Model--Based Design of Cancer Chemotherapy Treatment Schedules

Cancer is the name given to a class of diseases characterized by an imbalance in cell proliferation and apoptosis, or programmed cell death. Once cancer has reached detectable sizes ($10^{6}$ cells or 1 mm$^3$), it is assumed to have spread throughout the body, and a systemic form of treatment is needed. Chemotherapy treatment is commonly used, and it effects both healthy and diseased tissue. This creates a dichotomy for clinicians who need develop treatment schedules which balance toxic side effects with treatment efficacy. Nominally, the optimal treatment schedule --- where schedule is defined as the amount and frequency of drug delivered --- is the one found to be the most efficacious from the set evaluated during clinical trials. In this work, a model based approach for developing drug treatment schedules was developed. Cancer chemotherapy modeling is typically segregated into drug pharmacokinetics (PK), describing drug distribution throughout an organism, and pharmacodynamics (PD), which delineates cellular proliferation, and drug effects on the organism. This work considers two case studies: (i) a preclinical study of the oral administration of the antitumor agent 9-nitrocamptothecin (9NC) to severe combined immunodeficient (SCID) mice bearing subcutaneously implanted HT29 human colon xenografts; and (ii) a theoretical study of intravenous chemotherapy from the engineering literature.Metabolism of 9NC yields the active metabolite 9-aminocamptothecin (9AC). Both 9NC and 9AC exist in active lactone and inactive carboxylate forms. Four different PK model structures are presented to describe the plasma disposition of 9NC and 9AC: three linear models at a single dose level (0.67 mg/kg 9NC); and a nonlinear model for the dosing range 0.44 -- 1.0 mg/kg 9NC. Untreated tumor growth was modeled using two approaches: (i) exponential growth; and (ii) a switched exponential model transitioning between two different rates of exponential growth at a critical size. All of the PK/PD models considered here have bilinear kill terms which decrease tumor sizes at rates proportional to the effective drug concentration and the current tumor size. The PK/PD model combining the best linear PK model with exponential tumor growth accurately characterized tumor responses in ten experimental mice administered 0.67 mg/kg of 9NC myschedule (Monday-Friday for two weeks repeated every four weeks). The nonlinear PK model of 9NC coupled to the switched exponential PD model accurately captured the tumor response data at multiple dose levels. Each dosing problem was formulated as a mixed--integer linear programming problem (MILP), which guarantees globally optimal solutions. When minimizing the tumor volume at a specified final time, the MILP algorithm delivered as much drug as possible at the end of the treatment window (up to the cumulative toxicity constraint). While numerically optimal, it was found that an exponentially growing tumor, with bilinear kill driven by linear PK would experience the same decrease in tumor volume at a final time regardless of when the drug was administered as long as the {it same amount} was administered. An alternate objective function was selected to minimize tumor volume along a trajectory. This is more clinically relevant in that it better represents the objective of the clinician (eliminate the diseased tissue as rapidly as possible). This resulted in a treatment schedule which eliminated the tumor burden more rapidly, and this schedule can be evaluated recursively at the end of each cycle for efficacy and toxicity, as per current clinical practice.The second case study consists of an intravenously administered drug with first order elimination treating a tumor under Gompertzian growth. This system was also formulated as a MILP, and the two different objectives above were considered. The first objective was minimizing the tumor volume at a final time --- the objective the original authors considered. The MILP solution was qualitatively similar to the solutions originally found using control vector parameterization techniques. This solution also attempted to administer as much drug as possible at the end of the treatment interval. The problem was then posed as a receding horizon trajectory tracking problem. Once again, a more clinically relevant objective returned promising results; the tumor burden was rapidly eliminated

Design of of model-based controllers via parametric programming

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Predictive and Corrective Scheduling in Electric Energy Systems with Variable Resources

