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Resource Type: Expository Article

Primary Level: Undergraduate

Additional Level: High School

Minimodule: Cancer Modeling

Author: Alexander Bates

Abstract

In this assignment, you will explore classical models of cancer growth and analyze the behavior of these equations using qualitative techniques. As you work through the assignment, consider the following questions: How will you translate a real-world situation into a mathematical model? How can you solve the model you develop? Most importantly, how will you interpret the results in a way that ethically informs decisions in real-world scenarios?

Note: The information below was created with the assistance of AI.

Level of Mathematics
Overall Level:

Early Undergraduate (Primary) → Advanced High School (Upper Range)

Evidence:

Designed for:

“freshman-year first-semester mathematics course (presuming calculus)”
Introductory differential equations exposure

Mathematical sophistication:

First-order differential equations:

Exponential growth:
Logistic growth:

Parameter estimation from data
Qualitative analysis (increasing/decreasing behavior)

Interpretation:

Accessible to:

Advanced high school students (AP Calculus level)
Intro college (Calculus I / early Differential Equations)

Subject Matter
Core Mathematical Topics:

Differential Equations (Intro Level)

First-order ODEs
Exponential growth models
Logistic growth models

Mathematical Modeling

Translating real-world phenomena into equations

Data Analysis (Qualitative)

Fitting models to observed data

Parameter Estimation

Determining , from data

Supporting Topics:

Functions and graphs
Limits (qualitative behavior near carrying capacity)
Piecewise/model comparison

Application Areas

Primary Application:

Cancer biology / Tumor growth modeling

Predicting tumor volume over time

Broader Applications:

Medical decision-making

Screening strategies
Treatment planning

Population dynamics

Logistic growth (generalizable)

Epidemiology / biology modeling

Real-world impact:

Helps:

Predict tumor progression
Evaluate therapies
Inform ethical decisions (explicitly emphasized in module)

Prerequisites
Required Background:
Mathematics:

Algebra:

Manipulating equations

Functions and graphs
Basic calculus:

Derivatives
Exponential functions

Intro differential equations (minimal):

Understanding as rate of change

Recommended:

Experience with:

Graphing tools (Desmos, MATLAB, etc.)
Interpreting data tables (see Table 1, page 2)

Not required:

Advanced statistics
Linear algebra
Multivariable calculus

Correlation to Mathematics Standards

US Common Core (High School)
Strong alignment with:

HSA-CED (Create equations)

Building models from context

HSS-MD (Modeling with data)

Using data to refine parameters

HSA-REI (Solve equations)

Solving differential equations (intro form)

HSM (Modeling Standard)

Full modeling cycle:

Assumptions → equations → interpretation

AP Courses

AP Calculus AB/BC

Differential equations (growth/decay)
Exponential models
Logistic-type reasoning (conceptual)

AP Statistics (partial)

Data fitting (informal, visual—not statistical inference)

Undergraduate Standards
Aligned with:

Intro Differential Equations
Mathematical Modeling (entry-level)
Quantitative reasoning courses

Mathematical Practices (Process Standards)
This module strongly emphasizes:

MP4: Model with mathematics

Central focus (multiple models compared)

MP1: Problem solving

Iterative refinement (Problems 1–5)

MP3: Construct arguments

Justifying assumptions

MP2: Quantitative reasoning

Interpreting biological meaning of parameters

Pedagogical Features
Iterative Modeling Cycle:
Explicitly emphasized:

Transform (build model)
Solve (compute solution)
Interpret (evaluate and refine)

Visual & Data Components:

Page 2: Real tumor data table
Pages 8–11: Graphs comparing models vs. data

Exponential model overestimates growth
Logistic model fits better (Figure 4, page 10)

Ethical Component:

Explicit discussion of:
- Model limitations
- Applicability to humans

Ethical use in medicine