Spatial Tumor Growth Using Traveling Waves
College
College of Arts and Sciences
Department
Mathematics
Faculty Mentor
Honors Thesis Committee: Kristen Abernathy, Ph.D.; Zachary Abernathy, Ph.D.; Trent Kull, Ph.D
Abstract
Cancer can be found in many forms, including clumps of cancerous cells known as tumors. Malignant tumors can be found anywhere in the body; when tumors reach the point of angiogenesis, they can grow without restraint other than blood supply, and can be modeled using a system of partial differential equations (PDE). In this paper, we consider the PDE model of McGillan, which elaborates on the model by Gatenby and Gawlinski, the first model to incorporate the acid-mediation hypothesis. The McGillan model incorporates competition between tumor cells and healthy cells into the Gatenby and Gawlinski model. We modify the McGillan model by incorporating the cancer stem cell hypothesis, introducing and modifying an equation for the cancer stem cells. We then go on to numerically solve for four solutions or tumor growth states – a homogeneous invasion state, a heterogeneous invasion state, a cure state, and a stagnant state – all of which were also derived in the McGillan model. We then explore the stability of the solutions, some of which are bistable.
Previously Presented/Performed?
Joint Mathematics Meeting (JMM), Atlanta, Georgia, January 2017
Start Date
21-4-2017 1:30 PM
Spatial Tumor Growth Using Traveling Waves
DiGiorgio Campus Center, Room 220
Cancer can be found in many forms, including clumps of cancerous cells known as tumors. Malignant tumors can be found anywhere in the body; when tumors reach the point of angiogenesis, they can grow without restraint other than blood supply, and can be modeled using a system of partial differential equations (PDE). In this paper, we consider the PDE model of McGillan, which elaborates on the model by Gatenby and Gawlinski, the first model to incorporate the acid-mediation hypothesis. The McGillan model incorporates competition between tumor cells and healthy cells into the Gatenby and Gawlinski model. We modify the McGillan model by incorporating the cancer stem cell hypothesis, introducing and modifying an equation for the cancer stem cells. We then go on to numerically solve for four solutions or tumor growth states – a homogeneous invasion state, a heterogeneous invasion state, a cure state, and a stagnant state – all of which were also derived in the McGillan model. We then explore the stability of the solutions, some of which are bistable.
