Recovering Natural History: Modeling Cardiovascular Biomarkers in the Presence of Endogenous Medication Use

Thesis (Ph.D.)--University of Washington, 2016-08

The use of mathematical modelling in fire safety engineering

Fire is a form of uncontrolled combustion which generates heat, smoke, toxic and irritant gases. All of these products are harmful to man and account for the heavy annual cost of 800 lives and £1,000,000,000 worth of property damage in Britain alone. The new discipline of Fire Safety Engineering has developed as a means of reducing these unacceptable losses. One of the main tools of Fire Safety Engineering is the mathematical model and over the past 15 years a number of mathematical models have emerged to cater for the needs of this discipline. Part of the difficulty faced by the Fire Safety Engineer is the selection of the most appropriate modelling tool to use for the job. To make an informed choice it is essential to have a good understanding of the various modelling approaches, their capabilities and limitations. In this paper some of the fundamental modelling tools used to predict fire and evacuation are investigated as are the issues associated with their use and recent developments in modelling technology.

Temperature regulation of a pilot-scale batch reaction system via explicit model predictive control

In this paper, the temperature of a pilot-scale batch reaction system is modeled towards the design of a controller based on the explicit model predictive control (EMPC) strategy -- Some mathematical models are developed from experimental data to describe the system behavior -- The simplest, yet reliable, model obtained is a (1,1,1)-order ARX polynomial model for which the mentioned EMPC controller has been designed -- The resultant controller has a reduced mathematical complexity and, according to the successful results obtained in simulations, will be used directly on the real control system in a next stage of the entire experimental framework

The Control of a Mathematical Analog of a Tentacle

A 2-dimensional dynamic analog of squid tentacles was presented. The tentacle analog consists of a multi-cell structure, which can be easily replicated to a large scale. Each cell of the model is a quadrilateral with unit masses at the corners. Each side of the quadrilateral is a spring-damper system in parallel. The spring constants are the controls for the system. The dynamics are subject to the constraint that the area of each quadrilateral must remain constant. The system dynamics was analyzed, and various equilibrium points were found with different controls. Then these equilibrium points were further determined experimentally, demonstrated to be asymptotically stable. A simulation built in MATLAB was used to find the convergence rates under different controls and damping coefficients. Finally, a control scheme was developed and used to drive the system to several configurations observed in real tentacle.

A stylized software model to explore the free market equality/efficiency tradeoff

This paper provides an agent-based software exploration of the wellknown free market efficiency/equality trade-off. Our study simulates the interaction of agents producing, trading and consuming goods in the presence of different market structures, and looks at how efficient the producers/consumers mapping turn out to be as well as the resulting distribution of welfare among agents at the end of an arbitrarily large number of iterations. Two market mechanisms are compared: the competitive market (a double auction market in which agents outbid each other in order to buy and sell products) and the random one (in which products are allocated randomly). Our results confirm that the superior efficiency of the competitive market (an effective and never stopping producers/consumers mapping and a superior aggregative welfare) comes at a very high price in terms of inequality (above all when severe budget constraints are in play).

