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Uncertainty is inherent in experimentation, modeling, and analysis. Variations and errors in parameter estimates or physical processes are unavoidable and can affect the reliability of model predictions. Therefore understanding the role of uncertainty is embedded in the process of modeling and approximating the real world. In this manuscript we consider uncertainty propagation in a theoretical model of water/wastewater treatment. In dead-end microfiltration contaminated water is fed through a membrane that filters out colloids, bacteria, and protozoa. However, these particles foul the membrane reducing the filter productivity, which is alleviated by periodically reversing the flow, i.e. backwashing. We investigate how uncertainty in sensitive parameter estimates propagates to the estimates of the optimal amount of volume of water that is filtered in a fixed time period and the associated backwashing timing and duration. We find that the model provides conservative estimates for the total volume since the uncertainty is not propagated symmetrically with respect to over and underestimating specific measurable quantities. The uncertainty in the timing is more symmetric implying that there is essentially an equal amount of uncertainty for increasing or decreasing the frequency and duration of backwashing. We identified biofilm production as propagating the most uncertainty in the volume estimate. The fouling rate has the most effect on the timing estimates. Additionally we explored the affect of asymmetric parameter distributions and find that, for most parameters, asymmetry does not lead to increased asymmetry in predicted optimal regimes, implying that uncertainty in the skewness is likely not an issue. [ABSTRACT FROM AUTHOR]

One of the central problems in membrane implementation is that foulants deposit on surfaces of microfilters employed for water and wastewater treatment and reuse, reducing their productivity. In an effort to remove as much of the deposited materials as possible and maintain the flux, the direction of flow across these membranes is regularly and frequently reversed. However, backwashing only removes the loosely or reversibly bound materials necessitating the entire system to be taken off-line for chemical cleaning when fouling increases beyond a threshold value. We recently presented a mathematically rigorous methodology to determine timing and duration of backwashes so as to delay membrane chemical cleaning and reduce associated operational complexity and costs. By considering an optimal control formulation, using the flow direction as the control variable, the optimal timing and maximum volume of water that can be microfiltered in a given time period were predicted during dead-end filtration of bacterial suspensions. The value of these predictions depend on our ability to accurately and precisely estimate input parameters such as the clean membrane resistance, influent water quality estimates, and biofilm and exo-polymeric substance formation rates and effects. These estimates are inherently uncertain leading to uncertainty in the predictions, which can be rigorously quantified using Sobol indices. These provide a global method for assessing the sensitivity of model predictions (e.g. optimal timing and volume filtered) with respect to variations in the inputs. [ABSTRACT FROM AUTHOR]

Microfiltration and ultrafiltration are methods of removing colloidal impurities from water and wastewater. One of the major issues when dealing with the practical implementation of membranes is the reduction of water productivity as the foulants accumulate on the membrane surface or within the pores. Membrane regeneration by periodic backwashing is an effective method of reducing fouling; however, to date the timing and duration of the backwashing for effective fouling control is largely only empirically determined. In this manuscript, we present an optimal control formulation to determine the timing and duration for membrane regeneration by backwashing. In this formulation, we use the direction of the flow as the control variable and make predictions regarding the optimal protocol. We explicitly include irreversible attachment due to bacterial deposition and biofilm formation on the membrane and demonstrate that irreversible attachment of bacteria has important ramifications for the effective timing of hydraulic backwashes as well as the efficiency in producing clean water. In particular, we find that irreversible attachment and additional fouling due to exo-polymeric substance (EPS) production and biofilm formation decreases the maximum filtration volume. Additionally, as the effectiveness of membrane regeneration declines, the timing of the cycling is also altered. In general, including the role of EPS in biofouling substantially changes predictions of backwashing timing and implies important considerations for practical predictions. [ABSTRACT FROM AUTHOR]

Multi-phase or mixture models are often used to describe the dynamics of complex fluids. In this work, we use a general transformation to reduce the two-phase system of one spatial and time variable to a system of a single variable. Then we assess the behavior of solutions for the inviscid two-phase model with logistic growth. The growth rate widely impacts the behavior of the solution, producing either shocks or rarefactions. Increasing growth increases the frequency and spread of these waves, and eliminating growth reduces solutions to continuous traveling waves. This analysis generalizes a class of asymptotic/linear results for gel swelling as well as showing the extraordinary richness in the molding framework. [ABSTRACT FROM AUTHOR]

