4.3. Modeling of the Release Kinetics of the Devices
Detailed mathematical models have been developed for most of the drug profiles obtained from the devices described in section 4.2.
4.3.1 General Models
We are all familiar with the phenomenon that is observed when a drop of dye is added to a beaker and in time the color in the beaker becomes homogeneous, when the dye molecules have distributed themselves throughout the entire volume. This phenomenon is called diffusion, and it is due to the random movement of molecules in the solution. The tendency is for molecules to move from an area of high concentration to one of low concentration, that is they move against the concentration gradient. This is because initially there are far more molecules in the area of high concentration, and therefore the probability that they will move away is far greater than the probability that a molecule will move into that area of high concentration.
In the differential form Fick's Law becomes:
The term DiK/d is referred to as the permeability coefficient, P. It is a relatively easy parameter to measure, since from (4.2) it can be seen to be equal to the flux divided by the concentration difference across the membrane. Di, K and P all relate to the solute transport such as molecular size, polarity, and solubility in the polymer phase and to the structure of the polymer.
The radius is related to the molecular weight of the molecule, and the diffusion coefficient is inversely proportional to the cube root of the molecular weight.
For Insulin the diffusion coefficient is 8.2 x 10-7
4.3.2. Diffusion through polymers
These effects can be visualized in diagrammatic form in figure 4.3.1.
Figure 4.3.1. Blocking and detour effect of crystalline
regions in a polymer.
4.3.3. In vitro measurements
The mechanisms and rate of diffusion can be studied in the laboratory (in vitro) using special equipment known a diffusion cell. When a small molecule or peptide is involved the free volume theories of molecular motion can be invoked. For larger peptides (above 1000 molecular weight), and proteins the molecules diffuse through the polymer structure that is unique to the polymer in question. Proteins are said to diffuse by reptation, which as the name implies, involves a reptile-like movement through the available intermolecular space.
Measurements of the rate of diffusion are made in diffusion cells. Figure 4.3.2 shows the basic setup. The polymer of interest is cast into a membrane which is clamped between two chambers, which are filled with buffer. The solute of interest is introduced into the donor compartment, and samples of the buffer from the acceptor side are taken at subsequent time points. The compartments are kept well stirred, and sample size is kept small so that the reduction in acceptor chamber buffer volume is insignificant.
Figure 4.3.2. A typical diffusion cell