In radiation therapy, interfraction organ motion introduces a level of geometric uncertainty into the planning process. Plans, which are typically based upon a single instance of anatomy, must be robust against daily anatomical variations. For this problem, a model of the magnitude, direction, and likelihood of deformation is useful. In this thesis, principal component analysis (PCA) is used to statistically model the 3D organ motion for 19 prostate cancer patients, each with 8-13 fractional computed tomography (CT) images. Deformable image registration and the resultant displacement vector fields (DVFs) are used to quantify the interfraction systematic and random motion. By applying the PCA technique to the random DVFs, principal modes of random tissue deformation were determined for each patient, and a method for sampling synthetic random DVFs was developed. The PCA model was then extended to describe the principal modes of systematic and random organ motion for the population of patients. A leave-one-out study tested both the systematic and random motion model’s ability to represent PCA training set DVFs. The random and systematic DVF PCA models allowed the reconstruction of these data with absolute mean errors between 0.5-0.9 mm and 1-2 mm, respectively. To the best of the author’s knowledge, this study is the first successful effort to build a fully 3D statistical PCA model of systematic tissue deformation in a population of patients. By sampling synthetic systematic and random errors, organ occupancy maps were created for bony and prostate-centroid patient setup processes. By thresholding these maps, PCA-based planning target volume (PTV) was created and tested against conventional margin recipes (van Herk for bony alignment and 5 mm fixed [3 mm posterior] margin for centroid alignment) in a virtual clinical trial for low-risk prostate cancer. Deformably accumulated delivered dose served as a surrogate for clinical outcome. For the bony landmark setup subtrial, the PCA PTV significantly (p30, D20, and D5 to bladder and D50 to rectum, while increasing rectal D20 and D5. For the centroid-aligned setup, the PCA PTV significantly reduced all bladder DVH metrics and trended to lower rectal toxicity metrics. All PTVs covered the prostate with the prescription dose.

Auditory physiology is nearly unique in the human body because of its small-diameter neurons. When considering a single node on one neuron, the number of channels is very small, so ion fluxes exhibit randomness. Hodgkin and Huxley, in 1952, set forth a system of Ordinary Differential Equations (ODEs) to track the flow of ions in a squid motor neuron, based on a circuit analogy for electric current. This formalism for modeling is still in use today and is useful because coefficients can be directly measured. To measure auditory properties of Firing Efficiency (FE) and Post Stimulus Time (PST), we can simply measure the depolarization, or "upstroke," of a node. Hence, we reduce the four-dimensional squid neuron model to a two-dimensional system of ODEs. The stochastic variable m for sodium activation is allowed a random walk in addition to its normal evolution, and the results are drastic. The diffusion coefficient, for spreading, is inversely proportional to the number of channels; for 130 ion channels, D is closer to 1/3 than 0 and cannot be called negligible. A system of Partial Differential Equations (PDEs) is derived in these pages to model the distribution of states of the node with respect to the (nondimensionalized) voltage v and the sodium activation gate m. Initial conditions describe a distribution of (v,m) states; in most experiments, this would be a curve with mode at the resting state. Boundary conditions are Robin (Natural) boundary conditions, which gives conservation of the population. Evolution of the PDE has a drift term for the mean change of state and a diffusion term, the random change of state. The phase plane is broken into fired and resting regions, which form basins of attraction for fired and resting-state fixed points. If a stimulus causes ions to flow from the resting region into the fired region, this rate of flux is approximately the firing rate, analogous to clinically measuring when the voltage crosses a threshold. This gives a PST histogram. The FE is an integral of the population over the fired region at a measured stop time after the stimulus (since, in the reduced model, when neurons fire they do not repolarize). This dissertation also includes useful generalizations and methodology for turning other ODEs into PDEs. Within the HH modeling, parameters can be switched for other systems of the body, and may present a similar firing and non-firing separatrix (as in Chapter 3). For any system of ODEs, an advection model can show a distribution of initial conditions or the evolution of a given initial probability density over a state space (Chapter 4); a system of Stochastic Differential Equations can be modeled with an advection-diffusion equation (Chapter 5). As computers increase in speed and as the ability of software to create adaptive meshes and step sizes improves, modeling with a PDE becomes more and more efficient over its ODE counterpart.

