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162
The effective surface area
The effective surface areaA is lower than the total surface:
not all of the total surface is wetted
Wetting the surface is the first step in dissolution
The wettability may be estimated from the contact angle, height
of a drop on the surface etc.
The effective surface areaA is proportional to the specific
surface area Sv:
vSMA
163
Contact Angle
The contact angle is
a measure of the wetting of a liquid on a solid surface
is expressed in degrees, with 0 degrees being complete wetting and
180 degrees being absolute non-wetting
Contact angles on surfaces are influenced by many phenomena
interfacial tensions (viscosity of the fluid)
surface roughness (shape and size of the pores)
chemical heterogeneity
swelling
partial solution of low-molecular constituents in the polymer
material
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Contact Angle:
Methods of Measurement
Goniometry
a drop of the wetting liquid is placed on the solid sample
the image of the drop shape is investigated with the contact angle
being the tangent between the drop edge and the solid surface
Tensiometry
a sample of the solid is hung on a force balance and brought intocontact with the wetting liquid
the change in forces when the solid contacts the liquid are used to
calculate contact angles
good wetting
bad wetting
165
Specific surface area
The specific surface area of solid particles
is given in m2/g
increases with decreasing diameter of the particles
smaller particles exhibit higher dissolution rates
example: Griseofulvin absorption was supposed to be limited by dissolution rate
absorption of particles with a diameter of 2.7 m (2.5 m2/g) was twice
that of particles with a diameter of 10 m (0.4 m2/g)
0.5 1 1.5 2 2.5
1.0
1.5
2.0
2.5
relativea
bsorbedamount
specific surface / (m2/g)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
1.0
1.5
2.0
2.5
relativea
bsorbedamount
specific surface / (m2/g)
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Specific surface area
The specific surface area Sv may change for spherical particles
according to
or may remain nearly constant for needle-like particles
3/1= MkSv
constvS
167
The extended Noyes-Whitney equation
The effective surface areaA may be expressed as
with:
M: mass remaining to dissolve
t: time
z: parameter describing size change of particles
n: polydispersity proportional to the inverse of the width of the
size distribution curve;
for monodisperse particles: n = 1
for polydisperse systems: n < 1, n changes with time
Thus, the Noyes-Whitney equation becomes
1= nz tMkA
( ) 1= nzs tMcckdt
dM
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Drug release
Controlled release dosage forms:
drug release is the rate-limiting step (i.e., slower than drug
absorption)
sustained and controlled drug release
used for per oral, topical administration as well as in implants
169
Membrane controlled drug release
Assumptions:
drug concentration in solution c < cs
drug concentration in the dosage form < cs
Note:
If the resistance of the unstirred layer is considerably larger than
the resistance of the membrane, the diffusion through the unstirred
layer will be rate limiting
cons=
=
=
sP
s
cAk
chDAP
dtdM
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Matrix controlled release
Matrix controlled release occurs if the matrix is not degraded
during release
Matrix controlled release is observed for ointments, matrix
tablets
The embedded drug may be
completely dissolved
mainly suspended
within the matrix
171
Non-degradable porous matrix
The total mass Mfreleased from the matrix is given by
with
A: effective surface area
D: diffusion coefficient
V: volume of the matrix
M0: mass of the drug in the matrix at time t= 0
: tortuosity
tAkM rf =
=
VMDkr
02
H. Lapidus and N. Lordi.J. Pharm. Sci. 57:1292 (1968)
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Non-degradable, homogeneous matrix
173
Non-degradable, homogeneous matrix
Drug fully dissolved in matrix
Drug suspended in matrix
and forctotal>> cs:
with c0: initial concentration in matrix
D: diffusion coefficient in matrixcs: solubility in matrix
DtAcMf = 02
tccDM stotalf = 2
( ) tcccDAM stotalsf = 2
T. Higuchi.J. Soc. Cosm. Chem. 11:85 (1960) & T. Higuchi.J. Pharm. Sci. 50:874 (1961)
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The Weibull model
General empirical equation by Weibull (1951)
Adapted later to dissolution processes (1972)
Commonly used in dissolution studies
Disadvantages no kinetic fundament
no relation between parameters and dissolution rate
( )
=
a
TtMM
b
f exp10
a: time scale of the process, b: shape parameter,
T: location parameter
175
Partitioning
G. Camenisch, G. Folkers, and H. van de Waterbeemd. Review of theoretical passive drug
absorption models: Historical background, recent developments and limitations. Pharm.
