Matching Items (2)
Filtering by
- All Subjects: Hopf bifurcation
- All Subjects: Time Delay
- Creators: Baer, Steven M.
Description
Cancer is a worldwide burden in every aspect: physically, emotionally, and financially. A need for innovation in cancer research has led to a vast interdisciplinary effort to search for the next breakthrough. Mathematical modeling allows for a unique look into the underlying cellular dynamics and allows for testing treatment strategies without the need for clinical trials. This dissertation explores several iterations of a dendritic cell (DC) therapy model and correspondingly investigates what each iteration teaches about response to treatment.
In Chapter 2, motivated by the work of de Pillis et al. (2013), a mathematical model employing six ordinary differential (ODEs) and delay differential equations (DDEs) is formulated to understand the effectiveness of DC vaccines, accounting for cell trafficking with a blood and tumor compartment. A preliminary analysis is performed, with numerical simulations used to show the existence of oscillatory behavior. The model is then reduced to a system of four ODEs. Both models are validated using experimental data from melanoma-induced mice. Conditions under which the model admits rich dynamics observed in a clinical setting, such as periodic solutions and bistability, are established. Mathematical analysis proves the existence of a backward bifurcation and establishes thresholds for R0 that ensure tumor elimination or existence. A sensitivity analysis determines which parameters most significantly impact the reproduction number R0. Identifiability analysis reveals parameters of interest for estimation. Results are framed in terms of treatment implications, including effective combination and monotherapy strategies.
In Chapter 3, a study of whether the observed complexity can be represented with a simplified model is conducted. The DC model of Chapter 2 is reduced to a non-dimensional system of two DDEs. Mathematical and numerical analysis explore the impact of immune response time on the stability and eradication of the tumor, including an analytical proof of conditions necessary for the existence of a Hopf bifurcation. In a limiting case, conditions for global stability of the tumor-free equilibrium are outlined.
Lastly, Chapter 4 discusses future directions to explore. There still remain open questions to investigate and much work to be done, particularly involving uncertainty analysis. An outline of these steps is provided for future undertakings.
In Chapter 2, motivated by the work of de Pillis et al. (2013), a mathematical model employing six ordinary differential (ODEs) and delay differential equations (DDEs) is formulated to understand the effectiveness of DC vaccines, accounting for cell trafficking with a blood and tumor compartment. A preliminary analysis is performed, with numerical simulations used to show the existence of oscillatory behavior. The model is then reduced to a system of four ODEs. Both models are validated using experimental data from melanoma-induced mice. Conditions under which the model admits rich dynamics observed in a clinical setting, such as periodic solutions and bistability, are established. Mathematical analysis proves the existence of a backward bifurcation and establishes thresholds for R0 that ensure tumor elimination or existence. A sensitivity analysis determines which parameters most significantly impact the reproduction number R0. Identifiability analysis reveals parameters of interest for estimation. Results are framed in terms of treatment implications, including effective combination and monotherapy strategies.
In Chapter 3, a study of whether the observed complexity can be represented with a simplified model is conducted. The DC model of Chapter 2 is reduced to a non-dimensional system of two DDEs. Mathematical and numerical analysis explore the impact of immune response time on the stability and eradication of the tumor, including an analytical proof of conditions necessary for the existence of a Hopf bifurcation. In a limiting case, conditions for global stability of the tumor-free equilibrium are outlined.
Lastly, Chapter 4 discusses future directions to explore. There still remain open questions to investigate and much work to be done, particularly involving uncertainty analysis. An outline of these steps is provided for future undertakings.
ContributorsDickman, Lauren (Author) / Kuang, Yang (Thesis advisor) / Baer, Steven M. (Committee member) / Gardner, Carl (Committee member) / Gumel, Abba B. (Committee member) / Kostelich, Eric J. (Committee member) / Arizona State University (Publisher)
Created2020
Description
One explanation for membrane accommodation in response to a slowly rising current, and the phenomenon underlying the dynamics of elliptic bursting in nerves, is the mathematical problem of dynamic Hopf bifurcation. This problem has been studied extensively for linear (deterministic and stochastic) current ramps, nonlinear ramps, and elliptic bursting. These studies primarily investigated dynamic Hopf bifurcation in space-clamped excitable cells. In this study we introduce a new phenomenon associated with dynamic Hopf bifurcation. We show that for excitable spiny cables injected at one end with a slow current ramp, the generation of oscillations may occur an order one distance away from the current injection site. The phenomenon is significant since in the model the geometric and electrical parameters, as well as the ion channels, are uniformly distributed. In addition to demonstrating the phenomenon computationally, we analyze the problem using a singular perturbation method that provides a way to predict when and where the onset will occur in response to the input stimulus. We do not see this phenomenon for excitable cables in which the ion channels are embedded in the cable membrane itself, suggesting that it is essential for the channels to be isolated in the spines.
ContributorsBilinsky, Lydia M (Author) / Baer, Steven M. (Thesis advisor) / Crook, Sharon M (Committee member) / Jackiewicz, Zdzislaw (Committee member) / Gardner, Carl L (Committee member) / Jung, Ranu (Committee member) / Arizona State University (Publisher)
Created2012