Spatio-temporal systems with heterogeneity in their structure and behavior have two major problems. The first one is that such systems extend over very large spatial and temporal domains and consume a lot of resources to simulate that they are infeasible to study with current platforms. The second one is that the data available for understanding such systems is limited. This also makes it difficult to get the data for validation of their constituent processes while simultaneously considering their global behavior. For example, the valley fever fungus considered in this dissertation is spread over a large spatial grid in the arid Southwest and typically needs to be simulated over several decades of time to obtain useful information. It is also hard to get the temperature and moisture data at every grid point of the spatial domain over the region of study. In order to address the first problem, we develop a method based on the discrete event system specification which exploits the heterogeneity in the activity of the spatio-temporal system and which has been shown to be effective in solving relatively simple partial differential equation systems. The benefit of addressing the first problem is that it now makes it feasible to address the second problem.We address the second problem by making use of a multilevel methodology based on modeling and simulation and systems theory. This methodology helps us in the construction of models with different resolutions (base and lumped models). This allows us to refine an initially constructed lumped model with detailed physics-based process models and assess whether they improve on the original lumped models. For that assessment, we use the concept of experimental frame to delimit where the improvement is needed. This allows us to work with the available data, improve the component models in their own experimental frame and then move them to the overall frame. In this dissertation, we develop a multilevel methodology and apply it to a valley fever model. Moreover, we study the model's behavior in a particular experimental frame of interest, namely the formation of new sporing sites.

No description available.

