
-----------...--
(top)...
---
-T
-
g
-a??---.-s
----
T
-
T
--
I
-- SS--Tgg
?
551
-
Layer 2---3--s
g-
---
I-
ss
.
.-.
? -
..
- -I
--
s..-..
-. -. s..-..-I.-
Layer34.....
-s-
-------- S.. ..-.-S ---TS -s------
-f--
Lae
Layer
?T-
--
f--.-.
.-
. . .-.-.
5
S-u-
46
s~-ss
-----
-s--us---
(bo tom
---
-----a)s-e-
Layue 5.4
s-------g-e
)--s-
Ditiuino
legg-
---
rapdadrlaabedu
---
o
s-digacss-6-aa=sb
=.0
he
c
=.0
-103-
aimed at a slightly different problem
percolation
percolate,
surfaces
section
theory.
One difference
a cluster of pores
of
the matrix
3.1,
clusters
surfaces
are
connected
computation
and
that
for
(3)
lim P = 0
(4)
small
>
F
Limit
of
e.
the
The
to
surfaces,
the
the programs
percolation
must reach both
simulations
the
of
lattice
of appendix XX
probability
results
Moreover,
is
due
to
isolated and
fact
sites)
to
are
3.
P
shown in
One
as a
figure
5,
clear difference
is
e,
of N .
connecting
the
in order
for all N
and
E,
P
(3)
become
that,
to one of
can be compared with figure
regardless
In
that approached by
identified.
function of N
which
is
(lattice
(lattice).
Simple modifications
permit
than
to both
that
the fact
that as E-*O,
therefore
faces.
cannot belong
Inequality
in order for a
particle
it must be connected
to approach
Figures
F
as N
all
to at
(4)
particles
to clusters
is a consequence
to b-e connected
least one.
of
to both
P does
tend
and E increase, however.
6a,b show a comparison
of P and
F
for N =4 and
-104-
1.0-
0.84>1
---
0 . 68+
2 LAYERS
3 LAYERS
LAYERS
7 LAYERS
10 LAYERS
- - ._ 12 LAYERS
.. __ 15 LAYERS
-A-4
z
0 . 4-
E4
0 . 2--
0 . on
o.1
0.2
0.3
VOLUME LOADING
Figure V.5.
0.4
0.5
E
Results of computer simulation of percolation probability
as a function of volume loading c and the number of
lattice layers N .
-105-
4 LAYERS
1.0-0.8-,A
0.8-,'
a)
FRACTION RELEASED
-
F0
PERCOLATION FRACTION
P
0.4-0.2-
A
0.0 1 "- ,
I
i
-1
0.0 0.1 0.2 0.3 0.4 0.5
E
VOLUME FRACTION
15 LAYERS
1.0+
£
0.8-0.8-b)
0.4--
0.2-
7:
A
.. FRACTION RELEASED
.-
F
... PERCOLATION FRACTIONW
P
/3;
A
0.01
-
0.0 0.1 0.2 0.3 0.4 0.5
VOLUME LOADING
Figure V.6.
E
Comparison of total fraction released F. and percolation
probability P as a function of volume loading E for two
lattice depths N . a) N =4, b) N =15.
-106N =15,
respectively.
pronounced,
pore
that extend
surface effects are
through
lesser
in percolating
These
the
surface
and much drug can be released,
clusters
situated
For N =4,
the matrix.
and a greater
observations
through
For N =15,
the
fraction of drug
show that a membrane's
permeability is not necessarily equivalent
drug.
though not
is
clusters.
comparisons
for releasing
effects are quite
This may explain
that some
"face-to-face"
to
its
capacity
unpublished
spent EVAc matrices are
impermeable
to
drug.
It has
at
[1]
for a simple cubic
e=0.50, virtually all filled
a cluster
for
been shown
of
e=0.50,
infinite extent.
practically all
sites
This
in a
that
lattice belong
explains
drug is
lattice,
the
releasable,
result
to
that
regardless of
N,.
NPO
V.3.4
A
Effect
of
release
from matrix
rim
typical sustained release matrix used
experiments
depth
.15
means
that a
of
cm.
chapter
IV
was
For a drug
"true"
a
disk
of
particle size
representation
in a
diameter
lattice
sites.
are
still cubical,
is
somewhat
larger
sites
than
the 64x64x15
1.2
cm and
lattice should
120
lattice
the
of 90-106pm, this
cylinder with diameter of
(The
in
sites and
lattice
depth
however).
provided
be a
15
This
in
the
-107-
simulations described above.
300-425pm,
a cylinder-shaped
depth of 4
sites
64x64x4
used
lattice
above.
polymer disks,
was
only allowed
occur
through
in the
the
the
the
experiments
rims
64x64
sites and
than
simulations,
two
of
of
the
release
faces
of
the
depth L,
the
lattice.
ratio RA of areas
flat faces
(5)
RA
For
the
=
release
the matrix
(-xaL)/[2n(a/2)
case L=.15
the
cm,
by
results of
the
size
the
rim of
2
]
=
a=1.2
simulation
allow release
rim
to
the area
of
the
two
2L/a
cm,
predicts R A=0.25.
In
.
(5)
described
through
the
in
rim,
section
3.1,
which
might underpredict
as much as 25%.
Therefore,
modified
of
of diameter a and
combined is given by
words,
does not
the
of
the
it was
felt necessary to
simulations
the
lattice
lattice.
the worst
test whether
presented above are
and
the boundary
This was done
version of USEFUL2.
lattices were
is
smaller
through
the above
For a disk-shaped matrix
other
is
Furthermore,
while for
to
particle size
of diameter 20
This
IV, release was allowed
the
cubic
a drug
is appropriate.
lattice
of chapter
For
In
sensitive
to
condition imposed
using a
at
slightly
this version, cylindrical
generated with diameter 8 times
case, corresponding
the
to
the
the
example of
depth (this
the
-108-
previous paragraph).
Thus a was
this
that fit within
led
to a
lattice.
to
fit
the
lattice
For N,>8,
the
64x64xN
two
it was necessary
lattice.
cases where release
lattice sites.
The
can and
case of
release
than
However,
were
truncate
then
<8
64x64xN
the
cylinder
then
through
run for
the rim
compared for
the
it is
shown in figure 7.
through
the
rim
also seen that
the
tend
occur at
the
to
predictions
do not change
particles
likely
It
to
that are
be releasable
should
curves do not
outside
lower
the
be noted
sized cylindrical
The results of
simpler
modelling
64x64xN
purposes.
to
loadings.
by much.
releasable
through
that
represent the
predictions
through
effect
differences of up
It
the
is
the
16%
Thus
are
full
the
cases
rim.
the
but they
absolute
loadings,
the
rim are
of
with N >8,
the
matrices are
the
allowed.
simulations
adequate
indicate that
for
two
they lie
that would have been given if
the modified
also
the disk.
size matrices, and
matrices were
that
greater
seen,
At higher
through
seen
actually quite
the flat faces
for
is
does predict
case of no release
Relative
the
to
original
simulation was
small.
full
the
For N
cases.
release
drug
to be 8N .
cannot occur
The values of F
Typical results are
the
chosen
the present
-109-
1.04-
0.84-
0.84z
0
H
0.4--
0.24.
-A
0. O-L
U.U
-
I
0.1
N =4, rim coated
N =4, rim exposed
5.L,D
rim coatea
N =15 , rim exposed
i
0.2
0.3
0.4
0.5
VOLUME LOADING
Figure V.7.
Comparison of total fraction released F as a function
of volume loading 6 between lattices that allow release
through their rims (uncoated) and lattices that do.not
allow release through their rims (coated).
-110-
Simulation of desorption
V.4
from random
(release)
lattices
V.4.1
Introduction
The previous
the
final
total
section was
fraction of drug released
represented by a random
to how
the
drug would
not computed.
under
on
the
the
within
No assumptions
be released,
and kinetic
assumption that
lattice
The
pore
determine whether
sites
insight
into
serve
are
(i.e.
those
as
curves were
this
execute
section,
random walks
filled with
an approximation to diffusion
chapter
the
two purposes.
random pore
in release
experiments of
computed in
were made
space.
simulations
retardations
are
drug molecules
Such random walks
the
from a matrix
lattice.
Kinetic curves
available
"#$"'1s).
concerned with calculation of
shapes of
they
can
topology can produce
comparable
IV.
First,
Second,
to
those
the
the kinetic
seen in
simulations
curves
the
give some
seen in chapter
IV.
V.4.2
Description
Suppose
lattice
Then
the
cell
of
the
algorithm
a drug molecule originates
(r
c'40'z 0),
drug molecule
and
at
time
that cell
is
marked
can diffuse out
of
the
T=0
in
by a
lattice.
It is
-111-
diffusion
of
In
from site
molecule
the
to site
in
are
seen
that
dissolved,
and
the drug
tracer molecule.
to a
space available
this means
the clusters
simulation,
present
the
pore
thought of as a
the
cluster.
releasable
open
of
The
1 0 ).
(r 0 ,c 0
tracer molecule can be
the
"hopping"
random
that can diffuse
"tracer"
containing
cluster
the
within
freely
is a
the molecule
that
assumed
in all
pores
the
Physically
assumed
is
to be
to be
interact with each
that drug molecules do not
other.
A necessary input
time,"
or
time"
"residence
times are
identical).
in a pore
is
that
assumed
and
stochastic.
(r,c,Z)
(6)
6/Nrc.
A
site
surrounded
possible in
residence
in
the
time
the "hopping
is
in a pore
the
(the
residence
time
if N rc
then
is
is given
in
this
the
is
the
two
time
simulation it is
the
deterministic,
the number of neighboring pores
(r,c,2),
site
molecule at
=
matrix,
However,
Specifically,
sites).
T rc
of a molecule
the molecule's residence
bordering
sites
simulation
the
In a real
dependent only on
lattice
to
("$"
number of "$"
residence
time
for a
simulation by
*
by six other
simple
of 1.
cubic
On the
"$"
sites
(the
lattice) will
other
hand,
maximum
then have a
a "dead-end"
"$"
-112-
site,
which
is
connected
have a residence
The form
leave
passing
(6)
will be
A dummy
is
be
to
leave
time
number
time T
at
faces
the
the
to all
to
(r,c,Z)
the
are
the molecule
the
running
layers
lattice,
has
and the
is
molecule
time
Lf
left the
process
in
a molecule
the cube.
is
the
at
Its
before
0 and
lattice
N +1.
completing
situated
T,
at
success
to
and
the new
lattice
of
the
is
sites,
and
This allows
the
the random walk.
described.
site
sites N rc
The
outside
The dummy site
step can now be
tracer
updated.
of
surface
(with uniform probability),
moves
that
of attempts
simulation.
cy are first determined.
and
be justified
(6).
rc
sites,
site),
six
of neighboring lattice
random
added
the
simulation
T,
argued
representing
a "neighbor"
The basic
at
site
to
will
on a particular attempt will be equal
identified with
molecule
time will
can be
expected number
lattice
added
considered
that
the
success
so the
site,
site by making repeated attempts
given by equation
lattice
hopping
it
through one of
to N rc/6,
"$"
6.
for the
a cubic
probability of
might
time of
Intuitively,
chapter VI.
will
to only one other
and
Assume
(r,c,Z).
The
the residence
simulation then
one of
the
that
site.
T
the
lattice
chooses
neighboring
is
then
coordinates
1 is either 0 or N p+1, then
(i.e.
terminated.
moved
to
Otherwise,
the
the
dummy
-113-
simulation
step
The
on a
lattice.
Since
lattice.
the
completed,
F
,
the new
there are
The
lattice
were
distribution function
time,
at
the
was
final
64x64=4096
cumulative
at each
sites
per
times T required
After
site
"$"
layer,
e walks were performed
tabulated
p(T).
site.
started once
that approxima.tely 4096N
through the
leave
repeated
tracer molecule was
this means
to
is
as
a
all
for a
tracer
frequency
the walks were
fraction released as a
function of
using
computed
t
(7)
F
=
fp(T)dT
0
The programs
II.
The
the
simulations are
main driver routine
RRWALK calls
which are
for
the MACRO
is RRWALK,
section
subroutines, NABOR2 and RAND01.
which
backward, upper
number
sites
"$"
and
uniformly distributed
is
a FORTRAN
subroutine
of
neighbors
around
A
simulated
fraction of
is
have
There
NABOR2
left,
ESCAP2,
are
"$"
kinetic
in appendix
in FORTRAN.
and
COUNTD,
two other MACRO
scans
the
lattice
and
right,
forward,
RANDO1
returns a random
between 0 and
1.
Finally,
called NABPR which counts
drug released
obtained by
3.
lower neighbors.
a
coded
subroutines RANDM,
described in
identifies
listed
there
the number
site.
curve,
as
plotting F t,
a
which
shows
function of
the cumulative
a normalized
calculated using
(7),
time,
against
-114-
t 1/2IN
.
is
number of
the
The normalization by N
layers,
reflects
and not
the
the
fact that N
physical depth of
the
matrix.
V.4.3
Deterministic
As
a standard
algorithms
of random walk
performed.
This
site
the
has
lattice
that a
simulation
uniformly
(8)
1n =
1
It
perfect
(9)
FT
(n=O),
across
1/N
is
P
so
molecule
time
layers,
that
sink (absorber),
=0
the
time
is
step n.
is
and each
1.
Let fff
situated
At
the
at
be
a
start of
the
distributed
P
the
outside
of the
lattice
is a
so
for all
n.
transitions
A cell will receive
1/6
described.
so
1=1,.. .,N
,
E=1 was
sites have a "$"
the drug molecules are
all
Consider now
be
the hopping
tracer
also assumed
probability
lattice with
lattice
in layer 2 at
n
n
= 11
N +1
0
P
lattice.
all
six neighbors,
site
in a
computation will now
for s=l,
probability
results
for comparison, a deterministic
computation
Since
for comparison of
if any of
its
the
that can
tracer at
occur
in
step n+l
six neighbors has
the
the
with
tracer.
-115But four of
cell of
interest.
rn+l
(10)
the six neighbors are on
=
rI
equations
EI
+ 4Rin +
(8)-(10)
set
of difference
layer as
the
desorption from the
n+
constitute
that describe
fraction F
same
Thus
(1/6)(
Equations
the
the
probability
ful 1 cubic
released at
a
time
profile
lattice.
To
during
calculate
step n,
one
see how
different
can use
the
the equation
n
N
(11)
F
=
1
UI
-
n
A
walk
n=1
test was also made
is
from a continuum diffusion.
rearranged
(12)
In+l
I
which
is
(13)
The
to
discre
1
C
Equation (10)
random
can be
the form
11 =
a
Tt
into
the
(1/6)
( rn
tization
-
for
HU
n
1+1
+
the diffusion
equation
2
=
b
d
nd
(9
initial and boundary conditions
(14)
r(x,t)
=
1
(15)
f(0,t)
=
1(1,t)
0<x<I,
=
0
t>
0.
(8) and
t=0
,
(9) translate
to
-116The cumulative fraction F
and using
(13)-(15)
solving
released at
t
the
fact
t was found
time
by
that
1
1 -
Ft
(16)
frl(x,t)dx
0
The
solution
1 -
=
F
(17)
to
(13)-(16)
(8/n
2
1/6.
n=0
is
to
to note
interesting
is
to
related
that
in three
particle
choice of
the
(12)
has
there are
1/2
more
This
dimensions.
residence
the
coefficient
random walk in
of
to a coefficient of
lead
the difference
the
that
corresponding derivation
A
dimension would
for
[l/( 2 n+1)2]e-(2n+1)2,2t/6
)
t
It
is
time
one
[3].
The
reason
choices presented
point
given by
is closely
equation
(6).
and
Results
V.4.4
to
of
values
the
drug
a matrix
diameters
The
N
correspond
particle
depth of
300pm,
1.2
a
to
used
particle
in
was
and
100 m,
typically
lattice at
sizes
that are
the experiments.
mm, N,=4,7,12
170pm,
simulation
realization of
sizes
and e=.l,.2,.3,.4,.5.
N -4,7,12
performed for
Runs were
The
Discussion
correspond
to
respectively.
run
a given N
for only
and E.
one
The
close
Assuming
particle
-117-
led
lattice
is
However,
(8)-(11)
continuum
continuum solutions are
diffusion
converge
curves
should be
the
V.5
Total
gets
as N
emphasized
are
into
close
the
to each other determines
large.
In analyzing
should be made against
that
there
(No mention of
any of
the
fraction
data
of drug
is no
lengths
derivations.)
relative.
Comparison with
V.5.1
the
discrete random walk is.
the
comparisons
results.
went
comparisons
as
the
the
curve.
discrete
coefficients
How
solutions
discrete random walk and continuous
the
simulation results,
for
exact
as well
(17).
diffusion solution given by
expected,
scale
This
identical.
respective
for E=l,
the random walks
how good an approximation
It
e were
in figures 9a,b,c,
are shown
the
Also shown are
for
discrete and
exact
and
same N
that are almost
for N =4,7,12
results
respectively.
As
per
in figure 8.
shown
The
of
curves
yielded kinetic
it was
Moreover, preliminary
curves.
realizations with the
two
of runs
large number
the
smooth kinetic
to
in which
runs,
run,
E.
and
so
time,
hour of CPU
for different realizations
to run simulations
not practical
given N
took over an
for N =12
simulation
released
absolute
time
or diffusion
All
-118-
0.45-0.400.35-0.30-0.25--
*
0.20--
U
E
=0. 30 #1
o E =0. 30 #2
H
A JJA
Ajj
0 . 1540.10-4
A
E
=0. 10 #1
,
E
=0. 30
U
i
0 . 054-A
0. me%
00 6&
0.0
0.5
(SQRT TIME) /10
Figure V.8.
Pt-1Ar~
F
simul ated by RRWALK.
Two cases run at each volume loading.
IULUL~.I.LV~
-
#2
-119-
Figure V.9.
Simulated kinetics of random walk desorption from lattices
whose sites are filled at random with drug. Five volume
loadings (=.l=,.20,.3O,.40,.50) were tested. Results are
also shown for "straight" random walks [eqs.(3)-(ll)] and
for continuous diffusion [eq.(17)].
a) Np=4
b) Np=7
c) Np=12
-120-
RANDOM WALK DESORPTION:
4 LAYERS
1.00.9-
/-
rz4
0.80.70
0.6-
E-.4
0.5r-4
0.4U
0.30.20.10.0
0 .0
0.5
1.0
1.5
2.0
2.5
3.0
(SQRT TIME)/4
0.50
0.40
----
0.30
0.20
0.101
STRAIGHT RAND WALK
STRAIGHT DIFFUSION
Figure V.9a
-121-
RANDOM WALK DESORPTION:
7 LAYERS
1.00.9,'
0.8-
,'
----
-
-/
0.7-
-
z
0
0.60.50.4-
rQ
0.31
-
0.20.1-
n 0 IM
0.0
0.6
2.0
1.0
2.5
3.0
(SQRT TIME)/7
0.50-)
0.40
--*-
.c__
0.30
0.20
0.10-
?E
Figure V.9b
RAND WALK
-STRAIGHT
STRAIGHT DIFFUSION
-122-
RANDOM WALK DESORPTION:
12 LAYERS
1.0-4J
0.9--
rz4
w
C1
0.8-0.7--
z
06
0.5-E
0.4--
o
0.3-0.2-0.1-0.0-4
0.6
0.5
1.0
1.5
2.0
2.5
3.0
(SQRT TIME)/12
0.507
0.40
0.30
0.20
E
0.10 -
STRAIGHT RAND WALK
STRAIGHT DIFFUSION
Figure V.9c
-123-
VI.8
Figure
presents
In figure
was
approximately
1.2mm, which is
particle sizes
and 300-425pm
150-180pim,
values
of
For particle
12,
and
depth around
the
sizes
10a,bc,
(figures
were
of N
of
to model
of
IV.1).
(table
which most disks clustered
the
the value
F.
fraction
total
chosen assuming a matrix depth
N
respectively),
the
10 comparison is made
predictions.
90-106ptm,
for
disks of various
drug released from BSA
loadings.
data
4,
7, and
respectively.
The model does a poor job at
quantitatively, although the
the model and
sizes, and
data show
weaker
qualitative
as the
transitions
the
low-loading
curve
the
not be
should
drug
distribution
of drug
arrangement
loadings
indicates
dispersed
in
center of
the
the
polymer
reason
for
the
true at
low
loadings.
the matrix,
behavior
of
the
particles).
then
theoretical
the higher order details
of
amorphous
low
The disagreement at
that drug may not be homogeneously
the polymer.
Apparently, drug prefers
matrix, and avoids
seems
fraction of
lattice versus
(e.g.
low particle
in
dependent on
Both
trends agree.
dispersed
porosity)
(low
data
particle size increases.
total
especially
is homogeneously
If drug
the
This is
drug.
the
transitions at
sharp
The model always overpredicts
releasable
fitting
the
the
drug
unclear.
One
to be "sheltering"
inhomogeneity
is
surface.
the
In some
sense,
particles. The
might speculate
-124-
Figure V.10.
Comparison of results of the USEFUL2 simulation with
experimental results of Chapter IV.
a) BSA, 9 0-106pm
(Np=12)
b) BSA, 150-180pm (Np=7)
c) BSA, 300-425pm (Np=4)
-125-
BOVINE SERUM ALBUMIN
90-106um
1.1-1.0-0.9-0.8--
0.7--
0
0.6--
H
0.5--
0
0.3--
0.1-0.0
0.0
I
0.1
0.2
I
0.3
POROSITY E
._ THEORY (12 LAYERS)
DATA
Figure V.10a.
