A mathematical resolution to log transformations and the binning effect in applied processing of data in flow cytometry

This dissertation develops a new mathematical approach that overcomes the effect of a data processing phenomenon known as "histogram binning" inherent to flow cytometry data. A real-time procedure is introduced to prove the effectiveness and fast implementation of such an approach on real-world data. The histogram binning effect is a dilemma posed by two seemingly antagonistic developments: (1) flow cytometry data in its histogram form is extended in its dynamic range to improve its analysis and interpretation, and (2) the inevitable dynamic range extension introduces an unwelcome side effect, the binning effect, which skews the statistics of the data, undermining as a consequence the accuracy of the analysis and the eventual interpretation of the data. Researchers in the field contended with such a dilemma for many years, resorting either to hardware approaches that are rather costly with inherent calibration and noise effects; or have developed software techniques based on filtering the binning effect but without successfully preserving the statistical content of the original data. The mathematical approach introduced in this dissertation is so appealing that a patent application has been filed. The contribution of this dissertation is an incremental scientific innovation based on a mathematical framework that will allow researchers in the field of flow cytometry to improve the interpretation of data knowing that its statistical meaning has been faithfully preserved for its optimized analysis. Furthermore, with the same mathematical foundation, proof of the origin of such an inherent artifact is provided. These results are unique in that new mathematical derivations are established to define and solve the critical problem of the binning effect faced at the experimental assessment level, providing a data platform that preserves its statistical content. In addition, a novel method for accumulating the log-transformed data was developed. This new method uses the properties of the transformation of statistical distributions to accumulate the output histogram in a non-integer and multi-channel fashion. Although the mathematics of this new mapping technique seem intricate, the concise nature of the derivations allow for an implementation procedure that lends itself to a real-time implementation using lookup tables, a task that is also introduced in this dissertation.

A mathematical model for the maneuvering simulation of a propelled SPAR vessel

To predict the maneuvering performance of a propelled SPAR vessel, a mathematical model was established as a path simulator. A system-based mathematical model was chosen as it offers advantages in cost and time over full Computational Fluid Dynamics (CFD) simulations. The model is intended to provide a means of optimizing the maneuvering performance of this new vessel type. In this study the hydrodynamic forces and control forces are investigated as individual components, combined in a vectorial setting, and transferred to a body-fixed basis. SPAR vessels are known to be very sensitive to large amplitude motions during maneuvers due to the relatively small hydrostatic restoring forces. Previous model tests of SPAR vessels have shown significant roll and pitch amplitudes, especially during course change maneuvers. Thus, a full 6 DOF equation of motion was employed in the current numerical model. The mathematical model employed in this study was a combination of the model introduced by the Maneuvering Modeling Group (MMG) and the Abkowitz (1964) model. The new model represents the forces applied to the ship hull, the propeller forces and the rudder forces independently, as proposed by the MMG, but uses a 6DOF equation of motion introduced by Abkowitz to describe the motion of a maneuvering ship. The mathematical model was used to simulate the trajectory and motions of the propelled SPAR vessel in 10˚/10˚, 20˚/20˚ and 30˚/30˚ standard zig-zag maneuvers, as well as turning circle tests at rudder angles of 20˚ and 30˚. The simulation results were used to determine the maneuverability parameters (e.g. advance, transfer and tactical diameter) of the vessel. The final model provides the means of predicting and assessing the performance of the vessel type and can be easily adapted to specific vessel configurations based on the generic SPAR-type vessel used in this study.

