[Truncated abstract] The activity of the reproductive endocrine axis is the result of interactions among many organs and tissues, particularly the hypothalamus, pituitary gland and gonad. However, it depends on more than the communication between anatomical structures because it is also affected by genotype, internal factors (e.g., metabolic inputs) and external factors (e.g., photoperiod, socio-sexual cues, stress, nutrition). This multifactorial complexity makes it difficult to use animal experimentation to investigate the pathways and mechanisms involved. Therefore, in this study, I have turned to mathematical modelling. The general hypothesis was that, by modelling the hormonal feedback loop that links the hypothalamus, pituitary gland and gonad, I would be able to discover the critical control points in this homeostatic system. This would allow me to inform and direct research into the processes that control reproduction, including inputs from environmental factors. My studies began with the development of a model of the negative feedback loop through which testosterone controls the secretion of pulses of gonadotrophin-releasing hormone (GnRH) by the hypothalamus. The model incorporated two critical factors: testosterone concentration and a time delay in the inhibition of the activity of the GnRH 'pulse generator' by testosterone. The general assumptions were: i) there are two positive feedforward processes (GnRH pulses stimulate LH pulses, and, in turn, LH pulses stimulate testosterone secretion); ii) testosterone exerts negative feedback that reduces the frequency of GnRH pulses. The model incorporated a group of equations that represent the GnRH pulse generator, through which the inhibitory effect of testosterone acted to reduce GnRH pulse frequency. Simulations were run with various values for the time delay in feedback and, as model development progressed, the simulations were extended to include combinations of time delays and levels of sensitivity of the GnRH pulse generator to inhibition by testosterone. The output of the simulations showed clearly that a time delay in negative feedback, as well as the concentration of testosterone, can greatly affect the frequency of GnRH pulses and the shape of the GnRH secretory profile. Importantly, the effect of the time delay depends on the sensitivity of the pulse generator to testosterone. In addition, the simulations suggested two additional components that might be involved in the control of the GnRH pulse generator: i) a delay in the rate of adaptation to a change in steroid feedback; and ii) a minimum pulse interval (maximum frequency). These studies iii therefore suggest that the regulation of the activity of the GnRH pulse generator, and thus the frequency and profile of GnRH and LH pulses, requires interactions among these four components. These interactions should be tested in animal experimentation. In the next stage, I extended the model so I could test whether the feedback delay might involve the process of aromatization in which testosterone is converted to oestradiol at brain level. ... This information can be used to direct future experimental studies that will help us to understand the factors that underlie the dynamic behaviour of the hypothalamic and pituitary systems that control reproduction.

數學模型可被用於癌症之化療研究。一個著名的例子為學者Goldie和Coldman在1979年發表了第一個描述癌症化療中，腫瘤細胞突變率及其與治療藥物反應關聯性之數學模型。此一模型因對此問題之描述簡潔與優雅，廣為其他學者引用。Goldie和Coldman(經與Guaduskas合作)隨後於1982年利用此模型配合模擬方法說明在沒有交互抗藥性的治療中，就避免腫瘤細胞發生多重抗藥性突變而言，為何交替使用治療藥物為最佳治療方式。其後更在1983年，於考慮隨機特性下，推廣原有模型，並考慮此推廣模型之近似表示時，以嚴格數學方法證明其於1982年以模擬方法所得之結論。 然而，Goldie和Coldman之理論分析工作多集中於模型參數具有對稱結構之情形，而關於模型參數不具對稱結構時，文獻中少有理論分析之探討。於此一論文中，我們重新以多階段最佳化問題表達此一問題，並考慮模型參數不完全對稱下，最佳治療方式所應滿足之條件。根據我們提出的架構，可求得不完全對稱下最佳治療方式之解析解。此外，Goldie和Coldman關於模型參數具對稱結構之工作可視為我們架構下之一特例。因此，我們的架構提供Goldie和Coldman理論分析工作一個新的數學證明方法。本文除理論推導外，並以數值方法進行案例分析，以驗證我們工作之正確性。 / Mathematical models can be applied to study the chemotherapies on tumor cells. Espeically, in 1979, Goldie and Coldman proposed the first mathematical model to relate the drug sensitivity of tumors to their mutation rates. This pioneering work is subsequently referred by many scientists due to its simplicity and elegancy. The authors (jointly with Guaduskas) later used their model to explain why alternating non-crossresistant chemotherapy is optimal with simulation approach. Subsequently in 1983, they proposed an extended stochastic based model and provided a rigorous mathematical proof to their earlier simulation work when the extended model is approximated by its quasi-approximation. However, Goldie and Coldman’s analytic work on optimal treatments majorly focuses on process with symmetrical parameter settings. Little theoretical results on asymmetrical settings are discussed. In this thesis, we recast and restate Goldie, Coldman and Guaduskas’ model as a multi-stage optimization problem. Under an asymmetrical assumption, conditions under which a treatment policy can be optimal are derived. This framework enables us to consider some optimal policies on the model analytically. In addition, Goldie, Coldman and Guaduskas’ work with symmetrical settings can be treated as a special case of our framework. Base on the derived conditions, an alternative proof to Goldie and Coldman’s work is provided. In addition to the theoretical derivation, numerical results are included to justify the correctness of our work.