Hybrid mathematical and informational modeling of beam-to-column connections

The analysis of steel and composite frames has traditionally been carried out by idealizing beam-to-column connections as either rigid or pinned. Although some advanced analysis methods have been proposed to account for semi-rigid connections, the performance of these methods strongly depends on the proper modeling of connection behavior. The primary challenge of modeling beam-to-column connections is their inelastic response and continuously varying stiffness, strength, and ductility. In this dissertation, two distinct approaches—mathematical models and informational models—are proposed to account for the complex hysteretic behavior of beam-to-column connections. The performance of the two approaches is examined and is then followed by a discussion of their merits and deficiencies. To capitalize on the merits of both mathematical and informational representations, a new approach, a hybrid modeling framework, is developed and demonstrated through modeling beam-to-column connections. Component-based modeling is a compromise spanning two extremes in the field of mathematical modeling: simplified global models and finite element models. In the component-based modeling of angle connections, the five critical components of excessive deformation are identified. Constitutive relationships of angles, column panel zones, and contact between angles and column flanges, are derived by using only material and geometric properties and theoretical mechanics considerations. Those of slip and bolt hole ovalization are simplified by empirically-suggested mathematical representation and expert opinions. A mathematical model is then assembled as a macro-element by combining rigid bars and springs that represent the constitutive relationship of components. Lastly, the moment-rotation curves of the mathematical models are compared with those of experimental tests. In the case of a top-and-seat angle connection with double web angles, a pinched hysteretic response is predicted quite well by complete mechanical models, which take advantage of only material and geometric properties. On the other hand, to exhibit the highly pinched behavior of a top-and-seat angle connection without web angles, a mathematical model requires components of slip and bolt hole ovalization, which are more amenable to informational modeling. An alternative method is informational modeling, which constitutes a fundamental shift from mathematical equations to data that contain the required information about underlying mechanics. The information is extracted from observed data and stored in neural networks. Two different training data sets, analytically-generated and experimental data, are tested to examine the performance of informational models. Both informational models show acceptable agreement with the moment-rotation curves of the experiments. Adding a degradation parameter improves the informational models when modeling highly pinched hysteretic behavior. However, informational models cannot represent the contribution of individual components and therefore do not provide an insight into the underlying mechanics of components. In this study, a new hybrid modeling framework is proposed. In the hybrid framework, a conventional mathematical model is complemented by the informational methods. The basic premise of the proposed hybrid methodology is that not all features of system response are amenable to mathematical modeling, hence considering informational alternatives. This may be because (i) the underlying theory is not available or not sufficiently developed, or (ii) the existing theory is too complex and therefore not suitable for modeling within building frame analysis. The role of informational methods is to model aspects that the mathematical model leaves out. Autoprogressive algorithm and self-learning simulation extract the missing aspects from a system response. In a hybrid framework, experimental data is an integral part of modeling, rather than being used strictly for validation processes. The potential of the hybrid methodology is illustrated through modeling complex hysteretic behavior of beam-to-column connections. Mechanics-based components of deformation such as angles, flange-plates, and column panel zone, are idealized to a mathematical model by using a complete mechanical approach. Although the mathematical model represents envelope curves in terms of initial stiffness and yielding strength, it is not capable of capturing the pinching effects. Pinching is caused mainly by separation between angles and column flanges as well as slip between angles/flange-plates and beam flanges. These components of deformation are suitable for informational modeling. Finally, the moment-rotation curves of the hybrid models are validated with those of the experimental tests. The comparison shows that the hybrid models are capable of representing the highly pinched hysteretic behavior of beam-to-column connections. In addition, the developed hybrid model is successfully used to predict the behavior of a newly-designed connection.

Applying Mathematical Models to Study the Role of the Immune System in Chronic Myelogenous Leukemia

