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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 .
KNE denotes Introduction to Mathematical Oncology, the main text for the class.
[·] denotes an article, which can be found here . Note that links are also included directly in the Course Calendar.
| Lecture | Date | Topic | Readings | HW, quizzes, and additional notes |
|---|---|---|---|---|
| Week 1 | ||||
| 1 | Tue 01/16 | Introduction to course and cancer modeling | None |
Homework 1: Short description of who you are, why taking course, and what you'd like to see (math and/or biology). Please elaborate, at least a little. Due: Thursday 1/18. |
| 2 | Thu 01/18 | Introduction to molecular biology and cancer | Cancer background papers [1] and [2]. | Biological background slides |
| Week 2 | ||||
| 3 | Tue 01/23 | Finish biological background, begin review of ODEs (notation, phase portrait, etc.) | Notes and textbook from MATH 252 course, also internet and Wikipedia | Homework 2 (due 02/06) |
| 4 | Thu 01/25 | Continue review of ODEs (basic modeling, equilibrium) | See Lecture 3 above | |
| Week 3 | ||||
| 5 | Tue 01/30 | ODE review: conservation laws, derivation of logistic growth from limited nutrient system | See Lecture 3 above | Quiz 1 |
| 6 | Thu 02/01 | Finish ODE review: phase portrait Jacobian example, nullclines (global phase portrait), and other types of orbits. | Sections 2.1-2.3 in KNE. Survey on classical growth models: [3] |
Homework 3 (due 02/20) Note this is a compressed file which includes tumor data files. |
| Week 4 | ||||
| 7 | Tue 02/06 | 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. | Sections 2.1-2.3 in KNE. See [3] again (Lecture 6), as well as [4] for a different type of growth law (not discussed in class), but interesting (basically a type of exponential growth rescaled in mass and time). |
Introduction to MATLAB files used during the extra lecture from Friday, 2/2. |
| 8 | Thu 02/08 | Continue growth laws: exponential and logistic. | See readings from Lecture 7. | None |
| Week 5 | ||||
| 9 | Tue 02/13 | Finish classical growth laws: von Bertlanaffy and Gompertz. | Sections 2.1-2.3 in KNE. 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 |
| 10 | Thu 02/15 | Finish von Bertlanaffy growth laws. Introduced Gompertz model. | Resource [5] again, and Sections 2.2 and 2.3 in KNE. | None |
| Week 6 | ||||
| 11 | Tue 02/20 | 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 02/22 | Finished data fitting philosophy and techniques. | (for next week) Section 2.4 in KNE. See also Resources [8] and [6]. [8] is the orignal Gyllenberg-Webb model, and includes a complete analysis. | Homework 4 (due 03/08). Note this is a compressed file which includes data and MATLAB files. |
| Week 7 | ||||
| 13 | Tue 02/27 | 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. | See Resources [8] and [6]. [8] is the orignal Gyllenberg-Webb model, and includes a complete analysis. Section 2.4 in KNE | None |
| 14 | Thu 03/01 | Continue analysis of cell-cycle Gyllenberg-Webb model. Basic model properties and biological assumptions. Proof that populations always remain non-negative. | See Lecture 13 | Quiz 3 |
| Week 8 | ||||
| 15 | Tue 03/06 | Proof that Gyllenberg-Webb model displays sigmoidal dynamics as a general framework. Introduction to modeling of chemotherapy. | [8] and [6] again. Chapter 9 in KNE. | None |
| 16 | Thu 03/08 | Introduction to chemotherapy, and log-kill hypothesis. Mathematical representation of bolus injections and fractional kill. | Section 2.6 in KNE. Google "fractional-kill" for some basic introductions. Also Wiki on chemotherapy (lots of good information there). | Quiz 4 and Homework 5 (due 03/29) |
| Week 9 | ||||
| Spring Break! | ||||
| Week 10 | ||||
| 17 | Tue 03/20 | Continued on models of chemotherapy. Specifically, derived log-kill relation between rate and fractional population decrease. Also discussed Norton-Simon hypothesis. | See [10] for original paper introducing the Norton-Simon hypothesis, [11] for an update. | Homework 5 (due 03/29) |
| 18 | Thu 03/22 | Mathematical model of ovarian cancer treatment. Introduce question of sequencing chemotherapy and surgery. | [7] is the main source for this work. Also Section 2.6 in KNE. | None |
| Week 11 | ||||
| 19 | Tue 03/27 | Finish scheduling of surgery and chemotherapy work. | [7] again | Homework 6 (due 04/12) |
| 20 | Thu 03/29 | Introduction to tumor-immune system dynamics. Basic biology and model demonstrating immunostimulation, "sneaking through" phenomenon, and recurrence. | See [12] | (Very) Basics of immunology, Quiz 5 |
| Week 12 | ||||
| 21 | Tue 04/03 | 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. | [12], [13] | Homework 6 (due 04/12) |
| 22 | Thu 04/05 | 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). | [12], [13] | Project progress report is due on 04/13 (next Friday). For details of what is expected, see Project Information. |
| Week 13 | ||||
| 23 | Tue 04/10 | Continue model in Resource [12]. Non-dimensionalization, and proof of basic properities, including non-existence of periodic orbits. | [12] | Project summary is due on 04/13 (Friday) |
| 24 | Thu 04/12 | 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. | [12], [13] | Project summary due tomorrow. |
| Week 14 | ||||
| 25 | Tue 04/17 | Finish tumor-immune model (see Lecture 24). Introduced optimal control formulation. In particular, different objective functionals, and relation to the calculus of variations. | [14] | Homework 7 (due 04/30). Note this is a compressed file which includes MATLAB files. Quiz 6 |
| 26 | Thu 04/19 | 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. | [14] and [15] | None |
| Week 15 | ||||
| 27 | Tue 04/24 | Finish optimal control formulation and example. | [14] | None |
| 28 | Thu 04/26 | Model of drug resistance via gene amplification. Application of optimal control theory to study issue. Bang-bang vs. singular controls. Basic stochastic model of carcinogenesis, including relation between age-specific incidence and stages. | [15], [16], [17] | Homework 7 due on Monday 4/30 in my office. |