0.4
0.5
0.6
-126-
BOVINE SERUM ALBUMIN 150-180um
1.0-0.9-0.8-8
0.7
rz3
0.6
0.5-cA
0.4-0.3--
A
-4
0.2-0.1--
A
0.0-
0.0
I
0.1
0.2
0.3
0.4
0.5
POROSITY e
THEORY (7 LAYERS)
A
DATA
Figure V.10b.
0.6
0.7
-127-
BOVINE SERUM ALBUMIN 300-425um
1.00.9-8
0.8-0.7-0.6--
z
H
0.5-0.4--
E-
0
0.3-
0.1-
0.0 i
c .0
I
0.1
II
0.2
0.3
POROSITY 1
THEORY (4 LAYERS)
DATA
Figure V.10c.
i
0.4
0.5
-128that during
polymer
past
the
drying of
solution migrates
the
relatively
the
to
slabs
the
immobile
simulate
s, was
inhomogeneity in
defined, and
sites
was
(18)
s'(Z)
given
the slabs,
moving
particles.
USEFUL2 was
programmed
A
"surface
the fraction of drug-occupied
to
factor,"
lattice
=
-2),
X=2,..
the drug-filled
sites
.
.,N
distribution gives an overall volume
loading of E,
into
the
but
interior of
lattice.
The results of
shown
in
release
is
the
by
it shifts most of
the
of
the matrices.
E (N -2s)/(N
This
surface
drug
A simple modification of
after casting,
.03
figure
(table
for
than
E>.25
the
at E=.05
IV.3).
Thus,
simulation,
modified model
.
In fact,
original,
fits
for
for
case,
the
(BSA,90-106pLm) are
the
is 0.18,
This value was
"homogeneous"
The
simulation
For this
11.
fraction F.
.15~(.03/.18).
the
the
while
surface
used
N
predicted
set at
very well
the observed F
factor s was
for all
was
total
for
loadings.
set to
As
in
12.
e<.25, but poorly
E>.25, the modified model is worse
simpler model.
This
can
be explained.
By
-129-
BOVINE SERUM ALBUMIN
90-106um
1.11.0-8
o
0.9-'
0.8--
U
Ci2
3
0.7--
z
0.6--
0.
r4I
0.4-E'
0.30.2-0.1-0.0-
I
0.1
0.0.
I
0.2
0.3
0.4
0.5
POROSITY E
Figure V.11
Comparison of data and predictions of the total
fraction release F,., for two models. Data is
for BSA 90-1064m. "Homogeneous" model assumes
drug particles are placed uniformly on lattice.
"Surface" model assumes drug particle distribution
given by eq. (18) with s=0.15.
*
HOMOGENEOUS THEORY
SURFACE THEORY
DATA
0.6
-130-
pushing
drug-occupied
those sites
sites into
become more dense.
then reached
at a
the
The percolation
lower overall volume
interior becomes well connected.
high
that
at
least
cluster
can
surface
site,
sites
is
model and
very
lattice
higher
to
low.
Note
the
direct experimental
models
are
other
the
model
percolating
curves
a
drug-filled
for
surface
the homogeneous
cross very near e=.3, which
threshold
for a simple
this
too could
it is
is
cubic
disappear at
incorporated in
the author's
assessment of
the matrix
be
opinion
the
that more
the distribution of drug
is warranted
before more complex
results
drug
were
particle
obtained
for
simulations
complicated,
simple,
"surface"
model
homogeneous model nor
is able
However,
qualitative agreements
there are
function of
involving
sizes.
the
as a
is very
interior
through
quantitatively for
F
the
the probability
density of drug-filled
that
In summary, neither the
the more
Thus
attempted.
Similar
the
the
lattice,
threshold is
loading.
postulated inhomogeneities
However,
in
the
the
5).
loadings, and
particles
Now
outside
the percolation
(figure
modelling.
of
the
"surface"
to
Perhaps
site
even though
the
close
one
connect
very
interior of
total amount of drug
drug
loading and
on
to account
released F
the
.
behavior of
particle size.
These
-131
models
are a
first
step
towards understanding
the
drug particles embedded
the
amount of
in
the matrix, and
releasable drug.
leaving
preliminary
this
experiment,
by
the
model,
model.
are
topic,
in
accuracy of
the
in the
preparation
results
the
in appendix III.
of
that were well
application of
The changes
It is more
the EVAc
of chapters
different
themselves should
the model.
IV
should be
should be mentioned that a
produced results
experimental conditions by
the
it
This experiment, and
discussed
its effect on
data.
run under slightly
experimental conditions,
topology of
More complex models
preceded by more detailed microstructural
Before
the
and V and
led
the
the
not have altered
likely
matrices
in
fit
that a nuance
to
the difference
the results
presented
in appendix III.
V.5.2 Rate
of release
A very clear result is
cannot
be
To
this,
see
The curve
parallel
of
the
the
two
the
sole
cause
consider
does show
to
the
this
the retardation
the curve
is
to
the quotient
reach
occurs at
some
this
of
level,
occurs
t 1/2/12
at
1.25.
topology
of drug
for N =12,
some retardation,
straight random walk,
E=0.50,
of
the random pore
in
s=.50
the
release.
(figure 9c).
that it is not
straight random walk curve.
retardation
curves
that
A crude measure
time
say Ft=0.6.
it
takes
For
t 1/2/120.75.
Thus,
the
time
the
For
for
-132-
orders of
is approximately
factor
retardation
magnitude less
observed in
the
The shapes
loading are
loadings,
almost
along
reminiscent of
loading decreases,
the
and
the
there
the
curves
the
is an
straight random walk
the desorption of
virtually all
e,
surface
is
become
initial
curve.
particles.
releasable drug
as a
This
At low
released
immediately.
Of
the
to
As
At all N
over.
that lies
corresponds
retardation factors
simulated desorption curves
of the
experimental curves.
"burst"
is
experiments of chapter IV.
function of volume
more hunched
than the
This
(1.25/0.75)2<3.
early,
course,
"fast"
the
simulations were only designed
phase
of
release.
to model
-133-
REFERENCES
[1]
Frisch, H.L.
and Hammersley, J.M.,
processes and
1,
[21
894
Ottino,
J.M.,
transport in disordered
of percolation
introduction
37,
topics ,"
905
J.
Ind.
Soc.
Appl. Math.
(1963).
Mohanty, K.K.,
and
[31
re lated
"Percolation
and Davis,
composite
concepts,"
H.T.,
"Reaction
media:
Chem.
Eng.
Sci.
(1982).
Papoulis,
Stochastic
A.,
Probability, Random Variables, and
Processes, McGraw-Hill,
New York
(1965).
-134-
MODELLING THE
CHAPTER VI.
Overview of Chapter
VI.1
(eq.
from calculations
[1],
exists
EVAc itself
and since
tortuosity
release
of macromolecules
explain
In
diffusion are
hypothesis,
is
analyzed
and most
in
of
very highly
diffusion
lower
of
Finally,
for
The
attributes
first
retardation
the
powdered form when
the
matrix is
The
space
drug
is
coefficient of
filled
is
in
the
diffusion coefficient of
pores
the
must be
the pores
therefore viscous.
The
first cast,
initially with
solution inside
the drug
was
the retardation of
protein molecule.
pore
it
cannot
topology
the
concentrated, and
than the
2,
slow
the
release.
two explanations
in section
the
property of
the
powdered drug.
explaining
pore
to
I that
in chapter
investigated mathematically.
to a physicochemical
protein
random
retardation
chapter,
this
the
impermeable
the matrices.
from
that
VII
overall
the
in
protein
seem to be
sufficient for
is not
channel
in chapter
also argued
It was
relevant equation.
is
I.1 would
proteins such as BSA, equation
shown
the
it has been shown that a network of waterfilled
Since
water.
be expected
dimensions of
the
involving
the diffusion coefficient of dilute
and
matrix
I.1)
release from EVAc
that
shown
slower than would
orders of magnitude
is
matrices
it was
I and VI
chapters
In
pore
SLOWNESS OF RELEASE
Thus
the
may be much
the drug
in
its
-135The analysis
of section 2 will
show,
however,
concentration dependent diffusion, by
minor effect on
The
the rate of
the
itself,
retardation of
in
release
Section
3
is
in section 3 that equation I.1
seen
length
equation
sections 3 and 4,
to
focuses on geometric features
scales
I.1
has only a
drug release.
second hypothesis, analyzed
attributes
that
the pore
of
single
is not
greater than that of a single
should not be expected
structure.
pores.
It
relevant for
pore,
to hold for
so
that
porous EVAc
matrices.
VI.2
Effect of concentration dependence
on diffusion
coefficients
VI.2.1
Introduction
Diffusion coefficients of
macromolecules
proteins
and other
depend on concentration.
In general,
macromolecule
diffusivity decreases as a
function of
concentration
[2,3],
although at very
diffusivity may increase
low
with concentration,
electrostatic
repulsion between charged
[4-101.
increase is
This
solution is
and
when
Indeed,
the
far from
ionic
strength of
the
to
the
pH of
the
isoelectric point
solution
in chapter II
due
macromolecules
most marked when
the macromolecule's
it was remarked
concentrations
that
is
the
low
[7].
release
rate
-136-
of BSA
point
increases as
of BSA,
In all
with
pH moves
of
This
of
measurements,
studies of
it has been
constant.
This
Sections
2.2 and
and
2.3
review
It is
that
the
that while
tracer diffusivities decrease
increasing protein concentration,
behavior
diffusivities
concentration
The
release
is in
process is examined
simulations
the
sufficiently
is very
process.
function
of
dispute.
in section 2.4.
performed.
It
the
Numerical and
of desorption with various
concentration dependence are
there
as a
the
effect of concentration dependent diffusion on
analytical
provided
is
there is
significantly with
of mutual
at high
distinction
diffusion coefficients
shown in section 2.3
widespread agreement
the
experimental
In section 2.2
and "mutual"
is
the concentration dependence
respectively, of diffusion coefficients
"tracer"
elaborated.
[101.
release of
[1,12,13]
understanding
protein concentrations.
between
isoelectric
a diffusion equation that
section examines
theoretical
the
constant
that diffusion coefficients are
diffusion coefficients.
state
held
from EVAc polymers
simplifying assumption gives
linear.
away from
ionic strength
previous modelling
macromolecules
assumed
the
is
forms
found
of
that,
macromolecule's diffusion coefficient remains
large
over an appreciable concentration range,
little
These
retardation of
simulations
the
overall desorption
argue against
a
theory of
-137-
retardation based on
the high macromolecular
concentration in
the pores.
VI.2.2
Tracer and Mutual Diffusion Coefficients
The concentration dependence
coefficient
is
due
to
several
due
to
these
solution,
collisions
is
to
Einstein
thermal
showed
that
collisions of
macromolecule
opposed by
[14].
the viscous
The
force
drag of
characterized by a friction coefficient f.
Stokes-Einstein equation for
(1)
the diffusion
factors.
macromolecular diffusion is due
solvent molecules against the
of
the diffusion
the
The
coefficient is
D T= kT/f
where
k
is
Boltzmann's
temperature
of
the
where
v is
solution.
to
6 nvR,
of
the macromolecule.
The
constant and T
the
In
solvent
Stokes-Einstein
effective
can be kept
the Kelvin
the dilute
viscosity
equation
is
low macromolecule concentrations.
its form
is
limit, f is equal
and R
strictly
At higher
if f is allowed
interactions
between
valid
radius
only at
concentrations
for
the macromolecules
Then f increases with macromolecule
to decrease with- concentration.
the
to represent an
friction coefficient which accounts
hydrodynamic
is
[15].
concentration, causing
DT
-138DT
is called
corresponds
to
the
the
tracer diffusion coefficient, and
random motion of a
labeled macromolecule
in a solution of uniform concentration.
studies we are
interested
in
the mutual
coefficient DM, which is measured in a
solute gradients.
of
solute
gradient
DM is defined as
the
important in
solute
in
determining
In general DM
the
is not equal
flow
to regions of
replaced by
to D ,
The
the
It
for
concentration
is DM
that
is
two reasons.
lower concentration,
their volume
causing a
Macromolecules
countercurrent, reducing
the
are
net
of macromolecule.
second
difference between
diffusion coefficients
in concentrated regions
concentrated regions.
repulsion
by
flux
from regions of high
solvent molecules,
along in this
transport
quotient of the
rate of drug release.
countercurrent of solvent molecules.
dragged
solution containing
direction.
the
First, as macromolecules
concentration
that
in our
diffusion
in a chosen direction divided
of
must be
However,
and
is more
subtle.
of solute
This
is
tracer and
due
The
osmotic
is higher than
to
hard
to electrostatic repulsions
macromolecules.
Phillies
[41
proposed
Stokes-Einstein"
equation
for
the mutual
coefficient:
mutual
less
sphere
between
the
in
pressure
the charged
"generalized
diffusion
-139(2)
DM = kT(1-$ )
[--] T
In
this equation,
$ is
given by
mg/mi
the
the volume
, where c is
Oec/p
(mg/ml) and
/f
p is
the macromolecule
the density of
]T,P
for BSA).
is
solution, defined as the
pressure with addition of
constant
temperature and
(2) accounts
increasing
for solute
fraction of macromolecule,
the macromolecule
(p=1340
the osmotic compressibility
rate
of change
macromolecule
total
to
pressure.
the
The
[I]
of
in osmotic
solution at
factor
backflow, and clearly
solute concentration.
concentration, opposing
concentration
1-$
in
decreases with
increases with
3c TOP
effects of viscosity and
solute
backflow.
Various authors
both
the hard
osmotic
of
modulated
charge on
by
protein solutions,
The hard
.
first 'order, equal
is
to 1+8$.
two
proteins
[81).
macromolecules
Second,
by
First,
(Cl
there
is
the Debye
in
the Debye
the
is
the
can adsorb
the microions
strength increases,
to
the
is,
to
contribution
the average
pH and
the
strongly onto
screening of
solution.
As
pressure also
the
the
length decreases,
to osmotic
for
allowing theoretical
The electrostatic
there
electrostatic contribution
[4,7].
contributions
a macromolecule, determined by
ion concentration
expressions
sphere contribution to 3
factors.
chloride
ionic
have developed
sphere and electrostatic
pressure of
computation
[4,7,15]
and
the
decreases
-140VI.2.3
Experimental
Determinations
Coefficients
for Proteins
Keller et al.
[2]
concentration using
(pH=4.7)
a diffusion
coefficients were assessed by
the
test concentration
cell.
Mutual
measuring
to a
steady
and
from a source
gradients
Their results
First, both
features.
(figure
that DT and DM would be
surprising
isoionic
dispute
as
stated
above,
is equal
for the
1-$
(7]
locally with
f and
equation
14C-labeled
a
1)
show
two
that
it appears
are
conditions
tested.
it is
literature at present is
sphere contribution to
1+8$, which more
factor due
argue
to
the
solvent
than
backflow.
friction coefficient
the hard sphere component of
LC cannot be
(2).
The hard
to
test
indistinguishable, even at
theoretical
this matter.
Anderson and Reed
that
The
point.
on
compensates
covaries
of
Second,
the discussion of section 2.2,
In view of
in
of experimental
the range
chamber
were measured
tracer and mutual diffusion coefficients
identical under
rates of
tracer and mutual diffusion
coefficients decrease with concentration.
the
state
unlabeled BSA molecules, above
concentration pedestal.
that the
diffusion
sink chamber at zero
by establishing small countervailing
BSA molecules
diffusion
function of
Tracer diffusion coefficients
concentration.
salient
as a
through Millipore® membranes
transport
at
tracer and mutual
measured
coefficients of isoionic BSA
of Diffusion
al,
f
so
averaged separately
before
computing
the quotient of
a1
ac and
f,
By averaging
-141-
a Mutuol diffusion
0lTraer diffustes
r-~
0
6.
*H 4
00
0
Figure VI.l.
268 335
201
134
67
Concentration (mg/ml)
Tracer and mutual diffusion coefficients of BSA
for pH=4.7.
Reproduced from Keller et al.[11
Abscissa converted from volume fraction to
concentration.
-142Anderson and Reed were able
results
of Keller et al.
equality of DM and
electrostatic
et
justify
are
al.
theoretically the
Anderson and Reed's model
DT only for
effects
Keller
to
those
predict
conditions where
negligible.
obtained
an empirical
relation
for
the
BSA
diffusivity:
(3)
DT (c)
= DM (c)
where D =7.OxlO
-7
= D 0 tanhyc/yc
2
cm /sec
and
and
expression, with a different Do,
Anderson
diffusivity of
(4)
BSA,
velocities
All
above
studies
its
physiological
described
=
and
of BSA
6.5
D 0 (1-0)
a good
6.5
D 0 (l-c./p)
experimental
fit
to
the
data of
as a function of concentration.
of concentration
(7.4),
by Keller
concentration at
for hemoglobin.
is consistent with known sedimentation
isoionic point
pH
held
This
viz.
provides
Keller et al.,
also
for BSA.
proposed an alternative expression for the
D(c)
D T (c)
()=DM(c
This relation
BSA
[3]
y=.016ml/mg
it
dependence of D
indicate
that D
appears that D
et al.[9],
but D
>D
M- T
[5,9].
Near
behaves roughly as
increases
low BSA concentrations.
and D
of
M
T
This
with
behavior of D
M
-143has
been verified by quasielastic
centrifugation
[9],
Rayleigh
capillary
techniques
at pH=7.4
is
shown
concentration
least up
It
the
for
possible
increase
in DM
approximately equal
vs.c
with
for
to Do,
at
325 mg/ml.
In EVAc polymer
the macromolecules
which
that
is well
in
above
the diffusion coefficient suffers a
concentrations,
the
concentration approaches
entanglements
factors are
Stokes-Einstein
other
factors
considered
treatment.
[16]),
fraction of C
/p=
summary,
solubility.
such as
may play a significant
(C s=585 mg/ml
the pores
the
range
to
matrices
significant
At very
Neither of
in the generalized
Note
that
at
protein occupies a
585/1340 =
It is
excluded volume
role.
0.44
solubility
for BSA
solution volume
.
the mutual diffusion coefficient may
decrease significantly with concentration.
necessary
have not
diffusion coefficients have been measured.
high
EVAc
plot of D
that diffusion coefficients
1000 mg/ml,
as
In
that DM is
BSA above
drop
these
The
of a
and
and it can be assumed,
concentrations of
as high as
over which
and
2.
[10],
[5,81,
to c=240mg/ml.
been measured
can be
figure
should be noted
systems
An example
is not dramatic,
modelling purposes,
scattering
interferometry
[6,9].
in
light
see whether the
is due
to
the
region of high macromolecule
slowness of
fact
that
Thus
it is
protein release
diffusion
concentration.
occurs
We now
from
in a
turn
to
-144-
U
Q)
CN
9
]
A
8
(U
44)
a)
0
U
AA
7
o
A
6
0
0,
A
5
0
0 P.2-7.3
A PH.3-?.5
A PH7.5-7.6
* p.2-7.3Ifhqsnded bmIO
-G.Oin
Stus-Eie"M squoan
100
200
-
300
Concentration (mg/ml)
Figure
VI.2.
Mutual diffusion coefficient of
pH=7.4.
Reproduced from [5].
BSA,
for
-145
this
-
issue.
VI.2.4
Effect of concentration dependence
coefficients
VI.2.4.1
In
affect
on desorption
from slabs
Introduction
this section
it is
assumed
that concentration does
the diffusion coefficient of a macromolecule.
much concentration dependence affects
(release)
2.4.2
of diffusion
rate
from a
the steady
the overall desorption
slab will be assessed.
state diffusive
How
flux across
In section
a
slab, given a
concentration dependent diffusion coefficient,
is derived.
The steady
for
state
interpreting
case provides
some heuristics
transient desorption.
In section 2.4.3,
numerical and analytical calculations
transient diffusion
are made
for
(desorption) case.
VI.2.4.2 Effect of concentration dependence on
state
If
D(c)
steady
diffusion
the concentration dependent diffusion coefficient
is known,
flux across
then
it
a membrane
is
easy
to calculate
from a reservoir
concentration into a perfect sink
thickness
the
the
6 separates
concentrations
in
[17].
the diffusive
at a given
Assume a membrane of
two chambers
of a diffusion cell, with
the chambers
held at
C
and
0 (see
-146
figure
are
3).
Also assume
absorbed
in the
(5)
into
membrane,
porosity and
tortuosity effects
the diffusion coefficient.
in
]
'[D(c)
that
-
the
0
=
steady
At any depth x
state,
,
or
(6)
D(c)
= F
DX
S
Integrating
a constant.
6
(7)
over
x,
6
fD(c) -dx
3X
0
F fdx
=
S
0
Changing variables on
the
left hand
side
of
(7)
from x
to
c,
*
*
(8)
F
(C
)
C
[fD(c)dc]/6
=
0
But -F
is
the
flux
through
the
concentration independent case,
membrane.
i.e.
For
the
for all
D(c)=D(O)=D
0
(9)
The
F OS(C)
ratio
of
=
DC
the
/6
concentration
dependent flux to
is
concentration independent flux
then
C
*
(10)
R
(C*)
=
F
*
*
*
(C*)/F OS(C)
=
[fD(c)dc]/D 0 C
S
the
c,
-147-
Membrane
C =C*
I
c=o
H
6
Figure VI.3.
Schematic of test cell for measuring steady
state diffusion through a membrane of depth
6.
-148-
Now F
effective
(C*)
might be viewed as
F
where,
S
flux produced by an
diffusion coefficient D e(C)
concentration gradient C /6,
(11)
the
(C
)
D
=
using
acting on a
i.e.
eS (C )C /6
(8),
*
C
(12)
D
(C*)
=
[fD(C)dC]/C
*
.