<p>This report presents an analysis of a partial differential equation, resulting from population model with age structure. The existence and uniqueness of a solution to the equation are proved. We look at stability of the solution. The asymptotic behaviour of the solution is treated. The report also contains a section about the connection between the solution to the age structure population model and a simple model without age structure.</p>

The present study explored the spatial and temporal demographic trends in oyster population dynamics and their relationships to freshwater inflows and the pathogen Dermo (Perkinsus marinus) on three reefs (Shell, Mad Island, and Sammy?s) in West Matagorda Bay, Texas. The objectives were to design and link three population models that simulate oyster population dynamics and integrate the environmental factors that influence growth, reproduction and settlement of larvae among these three reefs. The following variables were evaluated: relative abundance of oyster spat, submarket- and market-size oysters, average weighted incidence of Dermo and percent Dermo infection (prevalence) in submarket- and market-size oysters and their relationships to environmental variables of salinity, temperature, flow and distance from freshwater sources. Using a 30-month continuous dataset, environmental variables accounted for 36% of the variation in Dermo-related variables among the three reefs, and were also positively correlated with distance from freshwater sources. The relative abundance of spat and dead oysters was related to peaks in freshwater inflows occurring 30 days prior to larval settlement. Using these spatial and temporal relationships among biological and environmental variables, and data from five years of monitoring three reefs in Matagorda Bay, an integrated Stella model was developed that simulated oyster population responses to stochastic environmental changes over a 50-year period. Although the geological and structural complexity of each reef appeared to be similar, the model showed the relationship of growth, spawning and spat set were related to hydrologic variation between different reefs and time periods. The model revealed that up-estuary reefs relied on the distribution of larvae from down-estuary reefs following mortality related to freshwater inflow. The model also indicated that loss of freshwater inflows to down-estuary reefs resulted in higher sustained Dermo infections, thus loss of spawning potential and subsequent distribution of larvae to up-estuary reefs. The three oyster populations in West Matagorda Bay provide spawning connectivity and function as an integrated resource for sustaining all oyster reef populations in this bay system. The model presented in this research provides a basis for understanding the population dynamics of WMB as well as a better understanding of the interaction among the reefs that sustain these populations. The model developed in this investigation provides a basis for developing oyster population models for other bay systems and for future research regarding hydrologic influences on oyster population dynamics.

This report presents an analysis of a partial differential equation, resulting from population model with age structure. The existence and uniqueness of a solution to the equation are proved. We look at stability of the solution. The asymptotic behaviour of the solution is treated. The report also contains a section about the connection between the solution to the age structure population model and a simple model without age structure.

I examined the population trends of northern fur seals (Callorhinus ursinus) using an age-specific metapopulation model that allowed migration between rookeries. Mortality and birth rates were modified to simulate future population trends. I also examined the early migratory behavior and habitat associations of pups from Bering Island (BI), Russia. I instrumented 35 pups with Mk10-AL satellite tags and stomach temperature telemeters which provided diving, foraging and location data. I hypothesized that some aspects of pup behavior from the stable BI population differed from the behavior of pups from the unstable Pribilof Islands (PI). The population model revealed that emigration did not contribute significantly to the current PI population decline. However, large source populations contributed significantly to population growth in newly colonized rookeries. A stabilization of the PI population was predicted with a 10 to 20 percent reduction in both juvenile and adult female mortality rates. The diving behavior of pups showed a general progression towards longer and deeper dives as pups aged, particularly between 1600 – 0400 (local time), that was similar to PI pup behavior. However, unlike pups from the PI, I found three main diving strategies among BI pups: 1) shallow daytime divers (mean depth = 3.56 m), 2) deep daytime divers (mean depth = 6.36 m) and 3) mixed divers (mean depth = 4.81 m). The foraging behavior of pups showed that most successful ingestion events occurred between 1600 – 0400, with successful ingestion events lasting 25.36 plus/minus 27.37 min. There was no significant difference among the three strategies in the depth of successful foraging dives. I also examined the foraging search strategies in adult females and pups. Both pups and adults conducted Levy walks, although pups foraged in smaller patches (1 km scales). Using a logistic model to determine habitat associations, I found that pup locations were positively correlated with increasing chlorophyll a concentrations, distances from shore, and sea surface temperatures, and were negatively correlated with depth. There was no significant relationship between all pup locations and the regions (peripheries or centers) or types (cyclonic or anti-cyclonic) of eddies, but ingestion event locations were related to mesoscale eddy peripheries.

