Mathematics and cancer research What are applications of mathematics in cancer research?
Unfortunately, I heard quite little about applications of mathematics, but I heard something about applications of physics, and let me put this story here, as it might be useful to be aware of it. Radiation therapy is well-known, but less well-known is proton therapy, which is much more rare and based on proton 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 that have proton accelerators. It is highly useful for cancer of sensitive tissues when it is dangerous to use other radiation therapy because it will destroy everything surrounding the malignancy.
A colleague of mine told me this really helped his father with eye cancer.
Moreover, proton therapy is so rare that the leading cancer experts were not aware of it and he become aware of it only through friends.
 A: 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.
A: 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.)
A: 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/
A: 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.
A: A book about mathematical models that describe the dynamics of tumor growth and the evolution of tumor cells has been recently published:
Dynamics of Cancer: Mathematical Foundations of Oncology

"Mathematically, the book starts with relatively simple ordinary differential equation models, and subsequently explores more complex stochastic and spatial models. Biologically, the book starts with explorations of the basic dynamics of tumor growth, including competitive interactions among cells, and subsequently moves on to the evolutionary dynamics of cancer cells, including scenarios of cancer initiation, progression, and treatment. The book finishes with a discussion of advanced topics, which describe how some of the mathematical concepts can be used to gain insights into a variety of questions, such as epigenetics, telomeres, gene therapy, and social interactions of cancer cells."

A: You can find some information here: Mathematical Biosciences Institute 
For example these...    1 2 
A: 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. 
A: 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. 
A: I attended a talk by Heiko Enderling on modeling cancer growth as a spatial process.
A: A recent-ish effort is "Mathematical Oncology": http://mathematical-oncology.org/index.html which has a blog : http://blog.mathematical-oncology.org  and an upcoming conference : http://mathematical-oncology.org/mathonc20/index.html
Also of interest is this "2019 Roadmap: https://www.ncbi.nlm.nih.gov/pubmed/30991381
A: Topological data analysis, used to predict the biological/physiological effects of different cancers from their genes.
Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival
Monica Nicolau, Arnold J. Levine, and Gunnar Carlsson,
PNAS 108(17), 2011
Abstract: 

High-throughput biological data, whether generated as sequencing,
  transcriptional microarrays, proteomic, or other means, continues to
  require analytic methods that address its high dimensional aspects.
  Because the computational part of data analysis ultimately identifies
  shape characteristics in the organization of data sets, the
  mathematics of shape recognition in high dimensions continues to be a
  crucial part of data analysis. This article introduces a method that
  extracts information from high-throughput microarray data and, by
  using topology, provides greater depth of information than current
  analytic techniques. The method, termed Progression Analysis of
  Disease (PAD), first identifies robust aspects of cluster analysis,
  then goes deeper to find a multitude of biologically meaningful shape
  characteristics in these data. Additionally, because PAD incorporates
  a visualization tool, it provides a simple picture or graph that can
  be used to further explore these data. Although PAD can be applied to
  a wide range of high-throughput data types, it is used here as an
  example to analyze breast cancer transcriptional data. This identified
  a unique subgroup of Estrogen Receptor-positive (ER+) breast cancers
  that express high levels of c-MYB and low levels of innate
  inflammatory genes. These patients exhibit 100% survival and no
  metastasis. No supervised step beyond distinction between tumor and
  healthy patients was used to identify this subtype. The group has a
  clear and distinct, statistically significant molecular signature, it
  highlights coherent biology but is invisible to cluster methods, and
  does not fit into the accepted classification of Luminal A/B,
  Normal-like subtypes of ER+ breast cancers. We denote the group as
  c-MYB+ breast cancer.

A: The AMS has a recent "Mathematical Moments" at https://www.ams.org/publicoutreach/mathmoments/mm164-cancer-research about cancer research.
A: There is plenty of research activity in this area. Here is an example of a recent workshop at IPAM on a specific topic: 
http://www.ipam.ucla.edu/programs/workshops/translating-cancer-data-and-models-to-clinical-practice/ .
A: Ricci flow is an analytic surgery. Prediction and control of cancer invasion Can be done by Ricci flow theory see this paper
Ricci Flow and Entropy Model for Avascular Tumor Growth and Decay Control
Tijana T. Ivancevic
https://arxiv.org/abs/0806.0691
We can apply discrete Ricci flow like Ollivier Ricci flow, to study cancer and tumors
https://www.nature.com/articles/srep12323
There are a list of discrete Ricci curvatures which based on we can define Ricci flow. Just we need to know which discrete Ricci curvature on the graph network is suitable for the tumor.
A: The Molab https://molab.es/ is the Mathematical Oncology Laboratory associated to the Spanish university Universidad de Castilla La Mancha, see also the sponsors from the web. See the labels Research and Projects.
On the other hand in recent days I heard on the Spanish radio RTVE a talk with María Soengas, see this Wikipedia, the podcast [1] refers details of her methodology*. While the mathematics should be an important tool in the study of her models I don't know what are the papers of her research group with more mathematical content.
I edit the post to enrich this about the work of scientists on (from Wikipedia) Mesenchymal stem cell and Oncolytic virus, in papers as [2] (I know it as open access from the website Nature) where you can to study their Mathematical model.
References:
[1] Optimismo, the podcast in Spanish language from the radio program Historias de la gente, Radio Televisión Española (RTVE) (28/01/2023) https://www.rtve.es/play/audios/historias-de-la-gente/historias-gente-optimismo/6788704/
[2] Khaphetsi Joseph Mahasa, Lisette de Pillis, Rachid Ouifki, Amina Eladdadi, Philip Maini, A-Rum Yoon and Chae-Ok Yun, Mesenchymal stem cells used as carrier cells of oncolytic adenovirus results in enhanced oncolytic virotherapy, Scientific Reports, volume 10, Article number: 425 (2020).
A: Katerina Stanková from Delft University does research on (metastatic) cancer treatment through game theory and dynamical systems theory.
A: 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. 
A: Raluca Eftimie has done a lot of numerical simulations of the solutions of the partial differential equations predicting the growth of certain brain tumors and the results have been extremely close to observation and in particular or direct immediate interest to neurosurgeons (who have used these numerical approximations to predict where they would have to intervene surgically).
See this paper for example.
A: 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.
A: The Wolfson Centre for Mathematical Biology (WCMB) at the University of Oxford does a great deal of work on cancer modelling:

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.

