From a MSc to a PhD
Hello there, my name is Niall McInerney and I have just begun the second year of my PhD studies in the University of Limerick.
Before I discuss what I am currently working on I will briefly explain the unanticipated trajectory which has led me into the life of postgraduate research. In September of 2013, I enrolled into the MSc in Mathematical Modelling offered here in UL by the Department of Mathematics and Statistics in association with MACSI.
Coming from a structural engineering/architecture background, I was worried that with only a handful of modules in linear algebra and multivariable calculus under my belt that I would be out of my depth. I quickly realised that my classmates were all in similar situations and most were treating the year as a conversion course be it from engineering/teaching/finance, you name it.
The Masters follows the ethos of MACSI and focuses on mathematical models of real world processes and teaching the techniques of modelling from the basics up. There are a wide variety of modules taught over the two semesters ranging from perturbation methods and numerical PDE’s to time series analysis and stochastic differential equations. This is followed by a summer dissertation of which is often in collaboration with a current MACSI project, although not exclusively. This is where I got my first experience of scientific research studying the asymptotics of sessile drops, where I was directly applying the skills and techniques acquired from a number of the MSc’s taught modules.
I would encourage anyone seeking to turn their career path in the direction of applied mathematics to consider taking this exciting and engaging course, as I found it to be an extremely enjoyable an interesting year.
Upon completion of the MSc, I was awarded departmental funding to pursue a PhD on a project entitled “Mathematical Modelling of Polymer Base Drug Delivery Devices’. Although completely dissimilar to my summer dissertation, the underlying mathematics still relies heavily on the perturbation analysis and numerical modelling/scientific computation skills developed over the previous year.
Mathematical modelling is an important tool in facilitating product development in the pharmaceutical industry, assisting is determining the geometries and configurations of these devices, as well as the desired drug loading concentrations. Polymer based controlled drug delivery devices are becoming increasingly popular in the pharmaceutical industry, due to the ability of chemists to synthesise polymers with desired properties to suit a required release profile. This can alleviate the disadvantages of traditional drug delivery, such as inaccurate dosages and patient compliance, minimising toxicity and maximising effectiveness.
Whilst there has been extensive studies of drug release mechanisms both from a theoretical and experimental point of view, there is little research into fully understanding the moving interface between the dissolved and loaded drug, especially in the case of swelling polymers where the polymer changes from a fully swollen to a fully collapsed state upon contact with an environmental fluid. In swelling controlled devices, a polymer matrix is initially in a dry glassy state with the drug molecules dispersed and unable to diffuse. The polymer swells upon contact with an environmental fluid (solvent), which then diffuses into the polymer (often hydrophilic hydrogels), creating a moving boundary separating the dry polymer from the now swollen rubbery polymer, within which the drug can now diffuse. There is also a second moving boundary at the edge of the polymer as a result of volume change.
There is a vast scope for developing simple approximate solutions which capture the key features of the model but are tractable and easy to implement. The first year of my PhD has involved developing simple 1D models describing polymer-penetrant-drug behaviour, initially linear diffusion with one moving boundary building up to nonlinear diffusion and two moving boundaries. In the next few months we hope to extend the analysis to the more realistic spherical geometries.