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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_5320_Библиотеки_им_академика_М_И_Перельмана
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d
d
d
d
C
t
J
x
−=
J
x
C
x
2
d
C
t
C
x
2
ddC
dCdxdCdydC
J
tS
CC
h
−
(
)
12
CC K
=×
https://t.me/med1917
Pharmaceutical Dosage Forms and Drug Delivery
in concentration with time, t (i.e., dC/ dt
J, per unit distance, x (i.e., dJ/ dx).
=− (4.3)
JJD × dC/ dx), with respect
to x, we obtain:
d
d
D
(4.4)
d
2
d
C(x, t) and J(x, t), respectively, to emphasize
that these parameters are functions of both distance, x, and time, t. Substituting dC/ dtJ/ dx, Ficks’s
second law of diffusion can be expressed as:
d
D
=
d
(4.5)
2
d
This equation represents diffusion only in one direction. To express concentration changes of diffusant
in three dimensions, Fick’s second law of diffusion is written as:
2
2
4.2.3 Diffusion Rate
D
=++
t
2
2
2
2
dz
gradient (dC/ dx) does not change with time. The second law refers to a change in the concentration of
diffusant with time at any distance (i.e., a nonsteady state). Diffusive transport from a dosage form is usually slow, leading to most of the drug transport happening under steady- state conditions. Therefore, it is
important to understand the diffusive conditions under a steady state.
4.2.3.1 Diffusion Cell
Figure 4.2 shows the schematic of a diffusion cell, with a diaphragm of thickness h and cross- sectional area
S separating the two compartments. A concentrated solution of the drug is loaded in the donor compartment
and allowed to diffuse into the solvent in the receptor compartment. The solvent in both compartments is
continuously mixed and sampled frequently to quantify drug transport across the membrane.
M
d
=×=×
1
d
D
(4.7)
in which (C1C2)/ h approximates dC/ dx. Concentrations C1 and C2 within the membrane (Figure 4.2) are
K
donor compartment (C
) or in the receptor compartment (C
donor
membrane/ solvent
) multiplied by the concentration in the
). Thus, we have:
receptor
1
donor membrane solvent/
(4.8)
CC K
2
=×
receptormembranesolvent/
K
C
C
donor receptor
12
d
t
KC
h
()
d
d
M
t
DSKC
h
DK
h
×
d
M
t
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Biopharmaceutical Considerations
FIGURE 4.2 Drug concentrations in a diffusion cell.
57
and
(4.9)
membrane solvent
==
/
C
(4.10)
C
Hence,
×× ×−
DS
d
M
=
membrane solvent donor receptor
/
C
(4.11)
Under sink conditions, the drug concentration in the receptor compartment is maintained much lower
than the drug concentration in the donor compartment, such that C
receptor
=
membrane solvent donor
/
(4.12)
P, in cm/ s, which is
P
=
membrane solvent/
(4.13)
as,
PSC=×× (4.14)
d
donor
S
rr
+
2
Srr=×××
π
outer inner
x rr=−
r
M
DK C
××
∆
(()
d
M
t
DSKC
h
d
d
µ
Ä
M
t
DSC
h
https://t.me/med1917
58
4.2.3.2 Spherical Membrane- Controlled Drug Delivery System
Pharmaceutical Dosage Forms and Drug Delivery
Following the same principles as outlined above for a diffusion cell, diffusive drug release from a spher-
membrane at the center point of its thickness and the linear distance of drug diffusion across the membrane, x. Given the inner- boundary radius of the membrane as r
membrane as r
, the surface area of the sphere at its mean radius is given by:
outer
π
outer inner
2
=××
4
and the outer- boundary radius of the
inner
(4.15)
The surface area of the sphere may be approximated by:
4
The linear distance for solute diffusion across the membrane is given by:
outer inne
(4.17)
Thus, the expression for the drug- release rate from a sphere is:
d
=××××
4
π
rr
t
d
outer inner
membrane solvent
/
rr
−
outer inner
(4.18)
C is the concentration gradient between the inside and the outside of the membrane.
4.2.3.3 Pore Diffusion
In microporous reservoir systems, drug molecules are released by diffusion through the solvent-
system, the pathway of drug transport is no longer straight but tortuous. The rate of drug transport
is directly proportional to the porosity, ε, of the membrane and inversely proportional to the tortuosity, τ
solvent (K
membrane/ solvent
) is no longer a factor, since drug dissolution in the membrane is not required.
