The viability of a drug delivery system which encapsulates chemotherapeutic drugs (Doxorubicin) in the hydrophobic core of polymeric micelles and triggers release by ultrasound application was investigated at an applied frequency of 500 kHz. The investigation also included elucidating the mechanism of drug release at 70 kHz, a frequency which had previously been shown to induce drug release. A fluorescence detection chamber was used to measure in vitro drug release from both Pluronic and stabilized micelles and a hydrophone was used to monitor bubble activity during the experiments. A threshold for release between 0.35 and 0.40 in mechanical index was found at 70 kHz and shown to correspond with the appearance of the subharmonic signal in the acoustic spectrum. Additionally, drug release was found to correlate with increase in subharmonic emission. No evidence of drug release or of the subharmonic signal was detected at 500 kHz. These findings confirmed the role of cavitation in ultrasonic drug release from micelles. A mathematical model of a bubble oscillator was solved to explore the differences in the behavior of a single 10 um bubble under 70 and 500 kHz ultrasound. The dynamics were found to be fundamentally different; the bubble follows a period-doubling route to chaos at 500 kHz and an intermittent route to chaos at 70 kHz. It was concluded that this type of "intermittent subharmonic" oscillation is associated with the apparent drug release. This research confirmed the central role of cavitation in ultrasonically-triggered drug delivery from micelles, established the importance of subharmonic bubble oscillations as an indicator, and expounded the key dynamic differences between 70 and 500 kHz ultrasonic cavitation.

This work is concerned with the study of flow stability and turbulence control - two old but still open problems of fluid mechanics. The topics are distinct and are (currently) approached from different directions and with different strategies. This thesis reflects this diversity in subject with a difference in geometry and, consequently, flow structure: the first problem is approached in the study of the flow in a toroidal pipe, the second one in an attempt to reduce the drag in a turbulent channel flow. The flow in a toroidal pipe is chosen as it represents the common asymptotic limit between spatially developing and helical pipes. Furthermore, the torus represents the smallest departure from the canonical straight pipe flow, at least for small curvatures. The interest in this geometry is twofold: it allows us to isolate the effect of the curvature on the flow and to approach straight as well as helical pipes. The analysis features a characterisation of the steady solution as a function of curvature and the Reynolds number. The problem of forcing fluid in the pipe is addressed, and the so-called Dean number is shown to be of little use, except for infinitesimally low curvatures. It is found that the flow is modally unstable and undergoes a Hopf bifurcation that leads to a limit cycle. The bifurcation and the corresponding eigenmodes are studied in detail, providing a complete picture of the instability. The second part of the thesis approaches fluid mechanics from a different perspective: the Reynolds number is too high for a deterministic description and the flow is analysed with statistical tools. The objective is to reduce the friction exerted by a turbulent flow on the walls of a channel, and the idea is to employ a control strategy independent of the small, and Reynolds number-dependent, turbulent scales. The method of choice was proposed by Schoppa &amp; Hussain [Phys. Fluids 10:1049-1051 (1998)] and consists in the imposition of streamwise invariant, large-scale vortices. The vortices are re-implemented as a volume force, validated and analysed. Results show that the original method only gave rise to transient drag reduction while the forcing version is capable of sustained drag reduction of up to 18%. An analysis of the method, though, reveals that its effectiveness decreases rapidly as the Reynolds number is increased. / <p>QC 20161004</p>

Significant progress has been made in understanding different scenarios for disease transmissions and behavior of epidemics in recent years. A considerable amount of work has been done in modeling the dynamics of diseases by systems of ordinary differential equations. But there are very few mathematical models that deal with the genetic mutations of a pathogen. In-fact, not much has been done to model the dynamics of mutations of pathogen explaining its effort to escape the host's immune defense system after it has infected the host. In this dissertation we develop an SIR model with variable infection age for the transmission of a pathogen that can mutate in the host to produce a second infectious mutant strain. We assume that there is a period of temporary immunity in the model. A temporary immunity period along with variable infection age leads to an integro-differential-difference model. Previous efforts on incorporating delays in epidemic models have mainly concentrated on inclusion of latency periods (this assumes that the force of infection at a present time is determined by the number of infectives in the past). We begin with reviewing some basic models. These basic models are the building blocks for the later, more detailed models. Next we consider the model for mutation of pathogen and discuss its implications. Finally, we improve this model for mutation of pathogen by incorporating delay induced by temporary immunity. We examine the influence of delay as we establish the existence, and derive the explicit forms of disease-free, boundary and endemic equilibriums. We will also investigate the local stability of each of these equilibriums. The possibility of Hopf bifurcation using delay as the bifurcation parameter is studied using both analytical and numerical solutions.