Metabolic systems tend to exhibit steady states that can be measured in terms of their concentrations and fluxes. These measurements can be regarded as a phenotypic representation of all the complex interactions and regulatory mechanisms taking place in the underlying metabolic network. Such interactions determine the system's response to external perturbations and are responsible, for example, for its asymptotic stability or for oscillatory trajectories around the steady state. However, determining these perturbation responses in the absence of fully specified kinetic models remains an important challenge of computational systems biology. Structural kinetic modeling (SKM) is a framework to analyse whether a metabolic steady state remains stable under perturbation, without requiring detailed knowledge about individual rate equations. It provides a parameterised representation of the system's Jacobian matrix in which the model parameters encode information about the enzyme-metabolite interactions. Stability criteria can be derived by generating a large number of structural kinetic models (SK-models) with randomly sampled parameter sets and evaluating the resulting Jacobian matrices. The parameter space can be analysed statistically in order to detect network positions that contribute significantly to the perturbation response. Because the sampled parameters are equivalent to the elasticities used in metabolic control analysis (MCA), the results are easy to interpret biologically. In this project, the SKM framework was extended by several novel methodological improvements. These improvements were evaluated in a simulation study using a set of small example pathways with simple Michaelis Menten rate laws. Afterwards, a detailed analysis of the dynamic properties of the neuronal TCA cycle was performed in order to demonstrate how the new insights obtained in this work could be used for the study of complex metabolic systems. The first improvement was achieved by examining the biological feasibility of the elasticity combinations created during Monte Carlo sampling. Using a set of small example systems, the findings showed that the majority of sampled SK-models would yield negative kinetic parameters if they were translated back into kinetic models. To overcome this problem, a simple criterion was formulated that mitigates such infeasible models and the application of this criterion changed the conclusions of the SKM experiment. The second improvement of this work was the application of supervised machine-learning approaches in order to analyse SKM experiments. So far, SKM experiments have focused on the detection of individual enzymes to identify single reactions important for maintaining the stability or oscillatory trajectories. In this work, this approach was extended by demonstrating how SKM enables the detection of ensembles of enzymes or metabolites that act together in an orchestrated manner to coordinate the pathways response to perturbations. In doing so, stable and unstable states served as class labels, and classifiers were trained to detect elasticity regions associated with stability and instability. Classification was performed using decision trees and relevance vector machines (RVMs). The decision trees produced good classification accuracy in terms of model bias and generalizability. RVMs outperformed decision trees when applied to small models, but encountered severe problems when applied to larger systems because of their high runtime requirements. The decision tree rulesets were analysed statistically and individually in order to explore the role of individual enzymes or metabolites in controlling the system's trajectories around steady states. The third improvement of this work was the establishment of a relationship between the SKM framework and the related field of MCA. In particular, it was shown how the sampled elasticities could be converted to flux control coefficients, which were then investigated for their predictive information content in classifier training. After evaluation on the small example pathways, the methodology was used to study two steady states of the neuronal TCA cycle with respect to their intrinsic mechanisms responsible for stability or instability. The findings showed that several elasticities were jointly coordinated to control stability and that the main source for potential instabilities were mutations in the enzyme alpha-ketoglutarate dehydrogenase. / Metabolische Systeme neigen zur Ausbildung von Fließgleichgewichten, deren Konzentrationen und Reaktionsflüsse experimentell charakterisierbar sind. Derartige Messungen bieten eine phänotypische Repräsentation der zahlreichen Interaktionen und regulatorischen Mechanismen des zugrundeliegenden metabolischen Netzwerks. Diese Interaktionen bestimmen die Reaktion des Systems auf externe Perturbationen, wie z.B. dessen asymptotische Stabilität und Oszillationen. Die Charakterisierung solcher Eigenschaften ist jedoch schwierig, wenn kein entsprechendes kinetisches Modell mit allen Ratengleichungen und kinetischen Parametern für das untersuchte System zur Verfügung steht. Die strukturelle kinetische Modellierung (SKM) ermöglicht die Untersuchung dynamischer Eigenschaften wie Stabilität oder Oszillationen, ohne die Ratengleichungen und zugehörigen Parameter im Detail zu kennen. Statt dessen liefert sie eine parametrisierte Repräsentation der Jacobimatrix, in welcher die einzelnen Parameter Informationen über die Sättigung der Enzyme des Systems mit ihren Substraten kodieren. Die Parameter entsprechen dabei den Elastizitäten aus der metabolischen Kontrollanalyse, was ihre biologische Interpretation vereinfacht. Stabilitätskriterien werden durch Monte Carlo Verfahren hergeleitet, wobei zunächst eine große Anzahl struktureller kinetische Modelle (SK-Modelle) mit zufällig gezogenen Parametermengen generiert, und anschließend die resultierenden Jacobimatrizen evaluiert werden. Im Anschluss kann der Parameterraum statistisch analysiert werden, um Enzyme und Metabolite mit signifikantem Einfluss auf die Stabilität zu detektieren. In der vorliegenden Arbeit wurde das bisherige SKM-Verfahren durch neue methodische Verbesserungen erweitert. Diese Verbesserungen wurden anhand einer Simulationsstudie evaluiert, welche auf kleinen Beispielsystemen mit einfachen Michaelis Menten Kinetiken basierte. Im Anschluss wurden sie für eine detaillierte Analyse der dynamischen Eigenschaften des Zitratzyklus verwendet. Die erste Erweiterung der bestehenden Methodik wurde durch Untersuchung der biologischen Machbarkeit der zufällig erzeugten Elastizitäten erreicht. Es konnte gezeigt werden, dass die Mehrheit der zufällig erzeugten SK-Modelle zu negativen Michaeliskonstanten führt. Um dieses Problem anzugehen, wurde ein einfaches Kriterium formuliert, welches das Auftreten solcher biologisch unrealistischer SK-Modelle verhindert. Es konnte gezeigt werden, dass die Anwendung des Kriteriums die Ergebnisse von SKM Experimenten stark beeinflussen kann. Der zweite Beitrag bezog sich auf die Analyse von SKM-Experimenten mit Hilfe überwachter maschineller Lernverfahren. Bisherige SKM-Studien konzentrierten sich meist auf die Detektion individueller Elastizitäten, um einzelne Reaktionen mit Einfluss auf das Stabilitäts- oder oszillatorische Verhalten zu identifizieren. In dieser Arbeit wurde demonstriert, wie SKM Experimente im Hinblick auf multivariate Muster analysiert werden können, um Elastizitäten zu entdecken, die gemeinsam auf orchestrierte und koordinierte Weise die Eigenschaften des Systems bestimmen. Sowohl Entscheidungsbäume als auch Relevanzvektormaschinen (RVMs) wurden als Klassifikatoren eingesetzt. Während Entscheidungsbäume im allgemeinen gute Klassifikationsergebnisse lieferten, scheiterten RVMs an ihren großen Laufzeitbedürfnissen bei Anwendung auf ein komplexes System wie den Zitratzyklus. Hergeleitete Entscheidungsbaumregeln wurden sowohl statistisch als auch individuell analysiert, um die Koordination von Enzymen und Metaboliten in der Kontrolle von Trajektorien des Systems zu untersuchen. Der dritte Beitrag, welcher in dieser Arbeit vorgestellt wurde, war die Etablierung der Beziehung zwischen SKM und der metabolischer Kontrollanalyse. Insbesondere wurde gezeigt, wie die zufällig generierten Elastizitäten in Flusskontrollkoeffizienten umgewandelt werden. Diese wurden im Anschluss bezüglich ihres Informationsgehaltes zum Klassifikationstraining untersucht. Nach der Evaluierung anhand einiger kleiner Beispielsysteme wurde die neue Methodik auf die Studie zweier Fließgleichgewichte des neuronalen Zitratzyklus angewandt, um intrinsische Mechanismen für Stabilität oder Instabilität zu finden. Die Ergebnisse identifizierten Mutationen im Enzym alpha-ketoglutarate dehydrogenase als wahrscheinlichste Quelle füur Instabilitäten.