Therefore,
d
=
membrane solvent donor
/
(4.12)
s donor
= (4.19)
Ds
d
d
µ
Ä
M
t
DSC
h
Srr=×××
π
outer inner
d
d
donor
M
t
dd
MPSC t=
d
d
M
t
kt
https://t.me/med1917
Biopharmaceutical Considerations
4.2.3.4 Determining Permeability Coefficient
59
s donor
= (4.19)
where surface area is given by:
C = C
C ≅ C
, we obtain:
donor
4
donor
C
receptor
, with the assumption that C
receptor
PSC= (4.20)
or
(4.21)
donor
P, can be obtained from the slope of a linear plot of M
versus t, provided that C
remains relatively constant.
donor
4.2.3.5 Lag Time in Nonsteady State Diffusion
A sustained- release dosage form may not exhibit a steady- state phenomenon from the initial time
of drug release. For example, the rate of drug diffusion across a membrane slowly increases to
steady- state kinetics (Figure 4.3
early stage and then becomes linear. This early stage is the condition of the non- steady state. Later,
the rate of diffusion is constant, the curve is essentially linear, and the system is at a steady state.
When the steady state portion of the line is extrapolated to the time axis, the point of intersection
represents the time of zero diffusion concentration if the system had been at the steady state all
along. This time period between the actual non- steady state and the projected steady- state time at
zero diffusion concentration is known as the lag time. This is the time required for a penetrant to
establish a uniform concentration gradient within the membrane that separates the donor from the
receptor compartments.
4.2.3.6 Matrix (Monolithic)- Type Nondegradable System
In a matrix- type polymeric delivery system, the drug is distributed throughout a polymeric matrix. The
drug may be dissolved or suspended in the polymer. Regardless of a drug’s physical state in the polymeric
matrix, the release of the drug decreases over time. In these systems, drug molecules can elute out of the
matrix only by dissolution in the surrounding polymer (if the drug is suspended) and by diffusion through
the polymer structure. Initially, drug molecules closest to the surface are released from the device. As
drug release continues, molecules must travel a greater distance to reach the exterior of the device. This
increases the diffusion time required for drug release. This increase in diffusion time results in a decrease
in the drug- release rate from the device with time.
In an insoluble matrix- type system, the drug- release rate decreases over time as a function of the square
= (4.22)
device
d
M
t
DSKC
h
d
d
cm
3
M
t
×××× ×
×
−−
14 10
92 3
.
s
d
cm
M
t
=⋅
..
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FIGURE 4.3 Drug diffusion rate across a polymeric membrane.
where k
is a proportionality constant dependent on the properties of the device.
device
Pharmaceutical Dosage Forms and Drug Delivery
Thereafter, the release rate usually declines exponentially. Thus, the reservoir system can provide constant release with time (zero- order release kinetics), whereas a matrix system provides decreasing release
with time (square root of time- release kinetics).
4.2.3.7 Calculation Examples
4.2.3.7.1 Drug- Release Rate
delivery system is often undertaken to simulate drug- release rate kinetics under different formulation
conditions, such as drug loading. The diffusion rate may be calculated by using experimental data in
D = 8.0(±4.7) × 10
cm2/ s and the
K
known. If the membrane thickness, h, of the device is 1.4 × 10
in the core, C
, is 0.02 g/ cm3, the tetracycline- release rate, dM/ dt, may be calculated as follows.
core
, drug- release rate can be calculated if the design parameters of the device are
membrane solvent donor
=
d
80 10 16810002
...
=
cm /s cm g/cm
−
2
cm and the concentration of tetracycline
/
(4.12)
31 10
.
−−⋅10
g/cm
=×
To obtain the results in micrograms per day,
d
=× ⋅× ×××
−
10 6
31 10 10 60 60 24 26 85
g/cm s¼g/gs/day ¼g /day
DK
h
×
05
×K
K
0025
d
d
ss
MtDS
h
CC
()
=−
()
d
CtDS
Vh
CC=−
()
M
t
DSC
h
d
d
C
t
DSC
Vh
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Biopharmaceutical Considerations
4.2.3.7.2 Partition Coefficient
2/ s, its permeability coef
cm /s
4
membrane solvent/
2
002..
(4.13)
/
membrane solvent
cm
P
cm/s
=
=
or
4.3 Dissolution
membrane solvent
/
.
cm/s
002
cm /s
4
.=×=05
0
2
cm
.
For most drugs, the rate at which the solid drug dissolves in a solvent (dissolution) is often the ratelimiting step in the drug’s bioavailability.
4.3.1 Noyes– Whitney Equation
S), the
thickness of the unstirred solvent layer on the particle surface (h
(D), and the concentration gradient, that is, the difference in the concentration of drug solution at the particle surface (Cs) and the bulk solution (C).