Mathematical modelling of integrin-like receptors systems

Nel presente lavoro, ho studiato e trovato le soluzioni esatte di un modello matematico applicato ai recettori cellulari della famiglia delle integrine. Nel modello le integrine sono considerate come un sistema a due livelli, attivo e non attivo. Quando le integrine si trovano nello stato inattivo possono diffondere nella membrana, mentre quando si trovano nello stato attivo risultano cristallizzate nella membrana, incapaci di diffondere. La variazione di concentrazione nella superficie cellulare di una sostanza chiamata attivatore dà luogo all’attivazione delle integrine. Inoltre, questi eterodimeri possono legare una molecola inibitrice con funzioni di controllo e regolazione, che chiameremo v, la quale, legandosi al recettore, fa aumentare la produzione della sostanza attizzatrice, che chiameremo u. In questo modo si innesca un meccanismo di retroazione positiva. L’inibitore v regola il meccanismo di produzione di u, ed assume, pertanto, il ruolo di modulatore. Infatti, grazie a questo sistema di fine regolazione il meccanismo di feedback positivo è in grado di autolimitarsi. Si costruisce poi un modello di equazioni differenziali partendo dalle semplici reazioni chimiche coinvolte. Una volta che il sistema di equazioni è impostato, si possono desumere le soluzioni per le concentrazioni dell’inibitore e dell’attivatore per un caso particolare dei parametri. Infine, si può eseguire un test per vedere cosa predice il modello in termini di integrine. Per farlo, ho utilizzato un’attivazione del tipo funzione gradino e l’ho inserita nel sistema, valutando la dinamica dei recettori. Si ottiene in questo modo un risultato in accordo con le previsioni: le integrine legate si trovano soprattutto ai limiti della zona attivata, mentre le integrine libere vengono a mancare nella zona attivata.

An investigation into the tension arising between natural language and mathematical language experienced by mechanical engineering students

This investigation is grounded within the concept of embodied cognition where the mind is considered to be part of a biological system. A first year undergraduate Mechanical Engineering cohort of students was tasked with explaining the behaviour of three balls of different masses being rolled down a ramp. The explanations given by the students highlighted the cognitive conflict between the everyday interpretation of the word energy and its mathematical use. The results showed that even after many years of schooling, students found it challenging to interpret the mathematics they had learned and relied upon pseudo-scientific notions to account for the behaviour of the balls.

Parsing mathematical constructs:results from a preliminary eye tracking study

The semantic model described in this paper is based on ones developed for arithmetic (e.g. McCloskey et al. 1985, Cohene and Dehaene 1995), natural language processing (Fodor 1975, Chomsky 1981) and work by the author on how learners parse mathematical structures. The semantic model highlights the importance of the parsing process and the relationship between this process and the mathematical lexicon/grammar. It concludes by demonstrating that for a learner to become an efficient, competent mathematician a process of top-down parsing is essential.

The reliability mathematical model of load-sharing parallel systems based on fatigue cumulative damage

Conventional reliability models for parallel systems are not applicable for the analysis of parallel systems with load transfer and sharing. In this short communication, firstly, the dependent failures of parallel systems are analyzed, and the reliability model of load-sharing parallel system is presented based on Miner cumulative damage theory and the full probability formula. Secondly, the parallel system reliability is calculated by Monte Carlo simulation when the component life follows the Weibull distribution. The research result shows that the proposed reliability mathematical model could analyze and evaluate the reliability of parallel systems in the presence of load transfer.

Mathematical Modeling of Perifusion Cell Culture Experiments

In perifusion cell cultures, the culture medium flows continuously through a chamber containing immobilized cells and the effluent is collected at the end. In our main applications, gonadotropin releasing hormone (GnRH) or oxytocin is introduced into the chamber as the input. They stimulate the cells to secrete luteinizing hormone (LH), which is collected in the effluent. To relate the effluent LH concentration to the cellular processes producing it, we develop and analyze a mathematical model consisting of coupled partial differential equations describing the intracellular signaling and the movement of substances in the cell chamber. We analyze three different data sets and give cellular mechanisms that explain the data. Our model indicates that two negative feedback loops, one fast and one slow, are needed to explain the data and we give their biological bases. We demonstrate that different LH outcomes in oxytocin and GnRH stimulations might originate from different receptor dynamics. We analyze the model to understand the influence of parameters, like the rate of the medium flow or the fraction collection time, on the experimental outcomes. We investigate how the rate of binding and dissociation of the input hormone to and from its receptor influence its movement down the chamber. Finally, we formulate and analyze simpler models that allow us to predict the distortion of a square pulse due to hormone-receptor interactions and to estimate parameters using perifusion data. We show that in the limit of high binding and dissociation the square pulse moves as a diffusing Gaussian and in this limit the biological parameters can be estimated.