0
Thus, R
(C)
is also
Apparently,
determined
the
the
solely by
the ratio
(C )
steady state rate
of
Calculation of
is not
is diffusion
coefficients do not exist
for
the
[15].
The
transient
dependent diffusion
situation must be
by computer.
Rather
desorption
than attempting
from
have
a
slab
chosen
to
in
theory
desorption of a slab with concentration
we
transport
transient concentration
Explicit analytical solutions
D(c),
D0
regime.
dependent diffusion:
analyzed
to
the slowest step, which
high concentration
VI.2.4.3.
of D
of
to
finite
solve
solve
the
depth
problem
with a
the semiinfinite
of
prescribed
desorption
-149-
case.
(Desorption is into a perfect sink.)
simpler
behavior
problem, as will be
for a
finite
slab
seen below,
is
virtually
early behavior for a semiinfinite
Within
the
and because
identical
a much
the early
to
the
slab.
The concentration dependent case
transformation.
This is
is
semiinfinite
solved
slab,
by
the
making
transient
diffusion equation
(13)
-
[D(c
with boundary
x-0, t)=o,
(15)
x=co,t)
the
(16)
holds.
0<x<w
conditions
(14)
and
1)5
C ,
initial
condition
c(x, t=0)=C
Setting
,
<x<w
n=x/2/D
D(c)=D(c)/D0 , equations
(17)
(18)
.=
1
c(0)
-c2 )n =0
*
(19)
[D
) d]
0
t (Boltzmann transformation) and
(13-16)
become
a
-150-
which is
an
ordinary differential
equation boundary value
problem.
First consider
Then
(17)
-
For
this
subject to
=
the
scheme
equation
it was
de'
- =
The
fourth order
is designed
=
details
the
necessary
into
has
the
solution
[12]
a
Runge-Kutta
(17)-(19) were
scheme
[16].
Because
system
to
transform
of
first
the
order
second order
equations:
c'
-(2nc'-c'
of
2dD
%
j)/D(c)
the numerical
having
flux out
(19),
for first order differential
concentration dependent case
Once
and
concentration dependent case,
(17)
dc
dp
(18)
C erf()
using a
equations,
(21)
32 c
~7
c(O)
solved
c.
=2ndn
which, when
(20)
for all
becomes
dc
(17')
the case where D(c)=D(O)=D0
of
solved
the
computations
for
are described
for c,
given any
semiinfinite
in
the
appendix IV.
expression
slab as a
for D(c),
function of
time
-151was
obtained by applying
(i.e.
the
inverse Boltzmann
transformation
x=2/'D -ta):
-FT
Dd
For
example,
(20)
and
(22)
(23)
-F OT=
C
As
the
(22)
derive
in
a
= (1/2)/D-0/td
x=
for
the
n=0
concentration
independent case,
yield
(D 0/jtt)1/2
steady
constant
state
effective
case,
F
and
FOT
can be
used
diffusion coefficient D eT(C
*
Se t
*
(24)
Then
and
)
F T(C
*
=
C
(D eT/ nt) 1/2
the ratio R T(C)
of DeT
to D
is given by
(24)]
(25)
RT (C *)
Using
(22)
(26)
R
(C*)
= DeT D
and
(23)
=
in
*2
=
(i/4C
[FT (C
)/FOT] 2
(25),
ad 2
)( Ti ) IT=0
[using
(22)
to
-152..
VI.2.4.4
The
[3]
were
were
Application
to specific diffusivity models
diffusivity models
studied, extrapolated
studied because
that at physiological
than
pH,
(It
Graphs of
equations
(3)
diffusivities
shapes
the
and
(4),
are
quite
are
[2]
1000 mg/ml.
the
and Anderson
These models
"worst case"
of
in section 2.3
the diffusion coefficients
are much
the Keller and Anderson
two model
are
al.
was concluded
those predicted by
models.)
curve
to
they provide
concentration dependence.
higher
of Keller et
diffusivities,
shown in figure 4a.
similar
up
to 325
somewhat different.
Anderson model predicts a lower
given by
The
mg/ml,
Above
diffusivity
two model
although
325 mg/ml,
than
the
the
the Keller
model.
The
) and R (C)
ratios RT (C
models are
shown in
Besides
models was
figure 4b.
tested in which D(c) behaves
approaching
in
the
0 faster at high
because
that
filter:
D(c)
Butterworth
=
similarly
low concentration regime,
"Butterworth" models,
(27)
the Keller and Anderson
the Keller and Anderson models, another
former models
of a
for
[l+(c/c 2 n -
concentrations.
the
form of D(c)
to
set of
the
but with D(c)
We
is
call
these
similar
to
-153-
REDUCED DIFUUZON COEFVICIENTS
1.01
0.0.K
0.6
0.6.
KELLE
O.S.
a)
0
0
MODEL
ANDERSON
MODEL
goo
1000
0.40.&
O.S
0
0.3
0.1*
0.0.
C04C
2
/m0
CalOZ~TAII
3(m/i
1.0
0.5.
KKELLE
STEADYS TATE KELLE
. TRANSIEDrIANDERSON
'STEADYT TATE ANDEMSON
. TRAM=IE
-
0.4
b)
U
Ca
0.+
0%
LA.
-
-
am
600
200
1000
a0
C* (ag/mi)
Figure
VI. 4a)
Model
concentration-dependent
coefficients,
and Anderson
b)
Values of R
Keller
given by Keller
[3]
(C )
(eqs.
and
2 and
RT (C)
and Anderson models.
diffusion
et al.[2J
3,
(see
respectively).
text)
for
-154-
the concentration such
is
where c2
coefficient drops near c 2 , and
comes as c gets
of
the
The
R
These
is
close
to
for c 2 =125 mg/ml and
run
for D(c)
that
are shown
in figure
The
5a.
share many
of D(c).
resulting
) for the Butterworth models are shown in
5b.
A
case
limiting
the
where
with n=m,
is
interest
of
diffusivity is
concentration c 2 , at which point
(Essentially, this means
diffusing
molecule.)
R T(C)
this case,
in
that c 2
There is
which
is
"Butterworth"
the
to D
equal
to
it drops
is
an analytical
[19]
C
<
c2
C
>
c2
*
RT (C
{c2 /[C
erf(a )*
where
*
/irn exp(ra
*2
*
)erf(r1 *) = c2 I(*2
2
up
case
to
zero.
of
the
solution
for
solubility
the
*
(29)
how
features of the Keller and Anderson models
and R S(C
figure
(28)
and n
large.
parameters give curves
curves for D(c)
(C)
0/2,
diffusion
also determines
"Butterworth" simulations were
n=2,3.
the
that determines how sharply
an exponent
zero D(c)
that D(c2 )D
-155REDUCD DIFIUN CGEFFICIETS
1.01
BDTTERWORTH MODELS:
C2 -125 mg/mi
0.64
a)
C
C
-
2
U
C
0.44
.................
0.2
I
(./
460
00
1000
CONENTATon (gn6/
1UTTIWtRTH MODEL3
C
=12566/21
b)
a.6.
\%
'I'
4
A
~
..-- ---
'I
Ua
0.44
-15
.2 TRANSIENT
STEADY STATE
3 TRANSIENT
3=2
...
........
3 STEADY STATE
2,<a;
c 2 = 125
C
.4
N
'.
I
o
600
400
i0
c*
Figure VI.5 a)
(1/al)
(n-I,
"Butterworth"
concentration deper dent
*
b)
Value of R
models.
(C
)
mg/ml) model
diffusion coefficient.
*
and IRT(C
)
for Butterworth
-156-
For
C >>c
the
2,
(30)
RT (C )
It is
easy
simplest asymptotic
~
to
/2)(c2/C
see
)
form
of
(29)
is
.
that for
the
steady state case,
*
(31)
R
(C)
C
<
c2
C
>
C
=
c
/C
.
VI.2.4.5 Discussion
The
plots
features.
reduction
in
the effective
even though
region.
curves
in
It
figures
in
there
diffusivity
in
is also
the
for
Figures
for
The
the
little
transient
that
the
that
D(c)
is
than in
shapes
the
much
the
the
of
in
the
the
shape of D(c)
lower at
"Butterworth" and Anderson models
the overall
4b and
relatively
much lower
apparent
fact
surprising
concentrated region of
the Keller model is not reflected
two models
is
4 and 5 depend mostly on
low values of c.
concentrations
in
that
the diffusion coefficient is
dilute
at
4 and 5 have some
First, it appears
desorption,
slab
in figures
high
than
prediction of
the
desorption rates.
5b indicate that the effective diffusion
--57-
coefficient is
functional
recalled
forms
that
case"
5a.
of D(c)
there
coefficients
4a and
reduced by a factor of at most 5,
strong
do not fall off
The results
estimates of
diffusivity on
example,
is
chosen.
the
However,
evidence
in
the
presented
the effect
the
constant up
to 400 mg/ml,
then
(28)
Because
equation
I.1
diffusion
diffusivity by a
the
is
release by
is not nearly sufficient
time
scale of
is
on
the
factor of
less
than
that
to
2.]
time given
the
by
effective
the concentration
than a factor
to account
salient feature
the
for
the
of 5.
observed
of
the graphs
that
slightly smaller effect on the
the
transient desorption
that
the concentration
the diffusion coefficient has
evidence
is
steady state case.
somewhat surprising
desorption
numerical
[For
predict a
diffusion coefficient for
dependence of
"worst
dependent
(29)
no more
concentration dependence has a
It
treated as
release.
second
than for
in figures
and
it appears
This
case
diffusion
diffusion coefficient is
inversely proportional
coefficient,
effective
manner shown
characteristic desorption
effect can retard
The
mutual
effective diffusion coefficient.
that
in
that
of concentration
it is assumed
the
it should be
should be
if
reduction
given
of the
that
slab.
this
At
so
present, we
is a robust
little effect
only have
prediction, and we
-158-
cannot offer an analytic proof
However, a
helpful.
Figure
6 diagrams
the
dilute.
left of x
But
steep
the drug
offset
so.
phenomenon may be
situation.
Imagine
highly concentrated at
that that region quickly
gradient offsets
at depth x
by
the diffusion coefficient
then the gradient near x*
falls
the gradient is
behavior very
dependence.
for
the
should always be
that
the
some depth
*
To
relatively high, so
This
the
the slab becomes
*
.
this
heuristic explanation of
protein in
x
that
robust
(Figure
a slab whose
model with C
the
6 is
1000
becomes very
large.
the
in the
actually
,
down that gradi'ent.
what makes
diffusivity
becomes very
low diffusivity at x
quickly
to changes
is
and
This
overall desorption
concentration
the concentration profile
is given by
the n=3
Butterworth
mg/ml.)
VI.2.5 Summary
In
this
section
it
diffusion coefficients
concentrations,
retardation
has
been
shown
can be very
low
that,
at high
this does not come close
of release.
from EVAc matrices
Therefore,
must be
the
attributed
although
to explaining
slowness of
to other
the
release
factors.
-159-
BUTTERWORTH: n=3 CONC. PROFILE
1.0-
LOW
DIFFUSIVITY
0.8--
0.6--
.0.4--
HIGH DIFFUSIVITY
.0.2-
0.0
0.0
0.1
0.2
0.3
0.4
r= x/2V(t
0
Figure
VI.6.
Concentration
of C
.
Note
profile
sharp
between regions
of
of
drop
slab
in
with high
value
concentration
low and high diffusivity.
-160-
VI.3
Models
VI.3.1
of
the effect of pore
Introduction
In
chapter III,
the model of Pismen
through porous media was
looks at diffusion over
described
two
(the
"macro"
Pore
topology effects are
These
diffusing
hopping
of
scale)
molecules
rate.
first
passage
second
Given
The
times
illustrated
diffusing
in
(the
scale.
It was assumed
"micro"
scale)
the
that
then
involves
distribution
through that
time is
pore can be
identified with
scale.
Figure
II.1 illustrates
geometry.
interconnected
It appears
tubes,
"hopping time"
molecule enters
pore
that pores,
between
The
speaking, when
a pore,
it must find an
and
effect
pores
Roughly
it can exit that pore
an
are varicose bodies,
via narrow "channels."
figure 7.
the
the macro
medium.
the effect of pore geometry on first
on the
However, because
porous
scale
to pore with a particular
molecule
the macro
The model
the whole
from pore
of a
each other
this might have
of
important on
scale
for diffusion
The first
the geometry of a pore,
feature of pore
to
scale
is examined.
rather than being
connected
length scales.
first passage
time on
[181
cursorily.
in section V.4.
"hop"
this section
important
before
the
passage times
hopping
In
The
pores.
calculated.
the
is
were considered
single
geometry
is
the
outlet
make further progress.
outlets are very narrow,
they are
-161--
Random walk path of
diffusing drug molecule
Conn( ecting
chann el
Pores
Polymer
matrix
Figure
VI.7.
Schematic of Brownian particle in a constricted
The particle bangs into the wall several
pore.
times before finding the pore exit.
difficult to
the
find.
pore wall
view
before
that these
basis
for
It
the
The molecule
it
finds
has numerous collisions with
an outlet.
slow diffusive release
(6],
[Figure
diameters are orders
of macromolecules
while
the
Models of
through narrow pores
In section 3.2
section
3.3,
this method
that permit analytical
of a pore
times
is
through
simulation
figure
VI.3.2
11.3
the pore.
performed,
of BSA
the order of
than the
is
~35 A
10 pm
is
for computing
described
in regions
is applied
to
It
In
simple
is
section 3.4
and
In
pore geometries
shown
that
the
size
first passage
a Monte Carlo
similiar
the results
passage
first
of arbitrary shape.
parameter determining
Development of first
VI.3.2.1
larger
here.
using a pore structure
is
themselves,
do not apply
solutions.
not the only
the
constricted
pores
the radius
on
are
restricted diffusion of molecules
a method
time distributions
the
of magnitude
are
[21,22]
passage
to
(e.g.
channel radii
II.1]).
the author's
from EVAc matrices.
that although the
channels are very narrow compared
diameters
is
constricted channels between pores
should be emphasized
the channel
It
to
that shown
are
discussed.
time
in
equations
Introduction
Consider a pore with geometry
shown in
figure 8.
It
is
-163-
n
3 c=O
9CDV2CI
Absorbing
boundary
c=0
Reflecting
boundary
Figure
VI.8.
Schematic of situation
for which first
passage
time distributions are solved.
Solute obeys
diffusion equation inside pore.
Two types of
Concenboundaries:
absorbing and reflecting.
tration
at
absorbing boundary is
Normal
zero.
solute
flux at
reflecting
boundary is zero.
-164 -
assumed
that the diffusion equation
DV 2 c
(32)
o tc =
holds
inside
molecule
the pore.
in water.
coefficient
boundaries
is
D
is
It is
the
assumed
independent of
are assumed.
diffusion coefficient of
One
that
is
reflecting,
corresponding
(33)
for
c = 0
Two
types of
perfectly absorbing and
to a pore outlet, while
conditions
diffusion
concentration.
corresponds
to
the
the
the
other is
pore walls.
perfectly
The boundary
the diffusion equation (32)
are
then
absorbing boundary,
,
and
(34)
nc = 0
where
n
(see
signifies
figure
8).
examples will
In section
equation
passage
the
the
Diffusion
the
derived.
time
occurs
and 3).
full
distribution
in d dimensions
first passage
In section 3.2.3,
for our
c at
the
The
made.
time distribution
the mean
latter
purposes and
equation.
boundary
(later
Two derivations are
equation is derived.
information
full
normal derivative of
use d=1,2
3.2.2,
is
sufficient
than
reflecting boundary,
,
first
often contains
is much
simpler
--165-
VI.3.2.2
First passage
Assume
position X
=
P(T;X)
time distribution
that a diffusing particle is
in the
interior of
the pore.
that
P(T;X)
is
a cumulative
(c.d.f.).
P(T;X)
passing
to
the
move
position X+x.
to
guaranteed
not have
X+x the
Define
limit.
displacement
<
distribution
In a short
time
By choosing
TI
that
the pore boundary between
time
.
t,
time and
the particle will
enough,
the
it can be
particles
times 0 and
will
t.
From
to an absorber before time T-t.
the probability
x after
an
function
t small
certainty
particle must get
be
time
can be derived by discretizing
with arbitrary
hit
Let Gt (x)
time 0 at
Prob[A Brownian particle starting at X reaches
absorbing boundary at
Note
placed at
density function
t.
Then
the
(p.d.f.)
of
the
following equation
holds:
(35)
P(T;X)
where V
the
(x)dx
interior of the
that Gt is
small
pore
is
fP(T-t;X+x)G
V
effectively
pore.
zero
(Again,
long before
t is chosen
X+x nears
boundary.)
For
direction,
a Brownian
i.e.
particle, G t(x)
is Gaussian
in
each
so
the
-166d
(36)
2
exp(-x i2/4Dt)//4-nDt]
Gt (x)
i=l
where x
is
The
i'th component of x.
the
integrand in (35)
d
= f[i(T-t;X)+P
V
P(T;X)
(37)
whreP
P
where
=p P
an
and
terms.
Plugging
dd
x +jP
P
(36)
into
The
linear
P(T-t;X)
so
(6)
(39)
of G
f
The
.
P(T;X)-P(T-t;X)
both sides
terms
vanish,
and
(40)
6 P =
2
DV
obtained.
first
P
2Gt(x)dx
x
integrals
in
to
h.o.t.
terms of
cross
due
+
the
(38)
the
Taylor
symmetry
are well known
[221,
becomes
Dividing
is
+
the integration
series vanish in
properties
obtains
(37)
the quadratic
terms and
h.o.t.]Gt(x)dx,
and h.o.t. denotes higher order
d
=
P(T;X)
(38)
+
x x.
1=1
j=1
i=1
=
P
in x:
is now expanded
passage
by
the
= DtV 2 P + h.o.t.
t and
letting
the higher
t-O,
order
result
p
The cumulative
time
probability
distribution of
for a Brownian particle
obeys
the
the
-167-
diffusion equation, with
the
particle
The
as
in figure 8 can
is not on the
required
"initial"
On
to
P(O;X) =
absorbing
(40)
also be derived.
pore boundary,
,
,
If
for X inside
a particle
residing in
the
absorbing
time
to reach an
is
for X on an absorbing boundary.
to obtain.
smooth enough so
continuous
unit normal
of
the wall
looks
to
wall's
tangent
the wall,
X will have
and
plane.
do so,
that at any point on
the wall
like
displacement x can be set
normal
To
vector exists.
region near a point X on
Locally,
the
the pore.
so a boundary condition
somewhat more difficult
is
time will
Therefore,
The boundary condition for a reflecting pore
wall
the
is
boundary,
1
in a pore
then a finite
the pore boundary requires no
P(O;X) =
(42)
0
other hand,
portion of
for
reach the absorbing boundary.
condition
the
diffusion coefficient as
itself.
diagrammed
(41)
same
initial and boundary conditions
molecule
be
the
the
a
flat
is
A
wall is
assume
the wall
blow-up of
shown
that
the
a
the pore
in figure 9.
surface,
and
in coordinates
such
that xI
is
run
"along"
the
other
A Brownian
Gaussian displacements
components
particle
starting
along all
the vector
at
components
point
-168-
Xe
.
n
X3
Figure
VI.9.
Local geometry near reflecting pore wall,
represented by tangent plane of wall.
x
is
normal to tangent plane, while x2 and x 3lie
on tangent plane.
-169-
except x .
The x 1
displacement will
be distributed according
to a reflected Gaussian distribution g 1 (x)
(43)
g 1 (x 1 )
= H(x 1 )exp(-x
where
H(- )
the
is
distribution
(44)
Heaviside
of all
G t(x)
2/4Dt)//Dt
step
1
viz.
,
function.
displacements
H(x 1)[exp(-x
=
1
[22],
is
The
joint
then
d
2
2
/4Dt)//4nDt]
i=2
Plugging
(45)
(44)
P(T;X)
Dividing
(46)
P(T-t;X)
=
through by
that
the
in
since
there
that
Therefore,
6
n
P
It has
+
= 2/D/irtP 1
infinity as
is no
"sticking"
1
P
t+O.
manipulation,
h.o.t.
TP
of
+ DV
in
2
P + h.o.t.
the right hand side of
However, the
,
the
boundary condition
=
some
t yields
becomes
P
after
2/Dt/nP 1 + DtV2 P +
limit,
the
=
obtains,
term multiplying
approaches
side,
(47)
(37)
[P(T;X)-P(T-t;X)]/t
Notice
(46)
into
at
which
left hand
is not
particle
to
infinite,
the wall.
the wall must
0
been shown
that,
for
the class
of pores
be
-17O0-
considered here,
first passage
equations
times are
the
corresponding
Only
through such pores.
the
initial
are different.
Mean
VI.3.2.3
first
Again assume a
in the
position X
time T(X)
After a short
there
as
same
the
for diffusion
conditions
average
equation and boundary conditions for
the
passage
times
pore.
interior
of
required
to reach
time
it must reach
the
Of
interest
to reach
time 0
is
the
to X+x.
From
absorber from X+x.
Averaging over all possible x yields
=
(48)
T(X)
where
Gt (x)
Expanding
in the
T
T(X)
fG
(x)T(X+x)dx
and V are
=
t +
fG
in the
defined as
integrand
(and
previous
using subscript notation
(x)[
(x)
d
dT x
+
+
dd
YYdTd
j=1
which,
(50)
af ter
T(X)
=
section.
of T),
derivatives
for partial
(49)
t +
(36)
evaluation
using
t + T(X) +
DtV 2 T +
at
the
There will be
the absorbing boundary.
mean time T(X+x) required
at
an absorbing boundary.
is displaced
particle
t the
placed
is
particle
diffusing
becomes
h.o.t.
