A semigroups theory approach to a model of suspension bridges

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Basic ingredients for mathematical modeling of tumor growth in vitro: Cooperative effects and search for space

Based on the literature data from HT-29 cell monolayers, we develop a model for its growth, analogous to an epidemic model, mixing local and global interactions. First, we propose and solve a deterministic equation for the progress of these colonies. Thus, we add a stochastic (local) interaction and simulate the evolution of an Eden-like aggregate by using dynamical Monte Carlo methods. The growth curves of both deterministic and stochastic models are in excellent agreement with the experimental observations. The waiting times distributions, generated via our stochastic model, allowed us to analyze the role of mesoscopic events. We obtain log-normal distributions in the initial stages of the growth and Gaussians at long times. We interpret these outcomes in the light of cellular division events: in the early stages, the phenomena are dependent each other in a multiplicative geometric-based process, and they are independent at long times. We conclude that the main ingredients for a good minimalist model of tumor growth, at mesoscopic level, are intrinsic cooperative mechanisms and competitive search for space. © 2013 Elsevier Ltd.

Mathematical modeling of a continuous alcoholic fermentation process in a two-stage tower reactor cascade with flocculating yeast recycle

Experiments of continuous alcoholic fermentation of sugarcane juice with flocculating yeast recycle were conducted in a system of two 0.22-L tower bioreactors in series, operated at a range of dilution rates (D (1) = D (2) = 0.27-0.95 h(-1)), constant recycle ratio (alpha = F (R) /F = 4.0) and a sugar concentration in the feed stream (S (0)) around 150 g/L. The data obtained in these experimental conditions were used to adjust the parameters of a mathematical model previously developed for the single-stage process. This model considers each of the tower bioreactors as a perfectly mixed continuous reactor and the kinetics of cell growth and product formation takes into account the limitation by substrate and the inhibition by ethanol and biomass, as well as the substrate consumption for cellular maintenance. The model predictions agreed satisfactorily with the measurements taken in both stages of the cascade. The major differences with respect to the kinetic parameters previously estimated for a single-stage system were observed for the maximum specific growth rate, for the inhibition constants of cell growth and for the specific rate of substrate consumption for cell maintenance. Mathematical models were validated and used to simulate alternative operating conditions as well as to analyze the performance of the two-stage process against that of the single-stage process.

A heat transfer model of the human upper limbs

A steady state multi-segmented heat transfer model of the human upper limbs was developed. The main purpose was to evaluate the impact of blood flow through superficial veins and subcutaneous vascular structures in the palm of the hands over the heat transfer between the limbs and the environment. The distinguishing feature of the model is the inclusion of a detailed circulatory network composed of vessels with diameter larger than 1 mm. The model was validated by comparing its results from exposures to a hot, a neutral, and a cold environment to experimental data presented in the literature. (C) 2011 Elsevier Ltd. All rights reserved.

Progress in the mathematical theory of quantum disordered systems

We review recent progress in the mathematical theory of quantum disordered systems: the Anderson transition, including some joint work with Marchetti, the (quantum and classical) Edwards-Anderson (EA) spin-glass model and return to equilibrium for a class of spin-glass models, which includes the EA model initially in a very large transverse magnetic field. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4770066]

Forced Desynchronization of Activity Rhythms in a Model of Chronic Jet Lag in Mice

We studied locomotor activity rhythms of C57/Bl6 mice under a chronic jet lag (CJL) protocol (ChrA(6/2)), which consisted of 6-hour phase advances of the light-dark schedule (LD) every 2 days. Through periodogram analysis, we found 2 components of the activity rhythm: a short-period component (21.01 +/- 0.04 h) that was entrained by the LD schedule and a long-period component (24.68 +/- 0.26 h). We developed a mathematical model comprising 2 coupled circadian oscillators that was tested experimentally with different CJL schedules. Our simulations suggested that under CJL, the system behaves as if it were under a zeitgeber with a period determined by (24 -[phase shift size/days between shifts]). Desynchronization within the system arises according to whether this effective zeitgeber is inside or outside the range of entrainment of the oscillators. In this sense, ChrA(6/2) is interpreted as a (24 - 6/2 = 21 h) zeitgeber, and simulations predicted the behavior of mice under other CJL schedules with an effective 21-hour zeitgeber. Animals studied under an asymmetric T = 21 h zeitgeber (carried out by a 3-hour shortening of every dark phase) showed 2 activity components as observed under ChrA(6/2): an entrained short-period (21.01 +/- 0.03 h) and a long-period component (23.93 +/- 0.31 h). Internal desynchronization was lost when mice were subjected to 9-hour advances every 3 days, a possibility also contemplated by the simulations. Simulations also predicted that desynchronization should be less prevalent under delaying than under advancing CJL. Indeed, most mice subjected to 6-hour delay shifts every 2 days (an effective 27-hour zeitgeber) displayed a single entrained activity component (26.92 +/- 0.11 h). Our results demonstrate that the disruption provoked by CJL schedules is not dependent on the phase-shift magnitude or the frequency of the shifts separately but on the combination of both, through its ratio and additionally on their absolute values. In this study, we present a novel model of forced desynchronization in mice under a specific CJL schedule; in addition, our model provides theoretical tools for the evaluation of circadian disruption under CJL conditions that are currently used in circadian research.

