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Mathematical and Statistical challenges in Cancer

 

A post-graduate course in the TCC programme - please register with graduate.studies@maths.ox.ac.uk to take this course.

Lecturer: Nigel Burroughs (Warwick Mathematics Institute)

Timetable: Thursdays 11am -1pm. Starting Jan 19th for 8 weeks.

TCC lecture room.

Examination (optional): An assessment essay (max 30 pages, due March 30th). Based on expanding upon one of the course topics or another of your own choosing on some mathematical/statistical topic relating to cancer (to be discussed with the lecturer). The essay will typically involve a critical reading of a number of research papers, but could involve research level analysis based on a research paper.


Content. Cancer is one of the largest medical challenges of the 21st century, affecting up to 1 in 3 people over their lifetime. Although treatments have improved significantly, incidence has also increased at roughly the same rate. Cancer is typically thought of as an age related disease, but there is increasing incidence of cancer in children. Cancer is caused by mutation, cells going through a cascade of mutations that give these cells abnormal behaviour - this includes an increased mutation rate, escape of the mechanisms that constrain growth and an ability to migrate throughout the body (leading to metastasis). Since these cells are effectively 'self', the immune system also has difficulty in developing an immune response against cancer cells. Cancer is thus a complex multi-facetted problem. This course will examine how mathematical and statistical approaches are aiding both an understanding of cancer, and the treatment of cancer.


The course will comprise 8 lectures (2 hours each) covering a selection of topics; lecture notes are emailed weekly to registered students.

Lecture Notes 19th Jan. Ch1 (Intro). Ch2 (Branching processes).

Lecture Notes 26th Jan. Ch3 (Mutation processes).

Provisional syllabus (this is only a guide as there may be insufficient time to cover all the topics).

1. An introduction to cancer, and why modelling (interdisciplinary approaches) are useful.

2. Mathematical modeling of evolution and mutation. Here we will look at branching process models of mutation and applications to cancer.

3. Mutation processes. Here we will look at how the genome is mutated during cancer and the different mutation signatures associated with different cancers.

4. Phylogeny of a cancer. Here we look at attempts at reconstructing cancer lineages across multiple tumours and how (driver) mutations affect cancer development.

5. The circadian clock (and cancer) and chronotherapy. Here we use ordinary differential equation models to examine how the circadian clock affects cancer treatment.

6. Tumour growth. Here we use partial differential equations to examine how tumours grow under resource competition.

Other possible topics include: 7. Metastasis (a Markov model), 8. Rare event and risk modelling: immune surveillance and rare events (theory of large deviations), 9. Cancer classification (survival curves and Bayesian classifiers), 10. Drugs and pharmacokinetics, 11. Clinical trials, iterative designs.

Prerequisites. This course will use a large variety of mathematical/statistical techniques so expertise in all areas is not expected. The course will use both deterministic models (e.g. circadian clock models and pharmacokinetic models (ODEs), tumour growth models (PDEs)), stochastic models (e.g. Markov chains, branching models), graph theory (phylogeny) and Bayesian statistics/likelihoods.

Literature. A variety of resources have been used.

Books:

Durrett Richard. Branching process models of cancer. Springer, 2015.

Research papers:

Philipp M. Altrock, Lin L. Liu & Franziska Michor. The mathematics of cancer: integrating quantitative models. Nature Reviews Cancer 15, 730–745 (2015) doi:10.1038/nrc4029.

Helen M. Byrne. Dissecting cancer through mathematics: from the cell to the animal model. Nature Reviews Cancer 10, 221-230 (March 2010) | doi:10.1038/nrc2808

Niko Beerenwinkel, Roland F. Schwarz, Moritz Gerstung and Florian Markowetz. Cancer Evolution: Mathematical Models and Computational Inference. Syst Biol (2015). 64 (1): e1-e25.
doi: 10.1093/sysbio/syu081