Wound healing is a complex process, in which a sequence of interrelated events at both the cell and tissue levels interact and contribute to the reduction in wound size. For diabetic patients, many of these processes are compromised, so that wound healing slows down and in some cases halts. In this thesis we develop a series of increasingly detailed mathematical models to describe and investigate healing of full thickness skin wounds. We begin by developing a time-dependent ordinary differential equation model. This phenomenological model focusses on the main processes contributing to closure of a full thickness wound: proliferation in the epidermis and growth and contraction in the dermis. Model simulations suggest that the relative contributions of growth and contraction to healing of the dermis are altered in diabetic wounds. We investigate further the balance between growth and contraction by developing a more detailed, spatially-resolved model using continuum mechanics. Due to the initial large retraction of the wound edge upon injury, we adopt a non-linear elastic framework. Morphoelasticity theory is applied, with the total deformation of the material decomposed into an addition of mass and an elastic response. We use the model to investigate how interactions between growth and stress influence dermal wound healing. The model reveals that contraction alone generates unrealistically high tension in the dermal tissue and, hence, volumetric growth must contribute to healing. We show that, in the simplified case of homogeneous growth, the tissue must grow anisotropically in order to reduce the size of the wound and we postulate mechanosensitive growth laws consistent with this result. After closure the surrounding tissue remodels, returning to its residually stressed state. We identify the steady state growth profile associated with this remodelled state. The model is used to predict the outcome of rewounding experiments as a method of quantifying the amount of stress in the tissue and the application of pressure treatments to control tissue synthesis. The thesis concludes with an extension to the spatially-resolved mechanical model to account for the effects of the biochemical environment. Partial differential equations describing the dynamics of fibroblasts and a regulating growth factor are coupled to equations for the tissue mechanics, described in the morphoelastic framework. By accounting for biomechanical and biochemical stimuli the model allows us to formulate mechanistic laws for growth and contraction. We explore how disruption of mechanical and chemical feedback can lead to abnormal wound healing and use the model to identify specific treatments for normalising healing in these cases.

The paper is focused on the Predator-Prey model modified in the case of harvesting one or both populations. Firstly there is given a short description of the basic model and the sensitivity analysis. The first essential modification is percentage harvesting. This model could be easily converted to the basic one using a substitution. The next modification is constant harvesting. Solving this system requires linearization, which was properly done and brought valuable results applicable even for the basic or the percentage harvesting model. The next chapter describes regulation models, which could be used especially in applying environmental policies. All reasonable regulation models are shown after distinguishing between discrete and continuous harvesting. The last chapter contains an algorithm for maximizing the profit of a harvester using econometrical modelling tools.

This thesis deals with the topic of partial differential equations parallel solutions. First, it focuses on ordinary differential equations (ODE) and their solution methods using Taylor polynomial. Another part is devoted to partial differential equations (PDE). There are several types of PDE, there are parabolic, hyperbolic and eliptic PDE. There is also explained how to use TKSL system for PDE computing. Another part focuses on solution methods of PDE, these methods are forward, backward and combined methods. There was explained, how to solve these methods in TKSL and Matlab systems. Computing accuracy and time complexity are also discussed. Another part of thesis is PDE parallel solutions. Thanks to the possibility of PDE convertion to ODE systems it is possible to represent each ODE equation by independent operation unit. These units enable parallel computing. The last chapter is devoted to implementation. Application enables generation of ODE systems for TKSL system. These ODE systems represent given hyperbolic PDE.

Thesis is dedicated to evaluation of roundoff impact on numerical integration methods accuracy and effectivity. Contains theoretical expectations taken from existing literature, implementation of chosen methods, experimental measurement of attained accuracy under different circumstances and their comparison with regard to time complexity. Library contains Runge-Kutta methods to order 7 and Adams-Bashforth methods to order 20 implemented using C++ templates which allow optional arbitrary-precision arithmetic. Small models with known analytic solution were used for experiments.