Microfiltration is used in a variety of industrial and municipal water purification settings where one of the main concerns is fouling from the particulate matter that is removed from the water. Our focus has been on developing a unified model that captures fouling behavior in a consolidated manner rather than describing individual blocking regimes using power law models. The unified model provides greater insights into fouling mechanisms so that a deeper understanding of flux decline can be obtained. Moreover, by characterizing both forwards and backwashing behavior together, mathematical theory is available to develop strategies that increase the effectiveness of microfiltration in conjunction with backwashing used to regenerate the filter. We present a very simplified model that was developed to provide details regarding the mathematical analysis and how optimal control theory can be used to predict the timing and duration of backwashing that will optimize the overall water flow through the membrane. We use optimal control theory to derive an analytic solution to the optimal problem and develop a strategy to implement the solution. The model estimates of forward operation are compared with experimental data for constant pressure filtration and indicate that the model is able to capture the basic processes. More interestingly, the optimal control solution and proposed implementation strategy are consistent with empirical demonstrations but provide mathematical evidence that the flux may be increased dramatically by precise timing of the forward and backwashing cycles. Model predictions can be evaluated during pilot-testing that often precedes microfilter regulatory approval and plant design. [ABSTRACT FROM AUTHOR]

Abstract: The instantaneous transmembrane pressure needs to be continuously increased to compensate for foulant accumulation during constant flux microfiltration. Herein, we compare predictions of a unified mathematical model and conventional blocking laws with laboratory data obtained during constant flux operation. Foulants employed included single species cultures of bacteria and coagulated natural colloids, both of which are known to form compressible cakes. The first principles model unifies fouling arising from pore blocking by individual cells, cake formation, as well as bacterial secretion of exopolymers. It also incorporates non-uniform spatial deposition patterns that have been observed during unstirred dead-end filtration. Mechanistically, these heterogeneities arise either from non-uniform membrane surface porosity or stochastic initial binding of foulants. Previous studies have shown that the initial patchy or uneven deposit morphology is magnified over longer time-scales during bacterial filtration by differential extracellular polymeric substances (EPS) secretion through quorum sensing. However, in this study, we are primarily interested in the averaged behavior (mainly flux and pressure). By spatially averaging the microscale variables we are able to compare with classical blocking law models. We show that blocking laws and the unified model both accurately model fouling under our experimental conditions. The unified model provides mechanistic insights into (bio)colloid deposition and associated fouling during constant flux microfiltration particularly since it is obtained excellent predictive agreement with experimental data using parameters taken exclusively from our recent study of constant pressure filtration. [Copyright &y& Elsevier]

Abstract: Microfiltration is used in a wide range of municipal and industrial settings to remove particulate matter including pathogenic microorganisms such as bacteria and protozoa. As more water is filtered at constant pressure, the accumulation of retained particles on the membrane decreases the filtration rate; a process commonly referred to as fouling. Mathematical treatment of flux decline has proved to be a useful tool in diagnosing filtration data even though the mathematical underpinnings are not completely understood. In particular, little is known about the transition between fouling phenomena (e.g. pore blocking to cake filtration). Moreover, less is known about the effect of extracellular polymeric substances (EPS) production by bacteria when they accumulate on a membrane over extended durations. In this manuscript, we develop a novel approach to model bacterial microfiltration by considering the effects of both differential binding and exopolymer production. Spatial gradients in bacteria concentrations initially occur due to the non-uniform membrane surface porosity and differential deposition caused by the stochastic nature of microorganism adhesion. These heterogeneities in bacterial deposition and associated pore blocking result in variable secretion of extracellular polymers. Long-term fouling is quantified as the cumulative resistance posed by both bound bacteria and EPS. We compare numerical simulations quantitatively and qualitatively to previously published experimental data and investigate variations of microbial deposition patterns across the membrane. We find substantial agreement between the model and experimental observations. We are also able to conclude that fluid dynamics must be important if the dominant variability is in the membrane structure, rather than in bacterial adhesion. However, variation in bacterial adhesion alone can also induce substantial spatial heterogeneity. [Copyright &y& Elsevier]