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.

The increasing yield of municipal solid waste (MSW) is one of the main by-products of modern society. Among various MSW treatment methods, plasma gasification in a fixed-bed melting reactor (PGM) is a new technology, which may provide an efficient and environmental friendly solution for problems related to MSW disposals. General objectives of this work are to develop mathematical models for the PGM process, and using these models to analyze the characteristics of this new technology. In this thesis, both experimental measurement and numerical analysis are carried out to evaluate the performance of both air gasification and air&steam gasification in a PGM reactor. Furthermore, parameter studies were launched to investigate the effect of three main operation parameters: equivalence ratio (ER), steam feedstock mass ratio(S/F) and plasma energy ratio (PER). Based on the above analysis, the optimal suggestions aiming at providing highest syngas calorific value, as well as system energy efficiency, are given. Six experimental tests were conducted in a demonstration reactor. These tests are classified into two groups: air gasification (case 1 and 2) and air&steam gasification (case 3 to 6). In all these cases, the plasma gasification and melting of MSW produced a   syngas with a lower heating value of 6.0-7.0 MJ/Nm3. By comparing the syngas yield and calorific value, the study found out that the steam and air mixture is a better gasification agent than pure air. It is also discovered that the operation parameters seriously influence the operation of the PGM process. A zero-dimensional kinetic free model was built up to investigate the influence of operation parameters. The model was developed using the popular process simulation software Aspen Plus. In this model, the whole plasma gasification and melting process was divided into four layers: drying, pyrolysis, char combustion&gasificaiton, and plasma melting. Mass and energy balances were considered in all layers. It was proved that the model is able to give good agreement of the syngas yield and composition. This model was used to study the influence of ER, S/F and PER on average gasification temperature, syngas composition and syngas yield. It is pointed out that a common problem for the PGM air gasification is the incomplete char conversion due to low ER value. Both increasing plasma power and feeding steam is helpful for solving this problem. The syngas quality can also be improved by reasonably feeding high temperature steam into the reactor.   In order to provide detailed information inside the reactor, a two-dimensional steady model was developed for the PGM process. The model used the Euler-Euler multiphase approach. The mass, momentum and energy balances of both gas and solid phases are considered in this model. The model described the complex chemical and physical processes such as drying, pyrolysis, homogeneous reactions, heterogeneous char reactions and melting of the inorganic components of MSW. The rates of chemical reactions are controlled by kinetic rates and physical transport theories. The model is capable of simulating the pressure fields, temperature fields, and velocity fields of both phase, as well as variations of gas and solid composition insider the reactor. This model was used to simulate both air gasification and air&steam gasification of MSW in the PGM reactor. For PGM air gasification, simulated results showed that when ER varies from 0.043 to 0.077, both the syngas yield and cold gas efficiency demonstrated a trend of increasing. This is explained mainly by the increase of char conversion rate with ER. However, the increase of ER was restricted by peak temperature inside the fixed-bed reactor. Therefore, it is not suggested to use only air as gasification in the PGM process. The influence of plasma power is not obvious when PER varies from 0.098 to 0.138.  The positive influences of steam addition on cold gas efficiency and syngas lower-heating-value are confirmed by the simulation results of PGM air&steam gasification. The main effect of steam addition is the rouse of water shift reaction, which largely accelerates the char conversion and final yields of hydrogen and carbon dioxide. The effect of steam injection is affected by steam feeding rate, air feeding rate and plasma power. Based on the above modeling work, Interactions between operation parameters were discussed. Possible operation extents of operation parameters are delimitated. The optimal points aiming at obtaining maximum syngas LHV and system CGE are suggested.

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.