This thesis describes applications of mathematical modelling to systems of demand analgesia for the relief of acute postoperative pain. It builds upon work described in the D.Phil. thesis of M.P. Reasbeck. Following major surgery, patients are given a hand-held button which they press when in need of pain relief. The relief is afforded by automatic intravenous infusion of opiates. New clinical demand analgesia hardware, PRODAC, has been developed and data have been collected with it in two major trials involving a total of 80 patients. Patients' drug requirements have been found not to be correlated with body weight, contrary to conventional teaching. The type of operation was also found to have no significant influence upon drug requirements. The performance of transcutaneous nerve stimulation (TNS) as a method of analgesia for acute postoperative pain has been studied and found to be poor. Reasbeck's mathematical model of patients in pain has been corrected and extended. The representation of pharmacokinetics has been enhanced by modelling the transfer of drug between blood plasma and analgesic receptor sites as a first-order process. The time constant of this process has been calculated for morphine using a novel method and found to be 12 minutes. On line estimation of 2nd order pharmacokinetic time constants has been found in simulation not to be feasible. New software has been used to tune the revised model to the clinical data collected with PRODAC. Model behaviour is now demonstrably life-like, which was not previously the case. Blood samples taken during demand analgesia have permitted a comparison between measured and estimated drug concentrations, with good results.

Although Twin Roll Casting (TRC) process has been used for almost 60 years in the aluminum industry, TRC of magnesium is relatively new. In TRC, molten metal is fed onto water-cooled rolls, where it solidifies and is then rolled. Solidification of the molten metal starts at the point of first metal-roll contact and is completed before the kissing point (point of least roll separation) of the two rolls. The unique thermo-physical properties inherent to magnesium and its alloys, such as lower specific heat and latent heat of fusion and larger freezing ranges (in comparison with aluminum and steel) make it challenging for TRC of this alloy. Therefore, a comprehensive understanding of the process and the interaction between the casting conditions and strip final quality is imperative to guarantee high quality twin roll cast strip production. A powerful tool to achieve such knowledge is to develop a mathematical model of the process. In this thesis, a 2D mathematical model for TRC of AZ31 magnesium alloy has been developed and validated based on the TRC facility located at the Natural Resources Canada Government Materials Laboratory (CanmetMATERIALS) in Hamilton, ON, Canada. The validation was performed by comparing the predicted exit strip temperature and secondary dendrite arm spacing (SDAS) through the strip thickness with those measured and obtained by experiments. The model was developed in two stages, first a thermal-fluid model was developed followed by validation and then a thermal-fluid-stress model was developed. This is the first time a comprehensive thermal-fluid-stress model has been developed to simulate the TRC process for magnesium alloys. The work has led to new knowledge about the TRC process and its effects on magnesium strip quality including the following: 1) Using ALSIM and ANSYS® CFX® commercial packages a 2D mathematical model of thermal-fluid-stress behavior of the magnesium sheet during TRC was successfully developed and validated. 2) An average value of 11 kW/m2°C for the Heat Transfer Coefficient (HTC) was found to best represent the heat transfer between the roll and the strip during TRC casting of AZ31 using the CanmetMATERIALS facility. 3) Modeling results showed that increasing casting speed, casting thicker strips and applying higher HTCs led to less uniform microstructure through thickness in terms of SDAS. 4) Simulations showed the importance of casting parameters such as casting speed and set-back distance on the thermal history and stress development in the sheet during TRC; higher casting speeds led to deeper sumps and higher exit temperatures as well as lower overall rolling loads and lower total strains experienced during TRC. 5) The effect of roll diameter on the thermal history and stress development in the strip was also studied and indicated how larger roll diameters increased the surface normal stress and rolling loads but had little effect on the mushy zone thickness. 6) The correlation between the mechanisms of center-line and inverse segregation formation and thermo-mechanical behavior of the strip was performed. The modeling results suggested that increasing the set-back distance decreases the risk of both defects. Moreover, increasing the roll diameter reduces the propensity to inverse segregation but has a minor effect for center-line segregation formation.