Chez les patients infectés par le VIH, les traitements antirétroviraux empêchent la réplicationvirale, ce qui est suivi, dans la plupart des cas, par une restauration de la population des lymphocytesT CD4+ (CD4). Néanmoins ce n’est pas le cas pour certains patients appelés patients àfaible réponse immunitaire. Des injections d’interleukine-7 (IL7) exogène, une cytokine impliquéedans l’homéostasie des CD4, sont considérées afin de maintenir les taux de CD4 au-dessus de500 cellules par μL, taux au-dessus duquel les patients ont une espérance de vie comparable auxpersonnes non infectées par le VIH. Les essais INSPIRE ont évalué l’effet d’injections répétéesd’IL7 chez les patients à faible réponse immunologique.Nous présentons plusieurs modèles mécanistes de l’effet des injections d’IL7 sur les CD4, quiincluent des effets aléatoires afin de tenir compte de la variabilité inter-individuelle. En utilisantces modèles avec une approche Bayésienne, les paramètres individuels d’un nouveau patient sontéchantillonnés, ce qui nous permet de faire des prédictions sur sa dynamique de CD4 et donc depersonnaliser le traitement. Nous proposons quatre protocoles adaptatifs permettant de limiter letemps passé sous 500 CD4 par μL, sans pour autant augmenter le nombre d’injections. Ces protocolesont été implémentés dans une application Shiny présentant une interface facile d’utilisation,et pourront être testés lors d’essais cliniques.Le réservoir viral, principalement constitué de CD4 quiescentes infectées, est la première barrièreà l’éradication du VIH. Les injections d’IL7 entrainent une augmentation du nombre deCD4 et donc du réservoir viral ; la question est alors de savoir si les injections provoquent denouvelles infections cellulaires ou si le réservoir augmente de la même façon que les CD4. Nousconcluons que si quelques patients ont présenté des dynamiques de marqueurs compatibles avecla survenue de nouvelles infections de cellules, ce n’est pas le cas de la majorité des patients. Laconfirmation de ces phénomènes et la caractérisation de potentiels patients à risque nécessite desdonnées supplémentaires mesurables dans un essai clinique. / In HIV infected patients, antiretroviral therapy suppresses the viral replication which is followedin most patients by a restoration of the CD4+ T cells (CD4) pool. However, it is not the case forsome patients called low immunological responders. Injections of interleukin-7 (IL7), a cytokineinvolved in the CD4 homeostasis, are considered in order to maintain the CD4 levels above 500cells per μL, the level at which life expectancy is similar to that of the non-infected. INSPIREtrials evaluated the effect of repeated injections of IL7 on low immunological responders.We present a few mechanistic models of the effect of IL7 injections on CD4, which includerandom effects to account for inter-individual variability. Using these models with a Bayesianapproach, the individual parameters of a new patient are sampled, which allows us to makepredictions about its CD4 dynamics and thus to personalize the treatment. We propose fouradaptive protocols that limit the time spent under 500 CD4 per μL, without increasing thenumber of injections. Those protocols are implemented into a Shiny app with an easy to useinterface, and they could be tested during clinical trials.The viral reservoir, mainly made up of quiescent infected CD4, is the main obstacle to HIVeradication. IL7 injections induce an increase of the level of CD4, hence of the viral reservoir ; thequestion is then to determine if the injections induce new cell infections or if the reservoir increasesin the same way as CD4. We conclude that while some patients presented marker dynamicsconsistent with the occurrence of new cell infections, this is not the case for the majority ofpatients. Confirmation of these events and characterization of potential at-risk patients requiresadditional measurable data in a clinical trial.

<p>Interleukin-15 (IL-15) is a cytokine that promotes maintenance and activation of cytotoxic immune cells. Therapeutic IL-15 stimulates these cells to fight cancer and chronic infections, such as Human Immunodeficiency Virus (HIV). Animal models of HIV have demonstrated that IL-15 agonists can suppress the virus, but this was transient and was not observed in all cohorts. We developed a mechanistic mathematical model of IL-15 therapy of HIV to explain these differences in efficacy and to explore solutions. First, the model was applied to evaluate mitigating factors, including immune regulation, viral escape, and drug tolerance, using Akaike Information Criterion.  We found that immune regulatory mechanisms could explain the viral rebound observed with continued IL-15 therapy. Next, the model was expanded to allow it to simultaneously explain both the transient viral suppression noted above and the lack of viral suppression observed in another animal cohort.  In this cohort, the model suggested that higher pre-treatment viral load came with higher activation of immune cells and a balancing regulatory inhibition of cytotoxicity. Finally, we conducted stability analysis at a range of IL-15 therapeutic strengths. While there was an ideal IL-15 strength, monotherapy could not maintain viral levels below what would clinically be considered to be safely controlled. Stable viral control in the model required the combination of IL-15 with blockade of key regulatory pathways. Immune therapy of complex diseases will likely require combinations of medicines that boost the immune response at multiple key points. Mathematical models like this can expedite development of these treatments.</p>