The present study explored the spatial and temporal demographic trends in oyster population dynamics and their relationships to freshwater inflows and the pathogen Dermo (Perkinsus marinus) on three reefs (Shell, Mad Island, and Sammy?s) in West Matagorda Bay, Texas. The objectives were to design and link three population models that simulate oyster population dynamics and integrate the environmental factors that influence growth, reproduction and settlement of larvae among these three reefs. The following variables were evaluated: relative abundance of oyster spat, submarket- and market-size oysters, average weighted incidence of Dermo and percent Dermo infection (prevalence) in submarket- and market-size oysters and their relationships to environmental variables of salinity, temperature, flow and distance from freshwater sources. Using a 30-month continuous dataset, environmental variables accounted for 36% of the variation in Dermo-related variables among the three reefs, and were also positively correlated with distance from freshwater sources. The relative abundance of spat and dead oysters was related to peaks in freshwater inflows occurring 30 days prior to larval settlement. Using these spatial and temporal relationships among biological and environmental variables, and data from five years of monitoring three reefs in Matagorda Bay, an integrated Stella model was developed that simulated oyster population responses to stochastic environmental changes over a 50-year period. Although the geological and structural complexity of each reef appeared to be similar, the model showed the relationship of growth, spawning and spat set were related to hydrologic variation between different reefs and time periods. The model revealed that up-estuary reefs relied on the distribution of larvae from down-estuary reefs following mortality related to freshwater inflow. The model also indicated that loss of freshwater inflows to down-estuary reefs resulted in higher sustained Dermo infections, thus loss of spawning potential and subsequent distribution of larvae to up-estuary reefs. The three oyster populations in West Matagorda Bay provide spawning connectivity and function as an integrated resource for sustaining all oyster reef populations in this bay system. The model presented in this research provides a basis for understanding the population dynamics of WMB as well as a better understanding of the interaction among the reefs that sustain these populations. The model developed in this investigation provides a basis for developing oyster population models for other bay systems and for future research regarding hydrologic influences on oyster population dynamics.