Freeze-drying microscopy in mathematical modeling of a biomaterial freeze-drying

Transplantation brings hope for many patients. A multidisciplinary approach on this field aims at creating biologically functional tissues to be used as implants and prostheses. The freeze-drying process allows the fundamental properties of these materials to be preserved, making future manipulation and storage easier. Optimizing a freeze-drying cycle is of great importance since it aims at reducing process costs while increasing product quality of this time-and-energy-consuming process. Mathematical modeling comes as a tool to help a better understanding of the process variables behavior and consequently it helps optimization studies. Freeze-drying microscopy is a technique usually applied to determine critical temperatures of liquid formulations. It has been used in this work to determine the sublimation rates of a biological tissue freeze-drying. The sublimation rates were measured from the speed of the moving interface between the dried and the frozen layer under 21.33, 42.66 and 63.99 Pa. The studied variables were used in a theoretical model to simulate various temperature profiles of the freeze-drying process. Good agreement between the experimental and the simulated results was found.

Exploration of the parameter space in an agent-based model of tuberculosis spread: emergence of drug resistance in developing vs developed countries

In this work we present an agent-based model for the spread of tuberculosis where the individuals can be infected with either drug-susceptible or drug-resistant strains and can also receive a treatment. The dynamics of the model and the role of each one of the parameters are explained. The whole set of parameters is explored to check their importance in the numerical simulation results. The model captures the beneficial impact of the adequate treatment on the prevalence of tuberculosis. Nevertheless, depending on the treatment parameters range, it also captures the emergence of drug resistance. Drug resistance emergence is particularly likely to occur for parameter values corresponding to less efficacious treatment, as usually found in developing countries.

A Cost-Effectiveness-Assessing Model of Vaccination for Varicella and Zoster

A decision analytical model is presented and analysed to assess the effectiveness and cost-effectiveness of routine vaccination against varicella and herpes-zoster, or shingles. These diseases have as common aetiological agent the varicella-zoster virus (VZV). Zoster can more likely occur in aged people with declining cell-mediated immunity. The general concern is that universal varicella vaccination might lead to more cases of zoster: with more vaccinated children exposure of the general population to varicella infectives become smaller and thus a larger proportion of older people will have weaker immunity to VZV, leading to more cases of reactivation of zoster. Our compartment model shows that only two possible equilibria exist, one without varicella and the other one where varicella arid zoster both thrive. Threshold quantities to distinguish these cases are derived. Cost estimates on a possible herd vaccination program are discussed indicating a possible tradeoff choice.

Effect of external fields in Axelrod's model of social dynamics

The study of the effects of spatially uniform fields on the steady-state properties of Axelrod's model has yielded plenty of counterintuitive results. Here, we reexamine the impact of this type of field for a selection of parameters such that the field-free steady state of the model is heterogeneous or multicultural. Analyses of both one- and two-dimensional versions of Axelrod's model indicate that the steady state remains heterogeneous regardless of the value of the field strength. Turning on the field leads to a discontinuous decrease on the number of cultural domains, which we argue is due to the instability of zero-field heterogeneous absorbing configurations. We find, however, that spatially nonuniform fields that implement a consensus rule among the neighborhood of the agents enforce homogenization. Although the overall effects of the fields are essentially the same irrespective of the dimensionality of the model, we argue that the dimensionality has a significant impact on the stability of the field-free homogeneous steady state.