.
+
h.o.t.Idx
a
-171-
t+0,
t, cancelling, and letting
Dividing by
the final
obtains
result
(51)
DV2T =
This
partial
conditions,
first
-1
so
passage
it is
time
simpler
than the
distribution.
section, it can also be
previous
conditions
for
(52)
=
0
T(X)
=
T(X)
initial
differential equation requires no
p.d.e.
the
for
Using arguments
shown that
the
of
full
the
boundary
are
(51)
,
X on an absorbing
boundary,
and
(53)
6
VI.3.3
n
-
passage
shown
not
times
reflecting boundary.
in model geometries
Introduction
this
section, equations
time are
that
the
, X on a
First passage
VI.3. 3.1
In
0
the
only
three
for
geometries.
the mean first
It will
distance over which a particle must
factor
The model
solved in
(51)-(53)
de termining
geometries are
its
first
passage
diagrammed in
figure
be
diffuse
time.
10.
In
is
-172-
77177-777Y
a.
straight
r= x
r= I
b.
cylindrical
:r=x
r= I
C.
spherical
/=,
I
r =1
Figure VI.10.
Geometries of model
first passage times
a)
b)
c)
pores for which mean
can be solved.
pore.
Straight
Cylindrically conical pore.
Spherically conical pore.
-173-
all
three
and
lie
geometries,
This allows
situations
scaled
that
the absorbing
parameter
to
properties
assumed
times
be varied.
is
pore whose
In the
only
on
rectangular,
the
time
first
distance
so
the
l-X,
length
pore
where X is
the
diffusivity, it
[To
the factor
is
translate
of diffusivity
a(l-X),
simply
a 2 /D.]
is
defined as
the mean
starting at coordinate
r in
X.
(figure
passage
r
into
32
V2=-2
and
boundary conditions are
units are
effect of geometric
examples, T(r;X)
is
calculated
starting point and
is given by
for a particle
Straight
The
for a particle
length scale
parameter
The mean
the
to those
following
pore whose
VI.3.3.2
Since
results below by
passage
between
to
is
(reflecting boundary)
boundary.
equal
time
of generality that D=1.
calculated here
in a
passage
independent of particle
without loss
multiply the
a
boundary is
physical
dimensional analysis.
the wall
the distance
reflecting,
coordinate
three dimensional
the mean first
from the absorbing
such
rear walls are
for various
to one
particle starting on
farthest
first
two and
to be reduced
In all cases,
D
side and
along natural manifolds
systems.
for a
the
time
10a)
for a straight
the pore.
the
The
appropriate
pore
depends
geometry is
equation and
-174-
(54 )
=
(55)
T(X;X)=
T
(56)
The
-1
,
0
and
,
=0
r=1
solution
of
(54)-(56)
is
[(X2 -r2 )-2(X-r)]/2
(57)
T(r;)
For a
particle
starting
(58)
T(l;)
(1-X)2 /2
=
=
at
r=l,
(55)
becomes
As X+0, T(l;X )+1 .
VI. 3.3.3
Cylindrically
The mean first
radial
position r in
so V2 =ir(r-),
(59)
(60)
(61)
r
and
)
-(r
r
=
T(X ;X)
=
r=1
0
0
passage
,
(51)-(53)
and
pore
nore
time for
the pore.
-1,
=
T)
conical
The
become
(figure
(fizure
this
case
geometry is
lob)
10b)
depends
on the
cylindrical,
-175-
The
(62)
solution of
is
(59)-(61)
2
-r
2
T(r; X)
=
(X
T(l;X)
=
(X 2 -1)/4
)/4
+
lnVr/X
so
(63)
As
Spherically conical
VI.3.3.4
Once
a gain,
radial
so
the
V2=- 13-(r23T
=r-)
(64)
1 r (r
_r
=r
(65)
T(X;X)
=
pore
(figure
the mean first passage
position r in
spherical,
(66)
(lnk)/2
T(l;X) ~ -(lnk)/2
X+O,
the
-
pore.
The
0,
and
= 0
r=1
The
(67)
so
solution of
T(r;X)
=
(64)-(66)
(X
2
-r
2
)/6
is
+
time depends only on
geometry is
and
andT (51)-( 53)
-1 P
=
(1/X-1/r)/3
10c)
become
-176-
(68)
As
=
T(l;X)
(X 2 -1)/6
X+O, T(l;X)
VI.3.3.5
Discussion
(1-X)
resulting
and
(68)
As
with identical
are
shown
times.
(log-log)
strongly in the case of
remains
the surface area of
constant.
The
find
vary
the
conical
absorber must be
pore,
followed by
cylindrically conical
since
the
effects)
VI.3.4
must
this
phenomenon
of the
X2.
greatest for
the
time
This
(58),
The
Thus
the
needed
is as
the
on
the
the
constricted
times
inlets
follows.
in
the
other
for
the
explanation is
incomplete,
itself
and possibly dimensional
outlets
to
spherically
in square pores with
and
pores.
time needed
be contributing.
First passage
is
straight pore
the absorber does not by
other geometrical,
(63),
difference
absorber
conical pore,
X and
pore.
surface area of
determine T;
length
spherically conical
surface are
respectively as
11.
the absorber in
cylindrically and spherically
hand,
Equations
in figure
partial explanation of
X+O,
fundamental
have been presented, with very different
mean first passage
seen most
A
(1/X-1)/3
1/3X
Three geometries
scales
+
-177-
RELATIVE MEAN FIRST PASSAGE TIMES
10000.0-
1000.0
............
SPHERICAL CONE
CONE
STRAIGHT PORE
100.04T(1;X)
10.04-
1 . 0-4
...............................
0 1 I-iI
0 001
.......
1
0.010
......
0.i00
i.6oo
x
Figure
VI.1.
Comparison of first
passage times as functions
of parameter X for pores with geometries
10.
shown in figure
-C
-178-
VI.3.4.1
Introduction
results
not
the
that
length
the
that
illustrate
only
factor determining
a crucial
diffusion
concept in understanding
pore geometries discussed
The
easy
analyze, are not realistic,
plausible
are
pore geometries
of a
scale
first passage
media.
to
to obtain analytical
purpose of section 3.3 was
The
times.
is
This
is
through porous
section 3.3, while
in
and
pore
in this
investigated
by
section more
the Monte
Carlo
method.
Figure
11.3
of
dimensions,
a conceptual model,
contains
the EVAc matrix pore structure.
water-filled pore
space
that are
bulging "pores"
via narrow
"channels" or "throats."
will
be
the
Section 3.4.2
parameters
within
conditions
for
the
(unlike
describes
performed
simple
first passage
shown how
using
section
instead.
to
a Monte
use
(In
this
to each other
chapter,
the important geometric
the cells,
the overall
in
connected
"cells"
term used.)
as well
no
so
3.3),
Section 3.4.3
time
as
problem.
the results
Carlo method,
of
first
boundary
the
time problem.
first passage
pore geometry is not simple,
exist
The
is modelled as a network of
containing
"throats"
in 2
closed
a Monte
form solutions
Carlo
contains a
simulation
solution of a
In section 3.4.4
section
3.4.3
passage
Since
it is
to compute,
times
in
the
more
is
-179-
geometries of section 3.4.2.
complicated
and 3.4.6
of the
results
In sections
3.4.5
presented and
simulations are
discussed.
VI.3.4.2
The
lattice has
diameter
to be at
the centers
of
the
pore bodies meet
Simulations
are
initiated
" outle
as
bisector of
shown
Brownian motion
in
figure
for the
by placing
the
bodies are
cells, and
boundaries.
throats
12.
of
bisector
the
called
are
is
The
to be
taken
the
the
the
two
to
passage
of
throats
other
reach
a
reaches
the particle
taken
time
the first
on
particle
computer then simulates
The
particle, until
two pores.
a
throat connecting
a perpendicular bisector of one of
either
the
The
ts."
perpendicular
pores,
12.
figure
cell
the
of
lattice
their respective
where
The points
the
The pore
..
throats are bisected by
connecting
in
shown
the
in
unit cell
the
taken from
typical example is
A
11.3.
assumed
setting
two adjacent pores,
Consider
figure
geometric
the
time through
the
pore.
Brownian motion
pores,
since
particle,
is executed
in a random walk
starting
that direction.
In
out in
fact,
through a connected
there
is no
one direction,
the diffusing
guarantee
will
pair of
that a
continue in
particle
may cross
-180-
K
BROWNIAN
MOTION
SIMULATION
STARTS
HERE
Figure VI.12.
Example
lattice
of
in
two adjacent pores
figure
11.3.
taken
from
-181--
original bisector any number of
the
and
throat
the pores
into one of
journey
way
its
make
cell
that outlets are
is assumed
resulting model
The
are
the
and
instead
12,
the
of
diameter of
z are
without
to
by using
width w
of
the
outlets
the
figure
are mirror
the
of
throats.
the
sizw
cell
overall
body is
pore
the central
on first passage
can scale
One
times
real
There
in
simulation
is
Note
that
in
figure
as
fixed at
2.
effect of
the
times can be assessed
confounding effect of a growing overall pore
the
dimension.
times
simulations
the
z
fixed
the
considered dimensionless, and
sizes
relative
a
of pores.
in
half-length
assuming
Thus w and
their
the
for
it
The parameters
number of outlets nT,
(throats),
the
Second,
centers of sides
in the
pore geometry
illustrated in figure 13.
through
two pores.
the
bisector between
boundary at the
pore
placing a reflecting
equivalent to
This is
boundary.
the
on
simulation
reflected
images,
that adjacent pores are mirror
the
in
is assumed
First, it
12.
figure
etc.
two restrictions are imposed
For simplicity,
model of
original
the
pore,
the other
into
to
return
and
It may even
times.
the
are
14.
for a
scale
the dimensionless
pore in a cell
factor
seven possible
Since
2
/(2+2z)
first
of fixed
passage
dimension&
2
outlet configurations,
configurations
images of configurations
6 and
7 in
that
as
shown
figure
1 and 3, respectively,
-182-
Mirror image
pore
Effective reflecting bisector
Throat length (z)
Throat
diameter
(w)
.9-
i bisector
nT 2
Absorbing
bisector
Figure VI.13.
Parameters for simulation of Brownian motion
through constricted pores.
n =Number of outlet
throats.
w=Outlet width.
z=Half-length
of
throat.
-183
-
I1
3
2
4
5
6
Figure VI.14.
Configurations
Pore outlet configurations.
6 and 7 are mirror images of I and 3,
in
so only 1-5 are tested
respectively,
simulations.
Brownian motion
-184-
they must yield
As
usual,
loss of
generality
Before
the diffusion ,coefficient is assumed without
1.
to be
is necessary
the next
in
the
to make
out
of a
at
two
two
dimensions whose
the origin.
Surround
the
origin by
Of
probability distribution of
"hitting
time
boundary
of
the
required
for
This
the
interest
is
time"
for the Brownian particle
the first
time.
Also
to
a square
the
which is
T,
reach the
of interest are
the mean T
that distribution and
the
probability distribution of
i.e.
the
point on
"hitting spot,"
Brownian
Once
The
particle makes
again,
full
is
square
with boundaries x=+l and y=+l.
the
derivations.
in
Consider a Brownian particle
path starts
Brownian motion
section.
First passage
VI.3.4.3
so
tested.
further describing
simulation, it
done
are
1-5
only configurations
time distributions,
first passage
same
the
the
its
the
boundary where
the
first contact.
diffusion coefficient is
probability distribution for T
imbedding procedure,
set
by
an
P(T,X)
is
the cumulative distribution
1.
is determined
illustrated
first,
to
function
in
figure
(c.d.f.)
15a.
of
-185P(T;x,y)ui
on boundary
4.
y-1!
a?
(0,0)
.
a)
P(0;z,y)s0
y
.1 ,
xe-i I
X-1
NO
on boundary
yeI
b)
.
ye
Figure
.1
(0,0)
1
x--1
VI.15a)
b)
,
a
Imbedding approach for determining
P (T)
cumulative probability distribution
See text for 1
time" T.
of the "hitting
exp lana tion.
Imbedding approach for determining mean
for explanation.
See text
time T .
hitting
-186the
time T
required
point X=(x,y) in
obtained
by
interior of
solving equations
whole boundary of
result
the
to reach the
the square
boundary,
starting from a
the square.
(40),
taken
(41)
and
P(T,X)
(42),
is
with the
to be absorbing.
The
is
P(TX)
(69)
=
1-(4/
)2[
2 n+1
-(2n+1)
2
n 2 T/4cos( 2 n+)n x]
n=o
- (2n+l ) 2 n
2n+
n=O
The
starting
at
the
T/4
e
desired cumulative distribution
particle
2
origin
cos(2n+l)ny]
of first hits
is given
for a
by
00
P
(70)
= P(TX=0)
(T)
=
e-(2n+l)
-/
2
n 2 T/4] 2
n=0O
The
p1 (T)
is
corresponding
probability
density
obtained by differentiating
-(2n+l)
2
_
n 02+
n=0
n
(70)
function
(p.d.f.)
with respect
to T:
n 2 T/4
2n+
n=O n+
(71)
)e-(2n+l)2,2T/4
T>0
p 1 (T)
0
The
,
special
summation
in
T=0
case for T=0
(71)
is needed because
is divergent at T=0.
the
second
-187-
Plots of P 1 (T)
and p 1 (T)
are
shown
in figures
16a,b,
respectively.
The mean hitting
imbedding
time T
is
into a more general
problem (figure
point X=(x,y),
the mean hitting
equations
and
(19)
T (X)
(72)
=
1/2
(20),
-
(x
2
+y
-1
n:0
The algebraic
(19),
while
conditions
T
(73)
2
)/4
1/2
-
-
3
(16/t
is
the
For any
solution of
)
+ cos(2n+1)7vcosh(2n+1)x
cos (2n+1) 2 cosh(2n+1)7y
37
(2n+1) cosh(2n+1)r2
is a particular
series portion is
(52).
-
time T(X)
15b).
viz.
portion of (72)
the
similarly obtained by
used
The desired value t
to
solution of
match the
=t(X=0)
is
boundary
then
S(-1)n
(16/IE 3 )
~ 0.295
Z= (2n+1) 3coah(2n+1)7
[Note:
of T,
T-
can also be
using
Given
equation (71)
of
the
First note
probability
of
be
without
supposed
lands
on
for
the
taking the
the
hitting
spot
that the
landing
on
loss
any
of
side given by
on
the
probability
the
boundary can
square
side
is
is
symmetric,
equal
generality
x=+l,
expected value
p.d.f. of T.]
a particular value of T,
distribution
derived.
computed by
-l<y<l
that
.
to
the
1/4.
so
be
the
It
particle
Define
can
-188-
.9.8.7a)
.5
.5-
. .3
.2
T
HITTING TIME
4
b)
3-
.2
a.
o.5
.2
1
T
HITTING 7IME
Figure
VI.16a) Cumulative
P
(T)
of hitting
particle
square
b)
distribution
time T
starting
box
function
at
of radius
(c.d.f.)
for Brownian
origin
and
ending
1.
Probability distribution
function pI(T),
derived
P
by
diffentiating
(T).
on
-189-
Q1 (y;T) =
lands
Prob{Particle
this
is
the
in
[-l,y)
at
time T,
and
first hit}
and
q 1 (y;T) =
To
associated
obtain q 1 (y;T)
describing
solved,
(75)
a one
the motion of
the
=
dQ 1 /dy.
dimensional
particle
with absorbing boundaries
initial "mass"
(74)
p.d.f.
concentrated at the
yy
t
c(+l,t)
=
diffusion equation
in the y
at y+l,
direction
and with the
origin:
9
0
for
t>0,
and
(76)
c(y,0)
where
6(- ) is the Dirac delta function, also known as
unit
impulse function.
particles
y=+l.
6(y)
diffusing
in
c(y,t) is
the
Then
q 1 (y;T)
=
c(y,T)/fc(y,T)dy
-1
"concentration"
y direction,
1
(77)
the
never having
the
of
hit
is
-190and
y
(78)
Qy(y;T)
=
fq
-1 1
(y;T)dy
The solution of
(79)
= I e-(2n+1) 2 .2 T/4cos( 2 n+l)ty/
n=0
c(y, t)
which, when plugged
into (77)
co
(80)
q
1
is
(76)-(78)
(y;T)
=
(n/4
and
(78),
-(2 n+)
(2n+1)y
2
)
2
yields
respectively
2 2
r T/4
n(-I)
-(2n+1) 22T/4
n
Q
2n+
eT
and
W
(81)
(2n+1)y+
n=O
Q 1 (y;T)
(_,)nlI
OD (-1)n
2Z: 2n+I
n=0
As
(82)
T+
e-(2nl)
2 2
Tr
T/4
- 2n+l) 22T/4
,
qI(y;T)
+
(n /4)cos(n y/2),
+
(1/2)(l+sin(ny/2)I
and
(83)
1(y;T)
A
plot
of
figure
17.
As expected,
more
q
concentrated
for several
toward
values
the most
of T
is
illustrated
probable values
the center
(y=0)
as T
in
of y become
approaches
-191-
T-.06
1.2
T-. 10
T-.
15
T-.295,-=
.4
-
.2
-1 .8
.6
-. 4
-2
0
.2
.
6
-
HITTING SPOT
Figure VI.17.
Probability distribution of "hitting spot" y,
assuming that particle lands at x=1, given the
hitting time T.
-192-
zero.
However,
it appears
greatly with T.
that
Indeed, q 1 (y;T=.295)
furthest deviation from those
for
the
by
y values
the values
relative
insensitivity
simplify
the
The
shown
of q1
simulation
above results
In the
to T
curve).
in figure
The
17
for
That is,
than
following
(and Q1 )
is
distribution of
the
concentrated near y=O
be more
plot of q (y;T=.06).
the
(dashed
(T)=.0155.
T will
does not vary
curve)
(dotted
two curves
for which P
the case T=.06,
than 2 percent of
less
function q 1
indistinguishable from qI(y;T=c)
visibly
is
the
is
is
indicated
section,
used
this
to
procedure.
can be
generalized
to
the case of a
particle with diffusion coefficient 1 executing Brownian
motion
p
p
,
p1,
q
p
in a
and Q
,
and
q1
square with boundaries x=+p and
respectively.
Ql,
squared,
length
(T)
= P
(T)
=
(84)
P
(85)
p
(86)
q
(y;T) =
(87)
Q
(y;T)
(88)
T
p
p
p
=
.
2 9
probability functions
be the
p
so
1
the
(T/p
p 1 (T/p
2
)
,
2
)
,
2 5p 'T
(y/p;T/p
2
)
2
)
relations
Let P ,
analogous
For diffusion,
following
q (y/p;T/p
= Q
y=+p.
time
hold:
to Pi,
scales as
-193(89)
q
=
D(y,T+)
(n/4)cos(ny/2p)
and
(90)
Q pD(y,T+
) =
(l/2)[l+sin(,ty/2p)]
VI.3.4.4 Description
of
Carlo Brownian motion
the Monte
simulation
this
In
Brownian motion procedure
by
performed
is
The
V.
in Appendix
in the
following paragraphs.
is
entered
assumed
the
that
time 0.
north
throat bisector allows one
the
central
over
particle from
pore
body.
a distance z.
particle
to
to
similar
(91)
to be a reflecting
t0
=
The symmetry across
the
traverse
those
.5z 2
of
to consider 'the north throat
The
boundary.
the distance
section
3.3.2,
to
is
to
the
reflected random walk
(mean)
can be
to
first step
the inlet
to
consists of a
The expected
figure 13
the
the bisector
This
to
is
position along
pore at a uniformly random
the
simulation is
has just
particle
Brownian
throat bisector at
move
should refer
reader
north
bisector
The actual
given.
the FORTRAN program called RSPASS, which
listed
It
the Monte Carlo
section a description of
be
time
required
for
the
shown, using methods
-194-
Because
the origin
uniformly
which
the
is
the
is also
at
position along
inlet
iterations
the Brownian particle is
at
the
start of
is
t
the
since
is
such
the box
the
(here
n-n
If X
largest radius such
Now
pore.
the
the
box for
=
T
(88)
The
the mean time
the
east
the
at
is
along
position
If X
to
pore
largest radius
the
of
pore
central
the
pore
body and
boundary, then p
box
particle
the
been shown
lies within
to hit a
the
is
the
point on
(88)]
[eq.
land
reflects the
particle makes
the
First
to be
the
side of
its
the
first hit
box (north,
chosen with equal probabilities,
that
side
according
is generated
) given by equations
(or c.d.f. Q
P
told
the
time was
random.
or west)
p.d.f. q
(90).
the
that half the
the box where
is generated
then
is
n
box of
A square
.
and
.295pn 2
point on
south,
first
on
for
n
position X n,
the boundaries
both
lies
run
where p
lies within
includes
"pore"
half-throats).
the
,
drawn around X
radius p n
that
the
X0 '
that after n
Suppose
illustrated
time
designated
interative
procedure,
the
the
distributed.
is
for an
18.
figure
bisector is
uniformly
the
point
initial
in
the
particle's
of arrival
as
set
the
step
first
position
particle on
the
distributed,
inlet after
The
of
(89)
and
to
and
p
lies
on
the pore
on a point
boundary and
outside
particle across
the
the pore,
pore
particle is
the
the
program
boundary.
The
simply
-195-
reflecting
boundary
starting point
xy 0 )
(X,y)
absorbing
X
x"
I+ yn+ If.01
boundary
CXmmi,ym+i)
(X.,y.)
reflect
absorbing
boundary
Figure
VI.18.
