4. Diffusion
4.1 Definition
• Diffusion is defined as a process of mass transfer of
individual molecules of a substance, brought about by
random molecular motion and associated with a
concentration gradient.
• Spreading out, mixing. The diffusion of gases and liquids
refers to their mixing without an external force.
4.2. Mechanisms
• Diffusion could occur:
– Throughout a
homogeneity).
single
bulk
phase
(solution’s
– Through a barrier, usually a polymeric membrane
(drug release from film coated dosage forms,
permeation and distribution of drug molecules in living
tissues, passage of water vapor and gases through
plastic container walls and caps, ultrafiltration).
4.2. Mechanisms
• Diffusion through a barrier may occur through
either:
– Simple molecular permeation (molecular diffusion)
through a homogenous nonporous membrane.
This process depends on the dissolution of the
permeating molecules in the barrier membrane.
– Movement
through
pores
and
channels
(heterogeneous
porous
membrane)
which
involves the passage of the permeating molecules
through solvent filled pores in the membrane
rather than through the polymeric matrix itself.
4.2. Mechanisms
A
B
Diffusion through a homogenous nonporous membrane
(A) and a porous membrane with solvent (usually water)
filled pores (B)
4.2. Mechanisms
• Both mechanisms usually exist in any system and
contribute to the overall mass transfer or diffusion.
• Pore size, molecular size and solubility of the permeating
molecules in the membrane polymeric matrix determine
the relative contribution of each of the two mechanisms.
4.2. Mechanisms
• By combining the two mechanisms for diffusion through
a membrane we can achieve a better representation of a
membrane on the molecular scale.
• A membrane can be visualized as a matted arrangement
of polymer strands with branching and intersecting
channels.
4.2. Mechanisms
• Depending on the size and shape of the diffusing
molecules, they may pass through the tortuous pores
formed by the overlapping strands of the polymer.
• The other alternative is to dissolve in the polymer matrix
and pass through the film by simple diffusion.
• Passage of steroidal molecules substituted with
hydrophilic groups in topical preparation through human
skin involves transport through the skin appendages
(hair follicles, sebum ducts and sweat pores in the
epidermis) as well as molecular diffusion through the
stratum corneum.
4.3. Related Phenomena and Processes
• Some processes and phenomena related to diffusion:
– Dialysis: A separation process based on unequal
rates of passage of solutes and solvent through
microporous membranes (Hemodialysis).
– Osmosis: Spontaneous diffusion of solvent from a
solution of low solute concentration (or a pure
solvent) into a more concentrated one through a
membrane that is permeable only to solvent
molecules (semipermeable).
Dialysis
Osmosis
Low salt
High salt
4.4. Fick’s First Law of Diffusion
• Flux J is the amount, M, of material flowing through a
unit cross section, S, of a barrier in unit time, t.
J = dM / S.dt
• The units of the Flux J are g.cm-2.sec-1.
(S)
4.4. Fick’s First Law of Diffusion
• According to the diffusion definition, the flow of material
is proportional to the concentration gradient.
• Concentration gradient represents a
concentration with a change in location.
change
of
• Concentration gradient is referred to as dc/dx where c is
the concentration in g/cm3 and x is the distance in cm of
movement perpendicular to the surface of the barrier (
i.e. across the barrier).
4.4. Fick’s First Law of Diffusion
J  dc/dx
J = -D*(dc/dx) Fick’s First Law
• In which D is the diffusion coefficient of the permeating
molecule (diffusant, penetrant) in cm2/sec.
• D is more correctly referred to as Diffusion Coefficient
rather than constant since it does not ordinarily remain
constant and may change with concentration.
• The negative sign in the equation signifies that the diffusion
occurs in the direction of decreased concentration.
• Flux J is always a positive quantity (dc/dx is always
negative).
• Diffusion will stop when the concentration gradient no longer
exists (i.e., when dc/dx = 0).
Diffusion and Molecular Properties
• Diffusion depends on the resistance to passage of a
diffusing molecule and is a function of the molecular
structure of the diffusant as well as the barrier material.
• Gas molecules diffuse rapidly through air and other
gases. Diffusivities in liquids are smaller and in solids
still smaller.
Diffusion and Molecular Properties
4.5. Fick’s Second Law of Diffusion
• Fick’s first law examined the mass diffusion across a unit
area of a barrier in a unit time (J) (i.e. rate of diffusion).
• Fick’s second law examines the rate of change of
diffusant concentration with time at a point in the
system (c/t) .
• The diffusant concentration c in a particular volume
element changes only as a result of a net flow of
diffusing molecules into or out of the specific volume
unit.
4.5. Fick’s Second Law of Diffusion
Input
C
Output
Volume
element
Bulk medium
4.5. Fick’s Second Law of Diffusion
• A difference in concentration results from a difference in
input and output.