One tool in efforts to tackle the ever growing problem of water scarcity is municipal wastewater reclamation to produce drinking water. Microfiltration (MF) is a central technology for potable reuse because it is highly effective in removing pathogenic protozoa, bacteria, and other colloids and for reverse osmosis pretreatment. However, as microfiltered materials accumulate at the membrane surface, its productivity is reduced requiring periodic removal of foulants. A mathematical model of MF is described in the context of hollow fiber filtration that focused on optimizing constant flux operation with backwashing. Design curves were also proposed for determining backwash timing. The model analysis is evaluated against real-world MF fouling for membranes that range in age from a few weeks to three years, observed at the world's largest water reuse facility operated by the Orange County Water District. The presented model compares well with the full-scale operational data, and model parameters accurately capture variations in fouling kinetics with membrane age, providing clues to changes in optimal regeneration timing and frequency as membrane performance declines over long time scales. • Comparison against three years of data provided by a full-scale water reuse facility. • Quantification of effects of aging on fouling and efficiency. • Parameter estimation for full-scale operation based on aging of microfilters. [ABSTRACT FROM AUTHOR]

Abstract: Dead-end microfiltration is an effective method for removing particulate matter including protozoa and bacteria in a variety of settings. As the filtered components accumulate on the filter, the flux declines during constant pressure operation necessitating periodic back-washing. It has been recently reported empirically that the filtered components are not homogeneously distributed on the filter. This must be accounted for in any mathematical theory describing the filtering process. We present a method for determining the coupled fluid dynamics and bacterial transport for dead-end filtration. Results of numerical simulation indicate that the model is able to closely capture experimental data; moreover, the model predicts that the coupled system induces the experimentally observed patchy deposition patterns through spatial variability in the pore structure and porosity of the membrane. [Copyright &y& Elsevier]

Abstract: Biofilms are well known for their extreme tolerance to antibiotics. Recent experimental evidence has indicated the existence of a small fraction of specialized persister cells may be responsible for this tolerance. Although persister cells seem to exist in planktonic bacterial populations, within a biofilm the additional protection offered by the polymeric matrix allows persister cells to evade elimination and serve as a source for re-population. Whether persister cells develop through interactions with toxin/antitoxin modules or are senescent bacteria is an open question. In this investigation we contrast results of the analysis of a mathematical model of the toxin/antitoxin hypothesis for bacteria in a chemostat with results incorporating the senescence hypothesis. We find that the persister fraction of the population as a function of washout rate provides a viable distinction between the two hypotheses. We also give simulation results that indicate that a strategy of alternating dose/withdrawal disinfection can be effective in clearing the entire persister and susceptible populations of bacteria. This strategy was considered previously in analysis of a generic model of persister formation. We find that extending the model of persister formation to include the toxin/antitoxin interactions in a chemostat does not alter the qualitative results that success of the dosing strategy depends on the withdrawal time. While this treatment is restricted to planktonic bacterial populations, it serves as a framework for including persister cells in a spatially dependent biofilm model. [Copyright &y& Elsevier]

Abstract: Almost all moist surfaces are colonized by microbial biofilms. Biofilms are implicated in cross-contamination of food products, biofouling and various human infections such as dental cavities, ulcerative colitis and chronic respiratory infections. The recalcitrance of biofilms to typical antibiotic and antimicrobial treatments is one focus of current investigations. Neither reaction-diffusion limitation nor heterogeneities in growth-rate explain the observed tolerance. Another hypothesis is that specialized ‘persister’ cells, which are extremely tolerant of antimicrobials, are the source of resistance. In this investigation, we describe the formation of ‘persister’ cells which neither grow nor die in the presence of antibiotics. We propose that these cells are of a different phenotype whose expression is regulated by the growth rate and the antibiotic concentration. Based on several experiments describing the dynamics of persister cells, we introduce a mathematical model that is used to describes the effect of a periodic dosing regiment. Results from our analysis indicate that the relative dose/withdrawal times are important in determining the effectiveness of such a treatment. A reduced model is also introduced and the similar behavior is demonstrated analytically. [Copyright &y& Elsevier]

Abstract: A mathematical model of the action of antimicrobial agents on bacterial biofilms is presented. The model includes the fluid dynamics in and around the biofilm, advective and diffusive transport of two chemical constituents and the mechanism of physiological resistance. Although the mathematical model applies in three dimensions, we present two-dimensional simulations for arbitrary biofilm domains and various dosing strategies. The model allows the prediction of the spatial evolution of bacterial population and chemical constituents as well as different dosing strategies based on the fluid motion. We find that the interaction between the nutrient and the antimicrobial agent can reproduce survival curves which are comparable to other model predictions as well as experimental results. The model predicts that exposing the biofilm to low concentration doses of antimicrobial agent for longer time is more effective than short time dosing with high antimicrobial agent concentration. The effects of flow reversal and the roughness of the fluid/biofilm are also investigated. We find that reversing the flow increases the effectiveness of dosing. In addition, we show that overall survival decreases with increasing surface roughness. [Copyright &y& Elsevier]

