Mathematical Models of Magnetic Nanoparticles in Hyperthermia and Targeted Delivery

Magnetic nanoparticles (MNPs), mainly iron oxide particles, have the advantage of being controllable by magnetic fields. MNPs show promise in biomedical applications, typically as carriers for biological or therapeutic entities or for their hyperthermic properties. Mathematical modelling assists in the design of MNP applications. However, the role of interparticle interactions is frequently ignored due to computational complexity, despite the general acceptance of the importance of interactions. Magnetic hyperthermia and magnetic drug delivery are two important clinical applications of MNPs where magnetic dipole interaction can be expected to have a significant role in the behaviour and thus be important in any potential medical applications. Good design of magnetic hyperthermia treatment approaches a thorough understanding of the complexities of the heating mechanisms. There are typically two mechanisms which lead to heating: Debye and Néel relaxation. Most models of hyperthermia consider only Debye relaxation and typically interparticle interaction is ignored. Targeted drug delivery aims to reduce the undesired side effects of drug usage by directing or capturing the active agents near a desired site within the body. This is particularly beneficial in, for instance, cancer chemotherapy, where the side effects of general drug administration can be severe. Although a number of mathematical models exist in literature, certain differences in the theoretical and experimental results have been noted. This thesis presents mathematical models of magnetic hyperthermia and magnetic delivery along with detailed analysis of three other mathematical models of magnetic interaction available in the literature. In this thesis, chapter 1 overviews some general information concerning the role of magnetic nanoparticles in biomedicine and the motivation for this work. Chapter 2 presents a mathematical model of hyperthermia which includes interparticle interactions, and offers empirical approximations to estimate the optimum heating for a chain of MNPs. Chapters 3–5 present replications and in some cases corrections of the models published by various authors. Chapter 6 presents a model investigating the aggregation of MNPs in parabolic flow. Here MNPs are considered whose initial positions are always above or below each other along the vertical axis of the vessel. A critical distance is then found between the MNPs within the vessel. If the MNPs begin their motion within this critical distance, then over time aggregation occurs. This critical distance is found to depend upon the initial position along the diameter of the vessel and also the fluid velocity. Analytic expressions for the upper and lower bounds are obtained and validated with the numerical results. Also, an empirical approximation of the critical distance is given, which gives close agreement with the numerical results.