Although tyrosine kinase inhibitors (TKIs) such as imatinib have transformed chronic myelogenous leukemia (CML) into a chronic condition, these therapies are not curative in the majority of cases. Most patients must continue TKI therapy indefinitely, a requirement that is both expensive and that compromises a patient's quality of life. While TKIs are known to reduce leukemic cells' proliferative capacity and to induce apoptosis, their effects on leukemic stem cells, the immune system, and the microenvironment are not fully understood. A more complete understanding of their global therapeutic effects would help us to identify any limitations of TKI monotherapy and to address these issues through novel combination therapies. Mathematical models are a complementary tool to experimental and clinical data that can provide valuable insights into the underlying mechanisms of TKI therapy. Previous modeling efforts have focused on CML patients who show biphasic and triphasic exponential declines in BCR-ABL ratio during therapy. However, our patient data indicates that many patients treated with TKIs show fluctuations in BCR-ABL ratio yet are able to achieve durable remissions. To investigate these fluctuations, we construct a mathematical model that integrates CML with a patient's autologous immune response to the disease. In our model, we define an immune window, which is an intermediate range of leukemic concentrations that lead to an effective immune response against CML. While small leukemic concentrations provide insufficient stimulus, large leukemic concentrations actively suppress a patient's immune system, thus limiting it's ability to respond. Our patient data and modeling results suggest that at diagnosis, a patient's high leukemic concentration is able to suppress their immune system. TKI therapy drives the leukemic population into the immune window, allowing the patient's immune cells to expand and eventually mount an efficient response against the residual CML. This response drives the leukemic population below the immune window, causing the immune population to contract and allowing the leukemia to partially recover. The leukemia eventually reenters the immune window, thus stimulating a sequence of weaker immune responses as the two populations approach equilibrium. We hypothesize that a patient's autologous immune response to CML may explain the fluctuations in BCR-ABL ratio that are regularly seen during TKI therapy. These fluctuations may serve as a signature of a patient's individual immune response to CML. By applying our modeling framework to patient data, we are able to construct an immune profile that can then be used to propose patient-specific combination therapies aimed at further reducing a patient's leukemic burden. Our characterization of a patient's anti-leukemia immune response may be especially valuable in the study of drug resistance, treatment cessation, and combination therapy.

Hydrometallurgical process modeling using advanced mathematical tools

Hydrometallurgical process modeling is the main objective of this Master’s thesis work. Three different leaching processes namely, high pressure pyrite oxidation, direct oxidation zinc concentrate (sphalerite) leaching and gold chloride leaching using rotating disc electrode (RDE) are modeled and simulated using gPROMS process simulation program in order to evaluate its model building capabilities. The leaching mechanism in each case is described in terms of a shrinking core model. The mathematical modeling carried out included process model development based on available literature, estimation of reaction kinetic parameters and assessment of the model reliability by checking the goodness fit and checking the cross correlation between the estimated parameters through the use of correlation matrices. The estimated parameter values in each case were compared with those obtained using the Modest simulation program. Further, based on the estimated reaction kinetic parameters, reactor simulation and modeling for direct oxidation zinc concentrate (sphalerite) leaching is carried out in Aspen Plus V8.6. The zinc leaching autoclave is based on Cominco reactor configuration and is modeled as a series of continuous stirred reactors (CSTRs). The sphalerite conversion is calculated and a sensitivity analysis is carried out so to determine the optimum reactor operation temperature and optimum oxygen mass flow rate. In this way, the implementation of reaction kinetic models into the process flowsheet simulation environment has been demonstrated.

A bidomain threshold model of propagating calcium waves

We present a bidomain fire-diffuse-fire model that facilitates mathematical analysis of propagating waves of elevated intracellular calcium (Ca) in living cells. Modelling Ca release as a threshold process allows the explicit construction of travelling wave solutions to probe the dependence of Ca wave speed on physiologically important parameters such as the threshold for Ca release from the endoplasmic reticulum (ER) to the cytosol, the rate of Ca resequestration from the cytosol to the ER, and the total [Ca] (cytosolic plus ER). Interestingly, linear stability analysis of the bidomain fire-diffuse-fire model predicts the onset of dynamic wave instabilities leading to the emergence of Ca waves that propagate in a back-and-forth manner. Numerical simulations are used to confirm the presence of these so-called "tango waves" and the dependence of Ca wave speed on the total [Ca]. The original publication is available at www.springerlink.com (Journal of Mathematical Biology)

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.

Sensory gating and its modulation by cannabinoids: electrophysiological, computational and mathematical analysis

Gating of sensory information can be assessed using an auditory conditioning-test paradigm which measures the reduction in the auditory evoked response to a test stimulus following an initial conditioning stimulus. Recording brainwaves from specific areas of the brain using multiple electrodes is helpful in the study of the neurobiology of sensory gating. In this paper, we use such technology to investigate the role of cannabinoids in sensory gating in the CA3 region of the rat hippocampus. Our experimental results show that application of the exogenous cannabinoid agonist WIN55,212-2 can abolish sensory gating. We have developed a phenomenological model of cannabinoid dynamics incorporated within a spiking neural network model of CA3 with synaptically interacting pyramidal and basket cells. Direct numerical simulations of this model suggest that the basic mechanism for this effect can be traced to the suppression of inhibition of slow GABAB synapses. Furthermore, by working with a simpler mathematical firing rate model we are able to show the robustness of this mechanism for the abolition of sensory gating.

