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.

4.3.1.1. Fick's first law
The flux (amount of solute i crossing a plane of unit surface area, normal to the direction of transport, in unit time) J, can be described by Fick's law which in the differential form is:

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.

4.3.1.2. Fick's second law
For unsteady state situations, where concentration is changing with time, we use Fick's second law, which assumes constant Di and constant boundaries (i.e. no polymer swelling):

4.3.1.3. Stokes Einstein equation
The diffusion coefficient D is a very important parameter in drug delivery. It is related to the size of the molecule (radius r) and the temperature (T absolute) and viscosity (h) of the liquid through which the species is diffusing. The relationship is known as the Stokes-Einstein equation:

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 cm2sec-1.

4.3.2. Diffusion through polymers

4.3.2.1. Effective diffusion coefficient
When we are considering diffusion through polymers, it is usual to use an effective diffusion coefficient Deff which is related to the diffusion coefficient in water-filled pores Dsw, and is given by equation (4.6):

4.3.2.2.Semi crystalline polymers
For diffusion in semi-crystalline polymers, a modified diffusion coefficient must be used. The crystalline areas are a barrier to diffusion, and can even block the passage of a large solutes.

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