ANNOUNCEMENTS. HW5 is posted. It is due by 12:00 on 3/11.

Transcription

1 ANNOUNCEMENTS HW5 is posted. It is due by 12:00 on 3/11.

2 Two Compartment Model for Drug Distribution from Blood to Peripheral Tissues Drug Injection Plasma/Blood Peripheral Tissues k 1 C 2 k e Mass Balance on Compartment 1: d = k 1 + C 2 k e Equation 5 Mass Balance on Compartment 2: dc 2 = k 1 C 2 Equation 6 t = 0 = C 0 = D/ where D = Mass of drug injected t = 0 C 2 = 0

3 Solve Equation 5 for C 2 : d = k 1 + C 2 k e C 2 = d + k 1 + k e (EQ. 7) Substitute back into Equation 6: dc 2 = d 2 + k 1 d 2 + k e d dc 2 = k 1 C 2 V d k 1 d 2 + k V e 1 d V = k 1 k 1 d 2 + k 1 + k V e 1 d 2 + d 2 t = 0 = C 0 = D/ where D = Mass of drug injected What is the second condition? [ k k e ] + k e = 0

4 Drug Injection Plasma/Blood Peripheral Tissues k 1 C 2 k e d 2 + d 2 What is the second condition? [ k k e ] + k e = 0 a. b. c. d d d = 0 = ( k k e )C 0 = ( k 1 + k e )C 0

5 d 2 + d 2 [ k k e ] + k e = 0 t = 0 = C 0 = D/ where D = Mass of drug injected d = ( k 1 + k e )C 0 The solution to this 2 nd order ODE and associated initial conditions is: [ ( ) + ( 1 α)exp( λ 2 t) ] = C 0 α exp λ 1 t ( ) + ( k k e ) 2 4 k e 2 λ 1 = k k e C 2 = C 0 ( ) ( k k e ) 2 4 k e 2 λ 2 = k k e Substituting (t) into Equation 7: ( )( λ 2 λ 1 ) α 1 α ( 1 α)λ 1 + αλ 2 ξ 21 [ [ ] exp ( λ 1t) exp( λ 2 t) ] α = λ 2 + k 1 + k e λ 1 λ 2 ξ 21 =

6 Effect of increasing rate of distribution from the blood to the peripheral organs Peripheral Tissues Plasma Compartment + k 1 + k 1 Peripheral Tissue

7 Concentration in the body The Goal: Constant over time!! Time (day)

8 Lecture 23 Outline: Membrane-based Drug Delivery Systems Reservoir based Delivery Systems -Design -Mathematical analysis -Parameters impacting release -Clinical scope

9 Transdermal Delivery via Reservoirs or Chambers ( The Patch ) Protective Impermeable Layer of Polymer Polyamide (nylon) (Band-Aids) Polyethylene Polypropylene Silicone elastomer Drug Reservoir (liquid or gel) Low MW, lipophillic mg (5-10 wt%) Rate-controlling Polymer Membrane Microporous polypropylene Microporous polyethylene Adhesive polyacrylates Drug Skin µm µm µm High drug loadings can be achieved. Inexpensive, non-invasive. Capillaries

10 Scopolamine Patch (1979) First transdermal patch approved in 1979 for motion sickness Works for ~ 3 days $2- $5 per patch Nitroglycerin patch Placed on the skin Prevents angina attacks $2- $5 per patch NicoDerm CQ patch NicoDerm Placed on the skin once every other day Useful in smoking cessation $2- $5 per patch Ortho Evra (2002) Placed on the abdomen, buttocks, upper arm, or upper torso Releases contraceptive drug for a week; need to change each week. Costs ~$34 per month

