1. 4. Diffusion

2. 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.

3. Diffusion could occur:
4.2. Mechanisms
Diffusion could occur:
Throughout a single bulk phase (solution’s homogeneity).
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. Diffusion through a barrier may occur through either:
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.

5. 4.2. Mechanisms A B
Diffusion through a homogenous nonporous membrane (A) and a porous membrane with solvent (usually water) filled pores (B)

6. 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.

7. 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.

8. 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.

9. 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).

10. Dialysis
Osmosis
Low salt
High salt

11. 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)

12. 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 change of concentration with a change in location.
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).

13. 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).

14. 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.

15. Diffusion and Molecular Properties

16. 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.

17. 4.5. Fick’s Second Law of Diffusion
Output
Input
Volume element
Bulk medium

18. 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

19. 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)

20. 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.

21. 4.6. Steady state Diffusion
Receptor
Compartment
Donor
Flux in
Flux out
Membrane
Flow of solvent to maintain sink condition
Diffusant
Solution
Pure
solvent

22. 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.

23. 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.

24. 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.

25. 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)

26. 4.6. Steady state Diffusionh
Thickness of barrier
Donor Compartment
Receptor Compartment Cr Cd C2 C1
High concentration of diffusant molecules

27. 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

28. 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

29. 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

30. 4.6. Steady state Diffusion
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
Amount Diffused

31. 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).

32. Drug Release

33. 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.

34. Monolithic System(Matrix) Vs Reservoir system

35. Reservoir systems could be represented again by diffusion cells….h
Thickness of membrane
Donor Compartment
Receptor Compartment Cr Cd C2 C1
High concentration of diffusant molecules Cd h Cr

36. 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).

37. Reservoir System
First order release
Steady state
Amount Diffused
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

38. Matrix type dosage forms
Figure – Drug eluted from a homogeneous polymer matrix

39. 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.

40. Matrix type dosage forms
Schematic presentation of the solid matrix and its receding boundary as the drug diffuses from the dosage form

41. 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.

42. Matrix type dosage forms
Higuchi (1960,1961) developed an equation to describe drug release from such matrix systems based on Fick’s 1st Law:
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:

43. Matrix type dosage forms
Example 13-6; Martin’s 6th ed.:
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.
(b) What is the instantaneous rate of drug release occurring at 120 min?

44. Summery od different drug release rates and mechanisms
Rx Order
Reservoir System with constant Cd
Reservoir
System with
decreasing Cd
Matrix System
Release Rate Eq
Zero
order
First
order
Higuchi

45. 5. Dissolution

46. 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?

47. Aqueous Diffusion Layer
5.2. Dissolution Rate
X = 0
X = h C Cs
Concentration
Solid Dosage Form
Matrix
Aqueous Diffusion Layer
Bulk Solution

48. 5.2. Dissolution Rate dM/dt = DS(Cs-C) / h
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 )

49. dC/dt = DS(Cs-C) / Vh dC/dt = DSCs / Vh
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

50. 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).

51. 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.

52. 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.

53. 5.2. Dissolution Rate
The saturation solubility of a drug is a key factor in the Noyes-Whitney 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.

54. 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.

55. 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.

56. 5.2. Dissolution Rate

57. 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.

58. 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.

59. 5.3. Dissolution of Solid Dosage Forms
Disintegration
Deaggregation
Dissolution
Drug in solution (in vitro or vivo)
Absorption
In vivo
Drug in blood, other fluids and tissues
Granules or aggregates
Tablet
Fine
Particles

60. 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.

61. 5.3. Dissolution of Solid Dosage Forms
Paddle
Disintegrating and Dissolving Tablet
Sampling Port
USP Dissolution Apparatus 2 (paddle)

63. 5.3. Dissolution of Solid Dosage Forms
USP Dissolution Apparatus 1 (basket)