The electric power sector is undergoing significant changes with the development of Smart Grid technologies and is rapidly influencing the way we consume energy. Demand Response (DR) is an important element in the emerging smart grid paradigm and is paving way for the more sophisticated implementation of Energy Hub Management Systems (EHMSs). Utilities are looking at Demand Side Management (DSM) and DR services that allow customers to make informed decisions regarding their energy consumption which in return, can help the energy providers to reduce their peak demand and hence enhance grid sustainability. Ice rinks are large commercial buildings which facilitate various activities such as hockey, figure skating, curling, recreational skating, public arenas, auditoriums and coliseums. These have a complex energy system; in which an enormous sheet of ice is maintained at a low temperature while at the same time the spectator stands are heated to ensure comfortable conditions for the spectators. Since indoor ice rinks account for a significant share of the commercial sector and are in operation for more than 8 months a year, their contribution in the total demand cannot be ignored. There is significant scope for energy savings in indoor ice rinks through optimal operation of their climate control systems. In this work, a mathematical model of indoor ice rinks for the implementation of EHMS is developed. The model incorporates weather forecast, electricity price information and end-user preferences as inputs and the objective is to shift the operation of climate control devices to the low electricity price periods, satisfying their operational constraints while having minimum impact on spectator comfort. The inside temperature and humidity dynamics of the spectator area are modeled to reduce total electrical energy costs while capturing the effect of climate control systems including radiant heating system, ventilation system and dehumidification system. Two different pricing schemes, Real Time Pricing (RTP) and Time-of-Use (TOU), are used to assess the model, and the resulting energy costs savings are compared. The expected energy cost savings are evaluated for a 8 month period of operation of the rink incorporating the uncertainties in electricity price, weather conditions and spectator schedules through Monte Carlo simulations. The proposed work can be implemented as a supervisory control in existing climate controllers of indoor ice rinks and would play a significant role in the enforcement of EHMS in Smart Grids.

Over time astrocytes have been thought to function in an auxiliary manner, providing neurons with metabolic and structural support. However, recent research suggests they may play a fundamental role in the generation and propagation of focal epileptic seizures by causing synchronized electrical bursts in neurons. It would be helpful to have a simple mathematical model that represents this dynamic and incorporates these updated experimental results. We have created a two-compartment model of a typical neuron found in the hippocampal CA1 region, an area often thought to be the origin of these seizures. The focus is on properly modeling the astrocytic input to examine the pathological excitation of these neurons and subsequent transmission of the signals. In particular, we consider the intracellular astrocytic calcium fluctuations which are associated with slow inward currents in neighbouring neurons. Using our model, a variety of experimental results are reproduced, and comments are made about the potential differences between graded and “all-or-none” astrocytes.

