4. Diffusion

04-May-21 4. Diffusion 1 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. 2 1 04-May-21 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). 3 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 04-May-21 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. • 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. 5 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. 6 3 04-May-21 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). 7 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. v= dM / dt J= dM dt.S • The units of the Flux J are g.cm-2.sec-1. (S) 8 4 04-May-21 4.4. Fick’s First Law of Diffusion Ja dC dx • 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). Hence, the unit of dc/dx is g/cm4 • dc/dx is always negative. c1 x1 > < c2 x2 9 4.4. Fick’s First Law of Diffusion Ja dC dx J= dM dC = -D dt.S 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). 10 5 04-May-21 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. 11 Diffusion and Molecular Properties 12 6 04-May-21 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). J = dM / S.dt = -D (dc/dx) • Fick’s second law examines the rate of change of diffusant concentration with time at a point in the system (dc/dt) . • 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. Input c Output Volume element Bulk medium 13 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 • • 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) 14 7 04-May-21 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. 15 4.6. Steady state Diffusion Diffusion Cell Donor Compartment Receptor Compartment Pure Diffusant Flux in Membrane Solution solvent Flux out Flow of solvent to maintain sink condition 16 8 04-May-21 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. 17 4.6. Steady state Diffusion • Therefore, the concentration in the receptor compartment (Cr) is always maintained at very low levels because of the sink condition. • In contrast, the concentration in the donor compartment (Cd) 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. • Overall: Cd >> Cr. 18 9 04-May-21 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. 19 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) 20 10 04-May-21 4.6. Steady state Diffusion Receptor Compartment Donor Compartment High constant Cd concentration Cd C1 C2 Cr Low constant concentration Cr h 0 Thickness of barrier dc/dx= c2-c1/h 21 4.6. Steady state Diffusion • In such systems (diffusion cells), Fick’s first law may be written as: J= dM ( C − C2 ) =D 1 S .dt 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 22 11 04-May-21 4.6. Steady state Diffusion Considering : K = C1 C2 = Cd C r dM ( C − C2 ) ( KCd − KCr ) =D 1 =D S .dt h h dM (C − C r ) = DSK d dt h J= 23 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 Cr  Cd , , then : dM (C − C r ) (C ) = DSK d = DSK d dt h 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 dM = PSCd dt 24 12 04-May-21 4.6. Steady state Diffusion dM / dt = PSCd If Cd remains relatively constant throughout time, then diffusion follows zero order kinetics. k0 =PS Cd dM / dt = k0 M = k0t M = PS Cd t P can be obtained from the slope of a linear plot of M versus t. Amount Diffused • dM = PSCd dt Time 25 4.6. Steady state Diffusion • If Cd changes appreciably with time, then P can be obtained from the slope of log Cd versus t. dM dM / dt = PSCd = PSCd dt dCd/ dt = (PS/Vd) Cd 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). 26 13 04-May-21 4.6. Steady state Diffusion Amount Diffused First order release Steady state log Cd = log Cdo − PSt 2.0303Vd Time 27 Measuring the intestinal permeability od drug using cell line in vitro Measuring the dermal permeability of drug using cell ski rings 28 14 04-May-21 Drug Release 29 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. 30 15 04-May-21 Monolithic System(Matrix) Vs Reservoir system Reservoir System Monolithic System(Matrix) Drug release from both systems (next slides) is assumed to be under steady state conditions: Cd, Cr and dC/dx are constant upon diffusion 31 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 Cr h Thickness of membrane 32 16 04-May-21 Reservoir System • Drug release from these systems follows generally zero order kinetics which can be presented by the following equation: dM = PSCd dt Steady state (Zero order) Slope= PSCd • This behavior is presented in the straight line presented in the following figure First order • 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 33 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 34 17 04-May-21 Reservoir System…… summary Rx Order Reservoir System with constant Cd Release Rate Eq M = K 0t Reservoir System with decreasing Cd log Cd = log Cdo − Zero order PSt 2.0303Vd First order 35 Matrix type dosage forms Figure – Drug eluted from a homogeneous polymer matrix 36 18 04-May-21 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. 37 Matrix type dosage forms Schematic presentation of the solid matrix and its receding boundary as the drug diffuses from the dosage form 38 19 04-May-21 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. 39 Matrix type dosage forms • Higuchi (1960,1961) developed an equation to describe drug release from such matrix systems based on Fick’s 1st Law: dM dQ DCs = = S .dt dt h Q Q= M/S Unit (g/cm2) 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: t 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: 40 20 04-May-21 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 * 360x10−6 * 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 360x10−6 * 0.02 * 1x10−3 = dt 2 * 120 dQ = 5.5 x10−6 g / cm 2 . min dt 41 Summery of 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 42 21