An introduction to the mathematical and computational techniques used in the study of cancer. The course will contain a survey of ways in which mathematical analysis has provided valuable biological insights into both concepts and data. Topics include models of tumor growth, evolution, metastases, and the emergence of drug resistance.
Since this is a special topics course, my philosophy towards MATH 495 differs from most standard curriculum courses that you probably have encountered up until this point at Rutgers. As there is not a prescribed set of material to be covered, we are free to choose topics as interest dictates. Thus, although I will begin the class with some standard (i.e. classical) material, as well as an introduction to the basic biology of cancer, as the semester progresses, I encourage you to inform me of any subjects you would particularly like to see (either biological or mathematical), and I do my best to incorporate them into the schedule. Furthermore, there will be no exams, but instead homework assignments, quizzes, and a project (see below in the Grades section for more details). As a special topics course, I don't believe exams are appropriate. However, the class will demand time and effort, as you will be required to learn some material independently, e.g. reading research papers and implementing models, as well as "filling in gaps" in details that I may skip. I will try not to assign anything extraordinarily difficult (such as open problems), but some of the material will be challenging. As such, please do not wait until the last minute to begin assignments, and don't hesitate to see me with any questions and/or concerns.
(MATH 244 OR MATH 252 OR MATH 292) AND (MATH 250). In laymen’s terms, you should have a good understanding of the standard engineering calculus sequence through ordinary differential equations, as well as linear algebra. Much analysis of models studied in this course will rely heavily on techniques from the previously math courses, so it is very important to have a good understanding of the prerequisites. Of course, I will review theory as we encounter it, but it most likely won’t be sufficient as a stand-alone treatment.
Some of the applications will deal with probability theory, specifically stochastic and branching processes. No prior knowledge of such topics will be assumed, except a basic notion of probability (i.e. an intuitive understanding of randomness). As we begin discussing probabilistic models, I will spend a lecture of two introducing the necessary tools, such as probabilistic formalism and the basics of stochastic processes. As classical cancer modelling assumes one of two regimes (deterministic vs. random), I believe in this course it is necessary to spend at least a some time studying this framework.
Besides a basic high school course, I will assume no prior biological background. I plan on spending a significant amount of time on the biology of cancer, and it will be important for the course that you understand the disease dynamics from the biological perspective. Even though this is a mathematics course, understanding the relevant biology is the first step in formulating useful models.
Formally, grades will be determined by homework, quizzes, and a project. The breakdown is as follows:
| Homework | 40% |
| Quizzes | 20% |
| Project | 40% |
Homework Homework will be assigned throughout the course, and collected periodically. Late homework will not be graded. Assignments will consist of a combination of theoretical and computational problems, and can be found on the Course Calendar. In general, you will have at least one week to complete assignments after distribution (on the public website), but in most cases you will have two weeks. Some assignments will require the use of computer software (in particular for plotting, ODE solving, and stochastic simulations). I recommend using MATLAB, although any programming software is acceptable (Python, Java, C/C++, etc). You will be expected to write your own code, as programming is an invaluable skill for mathematical biology (and science in general). However, outside of class I am happy to talk about programming, and for some basic tutorials, see the Resources section below. I am also willing to hold an extra class (outside of the normal lecture hours) to give a primer on MATLAB programming, if students would find that useful. Based on past experience, I strongly recommend learning MATLAB immediately, as to avoid becoming overwhelmed later in the course; in general programming expectations will increase as the course progresses. For accessing MATLAB, see Resources below. From context, it should be clear which problems require the use of a computer, and which should be completed by hand.
Note on collaboration You are encouraged to work together on homework assignments, but problem sets should be both written up and turned in individually. Furthermore, only submit work that is your own. Plagiarism is a serious offense, and will result in a score of zero for the assignment, as well as possible punishment from the University. You must also cite any reference you use and clearly mark any quotation or close paraphrase that you include. Such citation will not lower your grade, although extensive quotation might.
Quizzes Quizzes will be given regularly throughout the semester. In general, they will be in-class and shouldn't last more than 20 minutes. I will use them to assess the group's understanding of different topics covered, in that they will be useful in dictating both pace and material. You will be expected to interpret and analyze different models, and the material will be chosen from lecture and assigned readings. Their difficulty should be below that of the homework.