[pt] Esta dissertação tem o objetivo de aplicar os conceitos e ferramentas da Análise Real e Álgebra Linear num estudo sobre a teoria de existência, unicidade e estabilidade de soluções de sistemas de equações diferenciais ordinárias, considerando sistemas gerais parametrizados, lineares e não-lineares. / [en] This dissertation aims to apply the concepts and tools of Real Analysis and Linear Algebra to the theory of existence, uniquiness and stability of solutions of ordinary differential equations systems, considering general parametric, linear and non-linear systems.

<p>  </p> <p>A <em>digital twin</em> (DT), which keeps track of nuclear reactor history to provide real-time predictions, has been recently proposed for nuclear reactor monitoring. A digital twin can be implemented using either a differential equations-based physics model, or a data-driven machine learning model<strong>. </strong>The principal challenge in physics model-based DT consists of achieving sufficient model fidelity to represent a complex experimental system, while the main challenge in data-driven DT appears in the extensive training requirements and potential lack of predictive ability. </p> <p>In this thesis, we investigate the performance of a hybrid approach, which is based on physics-informed neural networks (PINNs) that encode fundamental physical laws into the loss function of the neural network. In this way, PINNs establish theoretical constraints and biases to supplement measurement data and provide solution to several limitations of purely data-driven machine learning (ML) models. We develop a PINN model to solve the point kinetic equations (PKEs), which are time dependent stiff nonlinear ordinary differential equations that constitute a nuclear reactor reduced-order model under the approximation of ignoring the spatial dependence of the neutron flux. PKEs portray the kinetic behavior of the system, and this kind of approach is the basis for most analyses of reactor systems, except in cases where flux shapes are known to vary with time. This system describes the nuclear parameters such as neutron density concentration, the delayed neutron precursor density concentration and reactivity. Both neutron density and delayed neutron precursor density concentrations are the vital parameters for safety and the transient behavior of the reactor power. </p> <p>The PINN model solution of PKEs is developed to monitor a start-up transient of the Purdue University Reactor Number One (PUR-1) using experimental parameters for the reactivity feedback schedule and the neutron source. The facility under modeling, PUR-1, is a pool type small research reactor located in West Lafayette Indiana. It is an all-digital light water reactor (LWR) submerged into a deep-water pool and has a power output of 10kW. The results demonstrate strong agreement between the PINN solution and finite difference numerical solution of PKEs. We investigate PINNs performance in both data interpolation and extrapolation. </p> <p>The findings of this thesis research indicate that the PINN model achieved highest performance and lowest errors in data interpolation. In the case of extrapolation data, three different test cases were considered, the first where the extrapolation is performed in a five-seconds interval, the second where the extrapolation is performed in a 10-seconds interval, and the third where the extrapolation is performed in a 15-seconds interval. The extrapolation errors are comparable to those of interpolation predictions. Extrapolation accuracy decreases with increasing time interval.</p>

