ODE-and PDE-based modeling of biological transportation networks

We study the global existence of solutions of a discrete (ODE based) model on a graph describing the formation of biological transportation networks, introduced by Hu and Cai. We propose an adaptation of this model so that a macroscopic (PDE based) system can be obtained as its formal continuum limit. We prove the global existence of weak solutions of the macroscopic PDE model. Finally, we present results of numerical simulations of the discrete model, illustrating the convergence to steady states, their non-uniqueness as well as their dependence on initial data and model parameters

Analysis of nonlinear poroviscoelastic flows with discontinuous porosities

Existence and uniqueness of solutions is shown for a class of viscoelastic flows in porous media with particular attention to problems with nonsmooth porosities. The considered models are formulated in terms of the time-dependent nonlinear interaction between porosity and effective pressure, which in certain cases leads to porosity waves. In particular, conditions for well-posedness in the presence of initial porosities with jump discontinuities are identified.Comment: 34 page

Mean-field optimal control for biological pattern formation

We propose a mean-field optimal control problem for the parameter identification of a given pattern. The cost functional is based on the Wasserstein distance between the probability measures of the modeled and the desired patterns. The first-order optimality conditions corresponding to the optimal control problem are derived using a Lagrangian approach on the mean-field level. Based on these conditions we propose a gradient descent method to identify relevant parameters such as angle of rotation and force scaling which may be spatially inhomogeneous. We discretize the first-order optimality conditions in order to employ the algorithm on the particle level. Moreover, we prove a rate for the convergence of the controls as the number of particles used for the discretization tends to infinity. Numerical results for the spatially homogeneous case demonstrate the feasibility of the approach

Rigorous continuum limit for the discrete network formation problem

Motivated by recent papers describing the formation of biological transport networks we study a discrete model proposed by Hu and Cai consisting of an energy consumption function constrained by a linear system on a graph. For the spatially two-dimensional rectangular setting we prove the rigorous continuum limit of the constrained energy functional as the number of nodes of the underlying graph tends to infinity and the edge lengths shrink to zero uniformly. The proof is based on reformulating the discrete energy functional as a sequence of integral functionals and proving their Γ-convergence towards a continuum energy functional.LMK was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/L016516/1 and the German National Academic Foundation (Studienstiftung des Deutschen Volkes)

Systemische Nebenwirkungen der zahnärztlichen Lokalanästhesie mit Articain und Lidocain - systematische Analyse von Nebenwirkungen