Mechanical Alterations in airway smooth muscle after interaction with indoor pollutant – A mathematical model.

The viscoelasticity of mammalian lung is determined by the mechanical properties and structural regulation of the airway smooth muscle (ASM). The exposure to polluted air may deteriorate these properties with harmful consequences to individual health. Formaldehyde (FA) is an important indoor pollutant found among volatile organic compounds. This pollutant permeates through the smooth muscle tissue forming covalent bonds between proteins in the extracellular matrix and intracellular protein structure changing mechanical properties of ASM and inducing asthma symptoms, such as airway hyperresponsiveness, even at low concentrations. In the experimental scenario, the mechanical effect of FA is the stiffening of the tissue, but the mechanism behind this effect is not fully understood. Thus, the aim of this study is to reproduce the mechanical behavior of the ASM, such as contraction and stretching, under FA action or not. For this, it was created a two-dimensional viscoelastic network model based on Voronoi tessellation solved using Runge-Kutta method of fourth order. The equilibrium configuration was reached when the forces in different parts of the network were equal. This model simulates the mechanical behavior of ASM through of a network of dashpots and springs. This dashpot-spring mechanical coupling mimics the composition of the actomyosin machinery of ASM through the contraction of springs to a minimum length. We hypothesized that formation of covalent bonds, due to the FA action, can be represented in the model by a simple change in the elastic constant of the springs, while the action of methacholine (MCh) reduce the equilibrium length of the spring. A sigmoid curve of tension as a function of MCh doses was obtained, showing increased tension when the muscle strip was exposed to FA. Our simulations suggest that FA, at a concentration of 0.1 ppm, can affect the elastic properties of the smooth muscle ¯bers by a factor of 120%. We also analyze the dynamic mechanical properties, observing the viscous and elastic behavior of the network. Finally, the proposed model, although simple, incorporates the phenomenology of both MCh and FA and reproduces experimental results observed with in vitro exposure of smooth muscle to FA. Thus, this new mechanical approach incorporates several well know features of the contractile system of the cells in a tissue level model. The model can also be used in different biological scales.

Mechanical alterations in airway smooth muscle after interaction with indoor pollutant - A mathematical model

The viscoelasticity of mammalian lung is determined by the mechanical properties and structural regulation of the airway smooth muscle (ASM). The exposure to polluted air may deteriorate these properties with harmful consequences to individual health. Formaldehyde (FA) is an important indoor pollutant found among volatile organic compounds. This pollutant permeates through the smooth muscle tissue forming covalent bonds between proteins in the extracellular matrix and intracellular protein structure changing mechanical properties of ASM and inducing asthma symptoms, such as airway hyperresponsiveness, even at low concentrations. In the experimental scenario, the mechanical effect of FA is the stiffening of the tissue, but the mechanism behind this effect is not fully w1derstood. Thus, the aim of this study is to reproduce the mechanical behavior of the ASM, such as contraction and stretching, under FA action or not. For this, it was created a two-dimensional viscoelastic network model based on Voronoi tessellation solved using Runge-Kutta method of fourth order. The equilibrium configuration was reached when the forces in different parts of the network were equal. This model simulates the mechanical behavior of ASM through of a network of dashpots and springs. This dashpot-spring mechanical coupling mimics the composition of the actomyosin machinery of ASM through the contraction of springs to a minimum length. We hypothesized that formation of covalent bonds, due to the FA action, can be represented in the model by a simple change in the elastic constant of the springs, while the action of methacholinc (MCh) reduce the equilibrium length of the spring. A sigmoid curve of tension as a function of MCh doses was obtained, showing increased tension when the muscle strip was exposed to FA. Our simulations suggest that FA, at a concentration of 0.1 ppm, can affect the elastic properties of the smooth muscle fibers by a factor of 120%. We also analyze the dynamic mechanical properties, observing the viscous and elastic behavior of the network. Finally, the proposed model, although simple, ir1corporates the phenomenology of both MCh and FA and reproduces experirnental results observed with ir1 vitro exposure of smooth muscle to .FA. Thus, this new mechanical approach incorporates several well know features of the contractile system of the cells ir1 a tissue level model. The model can also be used in different biological scales.

