Mathematical modeling has been instrumental to the development of natural sciences over the last half-century. Through iterated interactions between modeling and real-world exper- imentation, these models have furthered our understanding of the processes in biology and chemistry that they seek to represent. In certain application domains, such as the field of car- diac biology, communities of modelers with common interests have emerged, leading to the development of many models that attempt to explain the same or similar phenomena. As these communities have developed, however, reporting standards for modeling studies have been in- consistent, often focusing on the final parameterized result, and obscuring the assumptions and data used during their creation. These practices make it difficult for researchers to adapt exist- ing models to new systems or newly available data, and also to assess the identifiability of said models - the degree to which their optimal parameters are constrained by data - which is a key step in building trust that their formulation captures truth about the system of study. In this thesis, we develop tools that allow modelers working with biological or chemical time series data to assess identifiability in an automated fashion, and embed these tools within a novel online community resource that enforces reproducible standards of reporting and facilitates exchange of models and data. We begin by exploring the application of Bayesian and approximate Bayesian inference methods, which parameterize models while simultaneously assessing uncertainty about these estimates, to assess the identifiability of models of the cardiac action potential. We then demon- strate how the side-by-side application of these Bayesian and approximate Bayesian methods can be used to assess the information content of experiments where system observability is limited to "summary statistics" - low-dimensional representations of full time-series data. We next investigate how a posteriori methods of identifiability assessment, such as the above inference techniques, compare against a priori methods based on model structure. We compare these two approaches over a range of biologically relevant experimental conditions, and high- light the cases under which each strategy is preferable. We also explore the concept of optimal experimental design, in which measurements are chosen in order to maximize model identifia- bility, and compare the feasibility of established a priori approaches against a novel a posteriori approach. Finally, we propose a framework for representing and executing modeling experiments in a reproducible manner, and use this as the foundation for a prototype "Modeling Web Lab" where researchers may upload specifications for and share the results of the types of inference explored in this thesis. We demonstrate the Modeling Web Lab's utility across multiple mod- eling domains by re-creating the results of a contemporary modeling study of the hERG ion channel model, as well as the results of an original study of electrochemical redox reactions. We hope that this works serves to highlight the importance of both reproducible standards of model reporting, as well as identifiability assessment, which are inherently linked by the desire to foster trust in community-developed models in disciplines across the natural sciences.

Na modelagem caixa branca obtém-se um modelo para um processo a partir do equacionamento dos fenômenos físicos/químicos envolvidos. Estes modelos são para- metrizados, mas os valores dos parâmetros utilizados muitas vezes são desconhecidos. Nestes casos ´e necessário efetuar um procedimento de identificação paramétrica o que representa um problema altamente desafiador, com muitas questões teóricas e práticas em aberto, quando os parâmetros aparecem de forma n˜ao linear no modelo. O objetivo deste trabalho ´e apresentar e estudar um método capaz de determinar se uma estrutura de modelo predeterminada pode ser identificada e que possa ser utilizado em conjunto com algum outro método de identificação, para identificar o sistema. O método que será apresentado é baseado em álgebra diferencial e é conhecido como algoritmo de Ritt. O algoritmo de Ritt transforma uma estrutura de modelo polinomial predeterminada em regressões lineares nos parâmetros a partir das quais pode-se utilizar os métodos dos mínimos quadrados ou variáveis instrumentais para identificar o sistema. Apresentaremos alguns estudos de caso e faremos a análise de identificabilidade para cada um deles. Em alguns casos identificaremos o sistema e estudaremos a consistência e precisão das estimativas. / In white box modeling we obtain a model for a process from the equations of the physical/chimical phenomena involved. These models are parameterized, but the parameters used are often unknown. In these cases it is necessary to perform a parametric identification procedure which represents a highly challenging problem, with many theoretical and practical open questions when the parameters are non- linears in the model. The aim of this work is to present and study a method able to determine whether a predetermined model structure can be identified and that can be used in conjunction with another identification method to identify the system. The method that will be presented is based on differential algebra and is known as Ritt algorithm. The Ritt’s algorithm transforms a predetermined model structure in linear regression in the parameters from which one can use the least squares method or instrumental variables to identify the system. We will present some case studies and realise the analysis of identifiability for each case. For some cases we will identify the system, and then present a study for the consistency and precision of the estimates.