Articain und Lidocain sind die weltweit am häufigsten verwendeten Lokalanästhetika. Deshalb sollten im Rahmen dieser Arbeit die beiden Medikamente auf ihre Nebenwirkungen hin, insbesondere auf ihre systemischen Nebenwirkungen, untersucht und verglichen werden. Insgesamt standen dieser Untersuchung die Periodic Safety Update Reports (PSUR) vier verschiedener Lokalanästhetika (je zwei Articain- und Lidocainlösungen mit unterschiedlichen vasokonstriktorischen Zusätzen) zweier Hersteller zur Verfügung. Die untersuchten PSURs beinhalteten verschiedene Zeiträume zwischen 2000 bis 2010. Diese vier PSURs wurden im Rahmen dieser Arbeit ausgewertet und systematisch analysiert. Die Gesamtzahl der injizierten Lokalanästhesien dieser Untersuchung belief sich auf 1.751.139.111 Ampullen. Dabei traten 1.288 gemeldete Nebenwirkungen auf, was einer Melderate von 0,736 Meldungen pro einer Million Injektionen entsprach. Von den 1.288 Nebenwirkungsmeldungen konnten die meisten (n=577) den systemischen Nebenwirkungen mit umgerechnet 0,330 Meldungen pro einer Million Injektionen zugeordnet werden. Dem folgten lokale Nebenwirkungen (n=437), Anästhesieversager (n=257) und nicht zu klassifizierende Nebenwirkungen (n=17). Bei den systemischen Nebenwirkungsmeldungen traten folgende Symptome (in absteigender Reihenfolge) am häufigsten auf: Schwindel (n=36) > Hypersensitivitäten (n=35) > Synkope (n=33) > Palpitation (n=27) > Dyspnoe (n=26). Die meisten systemischen Nebenwirkungsmeldungen ließen sich als Störung des Nervensystems (n=169) klassifizieren, gefolgt von kardialen Störungen (n=77) und Störungen des Immunsystems (n=61). Dies bestätigte den Verdacht, dass systemische Nebenwirkungen nach einer Lokalanästhesie am häufigsten das Zentralnervensystem betreffen. Der Vergleich der Nebenwirkungsmeldeprofile der untersuchten Articain- und Lidocainlösungen zeigte eine etwas höhere Nebenwirkungsmelderate für Articainlösungen. Eine mögliche Begründung kann der Weber-Effekt liefern, da Articain noch nicht so lange auf dem Markt ist (Markteinführung in Deutschland seit 1975; in den USA im Jahr 2000) wie Lidocain (seit 1948). Ein Fazit dieser Arbeit ist, dass sich die lokalanästhetische Injektion zu einer sehr nebenwirkungsarmen Methode der Schmerzausschaltung entwickelt hat. Die Wahrscheinlichkeit einer Nebenwirkung lag dieser Untersuchung zufolge bei 0,00007 %. Allerdings muss an dieser Stelle erwähnt werden, dass bei etwa 95 % der auftretenden Nebenwirkungen keine Meldung realisiert wird. Bei der Wahl des Lokalanästhetikums ist es unter klinischen Gesichtspunkten bezüglich des möglichen Auftretens einer Nebenwirkung ohne Bedeutung, ob eine Articain- oder Lidocainlösung zur Anwendung kommt.Articaine and Lidocaine are the most commenly used local anesthetics in dentistry. Goal of this scientific evaluation was to analyse the adverse drug reactions with special emphasis on the systemic side effects of both drugs. Basis for the evaluation are the Perodic Safety Update Reports of four local anesthetics, two articaine and two lidocaine preparations of two manufacturers from 2000-2010 (Producer 1) and 2006-2009 (Producer 2). The adverse drug reaction reports (ADR's) are related to 1751 139 111 ampoules of articaine or lidocaine preparations sold during that period. Altogether 1.288 adverse reactions were submitted, 0,736 per 1 million local anesthetic units sold. The majority of the reported adverse reactions were systemic side effects (n=577) a relation of 0.33 per 1 million ampoules sold. 437 Local adverse reactions were reported, 257 reports were related to a reduced or insufficient efficacy of one of the local ansthetics. The most frequently reported systemic side effects were nausea (n=36), hypersensitivity (n=35) syncopes (n=33), palpitations (n=27) and dyspnoe (n=26). According to the results of our evaluation the majority of the reported systemic side effects are related to the central nervous system (n=169), cardio-vascular disorders (n=77) or immunologic disorders e.g allergic reactions. In order to reduce the risk for adverse drug reactions related to the application of local anesthetic and the vasoconstrictor a carefully done medical history with adequate risk assessment should be accomplished. In comparision of the two drugs articaine solutions showed a slightly higher number of reported systemic drug reactions than lidocaine. This might be related to the "Weber effect". Because articaine is the newer drug and was just recently introduced in many countries (USA in 2000, Lidocaine in 1948) the "Weber effect" may have contributed to the higher number of ADR reports after the use of articaine. The number of adverse reactions concerning the different "Organ classes" showed discrepancies between the different articaine and lidocaine products, sometimes the differences were up to hundred percent. The differences however don't proof a higher safety of one local anesthetic because the same anesthetic from the other manufacturer received up to 6 times less adverse reaction reports concerning systemic disorders per one million injections. The evaluation of the ADR reports proofed that local anesthesia with articaine or lidocaine products is a very safe method of pain control. The probability to produce adverse reactions with articaine or lidocaine is 0.00007 per injection according to the results of our Investigation. Even under the assumption of 95 percent underreporting of adverse drug reactions the total number of systemic side effects for the most commonly used local anesthetics in dentistry is extremly low in comparison to other pharmaceutical products.120 Blätte

Parallel-in-Time Solutions with Random Projection Neural Networks

This paper considers one of the fundamental parallel-in-time methods for the solution of ordinary differential equations, Parareal, and extends it by adopting a neural network as a coarse propagator. We provide a theoretical analysis of the convergence properties of the proposed algorithm and show its effectiveness for several examples, including Lorenz and Burgers' equations. In our numerical simulations, we further specialize the underpinning neural architecture to Random Projection Neural Networks (RPNNs), a 2-layer neural network where the first layer weights are drawn at random rather than optimized. This restriction substantially increases the efficiency of fitting RPNN's weights in comparison to a standard feedforward network without negatively impacting the accuracy, as demonstrated in the SIR system example

Detection of high codimensional bifurcations in variational PDEs

We derive bifurcation test equations for A-series singularities of nonlinear functionals and, based on these equations, we propose a numerical method for detecting high codimensional bifurcations in parameter-dependent PDEs such as parameter-dependent semilinear Poisson equations. As an example, we consider a Bratu-type problem and show how high codimensional bifurcations such as the swallowtail bifurcation can be found numerically. In particular, our original contributions are (1) the use of the Infinite-dimensional Splitting Lemma, (2) the unified and simplified treatment of all A-series bifurcations, (3) the presentation in Banach spaces, i.e. our results apply both to the PDE and its (variational) discretization, (4) further simplifications for parameter-dependent semilinear Poisson equations (both continuous and discrete), and (5) the unified treatment of the continuous problem and its discretisation