Although Twin Roll Casting (TRC) process has been used for almost 60 years in the aluminum industry, TRC of magnesium is relatively new. In TRC, molten metal is fed onto water-cooled rolls, where it solidifies and is then rolled. Solidification of the molten metal starts at the point of first metal-roll contact and is completed before the kissing point (point of least roll separation) of the two rolls. The unique thermo-physical properties inherent to magnesium and its alloys, such as lower specific heat and latent heat of fusion and larger freezing ranges (in comparison with aluminum and steel) make it challenging for TRC of this alloy. Therefore, a comprehensive understanding of the process and the interaction between the casting conditions and strip final quality is imperative to guarantee high quality twin roll cast strip production. A powerful tool to achieve such knowledge is to develop a mathematical model of the process. In this thesis, a 2D mathematical model for TRC of AZ31 magnesium alloy has been developed and validated based on the TRC facility located at the Natural Resources Canada Government Materials Laboratory (CanmetMATERIALS) in Hamilton, ON, Canada. The validation was performed by comparing the predicted exit strip temperature and secondary dendrite arm spacing (SDAS) through the strip thickness with those measured and obtained by experiments. The model was developed in two stages, first a thermal-fluid model was developed followed by validation and then a thermal-fluid-stress model was developed. This is the first time a comprehensive thermal-fluid-stress model has been developed to simulate the TRC process for magnesium alloys. The work has led to new knowledge about the TRC process and its effects on magnesium strip quality including the following: 1) Using ALSIM and ANSYS® CFX® commercial packages a 2D mathematical model of thermal-fluid-stress behavior of the magnesium sheet during TRC was successfully developed and validated. 2) An average value of 11 kW/m2°C for the Heat Transfer Coefficient (HTC) was found to best represent the heat transfer between the roll and the strip during TRC casting of AZ31 using the CanmetMATERIALS facility. 3) Modeling results showed that increasing casting speed, casting thicker strips and applying higher HTCs led to less uniform microstructure through thickness in terms of SDAS. 4) Simulations showed the importance of casting parameters such as casting speed and set-back distance on the thermal history and stress development in the sheet during TRC; higher casting speeds led to deeper sumps and higher exit temperatures as well as lower overall rolling loads and lower total strains experienced during TRC. 5) The effect of roll diameter on the thermal history and stress development in the strip was also studied and indicated how larger roll diameters increased the surface normal stress and rolling loads but had little effect on the mushy zone thickness. 6) The correlation between the mechanisms of center-line and inverse segregation formation and thermo-mechanical behavior of the strip was performed. The modeling results suggested that increasing the set-back distance decreases the risk of both defects. Moreover, increasing the roll diameter reduces the propensity to inverse segregation but has a minor effect for center-line segregation formation.

This thesis seeks to use mathematical and computational models to develop measures of clinically available data to deepen our understanding, and improve our treatments, of cancer. We consider two broad characteristics of cancer: heterogeneity, in the form of differences in cellular phenotype, and the physical microenvironment; and evolution, which has become accepted as a driver of tumour progression. To ensure that the conclusions drawn are as translatable as possible, we will attempt to use data types that are clinically available. Using a hybrid discrete-cell-based model in two spatial dimensions, we focus on these fundamental aspects of cancer, with the hope of generating new understanding and useful hypotheses to benefit current patients and oncologists. First, we model a tumour growing under the rules of the cancer stem cell hypothesis and a neutral model of evolution, and ask if we can infer the underlying biological proliferative structure. Specifically, we work toward predicting the symmetric division probability of our simulated tumours from clincally relevant observables, as this is a key driving parameter of tumour progression and therapeutic response. We focus on measures of clonal diversity, group size and shape, and a suite of statistical measures of the phylogenetic trees resulting from the tumour's evolution in different regions of parameter space. We find strikingly different patterns in these measures for changing symmetric division probability which hinge on the inclusion of spatial constraints. These results give us insight into differences between solid and liquid tumours, and also generate a number of actionable clinical and biological hypotheses. Second, we explicitly consider the physical microenvironment of tumours invading into healthy tissue, and model oxygen transport, uptake and cellular competition. We then explore the effect of spatial organisation of blood vessels within the tumour on tumour growth kinetics and cellularity. Finding wide variability in the distribution of oxygen across tumours dependent on both vascular organisation and density, we proceed to explore the utility of spatial measures of vessels on radiation efficacy. Our results offer a novel hypothesis as to the failure of vascular normalisation therapy and radiation, and a possible clinical solution.

