Mass Transport in Biological Systems

1
Mass Transport in Biological Systems

• OBJECTIVES
• To discuss how to model mass transport processes
  within the body.
• OUTLINE
• review Ficks first and second laws
• consider diffusion (reaction convection)
• apply these concepts to solute movement across
  cell membranes/capillary walls, protein
  adsorption

2
Mass Transport

• Can be achieved by both convection and diffusion.
  Mass transport is measured in terms of flux (
  amount of material crossing a unit area normal to
  the direction of transport per unit time).
• Most biological systems are dilute, so v is
  assumed to be the solvent velocity, and the
  transport of each component through the solvent
  can be studied as if the mixture was binary.

3
Ficks Laws of diffusion

• strictly valid only for dilute solutions

Ficks 1st law
slab
cylinder
Ficks 2nd law
sphere
4
Diffusion across a membrane Steady-state

• skin, buccal mucosa, cell membrane

Co
C
CL
L
distance, x
5
Example

• Scopolamine is available in a transdermal patch
  for the treatment of motion sickness. You are
  involved in the design of a new patch for this
  drug and need to first determine the diffusion
  coefficient of the drug in the outer layer of the
  skin. To do this, you set up a diffusion cell
  using excised human skin (thickness 100 ?m).
  The donor solution has a constant scopolamine
  concentration of 4.6 mg/ml. The concentration in
  the receptor solution is maintained at a
  concentration very much smaller than 4.6 mg/ml.
  The skin section you use is circular with a
  radius of 0.5 cm. The amount of the drug that
  has passed through the skin versus time is given
  in the following Table. Using this data and the
  fact that scopolamine has a partition coefficient
  (skin/water) of 17.4, calculate the diffusion
  coefficient of scopolamine in the skin.
• Total mass of scopolamine transported versus time

6
Diffusion Reaction The Krogh Capillary

• Consider oxygen delivery to tissue at
  steady-state. The oxygen is also being consumed
  in the tissue. This consumption is modeled using
  Michaelis-Menten kinetics.

r
z
2Ro
2Rc
7
O2 Delivery to Tissue
8
Example

• Given that the reaction rate of oxygen in
  skeletal muscle is 10-7 mol/(cm3 s), the
  capillary radius is 1.5 to 4 ?m, the diffusion
  coefficient of oxygen in muscle is 2(10-5) cm2/s,
  and the concentration of oxygen in plasma is
  4.05(10-8) mol/cm3, determine the maximum
  intercapillary radius. If the total oxygen
  content in the blood entering the capillary is 2
  (10-3)M and during exercise the average flow rate
  in the blood is 0.2 cm/s, determine the capillary
  distance at which the concentration of oxygen in
  the blood is zero.

9
Unsteady-State Diffusion

• Examples include injection of a drug into the
  muscle or sub-cutaneously followed by absorption
  into the blood stream, and the initial stages of
  protein adsorption to a biomaterial.
• Consider protein adsorption

C
at t 0, C Co
x 0