In the past decade, there has been sustained efforts around the globe in developing renewable energy-based generation in power systems. However, many renewables such as wind and solar are variable resources. They pose significant challenges to near real-time power system operations. This dissertation focuses on introducing and testing advanced scheduling algorithms for electric power systems with high penetration of variable resources. A novel predictive and optimal corrective look-ahead dispatch framework for real-time economic operation is proposed. This dissertation has four key parts. First, the basic framework of look-ahead dispatch is introduced. Different from conventional static economic dispatch, look-ahead dispatch is the fundamental function for future power system scheduling. Taking the whole dispatch horizon into account, look-ahead dispatch has a better economic performance in scheduling the resources in power systems. The decision-making of look-ahead dispatch is cost-effective, especially when handling with high penetration of variable resources. Second, we study the benefits of look-ahead dispatch in system security enhancement. An early detection algorithm is proposed to predict and identify potential security risks in the system. The proposed optimal corrective measures can be computed to prevent system insecurity at a minimized cost. Early awareness of such information is of vital importance to the system operators for taking timely actions with more flexible and cost-effective measures. Third, novel statistical wind power forecast models are presented, as an effort to reduce the uncertainty of renewable forecast to support the look-ahead economic dispatch and security management. The forecast models can produce more accurate forecast results by leveraging the spatio-temporal correlation in wind speed and direction data among geographically dispersed wind resources. Fourth, we propose a stochastic look-ahead dispatch (LAED-S) module to handle the high uncertainty in renewable resources. Even with state-of-the-art forecast technology, the near-real-time operational uncertainty from renewable resources cannot be eliminated. Given the uncertainty level, a conventional deterministic approach is not always the best option. The proposed LAED-S is able to judge whether a stochastic approach is preferred. The innovative computation algorithm of LAED-S leverages the progressive hedging and L-shaped method to produce good stochastic decision-making in a more efficient manner. Numerical experiments of a modified IEEE RTS system and a practical system are conducted to justify the proposed approaches in this dissertation. This framework can directly benefit the power system operation in moving from a static, passive real-time operation into a predictive and corrective paradigm

Controle preditivo multiobjetivo para processos com atraso

Tese (doutorado) - Universidade Federal de Santa Catarina, Centro Tecnológico. Programa de Pós-Graduação em Engenharia Elétrica.Esta tese apresenta contribuições para a melhoria da solução do controle de processos com atraso, através de estratégias de controle preditivo baseado no modelo (MPC) que incorporam aspectos como robustez, restrições e otimização multiobjetivo. As contribuições estão relacionadas à duas áreas: (i) controle de processos com atrasos dominantes e (ii) otimização multiobjetivo do controle. Para análise do efeito do atraso no comportamento do sistema em malha fechada, desenvolve-se uma nova formulação do controlador preditivo por matriz dinâmica (DMC), mostrando que este é composto por um controlador primário mais um preditor. Esta formulação permite avaliar analiticamente o efeito do atraso no DMC, comparado a outros controladores MPC, através de índices de robustez. Esta análise possibilita selecionar o algoritmo MPC mais adequado à implementação prática em processos com atraso, especialmente quanto à robustez do controle frente a variações paramétricas na planta. Do ponto de vista da otimização do controle, busca-se desenvolver estratégias de controle preditivo que consideram aspectos econômicos no projeto de controle, além das restrições técnicas e/ou operacionais que habitualmente constam nos requisitos de aplicações reais. As estratégias de controle propostas utilizam um nível supervisório para determinar as referências ótimas de controle para um nível regulatório. O nível supervisório baseia-se em um otimizador de múltiplos objetivos, projetado a partir de regras que integram heurística, lógica e dinâmica do processo por meio de expressões descritas através de lógica proposicional, ou por estruturação formulada através de regras de decisão. As soluções ótimas obtidas via otimização multiobjetivo funcionam como referências desejadas para o cálculo da lei de controle do algoritmo MPC. Diversas aplicações em plantas piloto e industriais ilustram os resultados obtidos com as diferentes estratégias desenvolvidas

Hybrid automaton based controller design for damage mitigation of islanded power systems