Submitted by Stéfany Moreira (stemellra@yahoo.com.br) on 2013-03-04T17:47:37Z No. of bitstreams: 1 DISSERTAÇÃO_PráticaModelagemMatemática.pdf: 719262 bytes, checksum: d633103113f5f4495fbce9e3b18c1927 (MD5) / Approved for entry into archive by Neide Nativa (neide@sisbin.ufop.br) on 2013-03-13T23:23:39Z (GMT) No. of bitstreams: 1 DISSERTAÇÃO_PráticaModelagemMatemática.pdf: 719262 bytes, checksum: d633103113f5f4495fbce9e3b18c1927 (MD5) / Made available in DSpace on 2013-03-13T23:23:39Z (GMT). No. of bitstreams: 1 DISSERTAÇÃO_PráticaModelagemMatemática.pdf: 719262 bytes, checksum: d633103113f5f4495fbce9e3b18c1927 (MD5) Previous issue date: 2011 / O presente trabalho apresenta uma pesquisa que aborda a prática de Modelagem Matemática como um cenário de investigação, na perspectiva da formação continuada de Professores de Matemática. Inicialmente, apresentamos algumas concepções de Modelagem Matemática, destacando algumas considerações para a prática docente, além de buscar relações com os cenários de investigação. Nossa metodologia de pesquisa contempla a elaboração e o desenvolvimento de Projetos de Modelagem Matemática relacionados aos preços de uma corrida de táxi e do combustível na bomba, implementados e avaliados por Professores de Matemática nos mais variados níveis de ensino, que cursaram uma disciplina de Modelagem Matemática no Mestrado Profissional em Educação Matemática da UFOP, em 2010. As Considerações Finais do nosso trabalho apontam que o desenvolvimento de Projetos de Modelagem Matemática evidencia a importância do Professor de Matemática: conhecer diversas perspectivas de Modelagem Matemática; vivenciar experiências de Modelagem Matemática em sua formação para desenvolver atividades de Modelagem em sala de aula; refletir sobre o papel das aplicações da Matemática relacionadas a problemas da realidade; valorizar a pesquisa, o tratamento da informação e o trabalho em grupo em sua prática pedagógica; transformar sua sala de aula em um ambiente propício àinvestigação de temas relevantes para os alunos; e saber trabalhar com outras áreas do conhecimento e em ambientes educacionais informatizados. __________________________________________________________________________________________ / ABSTRACT: This paper presents a study focusing on the practice of Mathematical Modeling as a research scenary, in view of the continuing education of Mathematics Teachers. Firstly, some concepts of Mathematical Modeling, highlighting some considerations for teaching practice, in addition seek relationships with research scenary. Our research methodology involves the preparation and development of projects related to Mathematical Modeling prices of a taxi ride and the fuel pump, implemented and evaluated by Mathematics Teachers at various levels of education, who attended a course in Mathematical Modeling at Professional Masters in Mathematical Education of UFOP in 2010. The final considerations of our work indicate that the development of Mathematical Modeling projects demonstrates the importance of Mathematics Teacher: knowledge of various perspectives of Mathematical Modeling, gain experience of Mathematical Modeling in their training to develop modeling activities in the classroom, reflect on the role of applications of Mathematics to problems related to reality, enhancing research, data processing and teamwork in their work; transform your classroom into an environment conducive to research topics relevant to students, and how to work with other areas of knowledge and computerized educational environments.

