1. 1 Review of Mass Transfer Fick’s First Law (one dimensional diffusion)  Jflux (moles/area/time)  At steady-state or any instant in time Fick’s Second Law  When concentration changes with time

2. 2 Example – Fick’s First Law SS valid if a high C 1 is maintained and C 2 remains << C 1 by removal of drug or large volume membrane C1C1 C2C2 h Determine amount of drug to pass through membrane in one hour D = 1 x 10 -10 cm 2 /s h = 2 x 10 -3 cm A = 10 cm 2 C 1 = 0.5 mol/L C 2 = 0

3. 3 Example Fick’s First Law

4. 4 Partitioning So far we have assumed the drug has equal affinity for solution and membrane. This is unlikely. Preference is indicated by partitioning Flux across membrane Cannot measure C m Partition Coefficient relates C m to C membrane C1C1 C2C2 h C m1 C m2 C 1 <C m1 the drug “prefers” the polymer

5. 5 Partition Coefficient A measure of relative concentrations in membrane vs. solution at equilibrium If both solvents (1 and 2) are the same, then Flux becomes The term [DK m /h] is the Permeability

6. 6 Partition Coefficient Sketch the profiles for a high K m and a low K m Sketch the profile for C2=0  Common situation: body acts as a sink to remove the drug C1C1 C2C2 h C1C1 C2C2 h high Km low Km C1C1 C 2 =0 h

7. 7 Example – Transdermal Delivery Digitoxin (used for heart failure; ointment) How much digitoxin can be delivered transdermally in one day Membrane control – skin acts as barrier membrane (stratum corneum, outermost layer) K m1 = 0.014  Between ointment and s.c. D = 5.2 x 10 -10 cm 2 /s  Through the s.c. h = 2x10 -3 cm  Typical thickness of s.c. A = 10cm 2  Covered by ointment C 1 = 0.01 mg/cm 3  Saturation C of drug in ointment

8. 8 Solution

9. 9 Some K values SteroidK Cortisol5.5e-3 Estradiol2e-1 Melengstiol acetate18.87 Norethindrone1.22 Norgestrel3.19 19-Norprogesterone33.3 Megesterol acetate35.7 Mestranol100 Progesterone22.7 Testosterone4.31 Reported by Sundaram and Kincl [93] in Kydonieus, Treatise in CDD

10. 10 Fick’s Second Law For one-dimensional unsteady-state diffusion How many IC’s and BC’s are needed?

11. 11 Finite source X=0 IC At t=0, C=C 0 BC1 x=L, C=0 BC2 X=0, dC/dx = 0 (symmetric) X=L Polymer containing drug X=-L Sink (large volume) Diffusion

12. 12 Finite source, reflective boundary IC At t=0, C=C 0 BC1 x=L, C=0 BC2 X=0, dC/dx = 0 (symmetric) X=L Polymer containing drug Impermeable boundary at x=0 X=0 Sink (large volume) Diffusion

13. 13 Solution method For cartesian (planar) systems  Separation of variables or Laplace Transforms  Error function solution We will investigate this later!