• The rate of change in the concentration of the diffusant with
time in the volume element (c/t) equals the rate of
change of the flux (amount diffusing) with distance (J/x)
in the x direction.
dc/dt = - dJ/dx
4.5. Fick’s Second Law of Diffusion
dc/dt = - dJ/dx
J = -D*(dc/dx)
•
Differentiating with respect to x
dJ/dx = D*(d2c/dx2)
•
substituting dc/dt from the top equation
 dc/dt = D*(d2c/dx2)
4.6. Steady state Diffusion

Diffusion Cells:
 In a diffusion cell, two compartments are separated by
a polymeric membrane.
 The diffusant is dissolved in a proper solvent and
placed in one compartment while the solvent alone is
placed in the other.
 The solution compartment is described as Donor
Compartment because it is the source of the diffusant
in the system while the solvent compartment is
described as the Receptor Compartment.
4.6. Steady state Diffusion
Donor
Compartment
Receptor
Compartment
Pure
Solution
solvent
Flux in
Membrane
Diffusant
Flux out
Flow of solvent
to maintain sink
condition
4.6. Steady state Diffusion
– As the diffusant passes through the membrane from
the donor compartment (d) to the receptor
compartment (r), the concentration in the donor
compartment (Cd) will fall while the concentration in
the receptor (Cr) will rise.
– However, to mimic the biological systems; the solution
in the receptor compartment is constantly removed
and replaced with a fresh solvent to keep the
concentration of the diffusant passing from the donor
compartment at a low level. This is referred to as the
Sink Condition.
4.6. Steady state Diffusion
• Therefore, the concentration in the receptor
compartment is always maintained at very low levels
because of the sink condition. This means that
Cr << Cd.
• In contrast, the concentration in the donor compartment
is kept very high or nearly constant (i.e. saturated
solubility). This could be ensured by having a reservoir
of precipitated or suspended drug for a long period of
time. So drugs diffuse to the receptor compartment will
be compensated by those dissolving from the
suspended particles.
4.6. Steady state Diffusion
• As both Cd and Cr are constant; concentration gradient
(dc/dx) is constant (but not zero). (Note that the
concentration in the two compartments is not the same).
• Furthermore, rate of diffusion (dM/dt) and consequently
flux (J=dM/S. dt) are constant (but not zero).
• When the system has properties that are not changing
with time, it is referred to be as in a steady state. Hence,
the rate of change in concentration
in the two
compartments with time (dc/dt) will become zero.
• Diffusion under such conditions is referred to as steady
state diffusion.
4.6. Steady state Diffusion
dc/dt = D*(d2c/dx2) = 0
• Since D is not equal to (0), then d2c/dx2 should be 0.
• Since d2c/dx2 is a second derivative, and is equal to (0)
the first derivative dc/dx should be a constant.
• This means that the concentration gradient dc/dx
across the membrane is constant (linear relationship
between concentration c and distance or membrane
thickness h)
4.6. Steady state Diffusion
Donor
Compartment
Cd
Receptor
Compartment
C1
High
concentration
of diffusant
molecules
C2
Cr
0
Thickness of
barrier
h
4.6. Steady state Diffusion
• In such systems (diffusion cells), Fick’s first law may
be written as:
J = dM / S.dt = D *(C1-C2)/h
• C1 and C2 are the concentrations within the
membrane and are not easily measured.
• However they can be calculated using the partition
coefficient (K) and the concentrations on the donor
(Cd) and receptor (Cr) sides which can be easily
measured
4.6. Steady state Diffusion
K = C1/Cd = C2/Cr
•
Replacing C1 and C2 with KCd and KCr
dM / S.dt = D (C1-C2)/h = D(K Cd -K Cr)/h
 dM / S.dt = DK(Cd - Cr)/h
 dM / dt = DSK(Cd - Cr)/h
4.6. Steady state Diffusion
• If the sink condition holds in the receptor compartment
 Cd>>Cr  0 and Cr drops out of the equation which
becomes
dM / dt = DSKCd /h
• The term DK/h is referred to as the Permeability
Coefficient or Permeability (P) and has the units of
linear velocity (cm/sec).
• The equation simplifies further to become
dM / dt = PSCd
4.6. Steady state Diffusion
Amount
Diffused
• If Cd remains relatively constant throughout time, P can
be obtained from the slope of a linear plot of M versus t.
M = PS Cd t
M = k0t
Time
4.6. Steady state Diffusion
• If Cd changes appreciably with time, then P can be
obtained from the slope of log Cd versus t.
log Cd = log Cd(0) - (PS/2.303Vd)t
• This eq. is first order (appreciable change in conc would
happen at the last stages of drug release). As in this
equation we used the conc. Term rather than M, we
divided by Vd (volume of donor).