Illustration of
Brownian motion
pore.
procedure
through
a
for
simulating
model
constricted
-196-
(possibly reflected) hitting point is
the
time
is
(92)
t
and
a new
n+1
The
one
of
= t +T
n
p
the
is
the number
reset to
is
s
the
of
=
m
less
(s
value
the pore,
where
far been executed
thus
the
final
reflecting
restarted,
m
in
(north)
time is not
but the
passage
times T
Pm
standard
computed for
When
the
the coefficient
the mean, defined by
/ fi)/TP
P
first passage
tolerance,
time has
(This criterion was
executed
sp,
the first
than a prescribed
accuracy.
were
is
through
algorithm described above.
variation of
the mean
and
the particle hits
The
program keeps a running mean TPm and
deviation
r
time T
particle hits
the
throat bisectors.
that have
the procedure
until
zero.
The
pore by
of walks
If
bisector,
(93)
process continues
first passage
t
of
commences.
the nonreflecting
pore.
to be Xn+l'
,
iteration
of
the
taken
updated by
iterative
is
then
for a
which are
been computed
not
single pore.)
the
the program assumes
final values
tested
The
to
until
program
of TPM and
that
sufficient
100
passages
then prints T
spm'
-197-
For each combination
and
width w,
figure
14,
standard
calculates
in the
points.
Neither
time nor
To
size.
be accurate,
in
geometry, which
Thus,
as
the
becomes
lattice points
described
here saves much
it fills
almost
the
long
A variant of RSPASS,
which
took
p-boxes
T
to
into
are
whole
in
iterations
the
size,
then
in a
fact randomly
lattice
if
the
single
the hitting
to
random walk
the
central pore
that
the
interpoint spacing
called RSPASSD, was
consideration
of grid
throat width.
example,
pore,
distance
its
feature of
to complete a
the
on grid
the number of required
the
For
time.
land
for a discrete
the
square of
the
grids.
Second,
the choice
take more
steps
the center of
jumps a
particle
usually
all walks
land
smallest
drawing boxes of maximal
By
located near
around
the
larger, and
inversely proportional
spacing.)
is
to
to
discretized.
decreases,
(The number of
accomplish.
the
Thus,
smaller.
becomes
this case
throat width
particle
lattice spacing
determined by
random walk must be
in
of T .
constrained
dependent on
the
shown
some advantages over
the
allows
space are
and accuracy are not
throat
random walks on discrete
is not
pore--it
estimate
here has
involving
techniques
anywhere
is
an
the present procedure
First,
speed
pore configurations
procedure described
The
is
the
for each of
RSPASS
throat half-length z and
of
procedure
the particle
box drawn
body, and
the
iteration.
also written,
times for
distributed, with a
the
198
distribution calculated using
(85),
lookups
and
table
50%
more computing
section
as
that
those
than does RSPASS,
19
z=.02,
and w=2.0,
set
to
0.05.
are
accurate
for
0.5,
The
of about 350
respective pore
The
for RSPASS
of
simulations,
set
values
of w
(2.,1.,.5,.2,.1,.05,.02)
configurations
For each
shown
in
(w,z),
figure
all
Therefore,
RSPASS
main
(.5,.l,.02).
to bring
it was
simulations.
the
For
seven
subsequent
and
in both cases
passages were needed
the
P was
programs
the
It was found
in
used were
tolerance
provided by
results
virtually identical.
of T
Parameters
0.1.
the estimates
to within 5%.
the
14a-e.
0.2,
same
was used.
Brownian motion procedure
in figure
to use RSPASS
next
the
the
P below .05 for a given pore geometry.
decided
of z
in
this reason, RSPASS
1.0,
Thus,
that an average
the
generally requires
For
and RSPASSD
shown
configurations
RSPASSD are
random variables
shows a comparison of values
generated by RSPASS
in
(87).
and
essentially the
are
the results of RSPASS
VI.3.4.5 Results of
and
on T
correct algorithm.
be shown
It will
time.
of RSPASSD.
Figure
rigorously
the generation of more
it requires
However,
the
fact represents
in
depend
(80),(81),(86)
by equations
a manner prescribed
RSPASSD
(70),(71),(84) and
hitting spot probabilities
that the
and
equations
five
14 were run.
was
and
run with
three values
pore
This
led
to
-199-
COFIGURATION 1
CWIGURATION
Ia
2
I
a
9
,o
'F
.31..3
a
COWI"URATION
wr
10;
I.
a
I
I
I
a
THMAT VIDTH
U
I
I
*3&.
.1
13
I
x.
THROAT VIMT
w
So -
wr
I I
I
I
-
a
9
THROAT VIDTH w
COWIGRATION
a
U
.
0
I
ON.Woodhmn.wo
I
.3
T'AT VIDTH
Figure VI.19.
I
........
I
w=.="
13
I
-.
V
I
THROAT VIDTH
I
w
a
Comparison of mean first passage times T
computed using RSPASS and REPASSD.
*
-
a-
RSPASS
RSPASSD
results.
results.
3
-200-
5X7X3=105
before, and
to bring
on
the
the average about 350
error
below
that
be 5% as
passages were
required
point.
passage consisted on
Each
chosen to
tolerance was
The error
runs.
order of
the
10 2-104
Table
consisting of box drawing and random jumping.
a distribution of stepsizes
(z=.0l,
w=.l).
.1.
.001 and
For w=.l,
be
with size
steps
single
step
with size
From T
.01.
Thus
a retardation
where
.5(2+2z)2
in a straight
have
2.
magnitude
.1
is
size
this
are between
.1 and
1.
lattice would
equivalent
to
100
steps
1 is equivalent to 104
is a very efficient procedure
factor due
to
pore geometry, R
,
using
RP = T P/[.5(2+2z)2
22
table
size
sample run
between
stepsize on a finite
of
(94)
The
stepsizes
1 gives
random walks.
calculated,
throats
the
there are also many
.01, while a step of
for executing
is
However,
for a
box radii)
that most of
an appropriate
A
.01.
Notice
(i.e.
steps
is
(w=2,
total
the
mean first passage
configuration 2)
length
values of TP,sP,
One
feature
as TP,
is
2+2z
and
(see
RP are
that s
time
for a particle
pore whose
section
pore
body and
3.3.2).
listed for all cases
is always
of
similar
implying a wide distribution of
values of
in
-201-
STEPSIZE
Configuration
Table
0.1-1.0
.01-.1
.001-.01
.0001-.001
<.0001
.217
.370
.325
.084
.003
2
.231
.378
.311
.077
.003
3
.199
.361
.350
.088
.003
4
.201
.359
.347
.091
.003
5
.117
.362
.364
.094
.003
VI.1
Frequency distribution of
of Brownian motion
z=.01,
w=.1
.
stepsizes for
simulation
through constricted pores.
-202-
z=0. 5
CONF.
s
nT
P
w=2.0
1
1
2
2
3
3 .408
4 .278
2 .099
1 .495
1 .433
2.886
3.443
1.771
1.224
0.982
0.757
0.951
0.466
0.332
0.318
w=1.0
1
1
2
2
3
4.723
5.396
2.645
2.208
1.810
3.982
4.026
1.879
1.823
1.185
1.050
1.199
0.588
0.491
0.402
w=0. 5
1
1
2
2
3
6.521
7.246
3.898
3.607
2.275
5.790
6.064
2.924
2.879
1.564
1.449
1.610
0.866
0.802
0.506
w=0. 2
11. 173
11. 295
6. 016
5. 966
4. 210
10.406
10.316
5.241
5.089
3.266
2.483
2.510
1.337
1.326
0.936
w=0. 1
18. 331
18. 305
9. 677
9. 355
6. 519
18.440
17.288
9.335
8.720
5.599
4.074
4.068
2.150
2.079
1.449
w=0.05
33.
29.
16.
16.
10.
431
357
039
367
861
32.642
28.422
15.367
17.732
9.313
7.429
6.524
3.564
3.637
2.414
w=0. 02
64.
72.
35.
34.
22.
223
044
416
345
202
63.047
73.079
34.332
34.200
22.164
14.272
16.010
7.870
7.632
4.934
Table
VI.2.
Mean first
deviations
R
passage times (T ),
and retardation
(s ),
Conf. = outlet configuration,
w = throat wid t h,1
z = throat hal f -depth,
nT = number of outlets.
standard
factors
(R
).
-203-
z=O. 1
CONF.
s
nT
P
w=2.0
1
1
2
2
3
1. 188
2. 385
0. 788
0. 415
0. 396
1. 369
1. 929
0. 671
0. 456
0. 361
0. 491
0. 986
0. 326
0. 171
0. 164
w=1 .0
1
2
2
1
1
0
274
932
581
023
814
2 .131
2 .572
1 .408
0 .877
0 .582
0 .940
1 .212
0 .653
0 .423
0 .336
2
2
3
2
R
1
2
2
2
3
3.572
4.211
1.899
1.589
1.250
3.115
3.523
1.581
1.484
0.921
1.476
1.740
0.785
0.657
0.517
w=0. 2
1
22
2
2
3
5. 515
5. 978
3. 186
2. 602
1. 911
5.194
4.902
2.897
2.390
1.549
2 .279
2 .470
1 .317
1 .075
0 .790
w=0. 1
2
1
1
2
2
3
7 .161
8 .072
3 .880
3 .517
2 .812
7.604
7.466
3.686
3.305
2.600
2. 959
3. 336
1. 603
1. 453
1. 162
w=0. 05
1
1
2
2
3
1 2. 087
1 0. 658
6. 165
4. 768
3. 895
12.068
10.029
5.553
4.564
3.202
4. 995
4. 404
2. 548
1. 970
1. 610
w=0.02
1
1
2
2
3
20. 411
18. 632
10. 043
9. 730
6. 528
20. 271
18.317
9.377
9.500
6. 658
8. 434
7. 699
4. 150
4. 021
2. 698
w=0. 5
Table
VI.2.
(Continued)
-204-
z=0. 02
CONF.
s
T
n
R
w=2.0
1
2
3
4
5
1
1
2
2
3
0.694
2.147
0.496
0.217
0.206
0.906
1.663
0.540
0.266
0.232
0.334
1.032
0.238
0.104
0.099
w=1.0
1
2
3
4
5
1
1
2
2
3
1.836
2.505
1.220
0.832
0.611
1.707
2.045
0.997
0.635
0.459
0.882
1.204
0.586
0.400
0.294
w=0.5
1
2
3
4
5
1
1
2
2
3
3.049
3.506
1.729
1.342
0.986
2.834
2.508
1.457
1.113
0.745
1.465
1.685
0.831
0.645
0.474
w=0.2
1
2
3
4
5
1
1
2
2
3
4.491
4.843
2.347
2.013
1.671
4.347
4.283
1.841
1.849
1.506
2.158
2.327
1.128
0.967
0.803
w=0.1
1
2
3
4
5
1
1
2
2
3
5.275
6.169
2.912
2.258
1.796
4.623
5.454
2.420
2.005
1.427
2.535
2.965
1.399
1.085
0.863
w=0.05
1
2
3
4
5
1
1
2
2
3
6.945
7.676
3.392
3.051
2.502
6.321
7.184
3.222
2.988
2.066
3.338
3.689
1.630
1.466
1.202
w=0.02
1
2
3
4
5
1
1
2
2
3
8.813
10.517
4.972
4.583
3.485
8.859
9.859
4.670
4.305
2.999
4.235
5.054
2.389
2.203
1.675
Table
VI.2
(Continued)
-205-
accuracy in estimating T
5%
achieve
the values of R
Many of
(large w,
throats
that are adjacent
particles
In
the
width w
inlet (north)
to the
"cut corners,"
to
subsequent analysis,
than or
less
to
equal
.2
length scales
macro and micro
sharp definition.
pores,
the
When
It was believed, prior
determining
outlets),
find
the
them.
are
(n =1),
harder
to
the
(i.e.
virtually
and
the
the diame ter
1/10
This
the
decoupling of the
if
to running
the
pores have
the
as wide as
simulation, that
important
the
narrower
the
the
throats
the dependence
for outlet
for configurations
parameter
in
(i.e.
be for a Brownian particle
3 and
of R
to
on w,
different symbols
throat configurations.
same
of
the
eliminates
throats are almost
In each case,
five
figure 20.
in
the most
shows
of z.
allows
This
side.
can occur only
it should
Figure 21
for each value
correspond
because
R,,
central pore body
be
w would
width
throat
the
shallow
to
less well defined.
is
lattice
the pore
less
only those cases with throat
Moreover,
cutting corners.
problem of
the
as illustrated
pore body) will be considered.
the
R
sides of
z) on
small
2 are
in table
displayed
there are wide outlets
This occurs where
than 1.
per configuration.
passages
times as many
25
require
would
1% accuracy
To achieve
.
to
required
passages are
explains why so many
This
TP.
The
values of
configurations
4 (n T=2).
This
1 and 2
is
-206-
Particle starts here
1
Z-
Throat
bisectors
Particle ends here
/
/
/
/
//
P~~~.
.mmm
m
m .ini. mm m mm
m
Figure VI.20.
mm
Diffusing particles can "cut corners"
are wide and shallow.
throats
if
-207-
for
especially true
z'I.5.
For a fixed value
R
for
If one compares,
decrease.
increases when nT and w
of z, R
fixed
for different values of NP,
on w
approximately parallel
on the
z,
dependences of
the
they are
that
sees
one
In fact,
log-log plot.
the
relation
(94)
R
)
(z,w,n
=
k(z,w)/N
,
where k(z,w).is an undetermined
This
quite well.
R
increases.
held fixed,
increases with w and N
z
As
function, holds
seen by comparing
can be
figures
An unsuccessful attempt was made
dependences.
(95)
=
R
where kz
figure
close
k
For z=.5,
/n
w
VI.3.4.6
The
the z and w
,
However, for
to describing
separate
the dependence
depends on z alone,
21a).
to
c.
and
21a, b,
(lines
works fairly well
z=.l
and
.02,
the dependence' of R
(95)
on
does
in
not come
z and w.
Discussion
results of
section 3.4.5
show
that one
can obtain
-208-
Figure VI.21. Retardation factors RP as a function of throat width w
and length z, for different throat configurations.
Lines are for equations Ry=Kz/n~w,
where nT is the
number of outlet throats, and Kz depends on z alone.
a)
z=.5
b)
z=.1
c)
z=.02
-209-
z =.5
100.0-
E
o
*
0
CONGIF.
CONFIG.
CONFIG.
CONFIG.
CONFIG.
1
2
3
4
5
10.0-
F34
0
1.04-
0.1 II
0.01
0.10
THROAT WIDTH
Figure 21a
1.00
v
-209a-
z
=.1
100.0-CONFIG. 1
CONFIG. 2
CONFIG. 3
o
*
CONFIG.
CONFIG.
4
5
10.0--
E0
1.0-
0.11
0.01
0.10
THROAT WIDTH
Figure 21b
1.00
w
-210-
z = .02
100.0--
ol
CONFIG. 1
CONFIG. 2
CONFIG. 3
o
CONFIG. 4
*
CONFIG. 5
10.0-C,
z
0.1
1.00
0.10
0.01
THROAT WIDTH
Figure 21c
w
-211arbitrarily high retardation
appropriate
choice
That R
outlet.
In
pore was
set
the
to
It
the equivalent of
1/n
effect of z and w are
an outlet.
z
that
expected
initially.
explanation.
"trap"
The
only find
throat.
central
there
pore
"slip
entered
the
an outlet, but
is
the
the
particle
back"
throat.
into
the
dependence of
manner.
This was not
a qualitative
be regarded
central
the
it reenters
particle can become
lost again.
in order
the
to
narrow
a random walk,
pore
Once
as a
particle must
traverse
is executing
the
z, R
the diffusing particle
progress
it must
the
because
increases.
body can
However, to make
Because
often
body,
However,
is now
For fixed
from which a Brownian particle must escape
make progress.
will
fixed, R
to
twicd.
unlike for nT,
increases with w
find an
from pore
on w varies with z in an as yet undetermined
As
the
section 3.4.5,
is expected,
the chance
to
This form
.
intertwined.
This
However,
time
results of
The
increases.
since
if one combines
1/nT factor must not be used
increases
w and nT'
for a particle
that
the
pore by
is obvious,
the
finds
has
z,
the mean hopping
section V.4 with
increasing w
not
it takes
should be noted
decreases as w
R
parameters
longer
section V.4,
justified.
then
the
should decrease with nT
fewer outlets,
results of
of
factors within a
body
the
The
after
it
central pore
longer the
it
-212-
largest retardation
The
considerably smaller
(table
experimentally
case
the
there
that
that one must
obtain retardations
of
in three
dimensions
Consider a cubic pore
diameter w=.2 has area
of
the
obtained
with
There
theoretical
a
grows much
retardation
difference
area
a
diameter
in
of
will
say
.02/2
=
for this
It appears
100.)
to
in order
constrictions
that
the
two dimensions, while
the
however,
than in
easier
2 (figure
(.2) 2=.04, which
two
consideration.
pore
faster with
of
dimension
the
of
100
it was
shown that
then does
pore.
the
rates
linked directly
to
that
factor
retardation
inner radius X
the oulets decrease
is
that deserves
for a cylindrically conical
could not be
the area
10.
In section 3.3
a
An outlet
1/100
of dimensionality
yields
dimensions.
22).
is
real
factor
obtain a constriction
To
is
is another aspect
factor
(We
Thus a constriction factor
conical
spherically
the
face.
throat is
factor of
of diameter
of
cubic
the
consistent with experimental
three-dimensional.
is
100
so
pore body.
severe
simulations were performed
system
retardation
simulated
It should be recalled,
observations.
is
This
retardations observed
is a constriction
impose very
2).
table
for w=.02,
diameter of the
the
(see
The
IV.2).
16 was obtained
factor of
1/100th
than
in the
factor obtained
approximately 16
simulation was
happen.
this will
that
the chance
the greater
throat,
as X decreases.
the
The
at which
-213-
.2
.2
I2
22
Area ratio=
Figure
VI.22.
-4
(.2/2)2
For cubic
square of
the throat
=1/100
the constriction
pore,
of the cube
the ratio
diameter.
is
the
factor
diameter to
-214-
Dimensionality
itself seems
to play a role.
that simulated Brownian motion should be
three-dimensional model pores
effects of
in
scale
scale simulations
just
Brownian
particles cannot find
pores.
In
the
lower
volume
of
lattice
lattice surface.
the
scale
due
is also
can be
VI.4
seen as
In effect,
this
study
three
of
getting
the
the
lost.
and
on
into
to
the macro
Thus
the
the micro
scale
same phenomenon.
retarding
identified:
drug
1.
High concentration
2.
Random pore
3.
Constricted pore geometry.
topology,
lattice
slowing release
factors
of
that
forward progress
retardation on
the macro scale
Combination of factors
In
been
loadings
that do not allow
two aspects
appropriate.
particles often wander
to particles
retardations seen on
"macro"
their way out of constricted
because at these
less well connected, and
the
is
the
between
greater retardation occurs at
"macro" case,
loadings
the
described and
the "micro" discussion has
theme
of
the
truly assess
in section V.4
simulations described
regions
to
the relationship
The main
is
executed through
constrictions on retardation.
Finally, a discussion of
"micro"
order
suggests
This
and
in
pores,
release have been
-215-
Factors
1 and
to explain
2 were
to be by
shown
themselves
the experimentally observed
insufficient
retardations.
Parameters can be chosen for factor 3 which yield
retardation
factor.
However,
simulations
indicate
that
physically
were
was
retardation
observed for
.50 w/w
in section
2 and
4.
these
factors are
multiplied
2
two-dimensional
may not be
in
achieved without
to yield
a retardation
factor
between
factor of approximately 3.
factor between
150/12=12.5
including the
6 and
and 150/6=25
the pore geometries.
of 150.
150
Factor 1
BSA-loaded EVAc matrices.
reasonable demand on
factor
of approximately
factors
independent of each other,
to yield a
constrictions
the original
those parameters
Factor 2 yields a
factor of between
much more
of
realistic.
Experimental
shown
the results
any desired
they
12.
can be
This
leaves
a
to be explained by
This
factor must be
1/nT factor twice.
the
As
geometric
This is
a
explanation than
-216-
REFERENCES
[1]
Bawa,
R.S.,
Controlled Release of Macromolecules
from
Ethylene-Vinyl Acetate Copolymer Matrices:
Microstructure and Kinetic Analyses, M.S.
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M.I.T.
(1981).
[21
Keller, K.H.,
mutual
Chem.
[3]
diffusion
75,
3925
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1883
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Anderson, J.L.
Fair,
488
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intermolecular
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J.
Chem. Phys.
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Rauh,
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Phys. Chem. 82,
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[8]
Phys.
solutions at high concentrations:
by quasielastic
65,
3240
J.
the concentration
Benedek, G.B.,
in protein
Anderson, J.L.,
64,
proteins
component solutions,"
diffusion as a function
(7]
"Tracer and
(1975).
Phillies, G.D.J.,
A
S.I
of macromolecular diffusion coefficients,"
diffusion I. Two
62,
[6]
"Prediction of
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[51
Eng.
of
Yum,
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Anderson, J.L.,
Ind.
and
coefficients
379
dependence
[4]
Canales, E.R.,
C.C.,
608
"Particle
and
ionic
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"Diffusion
finite concentrations,"
of
spherical
J.
Chem. Phys.
(1976).
B.D.,
Chao, D.Y.,
diffusion coefficients
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in bovine
measured by quasielastic
light
"Mutual
serum albumin
scattering,"
J.
solutions
Coll.
-217-
Sci.
Interf.
[9]
66,
323
(1978).
B.N.,
Kitchen, R.G. ,Preston,
"Diffusion of
serum albumin
J.
55,
Poly.
Sci.
[10]
DiLeo, A.
