3300 BMED Biotransport
BMED 3300 Biotransport
Lecture 2
August 24, 2011.
Professor Julie Babensee
Ficks First Law of Diffusion
v
J A = D ABC A
z
z=0
Diffusion
J A, z = D AB
dC A
dz
(For diffusion in the
z-direction only)
For dilute biological solutions, NA,z = JA,z
JA = molar flux in the z-direction relative to the molar average
velocity, [mol/cm2s]
dC A
= the concentration gradient in the z direction [mol/cm3/cm]
dz
DAB = mass diffusivity or diffusion coefficient for component A
diffusing through component [cm
diffusing through component B [cm2/s]
Diffusivity
Diffusivity
Units for diffusivity
J A, z
M
L2
1
=
=
D AB =
2
3 1/ L t
dC A / dz L t M / L
D = f (P, T, composition)
(P
For gases, D = 5x10-6 1 x 10-5 m2/s
For liquids, D = 10-10 10-9 m2/s
For solids, D = 10-14 10-10 m2/s
solids
10
Liquid
Liquid-Mass Diffusivity
Diffusivity
D AB B 7.4 108 ( B M B )1 / 2
=
0
T
V A.6
For nonelectrolytes in an dilute solutions
Where, DAB is the mass diffusivity of A diffusing through liquid solvent B in cm2/s
B is the viscosity of the solution in centipoises
T is the absolute temperature in Kelvin
is the absolute temperature in Kelvin
MB is the molecular weight of the solvent
VA is the molal (or molecular) volume of solute at normal boiling point in cm3/g mol
(See Table 24.4, 24.5, WWWR p. 439-440)
B is the association parameter for solvent B (see WWWR p. 440)
th
WWWR 440)
(Wilke and Chang, 1955)
Other empirical relationships for diffusivity as a function of molecular weight for
biological systems exist
Vast Differences in Extracellular and
Intracellular Fluid Composition
D. O. Cooney, Biomedical
Engineering Principles
An Introduction to Fluid, Heat
Fl
and Mass Transport
Processes, Marcel Dekker, Inc.,
1976.
Transport of Species Through the
Cell Membrane Occur by:
Ordinary passive diffusion
Facilitated diffusion (species aided by a carrier)
Active transport (carrier-aided transfer against a
concentration gradient, requiring biochemical energy)
Cell membranes are 80 100 thick
Postulated structure is a lipid bilayer with 8 pores
Permeability Permeability is used to describe solute transport (Ns, for example in gmoles/sec)
across a membrane such as a capillary membrane or a hemodialysis membrane.
Permeability is defined by the following expression using the
total membrane surface area S.
N S = Pm S (Chigh Clow )
(Chigh Clow) is the difference in the surface concentration
on either side of the membrane.
ith
th
Permeability is also equal to the effective diffusivity of the solute in the membrane, Dm,
times the oil/water partition coefficient of the species, K, (a measure of the relative
solubility of a solute within a membrane; K = C membrane / C interstital fluid), divided
di
by the thickness of the membrane, tm.
Pm =
KDm
tm
Permeability: Effects of Molecular/Pore Diameter
D. O. Cooney, Biomedical
Engineering Principles
An Introduction to Fluid, Heat
Fl
and Mass Transport
Processes, Marcel Dekker, Inc.,
1976.
Diffusion in Porous Membrane
A
A
Solvent B
Dm = f(DAB, pore morphology)
Dm
<< 1
D AB
Dm
1
=
D AB F
F = shape factor
= tortuosity; 1 < < 3
A
Permeability: Effects of Partition
Coefficient
W.M. Saltzman, Drug Delivery
Engineering Principles for Drug
Therapy, Oxford University
Press, 2001.
Permeability: Effects of Solute
Molecular Weight
W.M. Saltzman, Drug Delivery
Engineering Principles for Drug
Therapy, Oxford University
Press, 2001.