Na modelagem caixa branca obtém-se um modelo para um processo a partir do equacionamento dos fenômenos físicos/químicos envolvidos. Estes modelos são para- metrizados, mas os valores dos parâmetros utilizados muitas vezes são desconhecidos. Nestes casos ´e necessário efetuar um procedimento de identificação paramétrica o que representa um problema altamente desafiador, com muitas questões teóricas e práticas em aberto, quando os parâmetros aparecem de forma n˜ao linear no modelo. O objetivo deste trabalho ´e apresentar e estudar um método capaz de determinar se uma estrutura de modelo predeterminada pode ser identificada e que possa ser utilizado em conjunto com algum outro método de identificação, para identificar o sistema. O método que será apresentado é baseado em álgebra diferencial e é conhecido como algoritmo de Ritt. O algoritmo de Ritt transforma uma estrutura de modelo polinomial predeterminada em regressões lineares nos parâmetros a partir das quais pode-se utilizar os métodos dos mínimos quadrados ou variáveis instrumentais para identificar o sistema. Apresentaremos alguns estudos de caso e faremos a análise de identificabilidade para cada um deles. Em alguns casos identificaremos o sistema e estudaremos a consistência e precisão das estimativas. / In white box modeling we obtain a model for a process from the equations of the physical/chimical phenomena involved. These models are parameterized, but the parameters used are often unknown. In these cases it is necessary to perform a parametric identification procedure which represents a highly challenging problem, with many theoretical and practical open questions when the parameters are non- linears in the model. The aim of this work is to present and study a method able to determine whether a predetermined model structure can be identified and that can be used in conjunction with another identification method to identify the system. The method that will be presented is based on differential algebra and is known as Ritt algorithm. The Ritt’s algorithm transforms a predetermined model structure in linear regression in the parameters from which one can use the least squares method or instrumental variables to identify the system. We will present some case studies and realise the analysis of identifiability for each case. For some cases we will identify the system, and then present a study for the consistency and precision of the estimates.

Na modelagem caixa branca obtém-se um modelo para um processo a partir do equacionamento dos fenômenos físicos/químicos envolvidos. Estes modelos são para- metrizados, mas os valores dos parâmetros utilizados muitas vezes são desconhecidos. Nestes casos ´e necessário efetuar um procedimento de identificação paramétrica o que representa um problema altamente desafiador, com muitas questões teóricas e práticas em aberto, quando os parâmetros aparecem de forma n˜ao linear no modelo. O objetivo deste trabalho ´e apresentar e estudar um método capaz de determinar se uma estrutura de modelo predeterminada pode ser identificada e que possa ser utilizado em conjunto com algum outro método de identificação, para identificar o sistema. O método que será apresentado é baseado em álgebra diferencial e é conhecido como algoritmo de Ritt. O algoritmo de Ritt transforma uma estrutura de modelo polinomial predeterminada em regressões lineares nos parâmetros a partir das quais pode-se utilizar os métodos dos mínimos quadrados ou variáveis instrumentais para identificar o sistema. Apresentaremos alguns estudos de caso e faremos a análise de identificabilidade para cada um deles. Em alguns casos identificaremos o sistema e estudaremos a consistência e precisão das estimativas. / In white box modeling we obtain a model for a process from the equations of the physical/chimical phenomena involved. These models are parameterized, but the parameters used are often unknown. In these cases it is necessary to perform a parametric identification procedure which represents a highly challenging problem, with many theoretical and practical open questions when the parameters are non- linears in the model. The aim of this work is to present and study a method able to determine whether a predetermined model structure can be identified and that can be used in conjunction with another identification method to identify the system. The method that will be presented is based on differential algebra and is known as Ritt algorithm. The Ritt’s algorithm transforms a predetermined model structure in linear regression in the parameters from which one can use the least squares method or instrumental variables to identify the system. We will present some case studies and realise the analysis of identifiability for each case. For some cases we will identify the system, and then present a study for the consistency and precision of the estimates.