[11]
Doherty,
39
and Wells,
(1976).
(1983).
Benedek, G.B.,
"The effect
the diffusion of macromolecules,"
5426
[12]
Rhine,
W.D.,
The
Miller,
E.S.
Chem. Phys.
J.
Sustained Release of
and Peppas,
water-soluble
acetate
[14]
61,
bioactive
Thesis, M.I.T.
"Diffusional
N.A.,
from
agents
copolymers,"
Chem.
MS. Thesis, M.I.T.,
1981.
Eng.
Investigations on
Einstein, A.,
Macromolecules
Comm.
the
Russel,
"Brownian motion of
W.B.,
suspended
in liquids,"
(1979).
re lease
of
ethylene-v inyl
22,
(1983).
303
Theory of Brownian
Movement, Dover Publications, New York,
[15]
charge on
of
(1974).
from Polymeric Matrices, M.S.
[13]
solutions,"
in concentrated
Ph.D. Thesis, M.I.T.
P. and
J.D.,
1956
small particles
Ann. Rev. Fluid Mech. 13,
425
(1981).
[16]
Kozinski, A.A.
and Lightfoot,
ultrafiltration:
filtration,"
[17]
Crank,
J.
A.I.Ch.E. J.
Mathematics
Press, Oxford
[181
A general
E.N.
example
18,
1030
of Diffusion,
"Protein
of boundary
layer
(1972).
2nd ed.,
Clarendon
(1975).
Isaacson, E. and
Keller,
Methods, Wiley, New York
H.B.
Analysis
(1966).
of Numerical
-218-
[19]
Paul,
D.R. and McSpadden, S.K.,
from a polymer matrix,"
[20]
Pismen,
L.M.,
structure,"
[21]
[22]
Sci.
29,
J .A.
small
14,
pores,"
Biophys.
Deen, W.M. and
J.
Papoulis, A.,
Owen,
body,"
Smith, F.G.,
Memb. Sci.
J.E.,
"The
"Effective
particle in a
277-280
"Restricted
130
(1983).
a random
t ransport in
(1974).
charged microporous
217
of a
195,
(1976).
"Hindered diffusion of
12,
resistivity
33
(1974).
(1982).
Variables,
McGraw-Hill, New
A.I.M.E. Trans.
Brownian
122-7
Probability, Random
Processes,
Zwanzig, R.,
117A,
J.
polyelectrolytes in
Stochastic
[25]
Eng.
1,
porous media of
Anderson, J.L. and Quinn,
membranes,"
[24]
Memb. Sci.
"Diffusion in
Chem.
synthetic
[23]
J.
"Diffusional release
York
and
(1965).
fluid-filled
169-174
p orous
(1952).
diffusion coefficient or
two-dimensional
a
channel," Physica
-219-
was
a
gathered
previously
suggests
This
the
that
unknown variables
there are
and
physically during
For
solution may
be crucial
viscosity of
the
in the
variables
release
the
that affect
cast
process would be very valuable.
solvent evaporation
well.
quite
III)
thorough study of what happens
A
casting
rate of
(appendix
data
of
the matrices are
introduced when
behavior.
set
model fit
the
Moreover,
places.
in unexpected
discrepancies
there were
chapter V,
the model of
compared with
it
and when
show certain irregularities,
data of chapter IV
The
suggest further work.
V and VI all
Chapters VI,
FOR FURTHER WORK
SUGGESTIONS
CHAPTER VII.
example, the
polymer
the
formation
of pore
structure.
matrix
should be made.
chapter
of
results
Chapter
behavior
be
areas
two
conical
pores
understanding
the
be investigated
should also
indicates
in
three
the real
be
that
the
First,
of pursuit.
function of model
a
done partly because
It
in
in the
IV and V.
suggests
should
dimensional.
the
This may aid
retardation as
of
parameters
should
VI
distribution of drug
the
assessment of
A careful
pore
dimensions. This
pores
done because
are
three
the analysis of
dimensionality
may
play
an
-220-
important role
is
true
in determining
for cubic vs.
of
models are
square pores,
the approach
developed
show
that
However,
in
this
that contain parameters
of chapter VI
that pores are
these
If
this
it would be very
taken
are not accessible experimentally.
constriction model
factors.
theoretical and practical grounds.
interesting both on
A weakness
retardation
is
In
are
purely
is
that
that at present
particular,
based on SEM
connected via
observations
study
the
observations
throats.
narrow
and
qualitative,
constriction factors have not been assessed quantitatively.
The
development of experimental
determination of
would
the
be
study
media
the
an extremely
of EVAc
in general.
techniques
that allow
constriction parameters
important
controlled
of
breakthrough,
release
matrices,
the
section VI.3
not only
but
for
for
porous
-221-
ZERO ORDER RELEASE
I.
APPENDIX
they normally do not
the
within
(1)
3C
3t
shown
parameters are
- D
zero order
true
in
1
figure
(constant
the case of a slab
for
slab whose
Consider a
drug by diffusion.
that releases
geometric
provide
This is certainly
release.
rate)
often criticized
drug delivery devices are
Monolithic
because
FROM MONOLITHIC DEVICES
Assume
.
that
slab,
,
32C
ex
where
c(x,t)
t and
position
-L<x<L
the concentration
is
x in
matrix,
the
diffusion coefficient of
the
following
(2)
c(xt) =
(3)
c(+L,t) =
initial
the
the drug at time
(mg/ml) of
and D
drug in
is
the
the effective
Also assume
slab.
and boundary conditions
*
If Mt
C
t=O,
,
0
t>0.
,
the cumulative mass
is
the
area
of
(4)
Mt =
then
slab,
8LC
*
/n]
of drug
released
[1]
it is well known
exp[-D
2 2
2
(2n+l) 2n t/4L
through a unit
that
.
n=0
While M t/M
will
be
<0.6
released
(where
M
is
the
total
through a unit area),
drug
that
the asymptotic
form
amount
of
-222-
Slab
Release medium
Release medium
C(xO)UC*
C=0
C=0
C(Xt),t>0
C(L,t)=O
C (-Lt)=O
t>0
U
-L
0
t>O
L
X-/10
Figure
AI.l.
Geometric parameters of slab and concentration
profiles of drug solution for slab.
Release
is governed by simple diffusion and all solute
[eqs. (1)-(3)].
is dissolved.
-223-
M
(5)
t
holds.
= 2C
/M
t/1]
D
the release
Thus
zero order kinetics.
t1/2
release kinetics
the
slab case
In
circumstances
approximate
Case
than
transformation
(see
that
depends on
if D
kinetics are
although the
still
t1/2
from
in these
drug
1/2
t/.
Also,
drug
downward departure of
cases occurs
earlier than
[2].
in which it
shown
is
that
there are
several
achieve or
possible to
zero order release with matrix devices.
The
all drugs,
for
presented below are not suitable
methods
their
rates,
appendix it is
this
rather
spherical monolithic devices release
cylindrical and
in
shown
then the early
concentration,
originally at
can be
t1/2
to follow
a Boltzmann
Using
it
VI .2.4.3),
section
is said
and
limitations will be indicated.
1:
A dissolution
rate determining
This
slowly at
is
case
the
of
the
having dissolved
it
Once
O<x<x
with diffusion coefficient D
is
dissolution
for
(8)
tdiss= pd/Rd
9
x
.
Drug
rate
at
diffuses
sec).
time
2.
in figure
front
the
release.
step for drug
illustrated
position
dissolution is
front exists and
Rd mg/(cm 2
through
The
dissolves
the
region
characteristic
-224-
A
7
release
-release
medium~-
dissolved
drug
i-X
Figure AI.2.
-
/
/ ~
X
--00
7
dissolved
dru 9
---
medium
-
X
Concentration profile of slab for which release
is dissolution controlled [eqs. (8)-(10)].
A is drug loading.
-225-
where p
is
the
density
of the
diameter of a drug molecule
for diffusion
(9)
So
t diff=
(mg/cm 3)
drug
(cm).
The
and d
is
the
characteristic
time
is
f
e/D.
long as
(10)
tdiff <<tdiss
holds,
(11)
the release per unit area will
Mt= Rdt
.
This case
process
that
probably applies
many drugs
tdiff
Thus,
the
to
is
while
release
this case may
process, as
(i.e.
hold.
Just how
inequality
situation
Case
xf
2:
This
the
(10)
(10)
is
is
illustrated
the
release
in a matrix.
Note
which increases with time.
front reaches
the
holds
(10)
depends
the
i.e.
on
the
deeper
inequality
in case
Mixed dissolution and
case
start of
dispersed
reversed,
described below
the
be applicable right at
increases),
long
at
that are
dependent on xf,
matrix
the
be given by
tdiff>>td
start of
into
the
will not
drug.
is
,
When
the
3 becomes operative.
diffusion.
in
figure
3.
At all
depths x
-226-
release
release
-
medium
medium'
*
Figure AI.3.
dissolved drug.-
Eimw
EN
Concentration profile of slab in which
dissolution and diffusion are both
A is drug loading,
occuring [eq. (13)].
C is drug solubility.
-227-
there
in the matrix
exist two
and an undissolved phase.
phase
with diffusion coefficient D
is
(12)
.
De=
+
dissolved
drug and
the dissolved
CS
i.e.
is
maintained
throughout
a steady state will be
time
drug diffuses
f(C ,c)
the undissolved phase
a
a dissolved
The rate of drug dissolution
solubility concentration
=
then after
e
The
the concentration of
a function of
the drug's
If
phases of drug:
the
so
reached,
slab,
(12)
becomes
De
(13)
Thus
and
+
f(Cs,c)
profile of dissolved
the concentration
the
rate
release
to vanish
starts
kinetics will
phase will
(14)
td =
where
A
is
willihappen
over.
disappear at
(A-C
the
initial
states).
case where
Since c=0
the
)/f(CS,0)
concentration of drug
The
is constant.
(this
take
0.
=
the
is
undissolved phase
first at x-+L),
at x=+L,
surface at
constant,
the
t1/2
undissolved
time
,
drug
in
When
drug
the
loading
(i.e.
dissoved and
the
total
undissolved
-228= k(C -c)
,s)
f(C
(15)
received
with k constant, has
and
tested against
weight drugs
Xf,
rate
(figure 4a),
(16)
Mt =
where E
[D C
the
is
3
(mg/cm ),
The above
-
dt
matrix porosity
is
=
,
that
2itC D ar /(r
s e
f
A
is
front
[5]
state.
that
equation),
Higuchi
the
For
loading
drug
drug solubility.
the
equation predicts
[6]
steady
(the
At the
of diffusion.
the rate
2
)t
front.
in figure 4.
been shown
the
f
t1/2 kinetics.
hemisphere
total drug
equation:
following
(17)
it has
release from a coated
shown
can be
matches
(2A-EC
and C
considering
low molecular
certain
dissolution
to be in a quasi
is assumed
slabs
the
is diagrammed
situation
Diffusion
presented and
but dissolution is
low solubility,
dissolution
the
case have been
lately,
[3,4].
instantaneous at
This
considerable attention
the observed release of
Drug has
Case 3:
this
for
solutions
full
,
5
5
-a)
released
However, by
(figure
4b),
Qt obeys the
it
-229-
<solid drug>
Release medium
Release medium
a.
cS
I
/,
'-_
x -4
-
f__
dissolved drug
solid
b.
ru
Impermeable
coating
dissolved.--dru - -
r-a
mdm rrf
Release medium-
Figure
AI.4.
a)
Concentration profile of drug in slabs in
which drug dissolution rate at front x
matches diffusion rate [eq. (16)].
is drug solubility.
A is drug loading, C
b)
Drug configuration in coated hemisphere
with dissolution front at r=r .
-230-
radial
the radius of
is
where a
position of
(18)
time
radius of
[7]
BSA is
shown here
even when all
is
in solution.
can be
the
[8]),
it
is
drug is
the
limited
to
Since
it is
the
in
the
unlikely
the mechanism
possible
from a
released
zero
to approximate
solution.
case where
3.
of case
the
The
diffusion
constant.
The geometry of
Within
that
to
zero order
[7].
high (585 mg/ml
due
the actual
provide
device with zero order kinetics.
demonstration will be
5a.
to
is
where R
Hemispheric
[6].
in
that BSA
is demonstated
it
order release
coefficient
found
containing drug
constant release is
It will be
Details of
the hemisphere.
drugs with low solubility
solubility of
the
parentheses
r =R,
continue until
will
Hemispheric device
hemispheric
that
rf
becomes
been shown experimentally
have
release of
In
Clearly
term in
that equation (17)
release can be
course of
Case 4:
the
For r >>a
order release
outer
systems
front.
the
r f is
= 2nC D a
s e
dQ
dt
The zero
the
1, so
to
asymptotes
aperture and
the dissolution
time.
increases with
the small
the
situation is
hemispere,
the
spherical coordinates) holds,
illustrated
diffusion equation
i.e.
in
figure
(cast
in
- 231-
-.dissolved'mdug
a.
Release
medium
-~
u= RC*
t
u= r C
b.
--
U:aC
I----
rease
medium
r =O
Figure
AI.5.
r =a
r= R
hemisphere containing
in solution.
a)
Coated
is all
b)
Profile of uirC at
concentration.
initial
time
0.
drug
C*
is
that
-232-
=
(19)
+
D
c( r, t )=C
(20)
a<r<R
boundary conditions are
and
initial
The pertinent
ac
r~Tr
t=0,
and
(21)
c(a,t)=0
If
set
we
(22)
(23)
u(r,t)=rc(r,t),
=D
-
sink).
(perfect
t>0
,
then
(19)-(21)
become
32 u
2-
t=0
u(r,t)=rC
and
a<r<R,
and
(24)
u(a,t)=0
Now
the
(25)
1dt
The
mass
=
2Ta 2D
=
2jtaD
profile
through
rate
release
-(u/r)
e=D r
r=2a
D
the aperture
is
[(r -u)/r
r=a
given
by
I!!
e~r r~a
of u at
time 0
is
shown
in figure
5b .
One can
-233--
break
constant,
of
has constant
profile
the
amount of
time after the
be maintained
near r=a.
and work
at r=R
maintained,
slope C
its way towards
(25)
gives,
for
the
portion
For a considerable
.
in
the
profile will
While
r=a.)
slope will
the
started,
release has
(The decay
parts
the
"triangular"
top,
The
two solutions.
the
and adding
assumed
is
for each of
solving
solved by
can be
(25)
Because D
into two parts.
the profile
slope
the
start
is
of the
triangular part
profile,
(26)
The bottom,
a
.
d9(I)=2icaD C
e
dt
slab with
"rectangular" part of
a
constant
rectangular part of
early
times,
(27)
dQ(II)2C
When
d
easy
it is
For
to
the
see
that for
e
the
>>
the triangular
r=a.
profile
release
total
dQ(I)+dQ(II)=2aD C
e
dt
dt dt
(28)
condition.
initial
to
*[DX1/2
dt
Therefore
the
profile is analogous
the
,
portion
rate
is
+ 27aC
D
zero order release
of
the
profile
e
/nt]/
will hold
maintains
its
as
long as
slope
near
-234-
REFERENCES
[1]
Crank,
(1975)
[2]
of Diffusion, Clarendon, Oxford
Mathematics
J.,
.
Tanquary and R.E. Lacey,
Biology and Medicine, A.C.
[3]
Doelker,
Gurny, R.,
[4]
Pharm. Sci .
matrices," J.
J.
[6]
Pharm. Sci.
Rhine
W.D.,
"A
52,
new approach
V
ed.
[71
(1982).
1399
and Langer,
D.S.T.,
Hsieh,
to achieve zero-ord er kinetics
matrix
polymer
Bioactive
systems,"
Materials,
R.W.
in
Baker,
(1980).
Hsieh, D.S.T.,
controlled
Rhine, W.D. ,
release
macromolecules,"
[81
of
D.R.,
sustained action medication,"
.
from diffusion-control le d
Controlled Release
(1982).
27
(1963).
1145
Sukhatme,
71,
of
"Mechanism
Higuchi, T.,
3,
transport from dispersed
"Dissolution-controlled
[51
Paul,
S.K. and
Chandrasekaran,
f rom porous,
drugs
soluble
"Modelling of
N.A.,
Bio materials
polymers,"
hydrophobic
(1974).
Peppas,
and
E.,
of water
release
sustained
15
p.
Plenum, New York,
eds.,
of
in Experimental
o f Advances
vol 43
Agents,
Bioactive
Release
in Controlled
rates,"
and
mechanisms
release
"Controlled
Baker, R.W. and Lonsdale, H.K.,
Kozinski,
A.A.
J.
polymer
Pharm.
and
Langer, R.,
matrices
Sci.
72,
and Lightfoot, E.N.,
ultrafilltration" a general example
filtration,"
A.I.Ch.E.
J.
18,
1030
for
17
"Zero-order
micro(1983).
"Prote in
of
boundary
(1972).
and
-235-
PROGRAMS FOR CHAPTER V
APPENDIX II:
organization of
The
MACRO..
Lattices
a whole
ling,
This
A
quadwords.
are
32
"D"'s,
by
to a
actually
six
easier
to
so each
long,
use VAX
longwords,
to
while a
the
subroutine NABOR2.
The
The
lattices,
six
bit in array
zero bit corresponds
lattice of "$"'s,
subroutine ESCAP2.
consecutive
A one
lattices
array NEI
generated
contain
of
lattice
the
represents
the subroutine RANDM.
"D",
Array B corresponds
the
is
8 bytes
of storage.
and RRWALK, array A
generated
A correponds
using
Quadwords are
).
FORTRAN array
long.
bits
In USEFUL2
is
A quadword
therefore a REAL*8
some operations it
For
which
is
8X64 consecutive bytes
is
layer
64 bits
in a quadword.
stored
is
sites
VAX.
layer consists of 64 consecutive
lattice
dimension (64,N
the
on
structure
data
logical operations.
A
in FORTRAN.
REAL*8
with
lattice
allows parallel
should be
Since VAX quadwords are
.
are 64X64XN
row of
the
plus
VAX MACRO,
coded in VAX
is
much of which
following paragraphs
the
reader of
with
familiar
A
the code,
interpretation of
aid
to
also given
is
lattice representation
the
the
description of
A brief
in chapter V.
described
programs
the
listings of
appendix contains
This
to a
-
calculated from
in RRWALK is
from B
bits
that
in
the
are
-236-
set
if
the corresponding
bit in B has
forward, or backward neighbors,
left,
right,
respectively.
up,
down,
-237-
C
C
C
C
USEFUL2--MAIN DRIVER ROUTINE DETERMINING TOTAL RELEASABLE
DRUG AS FUNCTION OF VOLUME LOADING AND PARTICLE
SIZE
REAL*8 A(64,15),B(64,15)
INTEGER*4 DEPTHLOADNGSEEDKOUNTA,KOUNTBNITER
DIMENSION IDEPTH(10),LOAD(10),AVG(10,10)
DATA LOAD/5,10,15,20,25,30,35,40,45,50/
C
100
2
1
200
3
300
TYPE *,'GIVE ME NDEPTH,IDEPTH'
ACCEPT *,NDEPTH,(IDEPTH(I),I=1,NDEPTH)
TYPE *,'NOW GIVE ME SEED IN HEX'
ACCEPT 100,SEED
FORMAT(Z8)
DO 1 I=1,NDEPTH
DEPTH=IDEPTH(I)
TOTCEL=4096.*DEPTH
DO 1 J=1,10
LOADNG=10*LOAD(J)*1024/1000
AVLD=O.
SSQ=0 .
DO 2 K=1,10
CALL RANDM(A,LOADNG,DEPTH,SEED)
CALL ESCAP2(A,B,DEPTH,NITER)
CALL COUNTD(A,DEPTH,KOUNTA)
CALL COUNTD(B,DEPTH,KOUNTB)
TRUELD=KOUNTA/TOTCEL
AVLD=AVLD+TRUELD/10.
OUT=FLOAT(KOUNTB)/FLOAT(KOUNTA)
AVG(I,J)=AVG(I,J)+OUT
SSQ=SSQ+OUT*OUT
CONTINUE
AVG(I,J)=AVG(IJ)/1.
