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In this Thesis we develop mathematical models to analyze two proposed causative mechanisms for the ventricular expansion observed in hydrocephalus: cerebrospinal fluid pulsations and small transmantle pressure gradients. To begin, we describe a single compartment model and show that such simple one-dimensional models cannot represent the complex dynamics of the brain. Hence, all subsequent models of this Thesis are spatio-temporal. Next, we develop a poroelastic model to analyze the fluid-solid interactions caused by the pulsations. Periodic boundary conditions are applied and the system is solved analytically for the tissue displacement, pore pressure, and fluid filtration. The model demonstrates that fluid oscillates across the brain boundaries. We develop a pore flow model to determine the shear induced on a cell by this fluid flow, and a comparison with data indicates that these shear forces are negligible. Thus, only the material stresses remain as a possible mechanism for tissue damage and ventricular expansion. In order to analyze the material stresses caused by the pulsations, we develop a fractional order viscoelastic model based on the linear Zener model. Boundary conditions appropriate for infants and adults are applied and the tissue displacement and stresses are solved analytically. A comparison of the tissue stresses to tension data indicates that these stresses are insufficient to cause tissue damage and thus ventricular expansion. Using age-dependent data, we then determine the fractional Zener model parameter values for infant and adult cerebra. The predictions for displacement and stresses are recomputed and the infant displacement is found to be unphysical. We propose a new infant boundary condition which reduces the tissue displacement to a physically reasonable value. The model stresses, however, are unchanged and thus the pulsation-induced stresses remain insufficient to cause tissue damage and ventricular expansion. Lastly, we develop a fractional hyper-viscoelastic model, based on the Kelvin- Voigt model, to obtain large deformation predictions. Using boundary conditions and parameter values for infants, we determine the finite deformation caused by a small pressure gradient by summing the small strain deformation resulting from pressure gradient increments. This iterative technique predicts that pediatric hydrocephalus may be caused by the long-term existence of small transmantle pressure gradients. We conclude the Thesis with a discussion of the results and their implications for hydrocephalus research as well as a discussion of future endeavors.

In this Thesis we develop mathematical models to analyze two proposed causative mechanisms for the ventricular expansion observed in hydrocephalus: cerebrospinal fluid pulsations and small transmantle pressure gradients. To begin, we describe a single compartment model and show that such simple one-dimensional models cannot represent the complex dynamics of the brain. Hence, all subsequent models of this Thesis are spatio-temporal. Next, we develop a poroelastic model to analyze the fluid-solid interactions caused by the pulsations. Periodic boundary conditions are applied and the system is solved analytically for the tissue displacement, pore pressure, and fluid filtration. The model demonstrates that fluid oscillates across the brain boundaries. We develop a pore flow model to determine the shear induced on a cell by this fluid flow, and a comparison with data indicates that these shear forces are negligible. Thus, only the material stresses remain as a possible mechanism for tissue damage and ventricular expansion. In order to analyze the material stresses caused by the pulsations, we develop a fractional order viscoelastic model based on the linear Zener model. Boundary conditions appropriate for infants and adults are applied and the tissue displacement and stresses are solved analytically. A comparison of the tissue stresses to tension data indicates that these stresses are insufficient to cause tissue damage and thus ventricular expansion. Using age-dependent data, we then determine the fractional Zener model parameter values for infant and adult cerebra. The predictions for displacement and stresses are recomputed and the infant displacement is found to be unphysical. We propose a new infant boundary condition which reduces the tissue displacement to a physically reasonable value. The model stresses, however, are unchanged and thus the pulsation-induced stresses remain insufficient to cause tissue damage and ventricular expansion. Lastly, we develop a fractional hyper-viscoelastic model, based on the Kelvin- Voigt model, to obtain large deformation predictions. Using boundary conditions and parameter values for infants, we determine the finite deformation caused by a small pressure gradient by summing the small strain deformation resulting from pressure gradient increments. This iterative technique predicts that pediatric hydrocephalus may be caused by the long-term existence of small transmantle pressure gradients. We conclude the Thesis with a discussion of the results and their implications for hydrocephalus research as well as a discussion of future endeavors.

The traditional view of cancer asserts that a malignant tumour is composed of a population of cells, all of which share the ability to divide without limit. Within the last decade, however, this notion has lost ground to the emerging cancer stem cell hypothesis, which counters that only a (typically small) sub-population of so-called `cancer stem cells' has the capacity to proliferate indefinitely, and hence to drive and maintain tumour growth. Cancer stem cells have been putatively identified in leukemias and, more recently, in a variety of solid tumours including those of the breast and brain. The cancer stem cell hypothesis helps to explain certain clinically-observed phenomena, including the apparent inability of conventional anti-cancer therapies to eradicate the disease despite (transient) reduction of overall tumour bulk -- presumably these treatments fail to kill the underlying cancer stem cells. Herein, we develop stochastic and deterministic temporal models of tumour growth based on the cancer stem cell hypothesis, and apply these models to discussions of the treatment of glioblastoma multiforme, a common type of brain cancer believed to be maintained by cancer stem cells, and to the phenomenon of the epithelial-mesenchymal transition, a process thought to be important in generating cancer cells capable of metastasis.

The traditional view of cancer asserts that a malignant tumour is composed of a population of cells, all of which share the ability to divide without limit. Within the last decade, however, this notion has lost ground to the emerging cancer stem cell hypothesis, which counters that only a (typically small) sub-population of so-called `cancer stem cells' has the capacity to proliferate indefinitely, and hence to drive and maintain tumour growth. Cancer stem cells have been putatively identified in leukemias and, more recently, in a variety of solid tumours including those of the breast and brain. The cancer stem cell hypothesis helps to explain certain clinically-observed phenomena, including the apparent inability of conventional anti-cancer therapies to eradicate the disease despite (transient) reduction of overall tumour bulk -- presumably these treatments fail to kill the underlying cancer stem cells. Herein, we develop stochastic and deterministic temporal models of tumour growth based on the cancer stem cell hypothesis, and apply these models to discussions of the treatment of glioblastoma multiforme, a common type of brain cancer believed to be maintained by cancer stem cells, and to the phenomenon of the epithelial-mesenchymal transition, a process thought to be important in generating cancer cells capable of metastasis.