Niall McMahon

November 2012

A long time ago, I ended up with a small mathematical problem. In the paper that started off my research, my thesis supervisor Martin Crane described a neat trick for calculating the release rate of drug from a cylindrical tablet that consists of alternating layers of drug and non-medicinal material. At the end of the paper there are several nice expressions that can be used to get solutions for tablets consisting, in total, of 3 and 5 layers. Way back at the end of 2002, I thought: (1) can these expressions be generalised to deal with any number of layers and (2) what happens when the number of layers is very large? Does the drug release rate as predicted by Crane et al. tend to a limiting value?

*1-, 3- and 5-layer cylindrical tablets. Dark coloured layers consist of drug. Light coloured layers consist of inert materials.*

It turns out that the expressions *can* be easily generalised and the drug release rate *does *tend to a limiting value when the number of layers is very large. The generalisation is straightforward, with a pencil and paper and a little thinking, but the second part is more difficult. Some years ago, I wrote a small program that demonstrates that this series seems to converge to *0.5* as the number of layers, p increases. This means that the drug release rate predicted by the generalised solution probably converges to a fixed value. A closed mathematical solution was my ambition.

The first part of this work is outlined in a technical note, Generalising a Pohlhausen-Type Solution for Dissolution from Multi-Layer Drug Compacts.

The generalised solution involves summing an infinite series that looks a lot like a Lorenz curve.

Lorenz curves describe the distribution of wealth in a society. The mathematical similarity did not seem obvious at the outset but the generalised Pohlhausen solution is effectively a Lorenz curve. This observation provides a path to a solution. It's not quite worked through in detail but the similarities in mathematical description allow us to say that what's true for an equivalent Lorenz curve is true for the series at the heart of the generalised solution.

It's not so surprising in retrospect: the problem of drug dissolution rate from a multi-layer tablet in a steady flow is, in abstract, the same as wealth and population. Instead of wealth and population, we have drug dissolution rate and number of layers. Just as a nation's wealth is a fixed value, the overall dissolution rate from a dissolving compact is a single value. As each citizen contributes some of the nation's wealth, each layer contributes to some of the tablet's overall dissolution rate.

The second piece of work is outlined in a technical note, Lorenz Curves and Drug Dissolution.

__McMahon N M__, Crane M and Ruskin H J. Drug Dissolution Modelling. ERCIM News 82. July 2010.

__McMahon N M__, Crane M, Ruskin H J and Crane L J, The Importance of Boundary Conditions in the Simulation of Dissolution in the USP Apparatus. Simul Model Pract Th. 15 (3) (2007) 247-255.

Crane, M. and Crane, L. and Healy, A.M. and Corrigan, O.I. and Gallagher, K.M. and McCarthy, L.G., A Pohlhausen Solution for the Mass Flux From a Multi-layered Compact in the USP Drug Dissolution Apparatus, Simulation Modelling Practice and Theory (SIMPAT), Vol. 12/6 pp. 397-411, Elsevier, 2004.

I'd like to thank Professor Martin Clynes and the Irish National Institute for Cellular Biotechnology (NICB) for its support. Additional support from the Institute for Numerical Computation and Analysis (INCA) was invaluable. Prof. Michael Ryan of the School of Computing at DCU provided useful suggestions about how to proceed with this problem. Sandra Spillane of Dublin Institute of Technology provided useful help. Susan Lazarus and Aongus O' Cairbre at Dublin Institute of Technology advanced the solution for exponents of positive integers, as outlined in this document. Deirdre D'Arcy of the School of Pharmacy and Pharmaceutical Sciences at Trinity College Dublin provided useful experimental observations, as always.

This research was supported by the National Institute for Cellular Biotechnology.

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