Martin Heidegger's Mathematical Dialectic: Uncovering the Structure of Modernity

Martin Heidegger is generally regarded as one of the most significant—if also the most controversial—philosophers of the 20th century. Most scholarly engagement with Heidegger’s thought on Modernity approaches his work with a special focus on either his critique of technology, or on his more general critique of subjectivity. This dissertation project attempts to elucidate Martin Heidegger’s diagnosis of modernity, and, by extension, his thought as a whole, from the neglected standpoint of his understanding of mathematics, which he explicitly identifies as the essence of modernity.

Accordingly, our project attempts to work through the development of Modernity, as Heidegger understands it, on the basis of what we call a “mathematical dialectic.“ The basis of our analysis is that Heidegger’s understanding of Modernity, both on its own terms and in the context of his theory of history [Seinsgeschichte], is best understood in terms of the interaction between two essential, “mathematical” characteristics, namely, self-grounding and homogeneity. This project first investigates the mathematical qualities of these components of Modernity individually, and then attempts to trace the historical and philosophical development of Modernity on the basis of the interaction between these two components—an interaction that is, we argue, itself regulated by the structure of the mathematical, according to Heidegger’s understanding of the term.

The project undertaken here intends not only to serve as an interpretive, scholarly function of elucidating Heidegger’s understanding of Modernity, but also to advance the larger aim of defending the prescience, structural coherence, and relevance of Heidegger’s diagnosis of Modernity as such.

Investigating practitioner self-efficacy in an era of mathematical reform

The purpose of this study is to explore teacher self-efficacy at a time of radical mathematical reform. Project Maths – the new initiative which was rolled out nationwide in 2010 differs from previous attempts at innovation in that it targets a much closer connection between curriculum and pedagogy. Gone are the days of well-rehearsed routines where the role of the mathematician was essentially that of demonstrator. Teaching for understanding is now the main ‘official’ pedagogical focus, with emphasis on the practitioner playing the part of mediator between subject-matter and student. Mathematical instruction is not merely concerned with the transmission of knowledge and skills which is a particular pedagogical position to take. It is also an emotional practice (Hargreaves, 1998) that colours and expresses the feelings and actions of practitioners. While emotion plays a key role in teachers’ commitment to curricular reform, it is also shaped by the social and cultural contexts of mathematical change, alongside with the attitudes and beliefs of the mathematical teaching community. Inspired by Bandura’s theory of learning (1986), this investigation aims to shed light on the complex interplay between so-called mastery and vicarious experiences, social persuasion and physiological states. Vygotsky’s view of learning (1978) as a socio-cultural process is also drawn upon, as it provides a useful structure against which teacher self-efficacy and professional development can be examined. Finally, Hiebert’s theory (1986) is used to examine mathematics teaching self-efficacy and mathematics self-efficacy.

Improving the use of G-CSF during chemotherapy using physiological mathematical modelling : a quantitative systems pharmacology approach