O presente estudo se propôs a responder à questão de investigação: como a Modelagem Matemática pode contribuir como um meio do educando ser protagonista da sua aprendizagem, aspirando a sua autonomia? A pesquisa apresenta uma proposta de sequência de tarefas em um ambiente de aprendizagem de Modelagem Matemática, dividida em três partes, que são: o convite para realizar modelagem, uma experiência com a temática água e uma experiência com um tema de livre escolha. O estudo foi desenvolvido durante o segundo semestre de 2016, com uma turma de terceiro ano do Ensino Médio de uma escola municipal de Sapucaia do Sul, no horário regular de aula. Apoiada na teoria da Modelagem Matemática em uma perspectiva Sócio-crítica de Barbosa (2001), elaboração de perguntas em um ambiente de Modelagem Matemática de Sant’Ana e Sant’Ana (2009) e na Pedagogia da Autonomia de Paulo Freire (1996), e utilizando o estudo de caso como metodologia, o presente trabalho evidenciou que os estudantes podem ser ativos na construção dos seus conhecimentos. Além disso, percebeu-se, como resultados, que quando lhes é oportunizado um ambiente de liberdade e consideração, que respeita as suas escolhas, os alunos vão assumindo sua responsabilidade pela sua aprendizagem, construindo, aos poucos, suas preferências, suas opções e sua autonomia. / The present study aims to answer the research question: how can Mathematical Modeling contribute as a way for the learner to be the protagonist of their learning by aspiring to their autonomy? The research presents a task sequence proposal in a mathematical modeling-learning environment, divided into three parts. Such parts are the invitation to perform modeling, an experience with water theme, and an experiment with a theme of free student’s choice. The study was developed during the second semester of 2016, with a third year High School class from a municipal school in Sapucaia do Sul, at regular school hours. Based on the theory of Mathematical Modeling in a Socio-critical perspective of Barbosa (2001), elaboration of questions in an environment of Mathematical Modeling of Sant'Ana and Sant'Ana (2009) and in the pedagogy of autonomy of Paulo Freire (1996), and using the case study as a methodology, the present study showed that students can be active in the construction of their knowledge. In addition, it was noticed that, when it is offered an environment of freedom and consideration, respecting their choices, students will assume their responsibility for their learning, gradually building their preferences, their choices and their autonomy.

Submitted by MARINALDO ZAGO null (mzmarinaldozago@gmail.com) on 2016-02-24T01:20:19Z No. of bitstreams: 1 Dissertação Mestrado Marinaldo 2016 (Revista) (2).pdf: 1340836 bytes, checksum: ddd4c0614476082a2b54f2c78a5a9188 (MD5) / Approved for entry into archive by Ana Paula Grisoto (grisotoana@reitoria.unesp.br) on 2016-02-24T16:26:24Z (GMT) No. of bitstreams: 1 zago_m_me_sjrp.pdf: 1340836 bytes, checksum: ddd4c0614476082a2b54f2c78a5a9188 (MD5) / Made available in DSpace on 2016-02-24T16:26:24Z (GMT). No. of bitstreams: 1 zago_m_me_sjrp.pdf: 1340836 bytes, checksum: ddd4c0614476082a2b54f2c78a5a9188 (MD5) Previous issue date: 2016-01-29 / O ensino de Matemática vem passando por um grande conjunto de dificuldades. Isto pode ser comprovado com os resultados de alunos nas avaliações aplicadas em diferentes níveis e escalas, como o Saresp e o Enem.Um dos motivos que levaram a este quadro é o ensino de Matemática pautado por práticas tradicionais, pois, acaba sendo fator desmotivador para o ensino da disciplina. Uma metodologia alternativa, a Modelagem Matemática, pode ajudar a reverter este quadro, uma vez que trabalha com problemas reais, elencados do contexto dos próprios alunos, potencializando, deste modo, a motivação e o interesse, aguçando nos discentes a busca por soluções, mostrando as aplicações da Matemática no cotidiano. Neste trabalho foi realizada uma breve síntese das etapas envolvidas no processo de Modelagem Matemática, aplicada ao ensino de funções, seguido de seis propostas de atividades resolvidas para subsidiar o trabalho de professores pouco familiarizados com esta metodologia. Ainda, como forma de contribuir para uma importante atividade econômica do Município de Osvaldo Cruz-SP, foi realizada uma modelagem matemática com base em dados do setor de transportes, mais precisamente caminhões bitrens. Neste caso, trata-se da simulação de um exemplo mais complexo envolvendo o conceito de modelagem. Espera-se, por meio da divulgação desta pesquisa, contribuir para melhorar a qualidade do ensino de Matemática nas escolas públicas, tanto por meio de subsídio aos docentes, como pela proposição de metodologia com potencial de geração de envolvimento de alunos. / The teaching of mathematics has been experiencing a large set of difficulties. This can be proven by student results in assessments applied at different levels and scales, such as Saresp and Enem. One of the reasons that led to this situation is the teaching Mathematics guided by traditional practices, because it ends up being a demotivating factor to teach the subject. An alternative methodology, the Mathematical Modeling, may help change this situation, as it works with real problems, presented within students’ context, therefore increasing their motivation and interest, sharpening them in the search for solutions, showing the mathematics’ applications in everyday life. This work carried out a brief summary of the steps involved in the process of Mathematical Modeling, applied to teaching functions, followed by six proposals of solved activities to subsidize the work of teachers unfamiliar with this methodology. Also, in order to contribute to an important economic activity in Osvaldo Cruz-SP, mathematical modeling was carried out based on the transport industry data, more precisely b-train trucks. In this case, it was the simulation of a more complex example involving the concept of modeling. By this research we expect to help improve the quality of mathematics teaching in public schools, not only through subsidies to teachers, but also with the methodology proposition generating potential engagement of students.