Soybean aphid, Aphis glycines Matsumura, is an important pest of North American soybean. This dissertation identifies and addresses knowledge gaps, and integrates existing knowledge regarding distribution and abundance of this species. Early summer soybean colonization patterns by A. glycines were examined relative to landscape parameters, including density of overwintering hosts (buckthorn). An information-theoretic model selection approach was used to determine which landscape parameters were most influential in the distribution of colonizing aphids. Though buckthorn abundance best explained aphid colonization and population density, a density-dependent effect was observed. When aphid populations were low, more aphids were found in the vicinity of buckthorn, when aphid populations were higher, more aphids were found farther from buckthorn. Suction trap captures of migrating populations of A. glycines from 2005-2009 from a suction trap network covering much of central North America were examined. A model selection approach was used to determine the environmental triggers of summer and fall aphid flights, and spatial analysis and modeled wind trajectories were used to examine patterns in the abundance of alates. Two alate activity peaks were observed in fall. In summer, formation of alates was a function of field infestation. A tritrophic population model was built using DYMEX, a mechanistic lifecycle based modeling software package. The model incorporated soybean, A. glycines, and three natural enemy species, interacting based on phenological, physiological and functional response data available in the literature. The model was validated using Ontario field data, and several simulations were performed and are discussed. An evaluation of proposed control strategies for efficacy and impact on natural enemies and the environment was conducted. Two novel concepts are presented: the natural enemy unit, a standardization of the impact of predator guild on prey populations by the number of prey an individual predator can eat, and the selectivity index, where the selectivity of a pesticide is a function of the change in ratio of natural enemy units to prey before and after treatment. The selectivity index was inversely correlated with the Environmental Impact Quotient (EIQ), a theoretical measure of impact, validating EIQ's field applicability. / Natural Sciences and Engineering Research Council of Canada; The Keefer family trust; the Mary Edmunds Williams trust, the family of Fred W. Presant, and the University of Guelph provided scholarship and fellowship funds. Research was funded by a grant to Rebecca Hallett and Art Schaafsma from the Agriculture and Agri-Food Canada Pest Management Centre’s Pesticide Risk Reduction Program.

Elk (Cervus elaphus) in eastern Kentucky appear to have increased in number since reintroduction in 1997, but rugged landscapes and cryptic elk behavior have precluded use of typical population survey methods to accurately estimate population size. In December 2006, I used forward-looking infrared radiography (FLIR) to survey the elk population in eastern Kentucky. Elk locations identified by FLIR were used to create a landscape based model to estimate the density distribution of elk within a 7,088 km2 core area of the elk restoration zone. FLIR detected 76% of elk groups of < 10 individuals and 100% of elk groups of ≥ 10 individuals. The density of elk was positively associated with the amount of herbaceous area, herbaceous edge, herbaceous area weighted mean patch fractal dimensions, proximity to release sites, the number of elk released at each site and urban core area index, and negatively associated with road density. My model estimated the elk population at 7,001 (SE = 772, 95% CI = 5,488- 8,514) individuals within the core area, 53% of which were < 10 km from release sites. The predicted elk distribution pattern and abundance estimate derived from this model will be important for wildlife managers in successfully managing the Kentucky elk population.

Magister Scientiae - MSc / We review a number of compartmental models in epidemiology which leads to a nonlinear system of ordinary differential equations. We focus an SIR, SEIR and SIS epidemic models with and without vaccination. A threshold parameter R0 is identified which governs the spread of diseases, and this parameter is known as the basic reproductive number. The models have at least two equilibria, an endemic equilibrium and the disease-free equilibrium. We demonstrate that the disease will die out, if the basic reproductive number R0 < 1. This is the case of a disease-free state, with no infection in the population. Otherwise the disease may become endemic if the basic reproductive number R0 is bigger than unity. Furthermore, stability analysis for both endemic and disease-free steady states are investigated and we also give some numerical simulations. The second part of this dissertation deals with optimal vaccination strategy in epidemiology. We use optimal control technique on vaccination to minimize the impact of the disease. Hereby we mean minimizing the spread of the disease in the population, while also minimizing the effort on vaccination roll-out. We do this optimization for the cases of SIR and SEIR models, and show how optimal strategies can be obtained which minimize the damage caused by the infectious disease. Finally, we describe the numerical simulations using the fourth-order Runge-Kutta method. These are the most useful references: [G. Zaman, Y.H Kang, II. H. Jung. BioSystems 93, (2008), 240 − 249], [K. Hattaf, N. Yousfi. The Journal of Advanced Studies in Biology, Vol. 1(8), (2008), 383 − 390.], [Lenhart, J.T. Workman. Optimal Control and Applied to Biological Models. Chapman and Hall/CRC, (2007).], [P. Van den Driessche, J. Watmough. Math. Biosci., 7, (2005)], and [J. Wu, G. R¨ost. Mathematical Biosciences and Engineering, Vol 5(2), (2008), 389 − 391]. / South Africa