SIG=SQRT(SSQ/10.-AVG(IJ)*AVG(IJ))
TYPE 200,DEPTH,NITER,LOADNG/1024.,AVLDAVG(IJ),SIG
FORMAT(215,F5.2,3F7.3)
OPEN(UNIT=3,FILE='USEFUL2.OUT',TYPE='NEW',
FORM='FORMATTED')
1
DO 3 J=1,10
WRITE(3,300)LOAD(J)/100.,(AVG(IJ),I=1,NDEPTH)
FORMAT(10F7.3)
STOP
END
-238-
RANDOM MATRIX GENERATOR USING POWER RESIDUE METHOD
RON SIEGEL 10/28/82
CALL RANDM(A,IPERC,IDEPTHISEED)
A = OUTPUT ARRAY DIMENSIONED 64XIDEPTH
EACH ARRAY ELEMENT IS A QUADWORD (REAL*8)
CONTAINING 32 BITS WHICH ARE EITHER SET OR CLEARED
DEPENDING ON WHETHER THE CORRESPONDING CELL
IS OCCUPIED BY DRUG OR POLYMER
IPERC = VOLUME LOADING--BETWEEN 1 AND 1024
IDEPTH = DEPTH OF SLAB IN PARTICLE SIZE UNITS
ISEED = RANDOM NUMBER GENERATOR SEED WHICH IS
REPLACED AT THE END OF THE SUBROUTINE
.TITLE
RANDM
RANDM,CM<R2,R3,R4,R5,R6,R7,R8>
. ENTRY
A P=4
PERC P=8
DEPTH P=12
SEED P=16
MOVL
MOVL
MOVL
ASHL
MOVL
GENERATOR
MOVL
MULL2
INCL
THIS
IS THE
A P(AP),R2
@PERC P(AP),R3
@DEPTH_P(AP),R4
#7,R4,R4
@SEEDP(AP),R5
;ARRAY POINTER
;LOADING
;SLAB DEPTH
;TIMES 128 TO GET ARRAY SIZE
;SEED FOR RANDOM NUMBER
#69069,R6
R6 ,R5
R5
;MULTIPLIER INTO R6
;ONE CYCLE OF RANDOM
;NUMBER GENERATOR
MAIN LOOP
ALOOP:
MOVL
CLRL
#32 ,R1
R7
;BIT COUNTER IN LONGWORD
;GET RID OF GARBAGE IN R7
DLOOP:
MULL2
INCL
R6 ,R5
R5
;THIS IS THE RANDOM
;NUMBER GENERATOR
BICL3
ROTL
CMPL
ADWC
SOBGTR
#OX003FFFFFR5,R8 ;RESIDUE BETWEEN 0 AND 1023
;PUT INTO LOW ORDER
#10,R8,R8
R8,R3
;DETERMINE WHETHER OR NOT
;TO FILL (AND SHIFT)
R7,R7
R1,DLOOP
;MOVE TO NEXT CELL (BIT)
MOVL
R7,(R2)+
SOBGTR
R4,ALOOP
;PUT 32 CELLS
;LONGWORD
;MOVE TO NEXT
MOVL
R5,@SEEDP(AP)
;DONE--UPDATE RANDOM
RET
. END
INTO ARRAY
LONGWORD
;SEED
;AND GET OUT OF
HERE
NUMBER
-239-
ESCAP2
.TITLE
COUNTS NUMBER OF DRUG FILLED CELLS IN A CUBIC
LATTICE THAT CAN ACCESS THE SURFACE (TWO FACES) OF
THAT LATTICE
RON SIEGEL 10/14/81
MODIFIED FOR TWO SIDED RELEASE 2/17/83
CALL ESCAP2(A,B,IDEPTH,NITER)
A = INPUT ARRAY CONTAINING LATTICE IN QUESTION.
EACH ARRAY ELEMENT IS A QUADWORD (REAL*8)
AND THE DIMENSION IS 64XIDEPTH
CONTAINING A ROW OF 64 CELLS (BITS) WHICH ARE
SET TO ONE IF CELL CONTAINS DRUG.
B = OUTPUT ARRAY CONTAINING LATTICE IN WHICH THE
CELLS (BITS) ARE SET IF THE CORRESPONDING CELL
IN A IS SET AND CONNENCTED TO THE SURFACE OF
EACH ELEMENT IS A QUADWORD (REAL*8) AND
A.
EXTRA ROW
THE DIMENSION IS 64XIDEPTH.
CONVENIENCE.
FOR
IS FILLED WITH ZEROS
IDEPTH = DEPTH OF MATRIX (INTEGER*4)
NITER = NUMBER OF ITERATIONS REQUIRED
AP=4
B P=8
DEPTHP=12
NITER P=16
ESCAP2,©M<R2,R3,R4,R5,R6,R7,R8,R9>
.ENTRY
MOVL
MOVL
MOVL
MOVL
ASHL
A P(AP),RO
RO,R4
B P(AP),R1
@DEPTH P(AP),R2
#9,R2,R3
ADDL
ADDL
MOVL
CLRL
R3,R4
R1,R3
R3,R5
@NITERP(AP)
;ADDRESS
OF
INPUT LATTICE
;ADDRESS OF OUTPUT LATTICE
;DEPTH OF LATTICE
;MULTIPLY BY NUMBER OF BYTES
;PER LAYER
;R4 POINTW TO END OF A
;R3 NOW POINTS TO END OF B
;SO DOES R5
;INTERATION COUNT
COPY FIRST LAYER OF A TO FIRST LAYER OF B AND
ZERO OUT LAYER OF B AFTER END OF LATTICE
COPY:
DECL
DECL
MOVL
MOVZBL
MOVQ
MOVQ
SOBGTR
TSTL
BNEQ
RET
R2
R2
R2,SAVDEP
#64,R2
(RO)+,(R1)+
-(R4),-(R3)
R2,COPY
SAVDEP
PROCED
RO,SAVO
MOVL
R1,SAV1
MOVL
NOW ZERO OUT REST OF B
PROCED:
;DEPTH-1
;DEPTH-2
;WE'LL NEED THIS LATER
;64 ROWS PER LAYER
;FRONT SURFACE
;BACK SURFACE
;ONLY TWO
LAYERS?
;IF SO THEN WE'RE DONE
;POINT TO BEGINNING
;OF SECOND LAYER
-240-
ZEROB:
CLRQ
CMPL
BLSS
(Rl)+
R1 ,R3
ZEROB
NOW COMES THE MAIN PART
WE ITERATE UNTIL SATISFACTION ACHIEVED,
NO CHANGE IN THE B LATTICE
ITERLP:
INCL
ANDEM:
MCOML
I.E.
@NITERP(AP)-
;INCREMENT COUNT OF
;ITERATIONS
CLRW
DIFF
;THIS WILL BE SET IF B
;CHANGES AT ALL
MOVL
;START OF A MATRIX (LAYER 2)
SAVO,RO
MOVL
SAV1,Rl
;START OF B MATRIX (LAYER 2)
MOVL
;DEPTH-2 (THIS PROCESSING
SAVDEP,R2
;EXCLUDES TOP AN BOTTOM
;LAYERS)
INITIALIZE PROCESSING FOR LAYER
LAYER: MOVZBL
#64,R9
;COUNTER FOR ROWS
MOVQ
512(Rl),EDGE
;SAVE BEGINNING OF NEXT LAYER
CLRQ
512(Rl)
;SO WE CAN CLEAR IT
;(ELIMINATES EDGE PROBLEMS)
#1,SWITCH
;INDICATES WE'RE ON FIRST ROW
MOVZBL
ROW LOOP
ROW:
MOVQ
;ROW OF A
(RO)+,R3
(Rl),R5
;ROW OF B
MOVQ
MOVQ
R5,SAV64
;SAVE TO TEST LATER FOR
;CHANGES
#1,R5,R7
;LEFT NEIGHBORS
ASHQ
#-1,R5,R5
;RIGHT NEIGHBORS
ASHQ
BICL2
#0X80000000,R6
;GET RID OF SIGN BIT
BISL2
;LOGICALLY OR LEFT
R7,R5
BISL2
;AND RIGHT
R8,R6
BISL2
-512(Rl),R5
;AND DOWN ONE
BISL2
;LAYER AND
-508(R1),R6
;UP ONE
BISL2
512(Rl),R5
;LAYER
B ISL2
516(RI),R6
;AND BACKWARD
BISL2
8(Rl),R5
BISL2
12(Rl),R6
;ONE ROW
SOBGEQ
SWITCHANDEM
;AND UNLESS THE FIRST ROW,
;DO IT FORWARD
-8(RI),R5
BISL2
-4(Rl),R6
;ONE ROW
BISL2
MCOML
BICL2
BICL2
MOVQ
CMPL
BNEQ
CMPL
BEQL
SDIFF: MOVW
GOBACK: SOBCTR
R5 ,R5
R6 ,R6
R5 ,R3
R6 ,R4
R3 ,(R )+
R3,SAV64
SDIFF
R4,SAV64+4
GOBACK
#1 ,DIFF
R9,ROW
MCOML AND BICL TOGETHER
CONSITUTE A LOGICAL AND
;LOGICALLY AND FINAL RESULT
;WITH A MATRIX ENTRIES
;AND ADVANCE
;ANY CHANGES IN B MATRIX?
;SET
"DIFFERENT"
FLAG
;GO TO NEXT ROW OF
64
-241-
;RESTORE FIRST ROW OF NEXT
;LAYER
;GO TO NEXT LAYER OF 64X64
R2,LAYERJ
SOBGTR
;ALL DONE WITH ONE PASS THROUGH LATTICE
;ANY CHANGES?
DIFF
TSTW
;IF SO GO BACK AND DO IT
BNEQ
ITERJ
;AGAIN
;OTHERWISE, WE'RE DONE
RET
MOVQ
LAYERJ: JMP
ITERJ: JMP
SAVO:
SAVI:
SAVDEP:
SWITCH:
DIFF:
SAV64:
EDGE:
LONG
.LONG
.LONG
.LONG
.WORD
.QUAD
.QUAD
.END
EDGE, (RI.)
LAYER
ITERLP
0
0
0
0
0
0
0
-242COUNTD
.TITLE
COUNTS NUMBER OF ON
RON SIEGEL 10/19/81
BITS
IN A 3D LATTICE
COUNTD,@M<R2,R3,R4,R5>
.ENTRY
AP=4
DEPTH P=8
KOUNTP=12
CLRL
CLRL
MOVL
ASHL
NEWORD:
BEQL
TWOS:
INCL
MNEGL
BICL2
BRW
TSTEND: AOBLSS
MOVL
RET
.END
;COUNTER OF "ON" BITS
RO
;WORD COUNTER
R5
;STARTING ADDRESS
A P(AP),R1
#7,@DEPTHP(AP),R2 ;DEPTH MULTIPLIED BY
128
(R1)+,R3 ;GET NEXT WORD
MOVL
;ALL OUT OF ONE BITS?
TSTEND
;IF NOT, INCREMENT BIT COUNT
RO
;TWO'S COMPLEMENT AND
R3,R4
;WORD IN QUESTION TO ITSELF
R4,R3
;AND KEEP GOING
TWOS
;LOOP BACK TILL DONE
R2,R5,NEWORD
RO,@KOUNTP(AP) ;RESULT
-243-
C
C
C
C
RRWALK--MAIN DRIVER ROUTINE FOR RANDOM WALK
LATTICE SIMULATION
RON SIEGEL 1/5/84
IN RANDOM
REAL*8 A(64,15), B(64,16)
REAL*8 LFT(64,15) ,RGT(64,15) ,UP (64,15),DN(64,15)
REAL*8 FOR(64,15) ,BAK(64, 15) ,NE I(64,15,6)
INTEGER BIN(4000) ,DEPTH,DEPTH1
INTEGER*4 ISEED
DIMENSION NWHICH (6)
EQUIVALENCE ( LFT (1, 1) ,NEI(1, 1 ,1 )
EQUIVALENCE ( RGT (1, 1) ,NEI(1, 1 ,2 )
EQUIVALENCE CUP( 1,1 ),NEI(1 ,1 3) )
EQUIVALENCE ( DN( 1,1 ),NEI (1,1 4) )
EQUIVALENCE ( FOR (1, 1) ,NEI(1, 1 ,5 )
EQUIVALENCE ( BAK (1, 1) ,NEI(1, I ,6 )
C
C
9
10
11
12
13
TYPE *,'GIVE ME LOADING AND DEPTH'
ACCEPT *,LOAD,DEPTH
DEPTH1=DEPTH+1
TYPE *,'GIVE ME A SEED INTEGER'
ACCEPT *,ISEED
CALL INIT01(ISEED)
LOADNG=10*LOAD*1024/1000
CALL RANDM(A,LOADNG,DEPTH,ISEED)
CALL COUNTD(A,DEPTH,KOUNTA)
CALL ESCAP2(A,B,DEPTHNITER)
CALL NABOR2(B,LFT,RGT,UP,DN,FOR,BA K,DEPTH)
DO 30 IZ1=1,DEPTH
DO 31 IY1=1,64
DO 31 IX1=0,63
CALL NABPR(NEI,15,IX1,IYl,IZ1,NNAB ,NWHICH)
IF(NNAB.EQ.O)GO TO 31
IZ=IZl
IY=IYl
IX=IXL
KSTEP=O
T=O.
KSTEP=KSTEP+1
CALL NABPR(NEI,15,IX,IY,IZ,NNAB,NW HICH)
CUM=O.
RNAB=NNAB
T=T+6./RNAB
CALL RAND01(X)
DO 10 I=1,6
CUM=CUM+NWHICH(I)/RNAB
IF(X.LT.CUM)GO TO (11,12,13,14,15, 16),I
CONTINUE
IX=IX-1
GO TO 9
IX=IX+1
GO TO 9
IZ=IZ+1
-24 4-
14
15
16
20
25
31
30
35
5000
IF(IZ.EQ.DEPTH1)GO TO 20
GO TO 9
IZ=IZ-1
IF(IZ.EQ.O)GO TO 20
GO TO 9
IY=IY-1
GO TO 9
IY=IY+
GO TO 9
KWALK=KWALK+1
IT=T+.5
IF(IT.GT.4000)GO TO 25
BIN(IT)=BIN(IT)+
GO TO 31
KOVER=KOVER+1
CONTINUE
CONTINUE
OPEN(UNIT=2,FILE='RRWALK.OUT',TYPE='NEW',
FORM='FORMATTED')
1
COUNTA=KOUNTA
WRITE(2,5000)0,0.,0.
DO 35 IT=1,4000
T=IT
IF(IT.GT.1)BIN(IT)=BIN(IT)+BIN(IT-1)
WRITE(2,5000)ITSQRT(T),BIN(IT)/COUNTA
FORMAT(I5,2F7.3)
TYPE *,'KOVER/COUNTA=',KOVER/COUNTA
STOP
END
-245-
.TITLE
NABOR2
DETERMINE NEIGHBORS SHARING A FACE WITH EACH SUBCUBE
LATTICE.
SINK ON TWO FACES OF LATTICE.
RON SIEGEL 1/4/84
SUBROUTINE NABOR2(T,L,RU,DF,BDEPTH)
IN A
NABOR2,@M<R2,R3,R4,R5,R6,R7,R8>
.ENTRY
;TEST LATTICE
T P=4
;CELLS OF T WITH LEFT NEIGHBORS
L P=8
R P=12
;RIGHT
U P=16
;UP
D P=20
;DOWN
;LEFT
F P=24
;BACKWARDS
B P=28
;DEPTH OF LATTICE
DEPTHP=32
TAKE CARE OF LEFT AND RIGHT FIRST
;START OF TEST LATTICE
T P(AP),RO
MOVL
;START OF "LEFT" LATTICE
L~P(AP),R1
MOVL
;START OF "RIGHT" LATTICE
R~P(AP),R2
MOVL
;DEPTH OF LATTICE IN LAYERS
@DEPTHP(AP),R3
MOVL
;# ROWS PER LAYER
#64,R4
MOVZBL
LR1:
;ROW OF 64 CELLS
(RO)+,R5
MOVQ
LR2:
;LEFT NEIGHBORS
#1,R5,R7
ASHQ
;MCOML AND BICL3
R7,R7
MCOML
;CONSTITUTE LOGICAL "AND"
R8,R8
MCOML
;DEPOSIT LEFT NEIGHBORED
R7,R5,(Rl)+
BICL3
;INTO L ARRAY
R8,R6,(R1)+
BICL3
;DO AGAIN
R7,R5,R5
BICL3
;BUT INTO (R5,R6)
R8,R6,R6
BICL3
;AND PUT INTO LEFT NEIGHBOR
#-1,R5,R7
ASHQ
R7,(R2)+
MOVL
BICL3
#@X80000000,R8,(R2)+
;LOOP BACK IF STILL MORE ROWS
R4,LR2
SOBGTR
;LOOP BACK IF MORE LAYERS
R3,LR1
SOBGTR
UPS AND DOWNS
T P(AP),RO
MOVL
;START OF "UP" LATTICE
U P(AP),R1
MOVL
;START OF "DOWN" LATTICE
D~P(AP),R2
MOVL
;COPY FIRST LAYER OF
#64,R4
MOVZBL
;TEST LATTICE TO
(RO)+,(R2)+
MOVQ
UD1:
;"DOWN" LATTICE (I.E. CONNECT
R4,UD1
SOBGTR
;TO SINK)
;RESTORE RO TO BEGINNING OF
T_P(AP),RO
MOVL
;TEST LATTICE
@DEPTHP(AP),R3
MOVL
;ONLY DEPTH-1 COMPARISONS
R3
DECL
;BETWEEN LAYERS
#128 , R 4
MOVZBL
UD2:
512(RO) ,R5
MCOML
UD3:
BICL3
R5,(RO) ,(R1)+
R5 , (RO)+, (R2)+
BICL3
SOBGTR
R4,UD3
-246-
SOBGTR
MOVZBL
UD4:
MOVQ
SOBGTR
FORWARDS AND
MOVL
MOVL
MOVL
MOVL
BF1:
CLRQ
BF2:
R3,UD2
#64,R4
(RO)+,(R1)+
R4,UD4
BACKWARDS
T P(AP),RO
F P(AP),RI
B P(AP),R2
@DEPTHP(AP),R3
(Rl)+
CLRQ
504(R2)
MOVL
MCOML
BICL3
MOVL
SOBGTR
ADDL
ADDL
SOBGTR
#126 , R 4
(RO)+,R5
R5,4(RO) ,(RI)
(R1)+,(R2)+
R4,BF2
#8 ,RO
#8 ,R2
R3 , BF 1
RET
. END
;CONNECT TOP LAYER OF
;LATTICE TO
;SINK
"UP"
;FORWARD NEIGHBORS
;BACKWARD NEIGHBORS
;FIRST ROW HA'S NO FORWARD
;NEIGHBORS
;LAST ROW HAS NO BACKWARD
;NEIGHBORS
;GET TO START OF NEXT ROW
;ALL LATTICES
IN
-247-
C
C
C
1
SUBROUTINE NABPR--DETERMINES WH ICH NEIGHBORS OF A SITE ARE
FILLED
RON SIEGEL 1/4/84
SUBROUTINE NABPR(NEI,NEIDIM ,IX,IY,IZ,NNAB,NWHICH)
LOGICAL*4 BTT
INTEGER*4 IROW(2)
REAL*8 NEI(64,NEIDIM,6),ROW
DIMENSION NWHICH(6)
EQUIVALENCE (ROW,IROW(1))
IF(IX.LT.32) THEN
J=1
IX32=IX
ELSE
J=2
IX32=IX-32
ENDIF
NNAB=O
DO 1 I=1,6
ROW=NEI(IY,IZ,I)
BTT=BJTEST(IROW(J ) , IX32)
IF(BTT) THEN
NWHICH(I)=1
NNAB=NNAB+1
ELSE
NWHICH(I)=0
ENDIF
CONTINUE
RETURN
END
-248-
.TITLE
RANDO1
.ENTRY
RAND01,@M<>
SUBROUTINE RANDOL(X)
RETURNS RANDOM VALUE BETWEEN 0 AND 1
;COMPUTE RANDOM INTEGER
#69069,R,RO
MULL3
RO
;AS A LONGWORD
INCL
MOVL
ROR
;PUT BACK IN STORAGE
;ELIMINATE SIGN BIT
#@X80000000,RO
BICL2
;MAKE FLOATING POINT
CVTLF
RO,R1
;DIVIDE INTO LARGEST POSSIBLE
RMAX,R1,@4(AP)
DIVF3
;NUMBER
RET
.ENTRY
INIT01,@M<>
SUBROUTINE INIT01(ISEED)
SEEDS RANDOM NUMBER GENERATOR
@4(AP),R
MOVL
CVTLF
#@X7FFFFFFF.RMAX
R:
RMAX:
RET
.LONG
.LONG
.END
0
0
;THIS IS THE SEED
;THIS IS THE LARGEST
;POSSIBLE NUMBER
-249-
APPENDIX
were
to
set of experiments
the
Before
of the
the model
In
IV and V.
of chapters
and
by
out of
the slabs,
circular
face and around
circular
face exposed.
The
only
saline and
ultimately
clusters
set
to
to
"$".
"$"
extending all
small modification of
II).
incorporated
were
on one
leaving
then
into EVAc
the disks
After
rim with wax,
the
placed
one
in
release of BSA was monitored.
model of section V.3 was
the
For
only
=N,
if
the way
the
the
on
sites
lattice
drug-filled
set
case
a comparison between
they were coated
The disks
only change in
initially
the
the modified
section IV.2.
the method described in
unbuffered
to
corresponding modifications
150-180pm was
size
particle
were punched
this
For
is made.
theory
BSA,
the
briefly, and
the model are described
data
The
opposed
as
appendix,
this
experimental conditions and
to
gathered and compared
slightly different.
quite well,
fit
model
the
of data,
in chapter IV
fraction released.
total
experimental conditions were
set
described
data was
set of
a preliminary
run,
OF DATA
A SECOND SET
III.
fo
drug-filled
they
to
belonged
Z1=.
subroutine
(1=1)
layer
were
sites were
to
connected
This amounted
ESCAP2
that
(see
to
appendix
a
-250-
The
results of
for N =6
(the
1.05 mm)
are
fit
to
the release experiments
average
shown
the data.
depth of
in figure
1.
and
the model fit
these disks was approximately
The model
provides a good
-251-
EXHAUSTIVE RELEASE
1.0
A
.8
-o
&
£
-
65A
£&
&
A
..6
.4
0
&1-4
.2
0
.2
.4
.6
.8
1.0
POROSITY
Figure AIII.1.
Total fraction released for a preliminary
set of BSA disks released through only one
face.
Curve is fit of modified model that
only allows realease through one side.
PS=150-180 m, average depth=1.05mm.
Model
assumes 6 layers.
-252-
NUMERICAL PROCEDURE AND COMPUTER PROGRAM FOR
APPENDIX IV:
CONCENTRATION DEPENDENT DIFFUSION
In
this
appendix we describe
second order differential
the nonlinear
(1)dc
(
subject
dF
c(0)
=
0
(3)
c(")
=
*
C.
is
equation
(ODE)
dcl
the boundary conditions
(2)
D(c)
technique for
dTJ
daiLD(c)-i
--2 dni
to
the solution
bounded by 1.