Membrane Permeability
CA1
J A, x = D AB
C1
C C2
J ss = Dm 1
l
CA2
C2
x=0
x=l
dC A
= J ss
dx
Partition coefficient, K. between CA,ambient and CA,membrane
Depends on relative solubility of
species in ambient phase versus
polymer phase
If ideal solution, K f(C)
C
C
K= 1 = 2
C A1 C A2
K=
C1
C A1
J ss = Dm K
C A1 C A2
DK
J ss = m C
l
l
Two different in ways which polymer membrane permeability is represented
KDm = Pm
Is one way of defining permeability
[Dm] = cm2/s
[K] = dimensionless
[Pm] = cm2/s
I prefer this definition of permeability which is that
Pm =
Here permeability has units of cm/s
KDm
tm
Example: Permeability of Polymer Membrane to Insulin
A novel membrane for the encapsulation of islets as an insulin delivery system is
being developed in the lab that you are doing research project in. Your task is
being developed in the lab that you are doing a research project in. Your task is
to determine the permeability of this membrane to insulin. You decide to use a
diffusion cell as depicted in the figure below to determine the permeability of the
membrane to insulin. A 100 m thick sample of your polymer membrane is placed
between reservoir 1 and reservoir 2. At the start of your experiment (t=0) you place
th
(t
an insulin solution in saline into reservoir 1 with a saline solution with no insulin in
reservoir 2. You allow enough time for steady state to be set up and you make the
following measurements. You determine that the insulin concentration in reservoir 1
is 1 g/mL, the insulin concentration in reservoir 2 is 1 mg/mL, and the steady state
flux of insulin across the membrane is 3.6 x 10-5 g/cm2.s. What is the permeability
of the membrane to insulin?
Reservoir 2
Reservoir
Reservoir 1
CA1
CA2
X=
l
X=
0
Polymer
membrane
Solution
Solution
J SS = KDm
C
l
3
5 = KD (1 0.001 g/cm )
3.6 10
m
0.01 cm
KDm = 3.6 107 cm2 /s
KDm 3.6 107 cm2 /s
Pm =
=
= 3.6 105 cm/s
tm
0.01 cm
Example: Drug Delivery Device Safety
Example: Drug Delivery Device Safety
A simple drug delivery device uses a synthetic
simple drug delivery device uses synthetic
membrane to control the release of a drug from a
reservoir into the target tissue, typically inside the body.
Consider device with
Consider a device with a 3 cm2 of membrane, separating
cm
membrane separating
a drug in solution at a concentration of 1 mg/mL from the
body (total volume 50L), where the drug is essentially at
zero concentration The membrane diffusion coefficient
zero concentration. The membrane diffusion coefficient
is 2x10-6 cm2/s, the partition coefficient is 0.85 and its
thickness is 10 m. The rate of drug delivery should
never exceed
never exceed .0075 mg/s due to safety issues
mg/s due to safety issues
concerning drug toxicity. Is this drug delivery device
safe?
Solution
Solution
The rate of drug delivery is just the concentration
difference between both sides multiplied by the diffusion
coefficient
given information:
membrane thickness (10 m), l = 10 x 10-6 m = 0.001 cm
surface area = 3 cm2
diffusion coefficient, Dm = 2 x 10-6 cm2 / s
J A, x = D AB
dC A
= J ss
dx
J ss = KDm
C
l
= (0.85)(2 x106 )
(1 0)
0.001
= 0.0017 mg/cm2s
rate of drug delivery = Flux x surface area = 0.0017 x 3 = 0.0051 mg/s
0.0051 < 0.0075 mg/s therefore safe
Convective Mass Transfer
Convective mass transfer occurs between a moving fluid and
fl
a surface or between two relatively immiscible moving fluids
Fluid Flow
Fluid Flow
y
y
CA=CA
=(y)
x
CA
CAs-CA=[CAs-CA](y)
CAs > CA
Concentration and velocity profiles for a fluid
and velocity profiles for fluid
flowing past a solid surface
Convective Mass Transfer
Convective mass transfer described by:
N A = kc C A
Where, NA [e.g., moles/cm2s] is the molar mass transfer of species A
measured relative to fixed spatial coordinates
measured relative to fixed spatial coordinates
CA [e.g., moles/cm3] is the concentration difference between the
boundary surface concentration (CA,s) and the average
concentration of the fluid stream of the diffusing species (CA)
(C
kC [cm/s] is the convective mass transfer coefficient
Convective mass transfer occurs in the direction of decreasing
concentration