Spurred by increasingly unpredictable weather, high penetration of renewable resources and a period of focused US government policy, it is widely expected that microgrids within the electric distribution system will show exponential growth in the coming decade. Microgrids comprise of power generation, delivery and consumption assets within restricted electrical boundaries and under contiguous control oversight that enables holistic management of these assets. Microgrids can be islanded and operated independent of a larger electric power network, and as such, a primary function of microgrids is to enhance the energy reliability of the underlying loads. In this work, we focus on naval shipboard power systems. Apart from being islanded, in the true sense, resiliency and damage mitigation are key considerations in the design and operation of these power systems. Islanded power systems encompass a rich diversity of discrete and continuous dynamic behavior in multiple time-scales. A high penetration of devices with power electronics interface, low inherent system inertia, and high density of switching devices can lead to rapid disturbance propagation and system failure without advanced damage mitigation strategies. Hybrid systems formalism incorporates continuous dynamics as well as discrete switching behavior into a modeling and control framework, thus allowing a complete system description while crystallizing concepts of safety into system design criteria. We build on existing work to enhance a Dynamic Mixed Integer Programming (DMIP) model of a power system that combines continuous time differential algebraic models with switching dynamics synthesized into mixed integer inequalities. We use this model to derive an optimal system reconfiguration strategy to prevent voltage collapse of a benchmark shipboard power system. However, this methodology is restricted by the computational complexity of dynamic programming and scalability of non-automated processes. To overcome some of these limitations, we derive a hybrid automaton model of a power system as a Discrete Event System (DES) plant and controller. The DES plant consists of a switched continuous system with an interface. The system state space is categorized based on safety criteria and discrete control specifications are embedded as transition rules within the DES controller. The DES controller searches for feasible control policies that drive the system trajectories from unsafe states to safe states. We define metrics to quantify the performance of these policies, thus allowing the derivation of the most suitable policy for a set of design specifications and disturbance type. Applications in voltage control, frequency control and dynamic service restoration is presented on a benchmark power system with approximately forty continuous states and eighteen thousand discrete states. To enable the analysis, we build a computational framework based on efficienct symbolic computation tools in Mathematica and numerical integration tools in Matlab / Simulink so that the methodology can be replicated for a wide variety of applications. The framework is quite general, and may be expanded to problems beyond power systems.Ph.D., Electrical Engineering -- Drexel University, 201

Dynamic Modeling of Free Fatty Acid, Glucose, and Insulin During Rest and Exercise in Insulin Dependent Diabetes Mellitus Patients