In this Ph.D. thesis, we study regular and embedded solitons and generalized and degenerate Hopf bifurcations. These two areas of work are seperate and independent from each other. First, variational methods are employed to generate families of both regular and embedded solitary wave solutions for a generalized Pochhammer PDE and a generalized microstructure PDE that are currently of great interest. The technique for obtaining the embedded solitons incorporates several recent generalizations of the usual variational technique and is thus topical in itself. One unusual feature of the solitary waves derived here is that we are able to obtain them in analytical form (within the family of the trial functions). Thus, the residual is calculated, showing the accuracy of the resulting solitary waves. Given the importance of solitary wave solutions in wave dynamics and information propagation in nonlinear PDEs, as well as the fact that only the parameter regimes for the existence of solitary waves had previously been analyzed for the microstructure PDE considered here, the results obtained here are both new and timely.

Saccular aneurysms are the abnormal plastic deformation of veins and arteries that can lead to lethal thrombus genesis or internal hemorrhaging. Medication and surgery greatly reduce the mortality rates, but treatment is limited by predicting who will develop aneurysms. A common location for saccular aneurysm genesis is at the main middle cerebral artery (MCA) bifurcation. The main MCA bifurcation is comprised of the M1 MCA segment, parent artery, and two M2 segments, daughter arteries. Studies have found that the lateral angle (LA) ratio of the MCA bifurcation is correlated with aneurysm formation. The LA ratio is defined as the angle between the M1 and the larger M2 divided by the angle between the M1 and the smaller M2. When the LA ratio is equal to 1, perfectly symmetrical, no aneurysms are found at the MCA bifurcation. When the LA ratio is greater than 1.6, aneurysms are commonly found at the MCA bifurcation. In the research described here, varying MCA bifurcation angles were compared to uncover any changes to fluid flow and wall shear stress that could stimulate aneurysm growth. Eight pre-aneurysm MCA bifurcation models were created in SolidWorks® using 120 degrees, 90 degrees, and 60 degrees as the angle between the M1 and the larger M2. LA ratios of 1, 1.6 and 2.2 were then used to characterize the other branch angle (60 degrees with a LA ratio of 1 was excluded). These models were imported into COMSOL Multiphysics® where the laminar fluid flow module was used to simulate non-Newtonian blood flow. Fluid flow profiles showed little to no change between the models. Shear stress changed when the LA ratio was increased, but the changed varied between the 120, 90 and 60 degree models. 120 degree models had a 3.87% decrease in max shear stress with a LA ratio of 2.2 while the 90 degree models had 7.5% decrease in max shear stress with a LA ratio of 2.2. Each daughter artery had distinct areas of high shear stress when the LA ratio equaled 1. Increasing the LA ratio or decreasing the bifurcation angle caused the areas of shear stress to merge together. Increasing LA ratio caused shear stress to decrease and spread around the MCA bifurcation. The reduction in max wall shear stress for high LA ratios supports current aneurysm genesis hypothesizes, but additional testing is required before bifurcation geometries can be used to predicted aneurysm genesis.