Drug Release
Drug Release
• The release of drug from a delivery system and subsequent
bioabsorption involve factors of both dissolution and
diffusion.
• Diffusion is the main and most important mechanism
involved in drug release from dosage forms.
• Drug release occurs either from:
(i)
Reservoir
systems
with
zero
order
release
mechanism(e.g. Film coated dosage forms, microcapsule)
(ii) Matrix type dosage forms represent a very important group
of solid dosage forms where drug release is controlled by
dissolution and diffusion.
Monolithic System(Matrix) Vs Reservoir system
Monolithic System(Matrix)
Reservoir System
Reservoir System
Reservoir systems could be represented again by diffusion
cells….
Receptor
Compartment
Donor
Compartment
Cd
Cd
h
Cr
C1
High
concentration
of diffusant
molecules
C2
h
Thickness of
membrane
Cr
Reservoir System
• Drug release from these systems follows generally
zero order kinetics which can be presented by the
following equation:
• This behavior is presented in the straight dotted line
presented in the following figure
• If the excess solid in the dosage form is depleted, the
(Cd) decreases as the drug diffuses out of the system
and the release rate falls exponentially (First order
release).
PSt
log Cd  log Cdo 
2.0303Vd
Reservoir System
log Cd  log Cdo 
PSt
2.0303Vd
Amount Diffused
First order release
Steady state
So diffusion controlled release in
reservoir systems leads
generally to zero order release
kinetics but would end up with
first order kinetics after the
depletion of excess drug in the
reservoir
Time
Matrix type dosage forms
Figure – Drug eluted from a homogeneous polymer matrix
Matrix type dosage forms
• A matrix type dosage form is a drug delivery system in
which the drug is homogenously dispersed throughout a
polymeric matrix.
• The drug in the polymeric matrix is assumed to be
present at a total concentration A (mg/cm3). Part of the
drug is soluble in the polymeric matrix and the
concentration of the dissolved drug in the polymeric
matrix is Cs (mg/cm3). Cs is the solubility or saturation
concentration of the drug in the matrix.
Matrix type dosage forms
Schematic presentation of the solid matrix and its receding boundary as the drug
diffuses from the dosage form
Matrix type dosage forms
• To be released from the delivery
system, the drug molecules have to
dissolve and diffuse out from the
surface of the device.
• As the drug is released, the boundary
that forms between the drug and
empty matrix recedes into the tablet
and the distance for diffusion becomes
increasingly greater.
• Therefore, drug release will be faster
in the initial stages and become slower
later as the remaining drug molecules
should cross longer distances than the
first drug molecules. The release in
these systems is best described by
Higuchi.
Matrix type dosage forms
• Higuchi (1960,1961) developed an equation to
describe drug release from such matrix systems
based on Fick’s 1st Law:
Q
dM dQ DC s


S .dt dt
h
where dQ/dt is the rate of drug released per unit
area of exposed surface of the matrix.
• The amount of drug released (dQ) as the drug
boundary recedes by a distance of (dh) is given
by the approximate linear expression:
dQ = A.dh – ½(Cs.dh) = dh.(A –½Cs)
• The final form of the equation is known as the
Higuchi equation. It is represented as follows:
t
Matrix type dosage forms
Example 13-6; Martin’s 6th ed.:
(a) What is the amount of drug per unit area, Q, released from a tablet
matrix at time t = 120 min? The total concentration of drug in the
homogeneous matrix, A, is 0.02 g/cm3. The drug’s solubility, Cs, is 1.0
x 10-3 g/ cm3 in the polymer. The diffusion coefficient, D, of the drug in
the polymer matrix at 25°C is 6.0 x 10-6 cm2/sec or 360 x 10-6 cm2/min.
Q  2 DACs * t
Q  2 * 360 x106 * 0.02 * 1x103 * 120
Q  1.3x103 g / cm 2
(b) What is the instantaneous rate of drug release occurring at 120 min?
dQ

dt
ADC s
2t
dQ
360 x106 * 0.02 * 1x103

dt
2 * 120
dQ
 5.5 x106 g / cm 2 . min
dt
Summery od different drug release rates and
mechanisms
Rx Order
Reservoir
System with
constant Cd
Release Rate
Eq
M  K 0t
Zero
order
Reservoir
System with
decreasing Cd
log Cd  log Cdo 
First
order
PSt
2.0303Vd
Matrix System
QK t
Higuchi
5. Dissolution
5.1. Definitions
• A solution is defined as a mixture of two or more
components that form a single phase which is
homogenous down to the molecular level.
• Dissolution is the transfer of molecules or ions from a
solid state into solution.
• Dissolution rate is the rate at which a solid dissolves in
a solvent (change in mass divided by a change in time).
• So what’s the difference between solubility and
dissolution?
