To return to the main course page, please click here .
Note that this page will be updated continuously throughout the semester, so make sure to refresh your browser to see the most updated version.
Homework problems will be uploaded here, together with due dates (typically one to two week after uploading). Quizzes will also be posted once administered, and solutions to both will appear on Sakai .
| Lecture | Date | Topic | Readings | HW, quizzes, and additional notes |
|---|---|---|---|---|
| Week 1 | ||||
| 1 | Tue 01/17 | Introduction to course and cancer modeling | None |
HW: Short description of who you are, why taking course, and what you'd like to see (math and/or biology). Due: Thursday 1/19. |
| 2 | Thu 01/19 | Finish introduction, introduction to molecular biology and cancer | Cancer background papers [1] and [2] | Biological background slides |
| Week 2 | ||||
| 3 | Tue 01/24 | Finish biological background, begin review of ODEs (notation, phase portrait, etc.) | Notes and textbook from MATH 252 course, also internet and Wikipedia | Homework 1 (due 02/02) |
| 4 | Thu 01/26 | Continue review of ODEs (basic modeling, equilibrium, conservation laws) | See Lecture 3 above | Quiz 1 |
| Week 3 | ||||
| 5 | Tue 01/31 | ODE review: conservation laws and local phase portraits (Jacobian analysis/Hartman-Grobman Theorem) | See Lecture 3 above | None |
| 6 | Thu 02/02 | Finish ODE review: phase portrait example, nullclines (global phase portrait), and other types of orbits. | Survey on classical growth models: [3] | Homework 2 (due 02/21) Note this is a compressed file which includes tumor data files. |
| Week 4 | ||||
| 7 | Tue 02/07 | Begin study of classical growth models. Goals of fitting to clinical measurements (understanding mechanisms, prediction). General philosophy of fitting and prediction. Fundamental growth laws, including exponential and logistic. | See [3] again (Lecture 6), as well as [4] for a different type of growth law (not disussed in class), but interesting (basically a type of exponential growth rescaled in mass and time). | None |
| 8 | Thu 02/09 | Class canceled due to snow! | See [3] and [4]. Please read, at least to get main ideas. | Homework 2 due date extended (now due 02/21). I will continue with growth models, but move somewhat quickly to make up time, so please do some reading on your own. |
| Week 5 | ||||
| 9 | Tue 02/14 | Continue growth laws: exponential and logistic. | See readings from Lecture 7. | None |
| 10 | Thu 02/16 | Finish classical growth laws: von Bertlanaffy and Gompertz. | See [5]. This is a famous paper by A.K. Laird from 1964 analyzing tumor growth data and first using the Gompertz model in cancer. | Quiz 2 |
| Week 6 | ||||
| 11 | Tue 02/21 | Finish Gompertz growth. Introduce data fitting philosophy and techniques. | For more information on optimization, see the added resource here. This is a textbook, and contains much more than I would expect you to know, but is provided for those interested. | None |
| 12 | Thu 02/23 | More on data fitting, including derivation of the least-squares solution. | See textbook listed in Lecture 11. | Homework 3 (due 03/09) Note this is a compressed file which includes data files and a piece of MATLAB code. |
| Week 7 | ||||
| 13 | Tue 02/28 | Finish data fitting, including idea of iterative methods (Gauss-Newton). More biological (as opposed to phenomenological) growth models. | See textbook listed in Lecture 11. | Homework 3 (slightly) edited. |
| 14 | Thu 03/02 | General growth models, including necessary and sufficient conditions for sigmoidal growth from one-dimensional ODE. Introduction to basic cell-cycle model (proliferation and quiescence) as an explanation for sigmoidal growth. Basic model properties. | See Resources [8] and [6]. [8] is the orignal Gyllenberg-Webb model, and includes a complete analysis. | None |
| Week 8 | ||||
| 15 | Tue 03/07 | Proof that Gyllenberg-Webb model displays sigmoidal dynamics as a general framework. | [8] and [6] again | None |
| 16 | Thu 03/09 | Introduction to chemotherapy, and log-kill hypothesis. Mathematical representation of bolus injections and fractional kill. | Google "fractional-kill" for some basic introductions. Also Wiki on chemotherapy (lots of good information there). | Quiz 3 and Homework 4 (due 03/28) |
| Week 9 | ||||
| Spring Break! | ||||
| Week 10 | ||||
| 17 | Tue 03/21 | Continued on models of chemotherapy. Specifically, derived log-kill relation between rate and fractional population decrease. Also discussed Norton-Simon hypothesis. | See [9] for original paper introducing the Norton-Simon hypothesis, [11] for an update. | None |
| 18 | Thu 03/23 | Mathematical model of ovarian cancer treatment. Introduce question of sequencing chemotherapy and surgery. | [7] is the main source for this work. | Project paper selection is due 03/24 (tomorrow). |
| Week 11 | ||||
| 19 | Tue 03/28 | Finish scheduling of surgery and chemotherapy work. | [9] again | Homework 5 (due 04/11) |
| 20 | Thu 03/30 | Introduction to tumor-immune system dynamics. Basic biology and model demonstrating immunostimulation, "sneaking through" phenomenon, and recurrence. | See [10] | (Very) Basics of immunology. |
| Week 12 | ||||
| 21 | Tue 04/04 | Introduced biological background of model analyzing tumor-immune dynamics. Questions to be addressed (tumor dormancy, "sneaking through," and immunostimulation), as well as model formula from first principles. | [10] | None |
| 22 | Thu 04/06 | Further study of model introduced in Lecture 21. Reduction to planar system via quasi-steady state approximation. Proof that periodic orbits cannot exist for any parameter values (Bendixson's Criterion). | [10] | Project summary is due on 04/14 (next Friday). Start writing up your thoughts, and what you intend to reproduce. |
| Week 13 | ||||
| 23 | Tue 04/11 | Continue model in Resource [10]. Non-dimensionalization, and proof of basic properities, including non-existence of periodic orbits. | [10] | Project summary is due on 04/14 (Friday) |
| 24 | Thu 04/13 | Finish immune-tumor model. Study of global behavior via basins of attractions of steady states. Tumor dormancy, immunostimulation, and "sneaking through" as a result of nonlinear interactions and geometry of phase portrait. | [10] | Project summary due tomorrow. |
| Week 14 | ||||
| 25 | Tue 04/18 | Introduced optimal control formulation. In particular, different objective functionals, and relation to the calculus of variations. | [12] | Homework 6 (due 05/01) |
| 26 | Thu 04/20 | Finish optimal control formulation, including derivation of Hamiltonian and Pontryagin's Maximum Principle as a necessary condition. Examples to cancer studied, including applications to drug resistance. | [12] and [13] | None |
| Week 15 | ||||
| 27 | Tue 04/25 | Optimal control example, and introduction to model of drug resistance. | [13] | None |
| 28 | Thu 04/27 | Finish drug resistance model of gene amplification. Bang-bang vs. singular controls. Basic stochastic model of carcinogenesis, including relation between age-specific incidence and stages. | [14] and [15] | Homework 6 due on Monday 5/1 in my office |