La diminution des doses administrées ou même la cessation complète d'un traitement chimiothérapeutique est souvent la conséquence de la réduction du nombre de neutrophiles, qui sont les globules blancs les plus fréquents dans le sang. Cette réduction dans le nombre absolu des neutrophiles, aussi connue sous le nom de myélosuppression, est précipitée par les effets létaux non spécifiques des médicaments anti-cancéreux, qui, parallèlement à leur effet thérapeutique, produisent aussi des effets toxiques sur les cellules saines. Dans le but d'atténuer cet impact myélosuppresseur, on administre aux patients un facteur de stimulation des colonies de granulocytes recombinant humain (rhG-CSF), une forme exogène du G-CSF, l'hormone responsable de la stimulation de la production des neutrophiles et de leurs libération dans la circulation sanguine. Bien que les bienfaits d'un traitement prophylactique avec le G-CSF pendant la chimiothérapie soient bien établis, les protocoles d'administration demeurent mal définis et sont fréquemment déterminés ad libitum par les cliniciens. Avec l'optique d'améliorer le dosage thérapeutique et rationaliser l'utilisation du rhG-CSF pendant le traitement chimiothérapeutique, nous avons développé un modèle physiologique du processus de granulopoïèse, qui incorpore les connaissances actuelles de pointe relatives à la production des neutrophiles des cellules souches hématopoïétiques dans la moelle osseuse. À ce modèle physiologique, nous avons intégré des modèles pharmacocinétiques/pharmacodynamiques (PK/PD) de deux médicaments: le PM00104 (Zalypsis®), un médicament anti-cancéreux, et le rhG-CSF (filgrastim). En se servant des principes fondamentaux sous-jacents à la physiologie, nous avons estimé les paramètres de manière exhaustive sans devoir recourir à l'ajustement des données, ce qui nous a permis de prédire des données cliniques provenant de 172 patients soumis au protocol CHOP14 (6 cycles de chimiothérapie avec une période de 14 jours où l'administration du rhG-CSF se fait du jour 4 au jour 13 post-chimiothérapie). En utilisant ce modèle physio-PK/PD, nous avons démontré que le nombre d'administrations du rhG-CSF pourrait être réduit de dix (pratique actuelle) à quatre ou même trois administrations, à condition de retarder le début du traitement prophylactique par le rhG-CSF. Dans un souci d'applicabilité clinique de notre approche de modélisation, nous avons investigué l'impact de la variabilité PK présente dans une population de patients, sur les prédictions du modèle, en intégrant des modèles PK de population (Pop-PK) des deux médicaments. En considérant des cohortes de 500 patients in silico pour chacun des cinq scénarios de variabilité plausibles et en utilisant trois marqueurs cliniques, soient le temps au nadir des neutrophiles, la valeur du nadir, ainsi que l'aire sous la courbe concentration-effet, nous avons établi qu'il n'y avait aucune différence significative dans les prédictions du modèle entre le patient-type et la population. Ceci démontre la robustesse de l'approche que nous avons développée et qui s'apparente à une approche de pharmacologie quantitative des systèmes (QSP). Motivés par l'utilisation du rhG-CSF dans le traitement d'autres maladies, comme des pathologies périodiques telles que la neutropénie cyclique, nous avons ensuite soumis l'étude du modèle au contexte des maladies dynamiques. En mettant en évidence la non validité du paradigme de la rétroaction des cytokines pour l'administration exogène des mimétiques du G-CSF, nous avons développé un modèle physiologique PK/PD novateur comprenant les concentrations libres et liées du G-CSF. Ce nouveau modèle PK a aussi nécessité des changements dans le modèle PD puisqu’il nous a permis de retracer les concentrations du G-CSF lié aux neutrophiles. Nous avons démontré que l'hypothèse sous-jacente de l'équilibre entre la concentration libre et liée, selon la loi d'action de masse, n'est plus valide pour le G-CSF aux concentrations endogènes et mènerait en fait à la surestimation de la clairance rénale du médicament. En procédant ainsi, nous avons réussi à reproduire des données cliniques obtenues dans diverses conditions (l'administration exogène du G-CSF, l'administration du PM00104, CHOP14). Nous avons aussi fourni une explication logique des mécanismes responsables de la réponse physiologique aux deux médicaments. Finalement, afin de mettre en exergue l’approche intégrative en pharmacologie adoptée dans cette thèse, nous avons démontré sa valeur inestimable pour la mise en lumière et la reconstruction des systèmes vivants complexes, en faisant le parallèle avec d’autres disciplines scientifiques telles que la paléontologie et la forensique, où une approche semblable a largement fait ses preuves. Nous avons aussi discuté du potentiel de la pharmacologie quantitative des systèmes appliquées au développement du médicament et à la médecine translationnelle, en se servant du modèle physio-PK/PD que nous avons mis au point.

Using blogs to enhance the capacity of mathematical communication in High School

The aim of this investigation is to analyze the use of the blog as an educational resource for the development of the mathematical communication in secondary education. With this aim, four aspects are analyzed: organization of mathematical thinking through communication; communication of mathematical thinking; analysis and evaluation of the strategies and mathematical thought of others; and expression of mathematical ideas using mathematical language. The research was conducted from a qualitative approach on an exploratory level, with the case study method of 4 classrooms of second grade of secondary education in a private school in Lima. The observational technique of 20 publications in the blog of the math class was applied; a study of a focal group with a sample of 9 students with different levels of academic performance; and an interview with the academic coordinator of the school was conducted. The results show that the organization of mathematical thinking through communication is carried out in the blog in a written, graphical and oral way through explanations, schemes and videos. Regarding communication of mathematical thinking, the blog is used to describe concepts, arguments and mathematical procedures with words and examples of the students. The analysis and evaluation of the strategies and mathematical thinking is performed through comments and debates about the publications. It was also noted that the blog does not facilitate the use of mathematical language to express mathematical ideas, since it does not allow direct writing of symbols nor graphic representation.