Orientador: Maria Lúcia Lorenzetti Wodewotzki / Banca: Sandra Maria Pinto Magina / Banca: Cileda de Queiroz e Silva Coutinho / Banca: Antonio Carlos Simões Pião / Banca: Otávio Roberto Jacobini / Resumo: Os objetivos desse trabalho são o estudo teórico sobre os fundamentos da didática da Educação Estatística e sua integração com a Educação Crítica e com a Modelagem Matemática e aplicação dessa integração no microcosmo da sala de aula, com o desenvolvimento e a execução de projetos pedagógicos voltados para esse fim. Sobre os fundamentos teóricos da didática da Estatística, observamos que o planejamento da instrução deve possibilitar o desenvolvimento da literacia, do raciocínio e do pensamento estatístico, para que o ensino/aprendizagem seja feito com sucesso. A Modelagem Matemática e o trabalho com projetos servem como estratégia pedagógica utilizada para conceber os projetos de ensino que buscam desenvolver essas capacidades. A Educação Crítica aparece nos projetos com a problematização do ensino, o trabalho com dados reais, contextualizados, o estímulo ao debate e ao diálogo, a desierarquização e a democratização da sala de aula, o incentivo à capacidade crítica, a valorização do conhecimento reflexivo e a preparação do estudante para interpretar o mundo, praticar o discurso da responsabilidade social e a linguagem crítica, incentivando a liberdade individual, a ética e a justiça social. Conjugando essas idéias, emerge a concepção de Educação Estatística Crítica, que se mostra presente nos dois projetos aqui apresentados. / Abstract: The goals of this thesis are the theoretical study of the Statistic Education's didactical basis and its integration with the Critical Education and the Mathematical Modeling and application of this integration with the development of pedagogical projects toward this end. In the research of the theoretical basis of the Statistic's didactic, we observe that the instruction planning must be able to develop three capacities: literacy, reasoning and statistical thinking. The Mathematical Modeling and the work with projects are used as a pedagogical strategy to create the education projects looking for build up these capacities. The Critical Education is present in the projects with the problematization and the thematization, the real data manipulation, contextualized, the discussion stimulation, the non-hierarchyzation, the democratic values acquired in the classroom, the capacity stimulation of the students to be critical, the reflexive knowledge valuation and the student preparation to explain the world, the practicing speech of the social responsibility and the critical language, stimulating the individual freedom, the ethics and social justice. Putting together these ideas, the Critical Statistics Education's concept appears in this thesis in two projects, presented here. / Doutor

O presente estudo se propôs a responder à questão de investigação: como a Modelagem Matemática pode contribuir como um meio do educando ser protagonista da sua aprendizagem, aspirando a sua autonomia? A pesquisa apresenta uma proposta de sequência de tarefas em um ambiente de aprendizagem de Modelagem Matemática, dividida em três partes, que são: o convite para realizar modelagem, uma experiência com a temática água e uma experiência com um tema de livre escolha. O estudo foi desenvolvido durante o segundo semestre de 2016, com uma turma de terceiro ano do Ensino Médio de uma escola municipal de Sapucaia do Sul, no horário regular de aula. Apoiada na teoria da Modelagem Matemática em uma perspectiva Sócio-crítica de Barbosa (2001), elaboração de perguntas em um ambiente de Modelagem Matemática de Sant’Ana e Sant’Ana (2009) e na Pedagogia da Autonomia de Paulo Freire (1996), e utilizando o estudo de caso como metodologia, o presente trabalho evidenciou que os estudantes podem ser ativos na construção dos seus conhecimentos. Além disso, percebeu-se, como resultados, que quando lhes é oportunizado um ambiente de liberdade