Avec la complexité croissante des systèmes dynamiques qui apparaissent dans l'ingénierie et d'autres domaines de la science, l'étude des systèmes de grande taille composés d'un ensemble de sous-systèmes interconnectés est devenue un important sujet d'attention dans différents domaines, tels que la robotique, les réseaux de transports, les grandes structures spatiales (panneaux solaires, antennes, télescopes,...), les bâtiments,… et a conduit à des problèmes intéressants d'analyse d'identification paramétrique, de contrôle distribué et d'optimisation. L'absence d'une définition universelle et reconnue des systèmes qu'on appelle "grands systèmes", "systèmes complexes", "systèmes interconnectés",..., témoigne de la confusion entre ces différents concepts et la difficulté de définir des limites précises pour tels systèmes. L'analyse de l'identifiabilité et de l'identification de ces systèmes nécessite le traitement de modèles numériques de grande taille, la gestion de dynamiques diverses au sein du même système et la prise en compte de contraintes structurelles (des interconnections,...). Ceci est très compliqué et très délicat à manipuler. Ainsi, ces analyses sont rarement prises en considération globalement. La simplification du problème par décomposition du grand système en sous-problèmes de complexité réduite est souvent la seule solution possible, conduisant l'automaticien à exploiter clairement la structure du système.Cette thèse présente ainsi, une approche décentralisée d'identification des systèmes de grande taille "large scale systems" composés d'un ensemble de sous-systèmes interconnectés. Cette approche est basée sur les propriétés structurelles (commandabilité, observabilité et identifiabilité) du grand système. Cette approche à caractère méthodologique est mise en œuvre sur des applications thermiques des bâtiments. L'intérêt de cette approche est montré à travers des comparaisons avec une approche globale. / With the increasing complexity of dynamical systems that appear in engineering and other fields of science, the study of large systems consisting of a set of interconnected subsystems has become an important subject of attention in various areas such as robotics, transport networks, large spacial structures (solar panels, antennas, telescopes, \ldots), buildings, … and led to interesting problems of parametric identification analysis, distributed control and optimization. The lack of a universal definition of systems called "large systems", "complex systems", "interconnected systems", ..., demonstrates the confusion between these concepts and the difficulty of defining clear boundaries for such systems. The analysis of the identifiability and identification of these systems requires processing digital models of large scale, the management of diverse dynamics within the same system and the consideration of structural constraints (interconnections, ...) . This is very complicated and very difficult to handle. Thus, these analyzes are rarely taken into consideration globally. Simplifying the problem by decomposing the large system to sub-problems is often the only possible solution. This thesis presents a decentralized approach for the identification of "large scale systems" composed of a set of interconnected subsystems. This approach is based on the structural properties (controllability, observability and identifiability) of the global system. This methodological approach is implemented on thermal applications of buildings. The advantage of this approach is demonstrated through comparisons with a global approach.

Dense medium separation (DMS) plants are typically used to beneficiate run-of-mine (ROM) coal in coal metallurgy. These plants normally make use of a dense medium cyclone as the primary processing unit. Because of the deviations in the ROM quality, the production yield and quality become difficult to maintain. A control system could benefit such operations to maintain and increase product throughput and quality. There are many different methods for developing a control system in a metallurgical operation; however, what is most fundamental is the use of a mathematical model to design a controller. For this reason, a first principle dynamic mathematical model has been developed for a DMS circuit. Each unit operation is modelled individually, then integrated together to form the complete system. The developed DMS circuit dynamic model is then used to simulate the process. It is also found that most models developed for DMS operations typically make use of steady-sate analysis and that very little literature is available on dynamic models of this kind. Difficulties that arise when validating a model in metallurgical processes are insufficient measurement points or the challenges in measuring certain variables, such as physical properties (e.g. particle size) or chemical components (e.g. ash percentage). This paper also explains how the Runge-Kutta approximation can be used in simulating DMS unit processes with intermediate online measurements that may be available. This can ultimately assist in verifying the accuracy of the simulation. One of the other problems that can occur when developing models from first principles is the estimation of model parameters. Specifically when non-linear state-space relationships are developed, one must ensure that there is a unique solution for the parameters in question. A method employing parameter identifiability is also presented in this dissertation to illustrate its use. In addition the process of estimating parameters is explained and illustrated. Copyright / Dissertation (MEng)--University of Pretoria, 2010. / Electrical, Electronic and Computer Engineering / unrestricted