Mathematical Modeling to Assess the Drivers of the Recent Emergence of Typhoid Fever in Blantyre, Malawi

BACKGROUND: Multiyear epidemics of Salmonella enterica serovar Typhi have been reported from countries across eastern and southern Africa in recent years. In Blantyre, Malawi, a dramatic increase in typhoid fever cases has recently occurred, and may be linked to the emergence of the H58 haplotype. Strains belonging to the H58 haplotype often exhibit multidrug resistance and may have a fitness advantage relative to other Salmonella Typhi strains.

METHODS: To explore hypotheses for the increased number of typhoid fever cases in Blantyre, we fit a mathematical model to culture-confirmed cases of Salmonella enterica infections at Queen Elizabeth Central Hospital, Blantyre. We explored 4 hypotheses: (1) an increase in the basic reproductive number (R0) in response to increasing population density; (2) a decrease in the incidence of cross-immunizing infection with Salmonella Enteritidis; (3) an increase in the duration of infectiousness due to failure to respond to first-line antibiotics; and (4) an increase in the transmission rate following the emergence of the H58 haplotype.

RESULTS: Increasing population density or decreasing cross-immunity could not fully explain the observed pattern of typhoid emergence in Blantyre, whereas models allowing for an increase in the duration of infectiousness and/or the transmission rate of typhoid following the emergence of the H58 haplotype provided a good fit to the data.

CONCLUSIONS: Our results suggest that an increase in the transmissibility of typhoid due to the emergence of drug resistance associated with the H58 haplotype may help to explain recent outbreaks of typhoid in Malawi and similar settings in Africa.

New methods for movement technique development in cross-country skiing using mathematical models and simulation

This Licentiate Thesis is devoted to the presentation and discussion of some new contributions in applied mathematics directed towards scientific computing in sports engineering. It considers inverse problems of biomechanical simulations with rigid body musculoskeletal systems especially in cross-country skiing. This is a contrast to the main research on cross-country skiing biomechanics, which is based mainly on experimental testing alone. The thesis consists of an introduction and five papers. The introduction motivates the context of the papers and puts them into a more general framework. Two papers (D and E) consider studies of real questions in cross-country skiing, which are modelled and simulated. The results give some interesting indications, concerning these challenging questions, which can be used as a basis for further research. However, the measurements are not accurate enough to give the final answers. Paper C is a simulation study which is more extensive than paper D and E, and is compared to electromyography measurements in the literature. Validation in biomechanical simulations is difficult and reducing mathematical errors is one way of reaching closer to more realistic results. Paper A examines well-posedness for forward dynamics with full muscle dynamics. Moreover, paper B is a technical report which describes the problem formulation and mathematical models and simulation from paper A in more detail. Our new modelling together with the simulations enable new possibilities. This is similar to simulations of applications in other engineering fields, and need in the same way be handled with care in order to achieve reliable results. The results in this thesis indicate that it can be very useful to use mathematical modelling and numerical simulations when describing cross-country skiing biomechanics. Hence, this thesis contributes to the possibility of beginning to use and develop such modelling and simulation techniques also in this context.

Mathematical methods for the configuration of transportation systems with a focus on continuous and modular matrix conveyors

The lack of flexibility in logistic systems currently on the market leads to the development of new innovative transportation systems. In order to find the optimal configuration of such a system depending on the current goal functions, for example minimization of transport times and maximization of the throughput, various mathematical methods of multi-criteria optimization are applicable. In this work, the concept of a complex transportation system is presented. Furthermore, the question of finding the optimal configuration of such a system through mathematical methods of optimization is considered.