e consideração, que respeita as suas escolhas, os alunos vão assumindo sua responsabilidade pela sua aprendizagem, construindo, aos poucos, suas preferências, suas opções e sua autonomia. / The present study aims to answer the research question: how can Mathematical Modeling contribute as a way for the learner to be the protagonist of their learning by aspiring to their autonomy? The research presents a task sequence proposal in a mathematical modeling-learning environment, divided into three parts. Such parts are the invitation to perform modeling, an experience with water theme, and an experiment with a theme of free student’s choice. The study was developed during the second semester of 2016, with a third year High School class from a municipal school in Sapucaia do Sul, at regular school hours. Based on the theory of Mathematical Modeling in a Socio-critical perspective of Barbosa (2001), elaboration of questions in an environment of Mathematical Modeling of Sant'Ana and Sant'Ana (2009) and in the pedagogy of autonomy of Paulo Freire (1996), and using the case study as a methodology, the present study showed that students can be active in the construction of their knowledge. In addition, it was noticed that, when it is offered an environment of freedom and consideration, respecting their choices, students will assume their responsibility for their learning, gradually building their preferences, their choices and their autonomy.

No contexto de refino de petróleo, alquilação é o processo pelo qual se produz uma corrente de hidrocarbonetos chamada de alquilado, na faixa da gasolina, a partir de olefinas leves (propeno, butenos) e isobutano. A demanda por alquilado como corrente integrante do pool de gasolinas de uma refinaria tem aumentado ao longo das duas últimas décadas devido a diversos fatores, sendo o principal a adoção de normas mais restritivas para a qualidade de combustíveis automotivos. No presente trabalho foi desenvolvido e validado um modelo matemático fenomenológico capaz de representar adequadamente as reações químicas envolvidas no processo de alquilação catalítica que utiliza ácido fluorídrico como catalisador, bem como estimar o rendimento de produto e subprodutos da reação e a qualidade do alquilado, representada pelo teor de tri-metil-pentanos (TMP) e di-metil-hexanos (DMH) no alquilado. A construção do modelo foi baseada em balanços de massa e energia no conjunto reacional, utilizando mecanismos cinéticos disponíveis na literatura, sendo os seus parâmetros cinéticos estimados por regressão não linear a partir de literatura. O modelo foi validado com dados obtidos em uma unidade industrial de propriedade da Petrobrás, localizada em Cubatão. Os resultados obtidos permitem afirmar que o modelo apresentou boa resposta para a identificação do consumo de reagentes e rendimento de produtos. Sua resposta tendeu a subestimar a quantidade de frações pesadas do alquilado. / Alkylation is the process which light hydrocarbons (like butenes and iso-bute) are transformed into heavier hydrocarbons, called alkylate. The alkylate demand as a gasoline component has been growing since the middle 90s, mostly due to more severe quality requirements for gasoline. This work proposes a mathematical model that accounts for the main chemical reactions for a HF Alkylation Unit. The alkylate yield and quality, described by its composition, are also represented as well. This model as based on mass and energy balances for the alkylation reactor, using the most accepted kinetic mechanism available at literature. The model constants were estimated by non-linear regression with available data from literature. The model validation was performed comparing the simulation results with data collected at an industrial Alkylation unit at Cubatão, Brazil, without further parameter adjustment. The results show a good model response for the consumption of reagents and alky yield. The model, however, underpredicts the yield of heavier fractions.