Mathematical models of cognitive processes

The research activity carried out during the PhD course was focused on the development of mathematical models of some cognitive processes and their validation by means of data present in literature, with a double aim: i) to achieve a better interpretation and explanation of the great amount of data obtained on these processes from different methodologies (electrophysiological recordings on animals, neuropsychological, psychophysical and neuroimaging studies in humans), ii) to exploit model predictions and results to guide future research and experiments. In particular, the research activity has been focused on two different projects: 1) the first one concerns the development of neural oscillators networks, in order to investigate the mechanisms of synchronization of the neural oscillatory activity during cognitive processes, such as object recognition, memory, language, attention; 2) the second one concerns the mathematical modelling of multisensory integration processes (e.g. visual-acoustic), which occur in several cortical and subcortical regions (in particular in a subcortical structure named Superior Colliculus (SC)), and which are fundamental for orienting motor and attentive responses to external world stimuli. This activity has been realized in collaboration with the Center for Studies and Researches in Cognitive Neuroscience of the University of Bologna (in Cesena) and the Department of Neurobiology and Anatomy of the Wake Forest University School of Medicine (NC, USA). PART 1. Objects representation in a number of cognitive functions, like perception and recognition, foresees distribute processes in different cortical areas. One of the main neurophysiological question concerns how the correlation between these disparate areas is realized, in order to succeed in grouping together the characteristics of the same object (binding problem) and in maintaining segregated the properties belonging to different objects simultaneously present (segmentation problem). Different theories have been proposed to address these questions (Barlow, 1972). One of the most influential theory is the so called “assembly coding”, postulated by Singer (2003), according to which 1) an object is well described by a few fundamental properties, processing in different and distributed cortical areas; 2) the recognition of the object would be realized by means of the simultaneously activation of the cortical areas representing its different features; 3) groups of properties belonging to different objects would be kept separated in the time domain. In Chapter 1.1 and in Chapter 1.2 we present two neural network models for object recognition, based on the “assembly coding” hypothesis. These models are networks of Wilson-Cowan oscillators which exploit: i) two high-level “Gestalt Rules” (the similarity and previous knowledge rules), to realize the functional link between elements of different cortical areas representing properties of the same object (binding problem); 2) the synchronization of the neural oscillatory activity in the γ-band (30-100Hz), to segregate in time the representations of different objects simultaneously present (segmentation problem). These models are able to recognize and reconstruct multiple simultaneous external objects, even in difficult case (some wrong or lacking features, shared features, superimposed noise). In Chapter 1.3 the previous models are extended to realize a semantic memory, in which sensory-motor representations of objects are linked with words. To this aim, the network, previously developed, devoted to the representation of objects as a collection of sensory-motor features, is reciprocally linked with a second network devoted to the representation of words (lexical network) Synapses linking the two networks are trained via a time-dependent Hebbian rule, during a training period in which individual objects are presented together with the corresponding words. Simulation results demonstrate that, during the retrieval phase, the network can deal with the simultaneous presence of objects (from sensory-motor inputs) and words (from linguistic inputs), can correctly associate objects with words and segment objects even in the presence of incomplete information. Moreover, the network can realize some semantic links among words representing objects with some shared features. These results support the idea that semantic memory can be described as an integrated process, whose content is retrieved by the co-activation of different multimodal regions. In perspective, extended versions of this model may be used to test conceptual theories, and to provide a quantitative assessment of existing data (for instance concerning patients with neural deficits). PART 2. The ability of the brain to integrate information from different sensory channels is fundamental to perception of the external world (Stein et al, 1993). It is well documented that a number of extraprimary areas have neurons capable of such a task; one of the best known of these is the superior colliculus (SC). This midbrain structure receives auditory, visual and somatosensory inputs from different subcortical and cortical areas, and is involved in the control of orientation to external events (Wallace et al, 1993). SC neurons respond to each of these sensory inputs separately, but is also capable of integrating them (Stein et al, 1993) so that the response to the combined multisensory stimuli is greater than that to the individual component stimuli (enhancement). This enhancement is proportionately greater if the modality-specific paired stimuli are weaker (the principle of inverse effectiveness). Several studies have shown that the capability of SC neurons to engage in multisensory integration requires inputs from cortex; primarily the anterior ectosylvian sulcus (AES), but also the rostral lateral suprasylvian sulcus (rLS). If these cortical inputs are deactivated the response of SC neurons to cross-modal stimulation is no different from that evoked by the most effective of its individual component stimuli (Jiang et al 2001). This phenomenon can be better understood through mathematical models. The use of mathematical models and neural networks can place the mass of data that has been accumulated about this phenomenon and its underlying circuitry into a coherent theoretical structure. In Chapter 2.1 a simple neural network model of this structure is presented; this model is able to reproduce a large number of SC behaviours like multisensory enhancement, multisensory and unisensory depression, inverse effectiveness. In Chapter 2.2 this model was improved by incorporating more neurophysiological knowledge about the neural circuitry underlying SC multisensory integration, in order to suggest possible physiological mechanisms through which it is effected. This endeavour was realized in collaboration with Professor B.E. Stein and Doctor B. Rowland during the 6 months-period spent at the Department of Neurobiology and Anatomy of the Wake Forest University School of Medicine (NC, USA), within the Marco Polo Project. The model includes four distinct unisensory areas that are devoted to a topological representation of external stimuli. Two of them represent subregions of the AES (i.e., FAES, an auditory area, and AEV, a visual area) and send descending inputs to the ipsilateral SC; the other two represent subcortical areas (one auditory and one visual) projecting ascending inputs to the same SC. Different competitive mechanisms, realized by means of population of interneurons, are used in the model to reproduce the different behaviour of SC neurons in conditions of cortical activation and deactivation. The model, with a single set of parameters, is able to mimic the behaviour of SC multisensory neurons in response to very different stimulus conditions (multisensory enhancement, inverse effectiveness, within- and cross-modal suppression of spatially disparate stimuli), with cortex functional and cortex deactivated, and with a particular type of membrane receptors (NMDA receptors) active or inhibited. All these results agree with the data reported in Jiang et al. (2001) and in Binns and Salt (1996). The model suggests that non-linearities in neural responses and synaptic (excitatory and inhibitory) connections can explain the fundamental aspects of multisensory integration, and provides a biologically plausible hypothesis about the underlying circuitry.

