What are some applications of Mathematics to the medical field?
Tomography, which is now used heavily nowadays as a diagnostic tool, relies on rather deep mathematics to work properly. There is in particular the Radon transform and its inverse, which are useful for reconstructing a three-dimensional visualization of body parts from "slices" taken by a CAT scanner.
There are many topics you could choose from, the field of mathematical biology is vast. Here are a few ideas, in no particular order.
The Hodgkin-Huxley equations in neurobiology provide an incredibly accurate quantitative description of action potentials in neurons/myocytes/excitable cells.
The Logistic Equation is a simple model of population growth, and the Lotka-Volterra Equation describes population growth in a predator-prey situation.
David M. Eddy's, statistical work in public health prompted the American Cancer Society to change its recommendation for the frequency of Pap smears from one year to three years.
The Genetic Code is an interesting piece of combinatorics in itself, and I can not help but mention Genetic Algorithms which are a beautiful example of biology inspiring mathematics, rather than the converse.
Other potential topics are the application of mathematics to Genomics, Phylogenetics and the Topology/Geometry of proteins and macromolecules.