Acta Helv. 71(5):309-327, 1996
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Membrane permeation
Diffusion through
lipid bilayer (hydrophobic molecules)
pores (small, hydrophilic molecules)
Active transport (molecules with affinity to carrier)
Pinocytosis/Phagocytosis, cell drinking (large molecules)
177
Membrane permeation by diffusion
I. Small
hydrophobic
molecules
II. Small
hydrophilic
molecules
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Partitioning
Example 1: partition coefficient = 1
aqueousphase,
c0
aqueousphase,
c = c0
lipoidalmembrane
c
1 1 0,5 0,5
179
Example 2: partition coefficient = 10
Remember: flux is proportional to concentration gradient!
Assuming steady-state conditions:
Partitioning
aqueous
phase
aqueous
phase
lipoidal
membrane
cpartition
coefficient
= 10 1 0,5
10
5
cl
DPJ WatMembr = /.
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Partitioning
n2, v2, c2
n1, v1, c1n1,0 = ntotal,
v1, c1,0
181
Nernst distribution law
If two liquids (or solids) a and b are partially immiscible and if
there is a third component i present in both phases which behaves
individually as an ideal solute (i.e., if it is sufficiently dilute), the
ratio of its concentration c is independent of the individual values
ofc.
( )( )
constic
ic
b
a =
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Example: Nernst distribution law
( )
21
11
210,1
1
212
1
2
2
1
1
21
1
1
2
2
1
2
21
21
1
Pvv
v
n
v
Pvv
n
c
c
Pvv
n
Pv
nn
v
n
Pv
n
P
cc
n
v
v
n
c
cP
vvv
nnn
total
total
totaltotal
total
total
+=
+=
+=
==
=
==
+=
+=
183
Example: Nernst distribution law
The ratio of concentration before and after extraction is:
Example 1:
P= 10, v1 = 200 ml, v2 = 100 ml
Example 2:
P= 10, v1 = 200 ml, v2 = 200 ml
21
1
0,1
1
Pvv
v
c
c
+=
17.01200
200
10ml100ml200
ml200
0,1
1 =+
=c
c
09.02200
200
10ml200ml200
ml200
0,1
1 =+
=c
c
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Partitioning
In equilibrium state:
DrugOctanol DrugWater
Partitioning is a dynamic process (resulting from a dynamic
equilibrium)
POct: Partition coefficient
Usually given:POct
or logPOct
But:
water is not a physiological medium!
( )( )
[ ]Oct
Water
Octanol
Water
Octanol
]Drug[
Drug
Drug
DrugP
c
c==
185
Partition coefficient
The partition coefficient is the constant ratio of the
concentrations of the same molecular species in the two phases
of a heterogeneous system in equilibrium.
Most widespread predictor of drug absorption
Routinely determined for new chemical entities
In natural sciences, there are numerous partition coefficients:
Octanol / water ( =POct )
Cyclohexane / water
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Dependence of the partition coefficient on chemical
structure
Partition coefficients encode two major terms
cavity formation energies V
solute-solvent interaction energies
Example:
In the octanol/water system, includes
dipole-dipole interactions
H-bonding capacity of the solute
aVP =log
187
Partition coefficient
Chemical potential of a solute:
with i0
= activityai = mole fraction
i = activity coefficient, i < 1; ideal solution: i = 1
ci = mole fraction
Partitioning of a solute between two phases is defined by the
chemical potential difference
At equilibrium: chemical potential in both phases is equal
iii
iii
cRT
aRT
ln
ln
0
0
+=
+=
WatWat
0
WatOctOct
0
Oct lnln cRTcRT +=+
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Partition coefficient
Partition coefficient based on activities:
Partition coefficient based on concentrations:
Partition coefficient based on solubility:
===RTa e
c
c
a
aK
0Wat
0Oct
WatWat
OctOct
Wat
Oct
===RTc e
a
a
c
cK
0Wat
0Oct
Oct
Wat
WatOct
OctWat
Wat
Oct
Wat
Oct
s
sKs =
189
Correlation between partition coefficients from
different solvent systems
logPvalues from different solvent pairs are linearly
correlated [1,2]:
This is only valid, if the organic solvents have similar physical
properties!
Taking into account the hydrogen bonding capacity of the
solute:
1. R. Collander. The distribution of organic compounds between iso-octanol and water.
Acta Chem. Scand. 4:1085-1098, 1950
2. R. Collander. The partition of organic compounds between higher alcohols and water.
Acta Chem Scand 5:774 780 1951
dPbP
PdPb
+=
=
Octsolv
Octsolv
loglog
++= HOctcyclohex loglog IdPbP