5.2. Dissolution Rate
Aqueous Diffusion
Layer
Bulk Solution
Matrix
Solid Dosage Form
Concentration
Cs
C
X=0
X=h
5.2. Dissolution Rate
• Dissolution rate is described in quantitative terms by the
Noyes – Whitney equation:
dM/dt = DS(Cs-C) / h
•
Where
–
–
–
–
–
–
M is the mass of solute dissolved
dM/dt is the rate of dissolution (mass/ time)
D is the diffusion coefficient
S is the surface area of the exposed solid
Cs is the solubility of the solid
C is the concentration of the solute in the bulk solution at
time t
– h is the thickness of the diffusion layer (stagnant liquid film )
5.2. Dissolution Rate
• The previous equation is similar to Fick’s first law of
diffusion.
• The equation can be written in concentration forms as :
dC/dt = DS(Cs-C) / Vh
Where V is the volume of dissolution medium.
• When C << Cs (sink condition), the equation simplifies to
dC/dt = DSCs / Vh
5.2. Dissolution Rate
dC/dt = DSCs / Vh
Factors affecting dissolution rate?
• Solubility (Cs).
• Diffusion coefficient (D).
• Surface area (S) (i.e. particle size).
• Thickness of diffusion layer (h) (i.e. rate of agitation).
5.2. Dissolution Rate
• An aqueous diffusion layer or stagnant liquid film of
thickness h exists at the surface of a solid undergoing
dissolution.
• The aqueous diffusion layer represents a layer of solvent
in which the solute molecules exist in concentrations
ranging from Cs to C. Beyond the static diffusion layer, at
X greater than h, mixing occurs in the solution and the
drug is found at a uniform concentration, C, throughout
the bulk phase.
5.2. Dissolution Rate
• The change in concentration (concentration gradient,
dc/dx or (Cs-C)/h), in the diffusion layer is constant (i.e.
steady state conditions).
• The thickness of the diffusion layer can change with
mechanical agitation and stirring and this could affect the
dissolution rate.
5.2. Dissolution Rate
• The saturation solubility of a drug is a key factor in the NoyesWhitney equation.
• The driving force for dissolution is the concentration gradient
across the boundary layer. Therefore, the driving force
depends on the thickness of the boundary layer and the
concentration of the drug already dissolved.
• When the concentration of the dissolved drug, C, is less than
20% of the saturation concentration, Cs, the system is said to
operate under “sink conditions”. The driving force for
dissolution is greatest when the system is under sink
conditions.
5.2. Dissolution Rate
• The Noyes – Whitney equation assumes both h and S
are constant. This dissolution rate is known as intrinsic
dissolution rate.
• However, in many cases the static diffusion layer
thickness is altered by the force of agitation and the
surface area changes as the drug powder, granule or
tablet dissolves.
• The surface area, S, does not remain constant as
powder, granule, or tablet dissolves.
5.2. Dissolution Rate
• To determine the intrinsic dissolution rate experimentally,
both h and S are maintained constant.
• S is maintained constant by placing a compressed pellet
in a holder that exposes a surface of constant area.
• h is maintained constant by using a standard agitation
throughout the dissolution rate testing.
5.2. Dissolution Rate
5.2. Dissolution Rate
• The method used for the determination of the intrinsic
dissolution rate (constant h and S) adheres to the
requirements of the Noyes-Whitney equation. However:
– Although it provides valuable information on the
active drug itself, it does not give any on dosage
forms which are mixtures containing other material.
– Does not simulate the actual dissolution of material in
practice.
5.3. Dissolution of Solid Dosage Forms
• When a tablet or other solid dosage form is introduced into a
beaker of water or into the gastrointestinal tract, the drug
begins to pass into solution from the intact solid.
• The solid matrix disintegrates into granules and these
granules deaggregate in turn into fine particles.
• Dissolution could occur from the intact tablet, granules and
fine particles.
• Disintegration, deaggregation and dissolution may occur
simultaneously.
5.3. Dissolution of Solid Dosage Forms
Tablet
Disintegration
Granules
or
aggregates
Deaggregation
Fine
Particles
Dissolution
Drug in
Absorption
solution (in
vitro or vivo) In vivo
Drug in
blood, other
fluids and
tissues
5.3. Dissolution of Solid Dosage Forms
• Frequently, dissolution is the limiting or rate controlling
step in bioabsorption for drugs of low solubility, because
it is often the slowest of the various stages involved in
the release of the drug from is dosage form and passage
into systemic circulation.
• To simulate the drug dissolution from solid dosage forms,
paddle and basket dissoulution apparatus are used.
5.3. Dissolution of Solid Dosage Forms
Paddle
Sampling
Port
Disintegrating
and
Dissolving
Tablet
USP Dissolution Apparatus 2 (paddle)
5.3. Dissolution of Solid Dosage Forms
USP Dissolution Apparatus 1 (basket)
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