Over time astrocytes have been thought to function in an auxiliary manner, providing neurons with metabolic and structural support. However, recent research suggests they may play a fundamental role in the generation and propagation of focal epileptic seizures by causing synchronized electrical bursts in neurons. It would be helpful to have a simple mathematical model that represents this dynamic and incorporates these updated experimental results. We have created a two-compartment model of a typical neuron found in the hippocampal CA1 region, an area often thought to be the origin of these seizures. The focus is on properly modeling the astrocytic input to examine the pathological excitation of these neurons and subsequent transmission of the signals. In particular, we consider the intracellular astrocytic calcium fluctuations which are associated with slow inward currents in neighbouring neurons. Using our model, a variety of experimental results are reproduced, and comments are made about the potential differences between graded and “all-or-none” astrocytes.

A dynamic flight model of a jet trainer aircraft is developed in MATLAB-SIMULINK. Using a six degree of freedom mathematical model, non-linear simulation is used to observe the longitudinal and lateral-directional motions of the aircraft following a pilot input. The mathematical model is in state-space form and uses aircraft stability and control derivatives calculated from the aircraft geometric and aerodynamic characteristics. The simulation takes the changes in speed and altitude into consideration due to pilot input and demonstrates the non-linearity of the aircraft motion. The results from the simulation are compared with the results from flight characteristics manual of the actual aircraft to validate the mathematical model used. The simulation is carried out for a number of airspeed and altitude combinations to examine the effect of changing speed and altitude on the aircraft dynamic response.

Directional and horizontal drilling operations are increasingly conducted in all over the world, especially parallel to the growth of the technological developments in the industry. Common application fields for directional and horizontal drilling are in offshore and onshore when there is no way of drilling vertical wells. During directional and horizontal well drilling, many additional challenges occur when compared with vertical well drilling, such as limited weight on bit, harder hole cleaning, trajectory control, etc. This makes even harder to select the proper drilling parameters for increasing the rate of penetration. This study aims to propose a rate of penetration model considering many drilling parameters and conditions. The proposed model is a modified Bourgoyne &amp / Young&rsquo / s model which considers formation compaction, formation pressure, equivalent circulating density, and effective weight on bit, rotation of the bit, bit wear, hole cleaning, inclination, fluid loss properties and bit hydraulics. Also, a bit wear model is developed for roller cones and PDCs. The model performance is tested using field data obtained from several directional and horizontal offshore wells drilled at Persian Gulf. It is observed that the model can estimate rate of penetration with an error of &plusmn / 25 % when compared with the field data.

This research was a collaborative effort between the Air Force Research Laboratory (AFRL) and the University of Texas to examine the laser-tissue interaction of thermal lensing induced by continuous-wave, CW, near-infrared, NIR, laser radiation in the eye and its influence on the formation of a retinal lesion from said radiation. CW NIR laser radiation can lead to a thermal lesion induced on the retina given sufficient power and exposure duration as related to three basic parameters; the percent of transmitted energy to, the optical absorption of, and the size of the laser-beam created at the retina. Thermal lensing is a well-known phenomenon arising from the optical absorption, and subsequent temperature rise, along the path of the propagating beam through a medium. Thermal lensing causes the laser-beam profile delivered to the retina to be time dependent. Analysis of a dual-beam, multidimensional, high-frame rate, confocal imaging system in an artificial eye determined the rate of thermal lensing in aqueous media exposed to 1110, 1130, 1150 and 1318-nm wavelengths was related to the power density created along the optical axis and linear absorption coefficient of the medium. An adaptive optics imaging system was used to record the aberrations induced by the thermal lens at the retina in an artificial eye during steady-state. Though the laser-beam profiles changed over the exposure time, the CW NIR retinal damage thresholds between 1110-1319-nm were determined to follow conventional fitting algorithms which neglected thermal lensing. A first-order mathematical model of thermal lensing was developed by conjoining an ABCD beam propagation method, Beer's law of attenuation, and a solution to the heat-equation with respect to radial diffusion. The model predicted that thermal lensing would be strongest for small (< 4-mm) 1/e² laser-beam diameters input at the corneal plane and weakly transmitted wavelengths where less than 5% of the energy is delivered to the retina. The model predicted thermal lensing would cause the retinal damage threshold for wavelengths above 1300-nm to increase with decreasing beam-diameters delivered to the corneal plane, a behavior which was opposite of equivalent conditions simulated without thermal lensing. / text