Malfunctioning of the beta-cells of the pancreas leads to the metabolic disease known as diabetes mellitus (DM), which is characterized by significant glucose variation due to lack of insulin secretion, lack of insulin action, or both. DM can be broadly classified into two types: type 1 diabetes mellitus (T1DM) - which is caused mainly due to lack of insulin secretion; and type 2 diabetes mellitus (T2DM) - which is caused due to lack of insulin action. The most common intensive insulin treatment for T1DM requires administration of insulin subcutaneously 3 - 4 times daily in order to maintain normoglycemia (blood glucose concentration at 70 to 120 mg/dl). Although the effectiveness of this technique is adequate, wide glucose fluctuations persist depending upon individual daily activity, such as meal intake, exercise, etc. For tighter glucose control, the current focus is on the development of automated closed-loop insulin delivery systems. In a model-based control algorithm, model quality plays a vital role in controller performance. In order to have a reliable model-based automatic insulin delivery system operating under various physiological conditions, a model must be synthesized that has glucose-predicting ability and includes all the major energy-providing substrates at rest, as well as during physical activity. Since the 1960s, mathematical models of metabolism have been proposed in the literature. The majority of these models are glucose-based and have ignored the contribution of free fatty acid (FFA) metabolism, which is an important source of energy for the body. Also, significant interactions exist among FFA, glucose, and insulin. It is important to consider these metabolic interactions in order to characterize the endogenous energy production of a healthy or diabetic patient. In addition, physiological exercise induces fundamental metabolic changes in the body; this topic has also been largely overlooked by the diabetes modeling community.This dissertation takes a more lipocentric (lipid-based) approach in metabolic modeling for diabetes by combining FFA dynamics with glucose and insulin dynamics in the existing glucocentric models. A minimal modeling technique was used to synthesize a FFA model, and this was coupled with the Bergman minimal model to yield an extended minimal model. The model predictions of FFA, glucose, and insulin were validated with experimental data obtained from the literature. A mixed meal model was developed to capture the absorption of carbohydrates (CHO), proteins, and FFA from the gut into the circulatory system. The mixed meal model served as a disturbance to the extended minimal model. In a separate study, an exercise minimal model was developed to incorporate the effects of exercise on glucose and insulin dynamics. Here, the Bergman minimal model was modified by adding equations and terms to capture the changes in glucose and insulin dynamics during and after mild-to-moderate exercise.A single composite model for predicting FFA-glucose-insulin dynamics during rest and exercise was developed by combining the extended and exercise minimal models. To make the composite model more biologically relevant, modifications were made to the original model structures. The dynamical effects of insulin on glucose and FFA were divided into three parts: (i) insulin-mediated glucose uptake by the tissues, (ii) insulin-mediated suppression of endogenous glucose production, and (iii) anti-lipolytic effects of insulin. Labeled and unlabeled intra-venous glucose tolerance test data were used to estimate the parameters of the glucose model which facilitated separation of insulin action on glucose utilization and production. The model successfully captured the FFA-glucose interactions at the systemic level. The model also successfully predicted mild-to-moderate exercise effects on glucose and FFA dynamics. A detailed physiologically-based compartmental model of FFA was synthesized and integrated with the existing physiologically-based glucose-insulin model developed by Sorensen. Distribution of FFA in the circulatory system was evaluated by developing mass balance equations across the major FFA-utilizing tissues/organs. Rates of FFA production or consumption were added to each of the physiologic compartments. In order to incorporate the FFA effects on glucose, modifications were made to the existing mass balance equations in the Sorensen model. The model successfully captured the FFA-glucose-insulin interactions at the organ/tissue levels.Finally, the loop was closed by synthesizing model predictive controllers (MPC) based on the extended minimal model and the composite model. Both linear and nonlinear MPC algorithms were formulated to maintain glucose homeostasis by rejecting disturbances from mixed meal ingestion. For comparison purposes, MPC algorithms were also synthesized based on the Bergman minimal model which does not account for the FFA dynamics. The closed-loop simulation results indicated a tighter blood glucose control in the post-prandial period with the MPC formulations based on the lipocentric (extended minimal and composite) models

Robust constraint satisfaction: invariant sets and predictive control

Set invariance plays a fundamental role in the design of control systems for constrained systems sincethe constraints can be satisfied for all time if and only if the initial state is contained inside an invariantset. This thesis is concerned with robust set invariance theory and its application to guaranteeingfeasibility in model predictive control.In the first part of this thesis, some of the main ideas in set invariance theory are brought togetherand placed in a general, nonlinear setting. The key ingredients in computing robust controllable andinvariant sets are identified and discussed. Following this, linear systems with parametric uncertaintyand state disturbances are considered and algorithms for computing the respective robust controllableand invariant sets are described. In addition to discussing linear systems, an algorithm for computingthe robust controllable sets for piecewise affine systems with state disturbances is described.In the second part, the ideas from set invariance are applied to the problem of guaranteeing feasibilityand robust constraint satisfaction in Model Predictive Control (MPC). A new sufficient condition isderived for guaranteeing feasibility of a given MPC scheme. The effect of the choice of horizons andconstraints on the feasible set of the MPC controller is also investigated. Following this, a necessaryand sufficient condition is derived for determining whether a given MPC controller is robustly feasible.The use of a robustness constraint for designing robust MPC controllers is discussed and it is shownhow this proposed scheme can be used to guarantee robust constraint satisfaction for linear systemswith parametric uncertainty and state disturbances. A new necessary and sufficient condition as wellas some new sufficient conditions are derived for guaranteeing that the proposed MPC scheme isrobustly feasible.The third part of this thesis is concerned with recovering from constraint violations. An algorithm ispresented for designing soft-constrained MPC controllers which guarantee constraint satisfaction, ifpossible. Finally, a mixed-integer programming approach is described for finding a solution whichminimises the number of violations in a set of prioritised constraints