This dissertation focuses on the modelling, identification and the parameter estimation for the co-infection of HIV-1 and M. tuberculosis. Many research papers in this field focus primarily on HIV, but multiple infections are explored here, as it is common in many individuals infected by HIV. Tuberculosis is also responsible for the highest number of casualties per year in the group of HIV-infected individuals. A model is proposed to indicate the populations of both pathogen as well as key information factors, such as the overall infected cell population and antigen-presenting cells. Simulations are made to indicate the growth and decline in cell-type numbers for a specific individual. Such simulations would provide a means for further, well-founded investigation into appropriate treatment strategies. One previous such model developed by Kirschner is used to obtain a nominal parameter set. Furthermore, the nominal set is then used in conjunction with real-world samples provided by the National Institute for Communicable Diseases in South Africa, to solidify the credibility of the model in the practical case. This is achieved via simulations and employs parameter estimation techniques, namely the Nelder-Mead cost-function method. An identifiability study of the model is also done. Conclusions drawn from this study include the result that the treatment of M. tuberculosis does not affect the course of HIV-1 progression in a notable way, and that the model can indeed be used in the process of better understanding the disease profile over time of infected individuals. / Dissertation (MEng)--University of Pretoria, 2008. / Electrical, Electronic and Computer Engineering / MEng / unrestricted

abstract: Immunotherapy has received great attention recently, as it has become a powerful tool in fighting certain types of cancer. Immunotherapeutic drugs strengthen the immune system's natural ability to identify and eradicate cancer cells. This work focuses on immune checkpoint inhibitor and oncolytic virus therapies. Immune checkpoint inhibitors act as blocking mechanisms against the binding partner proteins, enabling T-cell activation and stimulation of the immune response. Oncolytic virus therapy utilizes genetically engineered viruses that kill cancer cells upon lysing. To elucidate the interactions between a growing tumor and the employed drugs, mathematical modeling has proven instrumental. This dissertation introduces and analyzes three different ordinary differential equation models to investigate tumor immunotherapy dynamics. The first model considers a monotherapy employing the immune checkpoint inhibitor anti-PD-1. The dynamics both with and without anti-PD-1 are studied, and mathematical analysis is performed in the case when no anti-PD-1 is administrated. Simulations are carried out to explore the effects of continuous treatment versus intermittent treatment. The outcome of the simulations does not demonstrate elimination of the tumor, suggesting the need for a combination type of treatment. An extension of the aforementioned model is deployed to investigate the pairing of an immune checkpoint inhibitor anti-PD-L1 with an immunostimulant NHS-muIL12. Additionally, a generic drug-free model is developed to explore the dynamics of both exponential and logistic tumor growth functions. Experimental data are used for model fitting and parameter estimation in the monotherapy cases. The model is utilized to predict the outcome of combination therapy, and reveals a synergistic effect: Compared to the monotherapy case, only one-third of the dosage can successfully control the tumor in the combination case. Finally, the treatment impact of oncolytic virus therapy in a previously developed and fit model is explored. To determine if one can trust the predictive abilities of the model, a practical identifiability analysis is performed. Particularly, the profile likelihood curves demonstrate practical unidentifiability, when all parameters are simultaneously fit. This observation poses concerns about the predictive abilities of the model. Further investigation showed that if half of the model parameters can be measured through biological experimentation, practical identifiability is achieved. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2020

The thesis presents fundamental characteristics of survival analysis in the case of competing risks and their relationships. In the case without regression, basic nonparametric estimates and a logarithmic likelihood function for parameter estimates is given. The main focus is on Cox's proportional hazards model (PH), a model with accelerated time (AFT) and a flexible regression model (FG) are also mentioned. The identifiability of the associated survival function is solved using copulas. Basics of copula theory and the measurement of dependence by correlation coefficients (Pearson, Spearman and Kendal) are described in a separate chapter. A substantial part of the theory is practically used in a generated case without regression.

Progress in individualized drug treatment is of increasing importance, promising to avoid much human suffering and reducing medical treatment costs for society. The strategy is to maximize the therapeutic effects and minimize the negative side effects of a drug on individual or group basis. To reach the goal, interactions between the human body and different drugs must be further clarified, for instance by using mathematical models. Whether clinical studies or laboratory experiments are used as primary sources of information, greatly influences the possibilities of obtaining data. This must be considered both prior and during model development and different strategies must be used. The character of the data may also restrict the level of complexity for the models, thus limiting their usage as tools for individualized treatment. In this thesis work two case studies have been made, each with the aim to develop a model for a specific human-drug interaction. The first case study concerns treatment of inflammatory bowel disease with thiopurines, whereas the second is about treatment of ovarian cancer with paclitaxel. Although both case studies make use of similar amounts of experimental data, model development depends considerably on prior knowledge about the systems, the character of the data and the choice of modelling tools. All these factors are presented for each of the case studies along with current results. Further, a system for classifying different but related models is also proposed with the intention that an increased understanding will contribute to advancement in individualized drug dosage. / <p>Report code: LiU-Tek-Lic-2007:41.</p>