Date of Award

2025

Degree Type

Open Access Dissertation

Degree Name

Mathematics, PhD

Program

Institute of Mathematical Sciences

Advisor/Supervisor/Committee Chair

Ali Nadim & Marina Chugunova

Dissertation or Thesis Committee Member

Alfonso Castro

Dissertation or Thesis Committee Member

Chiu-Yen Kao

Terms of Use & License Information

Terms of Use for work posted in Scholarship@Claremont.

Rights Information

© 2025 Belgacem Al-Azem

Keywords

Cancer tumor evolution, Mathematical biology dynamics, Cancer phenotype

Subject Categories

Applied Mathematics | Mathematics

Abstract

Several mathematical models of cancer tumor evolution are presented. The aim of this endeavor is to write a thesis that fosters numerical and analytical understanding of certain mathematical models in the area of oncology as well as to offer potential predictive tools of therapeutic and clinical applicability. In this work, we give a panoramic view of the cancer phenotype, for to understand cancer, we need to "see" its biophysics. Indeed, the discovery of oncogenes led to the thinking that cancer tumor is perhaps a genetic disease. But the role of angiogenesis and other microenvironment-related discoveries (such as the role of the immune system in oncogenesis) suggest that it is more likely that tumor develops due to genetic aberrations within a suitable microenvironment. The go-to mathematical model in oncology has been the Fisher-KPP model, named after Ronald Fisher, Andrey Kolmogorov, Ivan Petrovsky, and Nikolai Piskunov. We start with a variation of this model and design three different ones that include contributions from Greenspan Theory and Keller-Segel modeling framework as shown in concept map. In model I we maintain a strict observer position; we do not interfere, we simply observe the tumor evolution based on our assumptions. The model performs a “passive spread”: cancer cells behave like ink diffusing in water; they wander randomly, divide when nutrients are available, and die naturally. Every field (cancer cells, dead matter, nutrients) simply spreads out from where it is released and smooths itself over time. This is the picture you get if cells have no “intentional” motility—only Brownian jostling—and the micro-environment does not push or pull them in any preferred direction. In model II we still maintain neutrality but allow for the microenvironment to direct cells’ motility. Here cancer cells experience “active drift” and a sense of direction: cells ’feel’ pressure in packed regions and drift outward, and sniff out richer nutrient pockets and swim toward them. Together, these rules act like tiny GPS instructions layered on top of the earlier random walk. Now the model mimics real tumour cells that migrate up nutrient gradients and away from mechanical stress—behavior often seen in vivo during invasion and metastasis. In model III, we do away with neutrality and introduce directed motility, using chemotaxis agents, as well as localized cytotoxicities to eradicate the cancerous tumor. Greenspan-type avascular spheroids introduce nutrient diffusion, proliferation–quiescence–necrosis zoning, and fixed Dirichlet nutrient at infinity. Our model III sits here but simplifies Greenspan’s three-phase description (viable, quiescent, necrotic) to a logistic–necrotic duality and keeps a fixed spatial domain instead of a moving free boundary. Likewise, the Keller-Segel framework, originally proposed for Dictyostelium discoideum aggregation, couples the population density of motile cells, n ( x , t ), with the concentration of a signaling chemical, c ( x , t ). Our model III sits between those two models as shown in concept map. Our main contributions reside in the theory behind our model III–designing a mathematical model that utilizes chemotaxis in combination with cytotoxicity to draw cancer cells towards the core of the tumor domain and apply therapeutic agents to degrade the tumor or eradicate it. We humbly believe that chemotaxis is a fundamental process that can be strategically manipulated to improve cancer treatment outcomes without damaging unaffected or critically sensitive non-targeted parts of tissues.

ISBN

9798291577684

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