Analysis and applications to the cell interplay and control of low grade gliomas

Tumor-normal cell interplay defines the course of a neoplastic malignancy. The outcome of this dual relation is the ultimate prevailing of one of the cells and the death or retreat of the other. In this paper we study the mathematical principles that underlay one important scenario: that of slow-progressing cancers. For this, we develop, within a stochastic framework, a mathematical model to account for tumor-normal cell interaction in such a clinically relevant situation and derive a number of deterministic approximations from the stochastic model. We consider in detail the existence and uniqueness of the solutions of the deterministic model and study the stability analysis. We then focus our model to the specific case of low grade gliomas, where we introduce an optimal control problem for different objective functionals under the administration of chemotherapy. We derive the conditions for which singular and bang-bang control exist and calculate the optimal control and states

Nonlinear Control and Estimation of an Infammatory Immune Response

The immune response is a complex mechanism that can be triggered by biological or physical stresses on the organism. However an excessive and dys-regulated inflammatory response may lead to sepsis, a critical state, promoting tissue damage, organ dysfunction or even death.The main objective in this dissertation is to derive a strategy consisting of manipulating pro and anti-inflammatory mediators in order to direct the state of a virtual patient to a healthy equilibrium, after some disturbance from health due to infection. Two key challenges need to be addressed in solving such a problem: estimating the unmeasurable states of the inflammatory model as well as the model\u27s unknown rate parameters; and second, determining an appropriate strategy to effectively control the response.We initially study the nonlinear controllability, observability and identifiability of the inflammatory immune model. Then, we address the first challenge by comparing the performance of various nonlinear filters for state estimation in the presence of noise and incomplete information. For parameter estimation, a recently introduced approximate Markov chain Monte Carlo approach known as the Particle Metropolis- Hastings method is used. To control the highly nonlinear model, various model-based optimization approaches were investigated in which the control strategy is derived in terms of pro-inflammatory and anti-inflammatory response doses. Due to parameter variability and the difficult practical task of obtaining accurate state and parameter estimates in real time, a new model-free control methodology and its intelligent controllers is explored. The method does not rely on any precise modeling and the identification of each parameter of the inflammatory immune model is no longer needed for control design. The various methods are compared for performance to adequately control the responses in a diverse patient population as well as the clinical feasibility of the derived control protocol from the approach used

Model-Based External Forcing of Nonlinear Dynamics in Chemical and Biochemical Reaction Systems via Optimal Control

Ein ausf¨uhrliches, quantitatives Verständnis, welches durch Modellieren erzielt wird, sowie das Ermöglichen einer spezifischen externen Steuerung des zellularen Verhaltens sind allgemeine langfristige Ziele der modernen biowissenschaftlichen Forschung in der Systembiologie. Selbstorganisation ist möglicherweise ein allgemein gültiges Prinzip für die zelluläre Organisation, da viele dynamische Eigenschaften zellulärer Strukturen sowohl hinsichtlich ihrer Bildung, Aufrechterhaltung und Funktion diesem folgen. Die Steuerung selbstorganisierter Dynamiken eröffnet einen Weg zur Untersuchung von dynamischem Verhalten sowie zur Generierung des gewünschten Verhaltens. Um dieses Ziel zu verwirklichen, konzentriert sich diese Dissertation in erster Linie auf die gezielt orientierte Beeinflussung dieser Systeme durch optimale Steuerungsmethoden. Der Ansatz optimaler Steuerung bietet große Flexibilität hinsichtlich der Bestimmung der Zielfunktionen. Wir verwenden eine direkte, auf den Multiple-Shooting-Ansatz basierende numerische Optimiermethode, welche insbesondere auf nichtlineare selbstorganisierende Systeme verwendbar ist. Die vorliegende Arbeit zeigt, wie auf Modellen basierende optimale Steuerungsmethoden zum Erzeugen der gewünschten Systemdynamiken verwertet werden können. Im Fall des Circadischen Rhythmus und der Belousov-Zhabotinsky (BZ) Reaktion als Modellsysteme sind diese bezüglich der zeitabhängigen Steuerungsparameter nicht systemimmanent. Wir analysieren ein Circadisches Oszillatormodell des zentralen Uhrmechanismus für die Fruchtfliege Drosophila und zeigen, wie auf Modellen basierende optimale Steuerung, Phasenneueinstellung, Design von chronomodulierten Puls-Stimuli-Schemata zur Wiederherstellung des Circadischen Rhythmus in den Mutanten und optimale Phasensynchronisierung zwischen der Uhr und ihrer Umgebung erlaubt. Wir beziehen uns sowohl auf die optimalen Open-Loop- als auch auf die Rückkopplungssteuerungsmethoden. Circadische Rhythmen können das Timing und den Eintritt des Zellzyklus erheblich beeinflussen. Zur Untersuchung der auf Modellen basierenden optimalen Steuerungsszenarios sind ein detaillert gekoppelter Circadischer Zyklus und das Zellzyklusmodell f¨ur ein Säugetiersystem entwickelt worden. Erstergebnisse der numerischen Simulationen für den gekoppelten Circadischen Zyklus und das Zellzyklusmodell werden gezeigt. Insbesondere leicht zugängliche chemische Testrohrsysteme wie die BZ Reaktion sind für Untersuchungen der Steuerung selbstorganisierter Dynamiken sehr gut geeignet. Denn sie bieten ein Mittel für die Charkterisierung des Verhaltens, das für kompliziertere biologische Systeme relevant ist. Wir entwickeln ein ganz neuartiges detaillertes Modell für die lichtempfindliche BZ Reaktion, das auf einem Elementarreaktionsmechanismus beruht und reduzieren dieses aufgrund der Quasi-Steady-State- (QSSA) und partielle Gleichgewichtsnäherungen (PEA) explizit. Zur Stabilisierung instabiler stationärer Zustände sind systematische Analysen und auf Modellen basierende Steuerungen durchgeführt worden, woraus periodische Bahnen mit einer gewünschten Periode resultieren. Die Ergebnisse werden diskutiert und mit einem sehr einfachen 3-Variablen-Oregonator-Modell aus der Literatur verglichen