Komplexe Systeme wie das Klima der Erde bestehen aus vielen Komponenten, die durch eine komplizierte Kopplungsstruktur miteinander verbunden sind. Für die Analyse solcher Systeme erscheint es daher naheliegend, Methoden aus der Netzwerktheorie, der Theorie dynamischer Systeme und dem maschinellen Lernen zusammenzubringen. Durch die Kombination verschiedener Konzepte aus diesen Bereichen werden in dieser Arbeit drei neuartige Ansätze zur Untersuchung komplexer Systeme betrachtet. Im ersten Teil wird eine Methode zur Konstruktion komplexer Netzwerke vorgestellt, die in der Lage ist, Windpfade des südamerikanischen Monsunsystems zu identifizieren. Diese Analyse weist u.a. auf den Einfluss der Rossby-Wellenzüge auf das Monsunsystem hin. Dies wird weiter untersucht, indem gezeigt wird, dass der Niederschlag mit den Rossby-Wellen phasenkohärent ist. So zeigt der erste Teil dieser Arbeit, wie komplexe Netzwerke verwendet werden können, um räumlich-zeitliche Variabilitätsmuster zu identifizieren, die dann mit Methoden der nichtlinearen Dynamik weiter analysiert werden können. Die meisten komplexen Systeme weisen eine große Anzahl von möglichen asymptotischen Zuständen auf. Um solche Zustände zu beschreiben, wird im zweiten Teil die Monte Carlo Basin Bifurcation Analyse (MCBB), eine neuartige numerische Methode, vorgestellt. Angesiedelt zwischen der klassischen Analyse mit Ordnungsparametern und einer gründlicheren, detaillierteren Bifurkationsanalyse, kombiniert MCBB Zufallsstichproben mit Clustering, um die verschiedenen Zustände und ihre Einzugsgebiete zu identifizieren. Bei von Vorhersagen von komplexen Systemen ist es nicht immer einfach, wie Vorwissen in datengetriebenen Methoden integriert werden kann. Eine Möglichkeit hierzu ist die Verwendung von Neuronalen Partiellen Differentialgleichungen. Hier wird im letzten Teil der Arbeit gezeigt, wie hochdimensionale räumlich-zeitlich chaotische Systeme mit einem solchen Ansatz modelliert und vorhergesagt werden können. / Complex systems such as the Earth's climate are comprised of many constituents that are interlinked through an intricate coupling structure. For the analysis of such systems it therefore seems natural to bring together methods from network theory, dynamical systems theory and machine learning. By combining different concepts from these fields three novel approaches for the study of complex systems are considered throughout this thesis. In the first part, a novel complex network construction method is introduced that is able to identify the most important wind paths of the South American Monsoon system. Aside from the importance of cross-equatorial flows, this analysis points to the impact Rossby Wave trains have both on the precipitation and low-level circulation. This connection is then further explored by showing that the precipitation is phase coherent to the Rossby Wave. As such, the first part of this thesis demonstrates how complex networks can be used to identify spatiotemporal variability patterns within large amounts of data, that are then further analysed with methods from nonlinear dynamics. Most complex systems exhibit a large number of possible asymptotic states. To investigate and track such states, Monte Carlo Basin Bifurcation analysis (MCBB), a novel numerical method is introduced in the second part. Situated between the classical analysis with macroscopic order parameters and a more thorough, detailed bifurcation analysis, MCBB combines random sampling with clustering methods to identify and characterise the different asymptotic states and their basins of attraction. Forecasts of complex system are the next logical step. When doing so, it is not always straightforward how prior knowledge in data-driven methods. One possibility to do is by using Neural Partial Differential Equations. Here, it is demonstrated how high-dimensional spatiotemporally chaotic systems can be modelled and predicted with such an approach in the last part of the thesis.

<p>In this study, we concentrate on spin-1/2 systems. A series of tools using the Liouville space method have been developed for simulating of NMR of arbitrary pulse sequences.</p> <p>We have calculated one- and two-spin symbolically, and larger systems numerically of steady states. The one-spin calculations show how SSFP converges to continuous wave NMR. A general formula for two-spin systems has been derived for the creation of double-quantum signals as a function of irradiation strength, coupling constant, and chemical shift difference. The formalism is general and can be extended to more complex spin systems.</p> <p>Estimates of transverse relaxation, R₂, are affected by frequency offset and field inhomogeneity. We find that in the presence of expected B₀ inhomogeneity, off-resonance effects can be removed from R₂ measurements, when ||omega||<= 0.5 gamma\,B₁ in Hahn echo experiments, when ||omega||<=gamma\,B₁ in CPMG experiments with specific phase variations, by fitting exact solutions of the Bloch equations given in the Lagrange form.</p> <p>Approximate solutions of CPMG experiments show the specific phase variations can significantly smooth the dependence of measured intensities on frequency offset in the range of +/- 1/2 gamma\,B₁. The effective R₂ of CPMG experiments when using a phase variation scheme can be expressed as a second-order formula with respect to the ratio of offset to pi-pulse amplitude.</p> <p>Optimization problems using the exact or approximate solution of the Bloch equations are established for designing optimal broadband universal rotation (OBUR) pulses. OBUR pulses are independent of initial magnetization and can be applied to replace any pulse of the same flip angles in a pulse sequence. We demonstrate the process to exactly and efficiently calculate the first- and second-order derivatives with respect to pulses. Using these exact derivatives, a second-order optimization method is employed to design pulses. Experiments and simulations show that OBUR pulses can provide more uniform spectra in the designed offset range and come up with advantages in CPMG experiments.</p> / Doctor of Philosophy (PhD)