CC kS
=−
(4.23)
or
d
(4.24)
s
where dM/ dtD
2S is the surface area of the exposed solid, in cm2
k is the dissolution rate constant (k = D/ hh is the thickness of the unstirred layer at the solid
CsC is the drug concentration
in bulk solution at time t, in g/ mL.
The quantity, dC/ dt, represents the change in drug concentration in the bulk solution per unit time, or
the dissolution rate, and V is the volume of solution (mL). Thus, C = M/ V.
Under sink conditions, C << Cs
d
d
s
=
or
s
=
d
d
ss
MtDS
h
CC
()
=−
()
550
10
k
in
k
D
h
0
510
×
D
D =××=×
−−
34
..
in
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4.3.2 Calculation Example
Pharmaceutical Dosage Forms and Drug Delivery
Knowledge of dissolution rate constant, k, allows simulation of the rate of drug dissolution by using
different quantities of drug substance, changes in the particle size and surface area of the drug, and dissolution conditions, such as volume. A simulation of these results can assist in dissolution method development by minimizing the number of experiments needed under different conditions. The dissolution rate
For example, a preparation of drug particles weighing 550 mg and having a total surface area of
0.28 × 104 cm2
k can be calculated as follows:
CC kS
=−
(4.23)
or
mg
550
=×
min
mg
mincm
10
02810
0281015
k ../
=× ×× −
1
42
.
×
42
cm mg mL
×
.
mg/cm
15
1
262
3
−
500
mg
cm
262
500
3
mg
mL
.2201cm/m
=
00
The dissolution rate constant is related to the diffusion constant of the drug through the solvent (D) and
the diffusion layer thickness (h):
= (4.25)
can be calculated. Thus, if the diffusion layer’s thickness were 5 × 10
D)
would be given by:
0201
./cm mincm=
3
−
or
0 0201 510101 10
cm/min cm cm /m
4.3.3 Factors Influencing Dissolution Rate
2
summarized as follows:
1. Drug solubility: The greater the drug solubility, the greater the drug’s dissolution rate. This is
https://t.me/med1917
Biopharmaceutical Considerations
whereas those of acidic drugs are high.
2. Viscosity (of the dissolving medium): The greater the viscosity of the dissolving liquid, the lower
dissolving bulk medium and/ or the unstirred layer on the surface of the dissolving formulation
can be affected by the presence of hydrophilic polymers in the formulation, which dissolves to
form a viscous solution. In vivo, the viscosity may be affected by the food intake.
3. Diffusion layer’s thickness: The greater the diffusion layer’s thickness, the slower the dissolution
medium, both in vitro and in vivo. Hence, an increase in gastric and/ or intestinal motility may
increase the dissolution rate of poorly soluble drugs. For example, food and certain drugs can
4. Sink conditions: Removal rate of dissolved drugs by absorption through the GI mucosa and the
5. pH (of the dissolving medium): The drug dissolution rate is determined by the drug solubility in
the diffusion layer surrounding each dissolving drug particle. The pH of the diffusion layer has a
low solubility in the diffusion layer. If the pH in the diffusion layer could be increased, the solubility exhibited by the weak acidic drug in this layer (and hence the dissolution rate of the drug
have a relatively high solubility at the elevated microenvironmental pH in the diffusion layer due
to the strong counterion bases, KOH and NaOH, respectively. Thus, the dissolution of the drug
particles takes place at a faster rate.
Particle size and surface area
However, particle size reduction may not always be helpful in increasing the dissolution rate of a
drug and hence, its oral bioavailability. For example:
• Porosity
may have lower surface area compared with larger particles with greater porosity. The dissolution rate depends on the effective
porosity.
• In some cases, particle size reduction may cause particle aggregation, thus reducing the
effective surface area. To prevent the formation of aggregates, small drug particles are often
dispersed in polyethylene glycol (PEG), polyvinylpyrrolidone (PVP), dextrose, or surfactants
such as polysorbates. For example, micronized griseofulvin is dispersed in PEG 4000.
• In addition, certain drugs, such as penicillin G and erythromycin, are unstable
and do not readily dissolve in them. For such drugs, particle size reduction may increase not
7. Crystalline structure: Amorphous (noncrystalline) forms of a drug may have a faster dissolution rate compared with the crystalline forms. Some drugs exist in a number of crystal forms or
ution rates.
a. The dissolution rate of a drug from a crystal form is a balance between the energy required
to break the intermolecular bonds in the crystal and the energy released during the formation
dissolution rate.
b. Intrinsic dissolution rate
for its surface area. It is expressed in terms of mass per unit time per unit surface area.