Equation (1)
is
first recast
as a system of
first order
ODE's:
(4)
dc
d'n
dc'
Equations
two
(21c'-c
(4),
%
2dD
c)/D(c)
(2)
subject to
point boundary value
numerical
be
=
problem,
solution cannot be
converted
to an
and
initial
(3),
constitute a nonlinear
for which a simple
found.
However,
value problem by
the
problem can
replacing eq.
(3)
-253-
with
(3')
= S
c'(0)
where S
is chosen
(2),(3'),(4)
by the analyst.
converges,
which depends
The
(2),(3'),(4)
BDRK
is
the
At
c
n
is
set
is
the
stepsize
of
S
value
implements
scheme,
)ArI 0 , where An0
is
is
implemented
because the
program performs
the above
to
a
FORTRAN
that calls
the
fourth order
with variable
procedure,
the
specified by
stepsize.
stepsize
the
user.
The variable
system becomes very stiff
be proportional
versus
concentration.
initial
program BDRK
subprograms
of D(c)
to
procedure
for many values
may be generated.
the flux out of
this plot will
computation
C
small.
is proportional
FUNCTION
the
routine
concentration at the n'th step.
S, so that a plot of S versus C
The
particular
solved using
of the Runge-Kutta
to D(c
when D gets
The
is
finite difference
the n'th step
An
a
main driver
subroutine ODEsolver, which
Runge-Kutta
to
to
on S.
system
program BDRK.
as jp*c,
The solution
the
semiinfinite
loaded with
the
DIFF and
dDdC, which
provide
dD/dc.
slab,
the desired- plot of flux
must be
and
Since
user-supplied
Also, SUBROUTINE
the
ID,
which
-254-
passes
back a name for
passed back is
A
for
the
used
the model,
in output
listing of BDRK,
The name
filenames.
along with
Keller diffusivity model,
appendix.
must be provided.
the
is
subprograms necessary
provided
in this
-255-
C
C
C
C
C
C
COMPUTES RATIOS RT(C*) AND RS(C*).
RT IS COMPUTED BY INTEGRATING D(C).
RS IS COMPUTED BY SOLVING THE BOLTZMANN TRANSFORMED
DIFFUSION EQUATION WITH CONCENTRATION DEPENDENT
DIFFUSION COEFFICIENT.
1
3
2
1000
C
C
C
DIMENSION C(125),RATIOT(125),RATIOS (125),SL(125),D(125)
CHARACTER*6 MODEL
TYPE *,'GIVE ME DETA'
ACCEPT *,DETA
SLOPE=O.
OLDC=O.
OLDD=1.
RINT=0.
DO 1 I=1,100
SLOPE=SLOPE+30.
SL(I)=SLOPE
CALL ODESOLVER(DETASLOPE,CI,ETAS)
IF(CI.GT.1000.)GO TO 3
C(I)=CI
DI=DIFF(CI)
RATIOI=(SLOPE/CI)/(2./SQRT(3.14159))
RAT IOT (I)=RATIOI*RATIOI
D(I)=DI
RINT=RINT+.5*(DI+OLDD)*(CI-OLDC)
RATIOS(I)=RINT/CI
TYPE *,SLOPE,CI,DI,RATIOS(I),RATIOT(I)
OLDC=CI
OLDD=DI
II=I-1
CALL ID(MODEL)
OPEN(UNIT=1,FILE=MODEL//' .OUT',TYPE='NEW',
I
FORM='FORMATTED')
ZERO =0.
ONE= 1. 0
WRITE(1,1000)ZERO ,ONE,ONE,ONE
DO 2 I=1,II
WRITE(1, 1000)C(I) ,D(I),RATIOS(I),RATIOT(I)
CLOSE(1)
STOP
FORMAT(F7.2,F1O.5,2F10.4)
END
ODEsolver SOLVES DIFF
VARIABLE STEPSIZE
EQ.
USING
RUNGE-KUTTA
SUBROUTINE ODEsolver(dE,sl,z,etas)
e ta=0.
1
z=0.
y=sl
deta=diff(z)*dE
rk(eta,y, z ,de ta
call
,yl
,zl)
WITH
-256-
if(abs(z-zl)/deta.it.le-3)return
e ta=e ta+de ta
y=yl
z=zl
go to 1
END
C
C
RUNGE-KUTTA ONE-STEP
SUBROUTINE RK (X0,YOZOdE,YlZl)
CY1=dE*DER(XO,Y, Zo)
CZI=dE*YO
CY2=dE*DER(XO+dE*.5,YO+CY1*.5,ZO+CZ1*.5)
CZ2=dE*(YO+CYl*.5)
CY3=dE*DER(XO+dE*.5,YO+CY2*.5,ZO+CZ2*.5)
CZ3=dE*(YO+CY2*.5)
CY4=dE*DER(XO+dE,YO+CY3,ZO+CZ3)
CZ4=dE*(YO+CY3)
Yl=YO+(CY1+2.*CY2+2.*CY3+CY4)/6
Zl=ZO+(CZ1+2.*CZ2+2.*CZ3+CZ4)/6
RETURN
END
C
C
C
BOLTZMANN-TRANSFORMED DIFFUSION EQUATION
IN VECTOR FORM
FUNCTION DER (t,dC,C)
.s=dDdC(C)
f=Diff (C)
DER=-dC*(s*dC+2.*t)/f
RETURN
END
-257-
C
C
DIFFUSIVITY PROFILE GIVEN
BY KELLER,
SUBROUTINE ID(MODEL)
CHARACTER*6 MODEL
MODEL='KELLER'
RETURN
END
FUNCTION Diff (C)
gam=21. 3/1340.
gC=gam*C
IF (gC.EQ.0.) THEN
Diff=1.
ELSE
Diff=TANH(gC)/gC
END IF
RETURN
END
FUNCTION dDdC (C)
gam=2 1.3/1340.
gC=gam*C
IF (gC.EQ.0.) then
dDdC=0.
else
const=gam/gC
S=SECH(gC)
dDdC=const*(S*S-TANH(gC)/gC)
end if
RETURN
END
FUNCTION SECH(X)
IF (ABS(X).GT.30) then
SECH=O.
else
SECH=1/COSH(X)
endif
RETURN
END
CANALES, AND YUM
-258-
APPENDIX V.
This
and
its
The
appendix describes
subroutines.
tabulates
This
and
mean first passage
refer
RSPASS
which
in
accepts
to
is
provides
times of
listings
(throats).
SQUARE
and RNOS2.
user a random number generator
(array DIAM)
and
combinations are
half-lengths
tested.
information on the
and C2.
around
If a side of
the
contains an outlet
executes
the
particle
the
If
there
iterative routine
for a
For
determined
in RPASS
IS
two
ranges
sends
that
RSPASS also
to a throat,
the
(array DPTH),
central pore
SUBROUTINE SQUARE determines
pore body.
returns
and
sends
the
body,
then
is no
outlet on
then Cl(I)=l and C2(I)=-l.
RPASS
VI.3.4.
RSPASS
throat configuration, via
Cl(I)=-DIAM/2 and C2(I)=+DIAM/2.
that side,
reader
RPASS,
seed,
z
RSPASS
throat half-length and width information.
indexed by I,
The
the main driver routine for SUBROUTINE
throat widths w
arrays Cl
and
section VI.3.4.
from the
to RPASS
for RSPASS
particles diffusing
outlets
turn calls SUBROUTINEs
for which all
program
set of routines calculates
through pores with constricted
should
RSPASS
particles
particle
in
and
the
The MACRO
uniformly distributed
from I to 4,
the
is
radius
located
throats,
itself.
described
used
in
to
of a box
the
central
radius of
subroutine
random numbers
determine
in
the
box is
RNOS2
IP and
the
IS.
side of
-259-
the box on which a particle
and is
the
used
particle
where Q
stored
is
in
to determine
lands.
the
lands.
IP
position along
The position is
form
in RSPASS.
from 0 to
the
1023,
side on which
-l
IP+l
given by QI (1o253'
given by equation (VI.83).
table
ranges
Q1
is computed and
-260-
C
C
C
C
100
20
2
3
1
16
10
RSPASS--TABULATES MEAN FIRST PASSAGE TIMES FOR PORES
WITH THROATS OF VARIOUS WIDTHS (DIAM),
HALF-LENGTHS (DPTH), AND CONFIGURATIONS
LOGICAL*l FILE(11)
INTEGER*4 ISEED
DIMENSION T(1000),NARR(4),TB(4),KS(4,5)
DIMENSION DIAM(20),DPTH(10)
COMMON /PROBS/AHIT(0:1023)
COMMON /PARAMS/Cl(4),C2(4)
DATA KS/1,1,0,0, 1,0,1,0, 1,1,1,0, 1,1,0,1, 1,1,1,1/
TYPE *,'GIVE ME ISEED/NDEPTH,DPTH/NDIAM,DIAM'
ACCEPT *,ISEED
ACCEPT *,NDEPTH,(DPTH(IL),IL=1,NDEPTH)
ACCEPT *,NDIAM,(DIAM(ID),ID=1,NDIAM)
TYPE *,'FILENAME?'
ACCEPT 100,(FILE(I),I=1,10)
FORMAT(10A1)
TYPE *,'FRACTIONAL ERROR TOLERANCE'
ACCEPT *,TERROR
FILE(11)=O
OPEN(UNIT=2,FILE=FILE,TYPE='NEW',FORM='FORMATTEb')
DO 20 IP=0,1023
P=(IP+1)/1025.
AHIT(IP)=2./3.1415928*ASIN(2.*P-1.)
CALL RINIT(ISEED)
DO 50 IL=1,NDEPTH
DEPTH=DPTH(IL)
DO 50 ID=1,NDIAM
D=DIAM(ID)
D2=D/2.
DO 40 JC=1,5
DO 1 I=1,4
IF(KS(I,JC))2,3,2
Cl(I)=-D2
C2(I)=D2
GO TO 1
C1(I)=1
C2(I)=-1
CONTINUE
TT=O.
T2=0.
DO 16 I=1,4
NARR(I)=0
TB(I)=O.
I=0
I=1+1
CALL RPASS(DEPTHTIIS)
T(I)=TI
NARR(IS)=NARR(IS)+1
TB(IS)=TB(IS)+TI
TT=TT+TI
T2=T2+TI*TI
IF(I.LT.100)GO TO 10
-261-
1000
17
40
2000
50
TBAR-TT/I
TSTD=SQRT(T2/I-TBAR*TBAR)
FERROR=TSTD/SQRT(FLOAT(I))/TBAR
IF(FERROR.GT.TERROR)GO TO 10
TYPE 1000,DEPTH,D,JC,TBARTSTD,I
FORMAT('OLENGTH=',F5.3,', DIAM=',F6.3,', CONFIG.'
1
':
TBARTSTD=',2F10.3,' I=',I5)
DO 17 IS=1,4
IF(NARR(IS).EQ.O)GO TO 17
TB(IS)=TB(IS)/NARR(IS)
TYPE *,IS,NARR(IS),TB(IS)
CONTINUE
WRITE(2,2000)DEPTH,D,JC,TBAR,TSTD,NARR,TB
FORMAT(2F6.3,I3,2F8.3/4I5,4F8.3)
CONTINUE
STOP
END
,
12,
-262-
C
C
C
C
C
2
RPASS--COMPUTES FIRST PASSAGE TIME THROUGH PORE
WITH THROATS.
REFLECTION AT INLET THROAT BISECTOR ASSUMED.
SUBROUTINE RPASS(DEPTH,T-,ISIDE)
LOGICAL LSIDE
INTEGER*4 IR1,IR2,IR3,IS,ISIDE
REAL L
COMMON /CONST/CONST1,CONST2,PI4,PIPI
COMMON /PARAMS/Cl(4),C2(4)
COMMON /PROBS/AHIT(0:1023)
DATA PI/3.1415928653/
T=O
FIRST GET TO INLET
D2=DEPTH*DEPTH
T=T+.5*D2
YT=O.
CALL RNOS2(IR3, ISIDE)
XT=Cl(1)+(C2(1) -Cl(1))/1024.*IR3
ISIDE=.
C
C
WHEN IN AN THROAT...
C
10
Cll=Cl(ISIDE)
C21=C2 (IS IDE)
DELX=O.
DELY=O.
15
IF(YT.LT..0001)YT=.0001
L=AMIN1(DEPTH-YT,XT-Cll ,C21-XT)
IF(L.LT..000001)L=AMIN1 (DEPTH-YT,C21-C11)
T=T+. 295*L*L
CALL RNOS2(IP,IS)
XINC=AHIT(IP)*L
IF(XT-Cll.GT..000001)GO TO 16
GO TO (17,12,17,12),IS
17
XINC=ABS(XINC)
GO TO 19
16
IF(C21-XT.GT..000001)GO TO 19
GO TO (18,14,18,14),IS
18
XINC=-ABS (XINC)
19
GO TO (11,12,13,14),IS
11
YT=YT+L
XT=XT+XINC
IF(DEPTH-YT. GE..000001) GO TO 15
IF(ISIDE.EQ. 1)GO TO 2
RETURN
12
XT=XT+L
YT=YT+XINC
IF(YT)20,15,
13
14
YT=YT-L
XT=XT+XINC
IF(YT)20,15,
XT=XT-L
YT=YT+XINC
-263-
C
20
21
22
23
24
C
C
C
30
31
32
33
34
35
36
37
38
39
C
40
IF(YT)20,15,15
GET BACK INTO CENTRAL PORE BODY
LSIDE=.FALSE.
GO TO (21,22,23,24),ISIDE
Y=1.+YT
X=XT
GO TO 40
X=1.+YT
Y=-XT
GO TO 40
Y=-1.-YT
X=-XT
GO TO 40
X=-1.-YT
Y=XT
GO TO 40
WHEN
IN THE CENTRAL PORE BODY...
CALL SQUARE(X,Y,L,ISIDE,LSIDE)
T=T+.295*L*L
CALL RNOS2(IP,IS)
XINC=AHIT(IP)*L
GO TO (31,32,33,34),IS
DELX=XINC
DELY=L
GO TO 35
DELX=L
DELY=XINC
GO TO 35
DELX=XINC
DELY=-L
GO TO 35
DELX=-L
DELY=XINC
IF(LSIDE)GO TO (36,37,38,39),ISIDE
GO TO 40
DELY=-ABS(DELY)
GO TO 40
DELX=-ABS(DELX)
GO TO 40
DELY=ABS(DELY)
GO TO 40
DELX=ABS(DELX)
LSIDE=.FALSE.
X=X+DELX
Y=Y+DELY
IF(Y.LE..9999)GO TO 41
Y=1.
LSIDE=.TRUE.
ISIDE=1
IF(X.LT.Cl(l))GO TO 30
IF(X.GT.C2(1))GO TO 30
-264-
41
42
43
YT=O.
XT=X
GO TO 10
IF(X.LE..9999)GO TO
X=1.
LSIDE=.TRUE.
IS IDE=2
IF(Y.LT.Cl(2))GO TO
IF(Y.GT.C2(2))GO TO
YT=O.
XT=-Y
GO TO 1 0
IF(Y.GE .-. 9999)GO TO
Y=- 1.
LS IDE=. TRUE.
ISIDE=3
IF(X. LT .Cl(3))GO TO
IF(X.GT .C2(3))GO TO
YT=O.
XT=-X
GO TO 1 0
IF(X.GE .-. 9999)GO TO
LSIDE=. TRUE.
X=- 1.
ISIDE=4
IF(Y.LT .Cl(4))GO TO
IF(Y.GT .C2(4))GO TO
YT=O.
XT=Y
GO TO 10
END
42
30
30
43
30
30
30
30
30
SUBROUTINE SQUARE(X,Y ,L,ISIDE,LSIDE)
REAL L
INTEGER*4 ISIDE
LOGICAL LSIDE
COMMON /PARAMS/C(4), C2(4)
ABSX=ABS(X)
ABSY=ABS(Y)
C
TAKE CARE OF CORNER DEAD ENDS
IF(ABSX.LE..99)GO TO 30
IF(ABSY.LE..99)GO TO 30
X=X* .99
Y=Y* .99
ABSX=ABS(X)
ABSY=ABS(Y)
LSIDE=.FALSE.
GO TO 2
30
IF(LSIDE)GO TO 5
C'
IF IN INTERIOR OF CENTRAL PORE BODY...
IF(ABSX.GT.ABSY)GO TO 1
ISIDE=1
IF(Y.LT.O.)ISIDE=3
GO TO 2
-265-
1.
2
C
5
3
31
4
41
ISIDE=2
IF(X.LT.0. )ISIDE=4
L=AMIN1(1.-ABSX,1.-ABSY)
IF(L.NE.O.)RETURN
LSIDE=.TRUE.
IF ON BOUNDARY OF CENTRAL PORE BODY...
CC1=Cl(ISIDE)
CC2=C2(ISIDE)
GO TO (3,4,3,4),ISIDE
IF(X.GT.CC.)GO TO 31
L=AMIN1 (X+1. ,CCl-x)
RETURN
L=AMIN(1.-X,X-CC2)
RETURN
IF(Y.GT.CC1)GO TO 41
L=AMIN1(Y+1.,CCl-Y)
RETURN
L=AMIN1(1.-Y,Y-CC2)
RETURN
,END
-266-
STORE:
.TITLE
. ENTRY
IT-4
IS=8
RNOS 2
RNOS2 ,@M<R2>
MOVL
MOVL
MULL2
INCL
BICL3
ROTL
STORE,RO
#69069,R
RiRO
RO
#@XO03FFFFFROR2
#lO,R2,@IT(AP)
MULL2
INCL
BICL3
ROTL
INCL
MOVL
MOVL
RET
RlRO
RO
#@X3FFFFFFF,RO,R2
#2,R2,R2
R2
R2 ,@IS (AP)
RO,STORE
. ENTRY
MOVL
RET
.LONG
.END
RINIT,@M<>
@4(AP),STORE
0
-267-
BIOGRAPHICAL NOTE
The author, Ronald Alan Siegel, was born February 1,
1954, in Oakland, California.
He attended Condon Elementary
School, Roosevelt Jr. High School, and South Eugene High
School, all in Eugene, Oregon.
He graduated cum laude from
the University of Oregon in 1975, majoring in Mathematics.
That year, he was awarded the prize for Outstanding
Mathematics Student.
The author received his masters degree in Electrical
Engineering and Computer Science in 1979, and an Electrical
Engineers degree in 1980, both from M.I.T.
While at M.I.T., the author taught courses
circuit theory and sensory-motor physiology.
in electrical
The author worked as a computer programmer at Oregon
Research Institute, Eugene, Oregon, from 1969-1975, and at
Systems Control Inc., Palo Alto, California,
from 1975-1976,
as a research analyst.
In 1978 he worked as a research
engineer at Scientific Systems, Inc., Cambridge,
Massachusetts.
He has served as a consultant with Abbot
Laboratories since 1982.
He is currently Assistant Professor
of Pharmaceutics and Pharmaceutical Chemistry, School of
Pharmacy, University of California at San Francisco, San
Francisco, California.
The author
is a member
of Phi Beta Kappa.
Publications
1.
Siegel, R.A. and Colburn, H.S., "Internal and External
Noise in Binaural Detection," ACOUSTICAL SOCIETY OF
AMERICA, State College, PA, June, 1978
2.
Siegel, R.A. "Theoretical Aspects of Internal and
External Noise," ACOUSTICAL SOCIETY OF AMERICA,
Cambridge, MA, May, 1979
3.
"Internal and External Noise in Auditory Detection,"
M.S.E.E. Thesis, M.I.T., 1979
4.
"Perspectives on Internal
Noise,"
R.A. Siegel,
ACOUSTICAL SOCIETY OF AMERICA, Ottowa, Ontario,
May, 1981
5.
Siegel, R.A. and Colburn, H.S., "Internal and External
Noise in Binaural Detection," Hearing Research 11,
117-123
6.
Langer, R., Hsieh, D., Brown, L., Siegel, R., and
Bawa, R., "Recent Advances in Controlled Release
-268-
Systems for Macromolecules,"
Proceedings of the
8th International Symposium on the Controlled
Release of Bioactive Materials, 1981
7.
Siegel, R.A. and Langer, R. "Modelling of Controlled
Diffusion through Porous Materials using Monte
Carlo Methods,"
Proceedings of the 9th International
Symposium on the Controlled Release of Bioactive
Materials,
1982
8.
Siegel, R.A. and Langer, R.
"Simulation of Geometrical
and Topological Factors Affecting Controlled Release,"
A.I.Ch.E., Los Angeles, CA, December 1982
9.
Cohen, J.M., Siegel, R.A., and Langer, R. "Sintering
Technique for the Preparation of Polymer
Matrices for the Sustained Release of Macromolecules,"
Journal of Pharmaceutical Sciences, in press.
10.
Siegel, R.A., Cohen, J.M., Brown, L., and Langer, R.,
"Sintered Polymers for Sustained Macromolecular Drug
Release," in Recent Advances in Drug Delivery Systems
(Anderson, J. and Kim, S.W.,eds.), Plenum, in press
11.
Siegel, R.A and Langer, R., "Controlled Release of
Polypeptides and other Macromolecules," Pharmaceutical
Research, 1, 2-10.
12.
Bawa, R., Siegel, R.A., Marasca, B., Karel M., and
Langer, R., "An Explanation for the Sustained Release
of Macromolecules from Biocompatible Polymers,"
submitted.
Patent Application
J.M. Cohen, R.A. Siegel, and R. Langer, "Pressure Casting
Technique for Producing Polymer Matrices for the Sustained
Release of Macromolecules."