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12 Microporous Membranes 10-60% of the volume of the membrane is porous A film of oriented extruded polymer fibers is prepared. The film is stretched and some pieces of the fibers separate and stretch to form pores. NAweLTjJEeH57SvqMCVTBDgZKl4=&h=290&w=475&sz=71&hl=en&start=40&um=1&itbs=1&tbnid=qIvivLlmOxXJRM:&tbnh=79&tbnw=129&prev=/images%3Fq%3Dmicroporous %2Bpolypropylene%26start%3D21%26um%3D1%26hl%3Den%26client%3Dfirefox-a%26sa%3DN%26rls%3Dorg.mozilla:en-US:official%26ndsp%3D21%26tbs%3Disch:1

13 Mass Transfer Equations Governing Transdermal Drug Delivery Impermeable coating Drug Reservoir Rate-controlling membrane C p (x,t) C p,1 x = 0 C p,2 x = L Governing Equation Boundary Conditions c p t = D i:p 2 c p x 2 c p = c p,0 t = 0 0 < x < L c p = c p,1 t > 0 x = 0 c p = c p,2 t > 0 x = L Solution: c p (x,t) = c p,1 +(c p,2 c p,1 ) x L + 2 c p,2 cos(nπ ) c p,1 sin nπx exp D i:pn 2 π 2 t π n=1 n L L 2 + 4c p,0 1 ( 2m +1)πx sin exp D (2m π 2 t i:p +1)2 π m= 02m +1 L L 2 At Steady-State: c p (x) = c p,1 + ( c p,2 c p,1 ) x L

14 Concentration Profile for Drug inside of the Rate-Controlling Membrane of a Transdermal Delivery Device Impermeable coating Drug Reservoir Rate-controlling membrane C p (x,t) C p,1 x = 0 C p,2 x = L c p (x) = c p,1 + ( c p,2 c p,1 ) x L Below 3 drug concentration profiles are given for three different membranes. Assume that the diffusivity of the drug in each membrane and the dimensions of each membrane are equivalent (Surface area, A and Membrane thickness, L). Which membrane would result in the release of the most drug per time? C A B

15 C p (x) c p = c p,1 + ( c p,2 c ) x p,1 L C p,1 x = 0 C p,2 x = L A Device J A = D i:p dc A dx = moles cm 2 s J A Skin The amount of solute that passes through a surface with area (A) per unit time is W A. In drug delivery applications, this quantity is also called the release rate: dc W A = J A A = AD A i:p dx = moles s W A = AD c i:p ( c p,1 p,2) L = Quantity of drug released from chamber per unit time

16 Concentration Profile for Drug inside of the Rate-Controlling Membrane of a Transdermal Delivery Device Impermeable coating Drug Reservoir Rate-controlling membrane C p (x,t) C p,1 x = 0 C p,2 x = L c p (x) = c p,1 + ( c p,2 c p,1 ) x L Below 3 drug concentration profiles are given for three different membranes. Assume that the diffusivity of the drug in each membrane and the dimensions of each membrane are equivalent (Surface area, A and Membrane thickness, L). Which membrane would result in the release of the most drug per time? W A = AD ( i:p c p,1 c ) p,2 L Answer: B C B A

17 Effect of Design Properties on Cummulative Mass of Drug Released into Tissue from Transdermal Devices Increase: A, D AB Decrease: L 1 ( c p,1 c p,2 ) M t = AD AB t L Dose Released (mg) Decrease: A, D AB Increase: L Time (hours) Mass of Drug Released from the Reservoir Device Increases Linearly with Time The release rate (slope of the line) is constant with time

18 The Rate at which Drug Migrates through the Membrane is Described by an Effective Diffusion Coefficient D i:p = D Effective = D 0 τ D 0 τ ε The diffusion coefficient of drug in water. The tortuosity factor. It accounts for the fact that molecules must move through a more windy and sometimes constricted path as they diffuse through the aqueous environment. 1 τ =1 2 ( 3 1 ε )( 1 ε) 2 / 3 Fraction of the volume occupied by pores: V pores /V Total D Effective The diffusion coefficient of drug in a water-filled porous environment. D Effective < D 0