Review of Mass Transfer n Ficks First Law

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

Example – Fick’s First Law membrane n Determine amount of drug to pass through membrane in one hour n D h A C 1 C 2 h n n SS valid if a high C 1 is maintained and C 2 remains << C 1 by removal of drug or large volume n n = 1 x 10 -10 cm 2/s = 2 x 10 -3 cm = 10 cm 2 = 0. 5 mol/L =0 2

Example Fick’s First Law 3

Partitioning C 1 n n So far we have assumed the drug has equal affinity for solution and membrane. This is unlikely. Preference is indicated by partitioning membrane C 1<Cm 1 the drug Cm 1 “prefers” the polymer Cm 2 h C 2 n Flux across membrane n Cannot measure Cm Partition Coefficient relates Cm to C n 4

Partition Coefficient n n 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 [DKm/h] is the Permeability 5

Partition Coefficient n Sketch the profiles for a high Km and a low Km C 1 C 2 n C 2 h h high Km low Km Sketch the profile for C 2=0 q Common situation: body acts as a sink to remove the drug C 1 h C 2=0 6

Example – Transdermal Delivery n n n 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) n Km 1 = 0. 014 q n D q n h q n A q n C 1 q Between ointment and s. c. = 5. 2 x 10 -10 cm 2/s Through the s. c. = 2 x 10 -3 cm Typical thickness of s. c. = 10 cm 2 Covered by ointment = 0. 01 mg/cm 3 Saturation C of drug in ointment 7

Solution 8

Some K values Steroid K Cortisol 5. 5 e-3 Estradiol 2 e-1 Melengstiol acetate 18. 87 Norethindrone 1. 22 Norgestrel 3. 19 19 -Norprogesterone 33. 3 Megesterol acetate 35. 7 Mestranol 100 Progesterone 22. 7 Testosterone 4. 31 Reported by Sundaram and Kincl [93] in Kydonieus, Treatise in CDD 9

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

Finite source • IC Sink (large volume) X=L • At t=0, C=C 0 Diffusion • BC 1 X=0 Polymer containing drug X=-L • x=L, C=0 • BC 2 • X=0, d. C/dx = 0 (symmetric) 11

Finite source, reflective boundary • IC Sink (large volume) X=L • At t=0, C=C 0 Diffusion • BC 1 • x=L, C=0 X=0 Polymer containing drug Impermeable boundary at x=0 • BC 2 • X=0, d. C/dx = 0 (symmetric) 12

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