A bifurcation analysis is used to investigate the complex dynamics of two heavily loaded single-machine-infinite-busbar power systems modeling the characteristics of the BOARDMAN generator with respect to the rest of the North-Western American Power System and the CHOLLA# generator with respect to the SOWARO station. In the BOARDMAN system, we show that there are three Hopf bifurcations at practical compensation values, while in the CHOLLA#4 system, we show that there is only one Hopf bifurcation. The results show that as the compensation level increases, the operating condition loses stability with a complex conjugate pair of eigenvalues of the Jacobian matrix crossing transversely from the left- to the right-half of the complex plane, signifying a Hopf bifurcation. As a result, the power system oscillates subsynchronously with a small limit-cycle attractor. As the compensation level increases, the limit cycle grows and then loses stability via a secondary Hopf bifurcation, resulting in the creation of a two-period quasiperiodic subsynchronous oscillation, a two-torus attractor. On further increases of the compensation level, the quasiperiodic attractor collides with its basin boundary, resulting in the destruction of the attractor and its basin boundary in a bluesky catastrophe. Consequently, there are no bounded motions. When a damper winding is placed either along the q-axis, or d-axis, or both axes of the BOARDMAN system and the machine saturation is considered in the CHOLLA#4 system, the study shows that, there is only one Hopf bifurcation and it occurs at a much lower level of compensation, indicating that the damper windings and the machine saturation destabilize the system by inducing subsynchronous resonance. Finally, we investigate the effect of linear and nonlinear controllers on mitigating subsynchronous resonance in the CHOLLA#4 system . The study shows that the linear controller increases the compensation level at which subsynchronous resonance occurs and the nonlinear controller does not affect the location and type of the Hopf bifurcation, but it reduces the amplitude of the limit cycle born as a result of the Hopf bifurcation. / Ph. D.

Many biological populations reside in increasingly fragmented landscapes, which arise from human activities and natural causes. Landscape characteristics may change abruptly in space and create sharp transitions (interfaces) in landscape quality. How the patchiness of landscapes affects ecosystem diversity and stability depends, among other things, on how individuals move through the landscape. Individuals adjust their movement behavior to local habitat quality and show preferences for some habitat types over others. In this thesis, we focus on how landscape composition and the movement behaviour of individuals at an interface between patches of different quality affect the steady state of a single species. We consider a model of reaction-diffusion equations for the temporal evolution of the density of the population in space. Individual movement is described by a diffusion process, e.g., an uncorrelated random walk. Population net growth is encapsulated in the growth function that considers birth and death of individuals, including nonlinear effects that arise from competition and/or facilitation within the species. We consider the simplest case of two adjacent one-dimensional patches, e.g., two intervals on the real line that share one boundary point. Conditions are homogeneous within a patch but differ between patches. The movement behaviour of individuals between the two patches is incorporated into matching conditions of population flux and density at the interface between patches, i.e., the boundary point that the intervals share. These matching conditions turn out to be continuous in the flux but discontinuous in the density. Several authors have studied similar models recently. Most of these studies consider monostable dynamics on both patches, i.e., logistic growth. Under logistic growth, the net population growth rate is a strictly decreasing function of population density. Logistic population dynamics are very simple: the population extinction state is unstable and a positive steady state is globally asymptotically stable. In this work, we also include bistable dynamics, i.e., an Allee effect. Biologically, an Allee effect occurs when individuals cooperate at some level so that the net population growth rate is increasing with population density for at least some low or intermediate densities. Models with Allee growth typically exhibit bistability: there are two locally stable steady states, one at low density (possibly zero) and one at high density. This bistability makes mathematical analysis more challenging, but leads to more interesting results in return. Mathematically, most existing work on related models is based on linear stability analysis of the extinction state. We focus on the nonlinear models and specifically on positive steady states. We establish the existence, uniqueness and - in some cases - global asymptotic stability of a positive steady state. We classify the shape of these states depending on movement behaviour. We clarify the role of movement in this context. In particular, we investigate the following prior observation: a randomly diffusing population at steady state in a continuously varying habitat can exceed its carrying capacity. Our results clarify when and under which conditions this effect can arise in our two-patch landscape. The analysis of the model with an Allee effect on one of the two patches yields a rich and interesting structure of steady states. Under certain parameter conditions, some of these states are amenable to explicit stability calculations. These yield insights into the possible bifurcations that can occur in our system. Finally, we indicate how the model and analysis here can be extended to systems of reaction-diffusion equations on graphs that represent natural habitats with different geometries, for example watersheds.