Analysis and mathematical modeling of wave-structure interaction

The aim of this thesis, included within the THESEUS project, is the development of a mathematical model 2DV two-phase, based on the existing code IH-2VOF developed by the University of Cantabria, able to represent together the overtopping phenomenon and the sediment transport. Several numerical simulations were carried out in order to analyze the flow characteristics on a dike crest. The results show that the seaward/landward slope does not affect the evolution of the flow depth and velocity over the dike crest whereas the most important parameter is the relative submergence. Wave heights decrease and flow velocities increase while waves travel over the crest. In particular, by increasing the submergence, the wave height decay and the increase of the velocity are less marked. Besides, an appropriate curve able to fit the variation of the wave height/velocity over the dike crest were found. Both for the wave height and for the wave velocity different fitting coefficients were determined on the basis of the submergence and of the significant wave height. An equation describing the trend of the dimensionless coefficient c_h for the wave height was derived. These conclusions could be taken into consideration for the design criteria and the upgrade of the structures. In the second part of the thesis, new equations for the representation of the sediment transport in the IH-2VOF model were introduced in order to represent beach erosion while waves run-up and overtop the sea banks during storms. The new model allows to calculate sediment fluxes in the water column together with the sediment concentration. Moreover it is possible to model the bed profile evolution. Different tests were performed under low-intensity regular waves with an homogeneous layer of sand on the bottom of a channel in order to analyze the erosion-deposition patterns and verify the model results.

Effect of pulsatility on the mathematical modeling of rotary blood pumps

In this study, the effect of time derivatives of flow rate and rotational speed was investigated on the mathematical modeling of a rotary blood pump (RBP). The basic model estimates the pressure head of the pump as a dependent variable using measured flow and speed as predictive variables. Performance of the model was evaluated by adding time derivative terms for flow and speed. First, to create a realistic working condition, the Levitronix CentriMag RBP was implanted in a sheep. All parameters from the model were physically measured and digitally acquired over a wide range of conditions, including pulsatile speed. Second, a statistical analysis of the different variables (flow, speed, and their time derivatives) based on multiple regression analysis was performed to determine the significant variables for pressure head estimation. Finally, different mathematical models were used to show the effect of time derivative terms on the performance of the models. In order to evaluate how well the estimated pressure head using different models fits the measured pressure head, root mean square error and correlation coefficient were used. The results indicate that inclusion of time derivatives of flow and speed can improve model accuracy, but only minimally.