Dexamethasone and dexamethasone 21-phosphate were loaded into NDGA-polymerized collagen fibers and release rate studies were performed to calculate their diffusion coefficients. Dexamethasone loaded fibers were placed in a PBS solution for specified time intervals (1, 3, 6, 7, 12, 24, 30, and 48 hours) after which the eluant was removed and analyzed by capillary zone electrophoresis (CZE). CZE is a tool that can be utilized for quantitative analysis of chemical compounds. This data was incorporated into mathematical models to determine the diffusion coefficient. The diffusion coefficient (D) for dexamethasone in NDGA-polymerized collagen fibers is D = 1.86 x 10⁻¹⁴ m²/s. Similarly, dexamethasone 21-phosphate loaded fibers were placed into a PBS solution and analyzed using CZE at these specified intervals (15, 30, 45, 60, and 75 minutes). Applying this data to the mathematical model provided a diffusion coefficient for dexamethasone 21-phosphate in NDGA-polymerized collagen fibers of D = 2.36 x 10⁻¹³ m²/s. In an effort to control drug delivery from these fibers a polylactic-co-glycolic acid (PLGA) coating was applied to the fibers. This coating helped sustain delivery of dexamethasone 21-phosphate for over a 100 day period. CZE experiments were again conducted in conjunction with another mathematical model to characterize release. A semi steady-state diffusion coefficient was estimated to be D = 4.59 x 10⁻¹⁴ m²/s.

Doctor of Philosophy / Department of Industrial & Manufacturing Systems Engineering / David H. Ben-Arieh / Chih-Hang Wu / Patients who suffer from sepsis or septic shock are of great concern in the healthcare system. Recent data indicate that more than 900,000 severe sepsis or septic shock cases developed in the United States with mortality rates between 20% and 80%. In the United States alone, almost $17 billion is spent each year for the treatment of patients with sepsis. Clinical trials of treatments for sepsis have been extensively studied in the last 30 years, but there is no general agreement of the effectiveness of the proposed treatments for sepsis. Therefore, it is necessary to find accurate and effective tools that can help physicians predict the progression of disease in a patient-specific way, and then provide physicians recommendation on the treatment of sepsis to lower risk for patients dying from sepsis. The goal of this research is to develop a risk assessment tool and a risk management tool for sepsis. In order to achieve this goal, two system dynamic mathematical models (SDMMs) are initially developed to predict dynamic patterns of sepsis progression in innate immunity and adaptive immunity. The two SDMMs are able to identify key indicators and key processes of inflammatory responses to an infection, and a sepsis progression. Second, an integrated-mathematical-multi-agent-based model (IMMABM) is developed to capture the stochastic nature embedded in the development of inflammatory responses to a sepsis. Unlike existing agent-based models, this agent-based model is enhanced by incorporating developed SDMMs and extensive experimental data. With the risk assessment tools, a Markov decision process (MDP) is proposed, as a risk management tool, to apply to clinical decision-makings on sepsis. With extensive computational studies, the major contributions of this research are to firstly develop risk assessment tools to identify the risk of sepsis development during the immune system responding to an infection, and secondly propose a decision-making framework to manage the risk of infected individuals dying from sepsis. The methodology and modeling framework used in this dissertation can be expanded to other disease situations and treatment applications, and have a broad impact to the research area related to computational modeling, biology, medical decision-making, and industrial engineering.