Mathematical modelling of magnesium corrosion for orthopaedic implants

Magnesium (Mg) has grasped the attention of biomaterial researchers due to its desirable properties for orthopaedic implants. It is a biodegradable, lightweight structured metal with mechanical properties more comparable to the human bone than frequently used implant materials like titanium and stainless steel. However, the element corrodes rapidly in aqueous environments, which prevents its direct use as an implant material. In this thesis, novel mathematical models are presented to address the problem of Mg corrosion. In aqueous environments, a Mg implant reacts to form magnesium hydroxide (\ce{Mg(OH)2}), which can react further with bicarbonate ions to form magnesium carbonate (\ce{MgCO3}); these reactions are considered in the corrosion models developed in this work, and this is the first study to consider \ce{MgCO3}. A simple mass action model was derived first, which predicted the amount of Mg and its corrosion products over time, where an exponential decay of Mg was perceived. The backbone of this thesis is a PDE model for Mg corrosion, which considers distinct porous layers of \ce{Mg(OH)2} and \ce{MgCO3} surrounding a block of Mg with the advection and diffusion of \ce{H2O} and \ce{CO2} through porous media; this porous media assumption is a novel feature in comparison to other metal corrosion models. The model was derived and analysed in one spatial dimension for Cartesian, radically symmetric spherical and cylindrical geometries. Singularities resulting from the model at small time were handled using asymptotic analysis. The effect of the model parameters on key timescales was investigated, whereby porosity of the layers and reaction rates of \ce{H2O} and Mg were shown to have a significant effect. Furthermore, the porous media assumption on the Mg compound layers led to the prediction of a slightly faster corrosion of the original Mg block compared to that with different rates of advection. In addition to the above, corrosion from inside a single Mg pore was considered using the same modelling approach. The timescale for pore closure and the size of Mg corrosion at pore closure were of particular interest, and were affected by changes in the parameters. The pore closure time was found to be rapid in comparison to the degradation time of the implant. The final model in this work is a physiologically based pharmacokinetic (PBPK) model, which is used to explore the effects of a corroding Mg implant on blood serum levels; a high amount of Mg in the blood can cause complications. Values for the implant release rate of Mg and urine excretion rates were refined in the model, where it was highlighted that an Mg implantation must be carefully considered for patients, particularly those with reduced renal function.