The applications of inflatable membrane structures are increasing rapidly in the various fields of engineering and science. The geometric, material, force and contact non-linearities complicate this subject further, which in turn increases the demand for computationally efficient methods and interpretations of counter-intuitive behaviors noted by the  scientific community. To understand the complex behavior of membranes in biological and medical engineering contexts, it is necessary to understand the mechanical behavior of a membrane from a physics point of view.  The first part of the  present work studies the pre-stretched circular membrane in contact with a soft linear substrate. Adhesive and frictionless contact conditions are considered during inflation, while only adhesive contact conditions are considered during deflation. The peeling of membrane during deflation is studied, and a numerical formulation of the energy release rate is proposed. It is observed that the pre-stretch is having a considerable effect on the variation of the energy release rate. In the second part, free and constrained inflation of a cylindrical membrane is investigated. Adhesive and frictionless contact conditions are considered between the membrane and substrate. It is observed that the continuity of principal stretches and stresses depend on contact conditions and the inflation/deflation phase. The adhesive traction developed during inflation and deflation arrests the axial movement of material points, while an adhesive line force created at the contact boundary is responsible for a jump in stretches and stresses at the contact boundary. The pre-stretch produces a softening effect in free and constrained inflation of cylindrical membranes. The third part of the thesis discusses the instabilities observed for fluid containing cylindrical membranes. Both limit points and bifurcation points are observed on equilibrium branches. The secondary branches emerge from bifurcation points, with their directions determined by an eigen-mode injection method. The occurrence of critical points and the stability of equilibrium branches are determined by perturbation techniques. The relationship between eigenvalue analysis and symmetry is highlighted in this part of the thesis. / <p>QC 20150227</p>

Our goal in this thesis is to study the dynamics of the classical predator-prey model and the predator-prey model in the chemostat when a discrete delay is introduced to model the time between the capture of the prey and its conversion to biomass. In both models we use Holling type I response functions so that no oscillatory behavior is possible in the associated system when there is no delay. In both models, we prove that as the parameter modelling the delay is varied Hopf bifurcation can occur. However, we show that there seem to be differences in the possible sequences of bifurcations. Numerical simulations demonstrate that in the classical predator-prey model period doubling bifurcation can occur, possibly leading to chaos while that is not observed in the chemostat model for the parameters we use. For a delay differential equation, a prerequisite for Hopf bifurcation is the existence of a pair of pure imaginary eigenvalues for the characteristic equation associated with the linerization of the system. In this case, the characteristic equation is a transcendental equation with delay dependent coefficients. For our models, we develop two different methods to show how to find values of the bifurcation parameter at which pure imaginary eigenvalues occur. The method used for the classical predator-prey model was developed first. However, it was necessary to develop a more robust, less complicated method to analyze the predator-prey model in the chemostat with a discrete delay. The latter method was then generalized so that it could be applied to any second order transcendental equation with delay dependent coefficients. / Thesis / Doctor of Philosophy (PhD)

Genome-wide studies of diseases and chronic conditions frequently fail to uncover marked or consistent differences in RNA or protein concentrations. However, the developing field of kinetic proteomics has made promising discoveries in differences in the turnover rate of these same proteins, even when concentrations were not necessarily different. The situation can theoretically be modeled mathematically using bifurcation equations, but uncovering the proper form of these is difficult. To this end, we developed TWIG, a method for characterizing bifurcations that leverages information geometry to identify drivers of complex systems. Using this, we characterized the bifurcation and stability properties of all 132 possible 3- and 22,662 possible 4-node subgraphs (motifs) of protein-protein interaction networks. Analyzing millions of real world protein networks indicates that natural selection has little preference for motifs that are stable per se, but a great preference for motifs who have parameter regions that are exclusively stable, rather than poorly constrained mixtures of stability and instability. We apply this knowledge to mice on calorie restricted (CR) diets, demonstrating that changes in their protein turnover rates do indeed make their protein networks more stable, explaining why CR is the most robust way known to extend lifespan.