Short-term microfiltration (MF) fouling is commonly abated by periodically reversing the flow to remove foulants that weakly adhered to the membrane. Strong oxidants (i.e., chlorine) can be added to hydraulic backwash water to augment its efficacy—a process called chemically enhanced backwashing (CEB). Herein, we report a rigorous mathematical model for constant flux MF incorporating hydraulic backwashing and CEB, and validate it with laboratory data obtained using untreated and alum-coagulated water from the Foss Reservoir in Oklahoma, USA. We implemented an optimal control procedure and used it to predict MF behavior long past experimental timescales. We identified a frequency threshold beyond which the necessary transmembrane pressure (TMP) reached an asymptotic value, indicating a pseudo steady-state, periodic solution to the model when coupling hydraulic backwashing with CEB. We report differences in TMP saturation values and timescales by simulating transient MF of untreated and pretreated water. Numerical simulations revealed that the operating flux could be increased 10-fold after pretreatment (compared with raw water) before reaching the maximum manufacturer-recommended pressure for the hollow-fibers. The predicted higher flux and extended duration between cleaning-in-place demonstrated advantages of coagulation pretreatment under hydraulic backwashing and CEB. Model observations could guide decision making for CEB timing and frequency. [Display omitted] • Microfiltration model derived with hydraulic and chemically enhanced backwashing. • Quantitative comparisons with experiments for raw and coagulated surface water. • Predictions of long-term productivity at various flux and backwashing conditions. • Different chemically enhance backwashing thresholds and timescales for pressure plateaus identified. • Flux and pressure plateaus change linearly for coagulated water, non-linearly for raw. [ABSTRACT FROM AUTHOR]

Alopecia areata (AA) is an autoimmune disease, and its clinical phenotype is characterized by the formation of distinct hairless patterns on the scalp or other parts of the body. In most cases hair falls out in round patches. A well-established hypothesis for the pathogenesis of AA states that collapse of hair follicle immune privilege is one of the essential elements in disease development. To investigate the dynamics of alopecia areata, we develop a mathematical model that incorporates immune system components and hair follicle immune privilege agents whose involvement in AA has been confirmed in clinical studies and experimentally. We perform parameter sensitivity analysis in order to determine which inputs have the greatest effect on outcome variables. Our findings suggest that, among all processes reflected in the model, immune privilege guardians and the pro-inflammatory cytokine interferon- γ govern disease dynamics. These results agree with the immune privilege collapse hypothesis for the development of AA. [ABSTRACT FROM AUTHOR]

Highlights • A framework which addresses parameter identifiability using experimental data. • Sensitivity analysis; computational modifications for fast estimation of Sobol' indices. • An ODE model for growth and biofilm formation of Xylella fastidiosa. • Analysis from framework indicates addition of zinc inhibits nutrient flow to biofilm. Abstract Mathematical models are ubiquitous in analyzing dynamical biological systems. However, it might not be possible to explicitly account for the various sources of uncertainties in the model and the data if there is limited experimental data and information about the biological processes. The presence of uncertainty introduces problems with identifiability of the parameters of the model and determining appropriate regions to explore with respect to sensitivity and estimates of parameter values. Since the model analysis is likely dependent on the numerical estimates of the parameters, parameter identifiability should be addressed beforehand to capture biologically relevant parameter space. Here, we propose a framework which uses data from different experiment regimes to identify a region in the parameter space over which subsequent mathematical analysis can be conducted. Along with building confidence in the parameter estimates, it provides us with variations in the parameters due to changes in the experimental conditions. To determine significance of these variations, we conduct global sensitivity analysis, allowing us to make testable hypothesis for effects of changes in the experimental conditions on the biological system. As a case study, we develop a model for growth dynamics and biofilm formation of a bacterial plant pathogen, and use our framework to identify possible effects of zinc on the bacterial populations in different metabolic states. The framework reveals underlying issues with parameter identifiability and identifies a suitable region in the parameter space, sensitivity analysis over which informs us about the parameters that might be affected by addition of zinc. Moreover, these parameters prove to be identifiable in this region. [ABSTRACT FROM AUTHOR]