Economic-epidemiological analysis of tuberculosis : modelling the demographic-epidemiological implications of economic growth and public health investment.

Available from British Library Document Supply Centre-DSC:DXN048404 / BLDSC - British Library Document Supply CentreSIGLEGBUnited Kingdo

Numerical Solution of Optimal Control Problems with Explicit and Implicit Switches

This dissertation deals with the efficient numerical solution of switched optimal control problems whose dynamics may coincidentally be affected by both explicit and implicit switches. A framework is being developed for this purpose, in which both problem classes are uniformly converted into a mixed–integer optimal control problem with combinatorial constraints. Recent research results relate this problem class to a continuous optimal control problem with vanishing constraints, which in turn represents a considerable subclass of an optimal control problem with equilibrium constraints. In this thesis, this connection forms the foundation for a numerical treatment. We employ numerical algorithms that are based on a direct collocation approach and require, in particular, a highly accurate determination of the switching structure of the original problem. Due to the fact that the switching structure is a priori unknown in general, our approach aims to identify it successively. During this process, a sequence of nonlinear programs, which are derived by applying discretization schemes to optimal control problems, is solved approximatively. After each iteration, the discretization grid is updated according to the currently estimated switching structure. Besides a precise determination of the switching structure, it is of central importance to estimate the global error that occurs when optimal control problems are solved numerically. Again, we focus on certain direct collocation discretization schemes and analyze error contributions of individual discretization intervals. For this purpose, we exploit a relationship between discrete adjoints and the Lagrange multipliers associated with those nonlinear programs that arise from the collocation transcription process. This relationship can be derived with the help of a functional analytic framework and by interrelating collocation methods and Petrov–Galerkin finite element methods. In analogy to the dual-weighted residual methodology for Galerkin methods, which is well–known in the partial differential equation community, we then derive goal–oriented global error estimators. Based on those error estimators, we present mesh refinement strategies that allow for an equilibration and an efficient reduction of the global error. In doing so we note that the grid adaption processes with respect to both switching structure detection and global error reduction get along with each other. This allows us to distill an iterative solution framework. Usually, individual state and control components have the same polynomial degree if they originate from a collocation discretization scheme. Due to the special role which some control components have in the proposed solution framework it is desirable to allow varying polynomial degrees. This results in implementation problems, which can be solved by means of clever structure exploitation techniques and a suitable permutation of variables and equations. The resulting algorithm was developed in parallel to this work and implemented in a software package. The presented methods are implemented and evaluated on the basis of several benchmark problems. Furthermore, their applicability and efficiency is demonstrated. With regard to a future embedding of the described methods in an online optimal control context and the associated real-time requirements, an extension of the well–known multi–level iteration schemes is proposed. This approach is based on the trapezoidal rule and, compared to a full evaluation of the involved Jacobians, it significantly reduces the computational costs in case of sparse data matrices

International Conference on Mathematical Analysis and Applications in Science and Engineering – Book of Extended Abstracts

The present volume on Mathematical Analysis and Applications in Science and Engineering - Book of Extended Abstracts of the ICMASC’2022 collects the extended abstracts of the talks presented at the International Conference on Mathematical Analysis and Applications in Science and Engineering – ICMA2SC'22 that took place at the beautiful city of Porto, Portugal, in June 27th-June 29th 2022 (3 days). Its aim was to bring together researchers in every discipline of applied mathematics, science, engineering, industry, and technology, to discuss the development of new mathematical models, theories, and applications that contribute to the advancement of scientific knowledge and practice. Authors proposed research in topics including partial and ordinary differential equations, integer and fractional order equations, linear algebra, numerical analysis, operations research, discrete mathematics, optimization, control, probability, computational mathematics, amongst others. The conference was designed to maximize the involvement of all participants and will present the state-of- the-art research and the latest achievements.info:eu-repo/semantics/publishedVersio