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What are applications of mathematics in cancer research?

My answer.

Unfortunately I heard quite small about math, but I heard something about applications of physics. And let me put this story here, it might be useful to be aware. There is well-known radiation therapy. But less well-known is proton therapy http://en.wikipedia.org/wiki/Proton_therapy It is much more rare and based on protons accelerators used in particle physics. "The chief advantage of proton therapy is the ability to more precisely localize the radiation dosage when compared with other types of external beam radiotherapy." As far as I know, it is only practiced in physics research centers which have proton accelerators. It is highly useful for cancer of sensitive tissues when it is dangerous to use other radiation therapy cause it will destroy everything surrounding.

A colleague of mine told me this really helped his father with the eye cancer. Moreover this method is so rare, that the leading experts in cancer were NOT aware of it and he become known of it only through friends.

PS

The research institute where it happend is leading Russian research center ITEP (Institute for Theoretical and Experimental Physics). Note this: http://saveitep.org/

PSPS

If one is aware of new (may be on trial) methods for pancreatic cancer, please let me know: al dot chervov dog gmail dot com

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Fractal theory is a branch of mathematics that is very helpful in cancers detection. It is possible that, we model the cancer growth by some fractal structure. –  Shahrooz Janbaz Feb 5 '12 at 18:49

11 Answers 11

My colleague Emmanuel Grenier (École Normale Supérieure de Lyon) leads a research group on mathematics in medical sciences, in collaboration with surgeons and pharmacologists. One of their tasks concerns the control of angiogenesis, which is the way a cancer gets food and blood supply.

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The Centre for Mathematical Biology in Oxford does a great deal of work on cancer modelling. To quote their webpage "We are interested in modelling the dynamics of cancer progression and treatment from a number of different view points and on various spatial and temporal scales."

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A lot of maths (and related stats) goes into the development of techniques for screening in the detection of breast cancer using mamography. (I know of this from work by Reyer Zwiggelaar in Aberystwyth. He is in a computer science department but uses a lot of deep maths.)

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A variety of medical image reconstruction methods are very relevant to cancer research and diagnosis.

Prime examples are: Positron Emission Tomography (PET), CT, MRI, etc.; In particular, all of these famous and successful technologies depend on solutions to inverse problems where one must reconstruct an image from a set of noisy measurements.

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You can find some information here: Mathematical Biosciences Institute

For example these... 1 2

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There's a lot of work in statistical methods for designing clinical trials and analyzing molecular data. You might not consider statistics to be mathematics, but there are a lot of more core mathematical problems that come up in the execution of the statistics. For example, in the process of implementing Bayesian clinical trial methods, I've had to solve problems in special functions, numerical analysis, probability, optimization, etc.

Another area that comes to mind is optimization problems to determine how to maximize the radiation delivered to a tumor while minimizing the radiation delivered to healthy tissue.

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The Center for Mathematical Medicine at the Fields Institute at the University of Toronto does much mathematical research related to cancer. For example, see http://www.fields.utoronto.ca/programs/scientific/CMM/13-14/mathoncology/

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One can find some lecture on modern development of pharma-drugs (lecture was at public lectorium so it is quite understandable and interesting (imho)):

http://polit.ru/article/2011/03/22/cancercure/

Let me try to sketch a part and mention what is math-related. (Sorry, lecture is in Russian (try google.translate), but there are some slides inside in English, see also link to "Novartis" below).


Drug development process

Math can be used at step 3 (as far as I understood). Let me first give all steps.

1) Target discovery (find a protein or cascade or smth which are critical for cancer development)

2) Hit discovery (by brute force test 50.000-1.000.000 molecules whether they can kill target or not)

3) Lead-optimization - (assume on previous step you find "something" which can hit cancer, but you must care about that this "something" will not kill person also or it is stable enough to work in real life. At this stage one looks for certain modifications which can preserve the positive features and dismiss negative).

4) Trials on animals

5) Phase 1 trials (10-50 people just to test that they will not be killed by side effects)

6) Phase 2 trials (100-300 people determine dosation, effectiveness, safety)

7) Phase 3 trial (1000-3000 people determine: side effects, interaction and comparing with other drugs,

8) Registration

9) Post launch studies


This is based on presentation from "Novartis", part can be found at:

http://www.novartis.com/innovation/research-development/drug-discovery-development-process/index.shtml

"Novartis" is in particular famous for recent innovation of "Gleevec"(=Imatinib)

http://en.wikipedia.org/wiki/Imatinib

which is one of rare successes in the field of cancer drugs.)


So, (as far as I understood) at step 3 - "lead optimization" certain mathemaical modelling is sometimes possible. There is certain mathematical-based software which can try to predict some properties of moleculas based on their structure - so when people try to modify hiting molecula they sometimes use it.

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I attended a talk by Heiko Enderling on modeling cancer growth as a spatial process.

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Modelling cancer networks, with graph techniques, to be specific, use graph theory, such as network flow analysis, or web graphs to find where anomalies are, and predict the organs' behaviour are some of the areas where graph theory, can find use in. Other use of mathematical techniques, are those used in the field of knowledge management, knowledge discovery, and so on. Eigenvalues, with different measures show up bad genes from good, and could help in the isolation of cancerous areas. All this aids in the development of drugs that are used to cure cancer by studying their effects, on cancerous areas. For instance the web graphs could isolate and show up the drugs effects in time+cost effective ways.

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For the use of evolutionary equations in cancer development and therapy see this lab.

Edit:

The mission statement of the lab says: The research of our lab focuses on the evolutionary dynamics of cancer. Cancer emerges due to an evolutionary process in somatic tissue. The fundamental laws of evolution can best be formulated as exact mathematical equations. Therefore, the process of cancer initiation and progression is amenable to mathematical investigation. Current areas of research include cancer stem cells, evolution of drug resistance, and the dynamics of metastasis formation.

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I think evolutionary dynamics is a good answer, but can you expand this with slightly more detail? –  usul Nov 11 at 0:44
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There's even a TED talk by Franziska in case you want to relay this to your regular layman. –  Vít Tuček Nov 11 at 10:26

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