Project You will also be expected to complete a project, which will involve reading an article about mathematical oncology, reproducing some of the results, and presenting the work to the class. Or, if you have an idea for an original project, you are welcome to begin investigating that (such work could even lead to a published paper!). Note that if you choose the latter, I may point you to other work that's already been analyzed on the subject, if such work exists. You will be expected to understand most of the motivations/derivations, and any relevant biology (we may ask questions!). You should work in groups of approximately three, which you can form yourselves (if you want me to assign you to a group, send me an email); at some point there will be a deadline for group formation, so start thinking about them early. You can also choose the subject/paper, but it should be distinct from anything specifically covered in class. I can also assist in choosing topics. As the semester progresses, more information can be found on the Project Page.
This class has no definite schedule; however there are a few classical topics which I plan to cover. As mentioned previously, if there is additional material you would like to explore, please feel free to contact me. I will keep an updated class schedule on the webpage (Course Calendar), which will include topics covered, homework postings with due dates, extra notes, and any other pertintent information (such as project information). Below you will find a possible list of topics, along with a short explanation of each subject. Note that I probably won't have sufficient time for everything listed, so if something seems particularly interesting, let me know.
| Introduction to mathematical oncology | Various frameworks presented, not all of which are covered in the course (ODEs, PDEs, stochastic models, hybrid models) |
| Cancer biology overview | Basics of cell biology and the "Hallmarks of Cancer," as well as different treatments (chemotherapy, radiotherapy, immunotherapy, etc.) |
| Quick review of ODEs | Important tools from ordinary differential equations, including linear stability analysis (i.e. Jacobian analysis), nullclines, and Poincaré -Bendixson theory |
| Basic growth models | Single population models of tumor growth. Compare and contrast exponential, logistic, Gompertzian, and power growth laws, in relation to different clinical data sets. |
| Heterogeneity and cancer dynamics | Competition between clonal variants inside a single tumor. Also, how considerations of cell-cycle dynamics (dividing vs. quiescent cells) can influence treatment. Basic competition (i.e. selection) and mutation models. |
| Optimal control theory | The application of control theory techniques to understand "best" treatment protocols for chemotherapy. |
| Spatially structured tumor growth | Physical models of tumor growth in space, include necrosis and possibly angiogenesis (development of a blood supply). Basic partial differential equations and solution properties. |
| Models for chemotherapy and radiotherapy | Biological mechanisms by which killing agents act, and their implementation into mathematical models. |
| Primer on probability theory | Formalism of probability theory, Markov chains, and stochastic processes. Introduction of tools that will be directly applicable for model analysis (forward equations and equilibrium distributions). |
| Carcinogenesis | Tumor evolution and initiation from the stochastic perspective. |
| Drug resistance | Evolution of drug resistance from both the stochastic and deterministic regimes. Include classic Luria-Delbrü ck analysis and Goldie-Coldman model. |
Material will be taken from various sources, including the two mentioned textbooks above, as well as published articles. Specific articles will be posted in the Resources section below, and information can be found in the (Course Calendar). Slides will be uploaded there as well, but you are responsible for any material presented in class (as I will not always, or even often, use slides).
Students are expected to attend all classes; if you expect to miss one or two classes, please use the University absence reporting website to indicate the date and reason for your absence. An email is automatically sent to me. Please note that there will be no make-ups for quizzes or exams. If you have a major medical or personal problem and plan to miss an exam, please contact the instructor by email, with a note from the Dean's office to authenticate an absence that is supported by appropriate documentation.
All students in the course are expected to be familiar with and abide by the academic integrity policy. Violations of this policy are taken very seriously.
Full disability policies and procedures are indicated here. Students with disabilites requesting accommodations must present a Letter of Accommodations to the instructor as early in the term as possible.
I will use Sakai for email contact, as well as to post homework solutions. All enrolled students should have automatic access to the site after logging in to Sakai. Make sure to frequently check your email associated to your Sakai account.
I am including both resources directly relevant to the course (such as discussed scientific articles) as well as additional material that may be useful (such as MATLAB tutorials).