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This semester I'm charged with finding a senior capstone project for next year. I've given it a lot of thought and can't seem to find any interesting ideas that are appropriate for my level of mathematics:

Junior with A's in:

  • ODEs/PDEs
  • Linear Algebra
  • Topology
  • Organic Chemistry 1/2
  • Cellular Biology
  • Introduction to Biochemistry

And currently enrolled in Combinatorics and Complex Variables. I've approached several of my professors and asked for some project ideas but they've not given me anything at all. I'm not asking for you to give me a project to do without me having to do anything. I just want to get a feel for what some applications are to some of the more interesting fields, like Combinatorics and something that is somewhat related to all this, Chaos Theory.

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    $\begingroup$ I have converted the question to community wiki, as it's asking for a big list of possibilities and there is no single right answer. $\endgroup$ – Zev Chonoles Feb 15 '12 at 4:32

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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.

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  • $\begingroup$ So, that sounds really interesting. I know how to program a little, so producing a render from some supplied slices would be an awesome project. Do you know of any supplemental material to get me started? $\endgroup$ – MentatOfDune Feb 15 '12 at 0:37
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    $\begingroup$ @Mentat: this book should cover the theory nicely. For code, try searching around; you'll find stuff like this. $\endgroup$ – J. M. is a poor mathematician Feb 15 '12 at 0:46
  • $\begingroup$ I was aware of medical imaging, but the more and more I look into this stuff, the more I think this is what I'm going to do my project on. This is really awesome stuff. Thanks! $\endgroup$ – MentatOfDune Feb 15 '12 at 0:53
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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.

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You might be interested in the answers to a question about Math and cancer research at the MathOverflow site, https://mathoverflow.net/questions/87575/mathematics-and-cancer-research

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The Basic Local Alignment Search Tool (BLAST) is one of the most widely used bioinformatic programs, developed by mathematicians (Altschul, Gish, and Lipman) in the most cited paper of the 1990's (and the most cited biology paper of all time). BLAST is used in medical resequencing, and genome sequencing is having an increasing impact on medicine. The mathematical content is stochastic analysis.

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One area in which combinatorics interacts with the biological sciences is in the Ewens sampling formula of population genetics. Population genetics generally relies pretty heavily on mathematics.

And there's this book.

Statistics generally is heavily relied on in medical research.

Since you mention combinatorics, one area of statistics that relies on combinatorics and gets applied to medicine is design of experiments.

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  • $\begingroup$ I know statistics is heavily relied on, but I find the subject matter dry and boring. Population genetics is an interesting topic though, what would I do with that though? Trace a gene through its development? I'm not really sure. $\endgroup$ – MentatOfDune Feb 15 '12 at 0:55
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Related to J.M.'s answer, one really neat area is source localization of EEG. The idea is similar to other brain scanning methods such fMRI or PET, but instead of measuring blood flow in the brain or nuclear physics, EEG data are collected from a subject and electrical sources of these surface voltages are reconstructed using inverse problem approaches. Inverse problems are a venerable area of applied math, with a long history spanning many disciplines. While this is more interesting from a research standpoint instead of clinical, this is a really cool area of work for understanding brain activity.

For a good overview of different algorithms, check out this paper (warning: PDF).

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  • $\begingroup$ I can't read that paper. I don't have access to that journal. I'm trying to see if I can access it through the university system but no luck. It's a shame, I really wanted to read it. $\endgroup$ – MentatOfDune Feb 16 '12 at 4:23
  • $\begingroup$ @MentatOfDune Try searching through scholar.google.com. I thought it was the link to the PDF directly, but maybe not. I know there are freely available reviews on the subject around. $\endgroup$ – joshin4colours Feb 16 '12 at 5:20
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On top of what has already been said in other great answers: Bernoulli’s equation and the Navier–Stokes equations come up often when studying blood-flow.

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3D surface curves are used to model tumors to correctly apply heat treatments. This particular application is the motivation to Larson's Calculus's 13th Chapter: Multiple Integrals. He provides the examples:

$$\rho = 1+0,2\cdot\sin8\theta \cdot\sin\phi $$

$$\rho = 1+0,2\cdot\sin8\theta \cdot\sin4\phi $$

as models and asks to calculate their volume.

As a reference he leaves: "Heat Therapy for Tumors" Leah Edelstine-Keshet (UMAP Journal), Summer '91.

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Image processing rely heavily on mathematics and is much used to produce and analyze medical data. This book might give you some ideas: Mathematical problems in image processing: partial differential equations ... By Gilles Aubert, Pierre Kornprobst

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There exists an entire volume devoted to "Fuzzy Logic and Medicine". The link there can tell you more, as well as a google search.

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For application of chaos theory or non-linear dynamics to heart-rate, you can check the paper by A.L. Goldberger titled Non-linear dynamics for clinicians: chaos theory, fractals, and complexity at the bedside (link).

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I heard of a student at Sherbrooke University who made a program that modelled very accurately human brain tissue. It might have nothing to do with what you want, but it helped map vital tissues in brain surgeries and assisting doctors to take irreversible decisions accurately.

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  • $\begingroup$ That's actually like what I'm looking into doing now, the post about Tomography sparked my interest and I've been looking into it all day. $\endgroup$ – MentatOfDune Feb 16 '12 at 4:01
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If you're interested in applications of differential equations in the biological/medical area, I suggest looking at Clifford Taubes, "Modeling differential equations in biology" http://books.google.ca/books?id=Y464SAAACAAJ There are lots of things in there that would make a good project.

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Two devices I know of, the C-arm and the Gamma knife, both make use of some the things you've learned. I also read a while ago that fractals are used in cancer detection, since the fractal dimension of tumours is different from the fractal dimension of normal tissue.

I don't know what disciplines the profs you've talked to are in, but I would suggest trying to get in touch with engineering profs, or maybe computer science, whose research involves medicine/surgery. There's lots of math involved.

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