Review of Mass Transfer Ficks First Law (one dimensional diffusion) J flux (moles/area/time) At steady-state or any instant in time dC J D dx Ficks Second Law When concentration changes with time dC d 2C D 2 dt dx 1 Example Ficks First Law
membrane Determine amount of drug to pass through membrane in one hour D h A C1 C1 C2 h SS valid if a high C1 is maintained and C2 remains << C1 by removal of drug or large volume
= 1 x 10-10 cm2/s = 2 x 10-3 cm = 10 cm2 = 0.5 mol/L C2 = 0 2 Example Ficks First Law 3 Partitioning C1 So far we have assumed the drug has equal affinity for solution and membrane. This is unlikely. Preference is indicated by partitioning membrane C1
prefers the polymer Cm2 h C2 Flux across membrane J D dCm C Cm1 D m 2 dx h Cannot measure Cm Partition Coefficient relates Cm to C Cm1 Cm 2 K m1
K m2 C1 C2 4 Partition Coefficient A measure of relative concentrations in membrane vs. solution at equilibrium If both solvents (1 and 2) are the same, then K m1 K m 2 K m Flux becomes dCm C2 C1 J D DK m dx h The term [DKm/h] is the Permeability 5 Partition Coefficient
Sketch the profiles for a high Km and a low Km C1 C1 C2 C2 h h high Km low Km Sketch the profile for C2=0 Common situation: body acts as a sink to remove the drug C1 h C2=0 6 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) Km1 = 0.014 D h
A C1 Between ointment and s.c. = 5.2 x 10-10 cm2/s Through the s.c. = 2x10-3 cm Typical thickness of s.c. = 10cm2 Covered by ointment = 0.01 mg/cm3 Saturation C of drug in ointment 7 Solution 8 Some K values Steroid
100 Progesterone 22.7 Testosterone 4.31 Reported by Sundaram and Kincl  in Kydonieus, Treatise in CDD 9 Ficks Second Law For one-dimensional unsteady-state diffusion dC d 2C D 2 dt dx How many ICs and BCs are needed? 10 Finite source
Sink (large volume) IC Diffusion X=L BC1 X=0 Polymer containing drug X=-L At t=0, C=C0 x=L, C=0 BC2
X=0, dC/dx = 0 (symmetric) 11 Finite source, reflective boundary Sink (large volume) IC Diffusion X=L BC1 X=0 Polymer containing drug Impermeable boundary at x=0 At t=0, C=C0
x=L, C=0 BC2 X=0, dC/dx = 0 (symmetric) 12 Solution method For cartesian (planar) systems Separation of variables or Laplace Transforms Error function solution We will investigate this later! 13
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