Controlled release drug delivery via polymeric microspheres: a neat application of the spherical diffusion equation

ABSTRACT

Various applied mathematics undergraduate skills are demonstrated via an adaptation of Crank's axisymmetric spherical diffusion model. By the introduction of a one-parameter Heaviside initial condition, the pharmaceutically problematic initial mass flux is attenuated. Quantities germane to the pharmaceutical industry are examined and the model is tested with data derived from industry journals. A binomial algorithm for the acceleration of alternating sequences is demonstrated. The model is accompanied by a MAPLE worksheet for further student exploration.

KEYWORDS:

• PDE
• diffusion
• spherical
• controlled release
• model
• MAPLE
• Heaviside
• pharmaceutical

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. Aka zero-order release kinetics.

2. PolyEthyleneGlycol and Poly(Lactic-co-Glycolic Acid).

3. Picture a mass of polymer chains like reptiles slithering over one another.

4. The more traditional way of modelling this is by a Stefan-type moving boundary problem, which is considerably more involved than this approach.

5. The statement of orthonormality, with weight function w(r) = 1, is $${\int }_0^{1}2sin{\lambda }_{n}rsin{\lambda }_{m}rdr=\delta (m,n)\text{(Kronecker}\text{delta)}.$$

6. This gives the remaining mass in the delivery device.

7. Also contains worksheets for 1D Cartesian (wire) and 1D Polar (disk) diffusion models.

8. This was later extended to spherical and cylindrical geometries.