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Pharmaceutical Dosage Forms and Drug Delivery
Drug forms that have higher intrinsic dissolution rates are expected to have higher dissolution rates.
c. The greater strength of a crystalline polymorph, sometimes evident by its high melting point
and sometimes by the rank order, correlates with its lower intrinsic dissolution rate.
d.
tend to have higher intrinsic dissolution rates.
8. Temperature: An increase in temperature leads to greater solubility of a solid, with positive heat
of the solution. The heat of the solution indicates the release of heat on dissolving. Positive heat of
dissolve at a more rapid rate if the system is heated. Therefore, in vitro dissolution studies are
in vivo dissolution conditions.
9. Surfactants: Surface- active agents increase the dissolution rate by (a) lowering the interfacial
tension, which lowers the contact angle of the solvent on the solid surface and increases wetting
of the drug particle and penetration of the solvent inside the dosage form, and (b) increasing the
saturation solubility of the drug in the dissolution medium. Surfactants such as sodium lauryl sulfate (SLS) and Triton X- 100 frequently achieve sink conditions and rapid dissolution during in
vitro dissolution method development.
4.4 Absorption
Bioavailability is the fraction of an ingested dose of a drug that is absorbed into the systemic circulation,
compared with the same dose of the compound injected intravenously, which is directly injected into the
systemic circulation. The bioavailability of a drug is determined during new product development.
Bioequivalence, on the other hand, is a comparison of relative bioavailability of two dosage forms in
terms of the rate and extent of the drug levels achieved in the systemic circulation and the maximum drug
concentration reached. Generic drugs are required to satisfy statistical criteria of bioequivalence to the
branded version before they can be considered equivalent.
In the case of oral dosage forms, drug bioavailability depends on the rate and extent of drug absorption from the GI tract. Drug absorption from the gut depends on many factors, such as the drug’s solu-
drug’s particle size and surface area. Thus, an interplay of the physicochemical properties of the drug
and the physicochemical properties of the GI tract determines the outcome of factors that determine
drug absorption.
Drug absorption is affected not only by the properties of the drug and its dosage forms but also by the
nature of the biological membranes. Drugs pass through living membranes by the following processes
(Figure 4.4):
1. Passive diffusion
a. Simple diffusion
b. Facilitated diffusion
i. Channel- mediated transport
ii. Carrier- mediated transport
2. Active transport
route across the epithelial cell barrier. The surface lining of the GI tract consists of epithelial cells
attached by tight junctions formed through their membranes. Drug transport across the tight junctions
between cells is known as paracellular transport
water accompanying water- soluble drug molecules. Drug transport by absorption into the epithelial cell
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Biopharmaceutical Considerations
FIGURE 4.4 An illustration of the main transport processes across cellular membranes.
from the gut’s lumen side, followed by release of the drug molecule from the epithelial membrane on the
other side of the epithelial cell into the systemic circulation, is known as transcellular transport.
4.4.1 Passive Transport
Passive transport can be divided into simple diffusion, carrier- mediated diffusion, and channel- mediated
diffusion (Figure 4.4).
4.4.1.1 Simple Diffusion
tails in the center and hydrophilic heads facing the aqueous environment on either side. Therefore, hydrophobic lipid- soluble drugs of low molecular weight can pass through membranes by simple diffusion.
Passive transport by simple diffusion is driven by differences in drug concentration on the two sides of
the membrane. In intestinal absorption, for example, the drug travels by passive transport from a region of
high concentration in the GI tract to a region of low concentration in the systemic circulation. Given the
instantaneous dilution of the absorbed drug, once it reaches the bloodstream, sink conditions are essentially maintained at all times.
4.4.1.2 Carrier- Mediated Transport
Carrier- mediated transport is a passive diffusion process that involves the facilitation or increase of
diffusion rate by involving a carrier protein embedded in the biological membrane. It differs from active
transport in that the drug moves along a concentration gradient (i.e., from a region of high concentration
use energy, such as adenosine triphosphate (ATP), to transport the drug. Carrier- mediated transport is saturable, structurally selective for the drug, and shows competition kinetics for drugs of similar structures.
Carrier- mediated transport does not require the substrate to be lipophilic: both hydrophilic and lipophilic
solutes can be transported in this manner.
Amino acid transporters, oligopeptide transporters, glucose transporters, lactic acid transporters, phosphate transporters, bile acid transporters, and other transporters facilitate drug transport across the GI
tract, especially the small intestine. Transporters
transport the molecules (e.g., glucose) across the membrane. Transporters bind to the molecule, transport
it across the membrane, and then release it on the other side. The transporter remains unchanged after
the process.
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