Calculus 1 Project Topics
Project 1: Tumor growth and response to irradiation
Objectives: Developing a mathematical model for tumor growth to study dynamics of a tumor size during various treatment plans. Simple starting point will be exponential growth model and to introduce the action of the treatment to the ‘differential equation’. Depending on various types and various frequencies of treatments we will have different dynamics for the tumor growth.
(1)Ideal tumor growth model without irradiation.
(2)Tumor growth under time dependent irradiation. Expectation: to improve the exponential model by introducing negative ‘rate of change’ of irradiation.
(3)We can find values for constant from literature of modern research. Otherwise we can simply propose some values and observe how the dynamics would work.
(4)Here denotes the volume (size) of the tumor and denotes the treatment plan and the independent variable denotes the time in days.
(5)Considering various irradiation treatment plans, say constant continuous irradiation, daily, or weekly irradiation with single dose. It is then important to check ‘obvious’ positive and negative effects of each type of treatment plan.
(6)One can then consider time-varying treatment dosage and see how it would affect the system.
(7)Using numerical methods (first order ‘gradient approximation’) to qualitatively for various irradiation plans If possible try to find solutions for analytically.
(8)Additional topics: ‘Stability of the system’.
(9)What if the irradiation plan depends on the size of the tumor? i.e.
(10)Validating our observations with clinical data if time permits. This step needs two parts. We first need to validate the tumor growth model without irradiation. Then to compare with the clinical data under irradiation plans.
(Required background knowledge (that you may not have by now) and unfamiliar technical terms should not scare you. One part of the project is to get familiar with them. After all this is a project!)
Project 2:Improving athletic performances by mathematical models. (Maximizing the distance the range of a long jump.)
Objective: In physics, we study maximum range of a projectile. (In calculus, this is an easy application.) However, we cannot study real world objects as particles. We need to make these models realistic and it makes mathematical models complicated. An athlete will not be considered as a particle in this consideration any longer. Furthermore, how the air resistance or air flow will affect the performance of an athlete? Objective of this study is to build a mathematical model that describes a real world event and to use that to improve the quality of a real world application.
This project is based on improving athletic achievements by introducing a mathematical model.
(1)Under ideal case, consider the athlete is denoted with negligible dimensions (i.e. the athlete is denoted by a point). Then maximum range will be produced under angle. (neglect air resistance.)
(2)However, athlete’s body structure, dimensions and flexibility play a considerable role.
(3)Trajectory defined by contact point should be tracked. For a good jump, mostly this will be toes.
(4)We need to find the angle of takeoff and the and the position at the landing to maximize the range.
Project 3: Writing computer programs by using Mathematica.
Objectives: This project has two objectives. Writing computer programs to find roots (approximately) of an equation of the form will be the first objective. Second objective will be to write a program to find the area (approximately) trapped between a graph and X-axis. This is called ‘numerical integration’. For both objective ‘error analysis’ and a study of ‘applicability’ is essential. Computer programs should be written in a way that you can input any function and get the desired output.
(1)Writing efficient computer programs with fast convergence rate, to find the area under a given graph. First one can try Riemann sums. What kind of geometric improvements you can make?
(2)What are the other methods you can propose? (Ex: Trapezoidal and Simpson rules)
(3)Writing a computer program to find zeros of a given function. Analyzing the cases where the computer program might fail. (Blind spots: Not all functions will obey computer algorithms. Need to explore the limitations.)