An introduction to mathematical models of coagulation-fragmentation processes: a discrete deterministic mean-field approach

We summarise the properties and the fundamental mathematical results associated with basic models which describe coagulation and fragmentation processes in a deterministic manner and in which cluster size is a discrete quantity (an integer multiple of some basic unit size). In particular, we discuss Smoluchowski's equation for aggregation, the Becker-Döring model of simultaneous aggregation and fragmentation, and more general models involving coagulation and fragmentation.

A two-fluid model for tissue growth within a dynamic flow environment

We study the growth of a tissue construct in a perfusion bioreactor, focussing on its response to the mechanical environment. The bioreactor system is modelled as a two-dimensional channel containing a tissue construct through which a flow of culture medium is driven. We employ a multiphase formulation of the type presented by G. Lemon, J. King, H. Byrne, O. Jensen and K. Shakesheff in their study (Multiphase modelling of tissue growth using the theory of mixtures. J. Math. Biol. 52(2), 2006, 571–594) restricted to two interacting fluid phases, representing a cell population (and attendant extracellular matrix) and a culture medium, and employ the simplifying limit of large interphase viscous drag after S. Franks in her study (Mathematical Modelling of Tumour Growth and Stability. Ph.D. Thesis, University of Nottingham, UK, 2002) and S. Franks and J. King in their study Interactions between a uniformly proliferating tumour and its surrounding: Uniform material properties. Math. Med. Biol. 20, 2003, 47–89). The novel aspects of this study are: (i) the investigation of the effect of an imposed flow on the growth of the tissue construct, and (ii) the inclusion of a chanotransduction mechanism regulating the response of the cells to the local mechanical environment. Specifically, we consider the response of the cells to their local density and the culture medium pressure. As such, this study forms the first step towards a general multiphase formulation that incorporates the effect of mechanotransduction on the growth and morphology of a tissue construct. The model is analysed using analytic and numerical techniques, the results of which illustrate the potential use of the model to predict the dominant regulatory stimuli in a cell population.

Numerical and analytical study of an atherosclerosis inflammatory disease model

We study a reaction–diffusion mathematical model for the evolution of atherosclerosis as an inflammation process by combining analytical tools with computer-intensive numerical calculations. The computational work involved the calculation of more than sixty thousand solutions of the full reaction–diffusion system and lead to the complete characterisation of the ωω-limit for every initial condition. Qualitative properties of the solution are rigorously proved, some of them hinted at by the numerical study

Zig-zagging in geometrical reasoning in technological collaborative environments: a Mathematical Working Space-framed study concerning cognition and affect

This study highlights the importance of cognition-affect interaction pathways in the construction of mathematical knowledge. Scientific output demands further research on the conceptual structure underlying such interaction aimed at coping with the high complexity of its interpretation. The paper discusses the effectiveness of using a dynamic model such as that outlined in the Mathematical Working Spaces (MWS) framework, in order to describe the interplay between cognition and affect in the transitions from instrumental to discursive geneses in geometrical reasoning. The results based on empirical data from a teaching experiment at a middle school show that the use of dynamic geometry software favours students’ attitudinal and volitional dimensions and helps them to maintain productive affective pathways, affording greater intellectual independence in mathematical work and interaction with the context that impact learning opportunities in geometric proofs. The reflective and heuristic dimensions of teacher mediation in students’ learning is crucial in the transition from instrumental to discursive genesis and working stability in the Instrumental-Discursive plane of MWS.