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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 01/22 | Introduction to course and cancer modeling | None |
Intro HW: 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: Tuesday 1/29. |
| 2 | Thu 01/24 | No class! Please review your old ODE material and look into MATLAB. | Please review ODE material, and see the following two sources on phase plane analysis: intro to nullclines and phase plane examples. Also see Section 2.2 and 2.4 in Prof. Sontag's notes | Homework 1 (due 02/12) |
| Week 2 | ||||
| 3 | Tue 01/29 | Introduction to molecular biology and cancer | Cancer background papers [1] and [2]. | Biological background slides |
| 4 | Thu 01/31 | Finish biological background, review of ODEs | Notes and textbook from MATH 252 course, also internet and Wikipedia | Two sources on phase plane analysis: intro to nullclines and phase plane examples. Also see Section 2.2 and 2.4 in Prof. Sontag's notes |
| Week 3 | ||||
| 5 | Tue 02/05 | Finish ODE review: conservation laws, phase portrait Jacobian example, nullclines (global phase portrait), and other types of orbits. | Sections 2.1-2.3 in KNE. |
Homework 2 (due 02/26) Note this is a compressed file which includes tumor data files. |
| 6 | Thu 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. | Sections 2.1-2.3 in KNE. See [3] for a survey of growth models, 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). |
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| Week 4 | ||||
| 7 | Thu 02/12 | Continue growth laws: exponential and logistic. | See readings from Lecture 6. | Files related to MATLAB intro given on Friday 2/8 |
| 8 | Thu 02/14 | 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. |
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| Week 5 | ||||
| 9 | Thu 02/19 | Finish Gompertz equation. Introduce data fitting philosophy and techniques. Introduction to Information Theory and model selection criterion. | 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. | Quiz 1 |
| 10 | Thu 02/21 | More on data fitting, including least-squares and Newton methods. Basics of fitting a specific model to a data set. | (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. | None |
| Week 6 | ||||
| 11 | Tue 02/26 | Finish data fitting. 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 | Homework 3 (due 03/14) Note this is a compressed file which includes data and MATLAB files. New version of phase plane plotter. |
| 12 | Thu 02/28 | Continue analysis of cell-cycle Gyllenberg-Webb model. Basic model properties and biological assumptions. Proof that populations always remain non-negative. | See Resources [8] and [6]. [8] is the orignal Gyllenberg-Webb model, and includes a complete analysis. Section 2.4 in KNE | Quiz 2 will be 3/5 (Tuesday). Look at basic growth models. |
| Week 7 | ||||
| 13 | Tue 03/05 | 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. | Quiz 2 |
| 14 | Thu 03/07 | 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). | Homework 4 (due 03/28) |
| Week 8 | ||||
| 15 | Tue 03/12 | Continued on models of chemotherapy. Specifically, derived log-kill relation between rate and fractional population decrease. Also discussed Norton-Simon hypothesis. Introduced adjuvant and neoadjuvant therapy. | See [10] for original paper introducing the Norton-Simon hypothesis, [11] for an update. | Homework 4 (due 03/28) |
| 16 | Thu 03/14 | Mathematical model of ovarian cancer treatment. Mathematical analysis of scheduling of surgery and chemotherapy for ovarian cancer. | [7] is the main source for this work. Also Section 2.6 in KNE. | None |
| Week 9 | ||||
| Spring Break! | ||||
| Week 10 | ||||
| 17 | Tue 03/26 | Finish scheduling of surgery and chemotherapy work. | [7] again | Homework 5 (due 04/11) |
| 18 | Thu 03/28 | Introduction to tumor-immune system dynamics. Basic biology and model demonstrating immunostimulation, "sneaking through" phenomenon, and recurrence. | See [12] | (Very) Basics of immunology, |
| Week 11 | ||||
| 19 | Tue 04/02 | 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] | Project progress report is due on 04/19 For details of what is expected, see Project Information |
| 20 | Thu 04/04 | Further study of model introduced in Lecture 19. Reduction to planar system via quasi-steady state approximation. Non-dimensionalization of model. | [12], [13] | Quiz 3 | .
| Week 12 | ||||
| 21 | Tue 04/09 | Further discussion of non-dimensionalization. Different types of dynamics, including the impossibility of periodic orbits (Bendixson's Criterion). | [12], [13] | Homework 6 (due 04/30). Note this is a compressed file which includes MATLAB files. |
| 22 | Thu 04/11 | Proof of Bendixson's Criterion and application to tumor-immune 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 progress report is due on 04/19 (next Friday). For details of what is expected, see Project Information. |
| Week 13 | ||||
| 23 | Tue 04/16 | Finish tumor-immune model. Global bifurcations (heteroclinic) leading to "sneaking through" and immunostimulation. | [12], [13] | Homework 6 (due 04/30). Note this is a compressed file which MATLAB files. |
| 24 | Thu 04/18 | Biology and models of carcinogenesis. Basic review of some elementary probability, including the Poisson process. | [14], [15] | Project progress report is due on 04/19 (tomorrow). For details of what is expected, see Project Information. |
| Week 14 | ||||
| 25 | Tue 04/23 | More background on probability theory. Difference between discrete and exponential time models of systems. | [16] | Quiz 5 will be on Tuesday, 5/30. |
| 26 | Thu 04/25 | Proof of expoentnial waiting time, and equivalent formulation of continuous-time Markov chains. Examples from pure-birth process, including relationt to exponential growth. | [14], [16] | Homework 7 (due 05/7). Note this is a compressed file which contains a MATLAB file. |
| Week 15 | ||||
| 27 | Tue 04/30 | Basic stochastic model of carcinogenesis, including relation between age-specific incidence and stages/mutational events. | [14], [15] | Quiz 5 |
| 28 | Thu 05/02 | Two-stage clonal expansion (TSCE) model of carcinogenesis. Explicit model of malignant transformations of normal (healthy) stem cells. | [15] | Homework 7 (due 05/7). Note this is a compressed file which contains a MATLAB file. |