A Model of Gastric Mucosal pH Regulation: Extending Sensitivity Analysis Using Sobol' Indices to Understand Higher Moments.

Several recent theoretical studies have indicated that a relatively simple secretion control mechanism in the epithelial cells lining the stomach may be responsible for maintaining a neutral (healthy) pH adjacent to the stomach wall, even in the face of enormous electrodiffusive acid transport from the interior of the stomach. Subsequent work used Sobol' Indices (SIs) to quantify the degree to which this secretion mechanism is "self-regulating" i.e. the degree to which the wall pH is held neutral as mathematical parameters vary. However, questions remain regarding the nature of the control that specific parameters exert over the maintenance of a healthy stomach wall pH. Studying the sensitivity of higher moments of the statistical distribution of a model output can provide useful information, for example, how one parameter may skew the distribution towards or away from a physiologically advantageous regime. In this work, we prove a relationship between SIs and the higher moments and show how it can potentially reduce the cost of computing sensitivity of said moments. We define γ -indices to quantify sensitivity of variance, skewness, and kurtosis to the choice of value of a parameter, and we propose an efficient strategy that uses both SIs and γ -indices for a more comprehensive sensitivity analysis. Our analysis uncovers a control parameter which governs the "tightness of control" that the secretion mechanism exerts on wall pH. Finally, we discuss how uncertainty in this parameter can be reduced using expert information about higher moments, and speculate about the physiological advantage conferred by this control mechanism.

Bayesian estimation of Pseudomonas aeruginosa viscoelastic properties based on creep responses of wild type, rugose, and mucoid variant biofilms.

Pseudomonas aeruginosa biofilms are relevant for a variety of disease settings, including pulmonary infections in people with cystic fibrosis. Biofilms are initiated by individual bacteria that undergo a phenotypic switch and produce an extracellular polymeric slime (EPS). However, the viscoelastic characteristics of biofilms at different stages of formation and the contributions of different EPS constituents have not been fully explored. For this purpose, we develop and parameterize a mathematical model to study the rheological behavior of three biofilms - P. aeruginosa wild type PAO1, isogenic rugose small colony variant (RSCV), and mucoid variant biofilms against a range of experimental data. Using Bayesian inference to estimate these viscoelastic properties, we quantify the rheological characteristics of the biofilm EPS. We employ a Monte Carlo Markov Chain algorithm to estimate these properties of P. aeruginosa variant biofilms in comparison to those of wild type. This information helps us understand the rheological behavior of biofilms at different stages of their development. The mechanical properties of wild type biofilms change significantly over time and are more sensitive to small changes in their composition than the other two mutants.

Exploring how ecological and epidemiological processes shape multi-host disease dynamics using global sensitivity analysis.

We use global sensitivity analysis (specifically, Partial Rank Correlation Coefficients) to explore the roles of ecological and epidemiological processes in shaping the temporal dynamics of a parameterized SIR-type model of two host species and an environmentally transmitted pathogen. We compute the sensitivities of disease prevalence in each host species to model parameters. Sensitivity rankings are calculated, interpreted biologically, and contrasted for cases where the pathogen is introduced into a disease-free community and cases where a second host species is introduced into an endemic single-host community. In some cases the magnitudes and dynamics of the sensitivities can be predicted only by knowing the host species' characteristics (i.e., their competitive abilities and disease competence) whereas in other cases they can be predicted by factors independent of the species' characteristics (specifically, intraspecific versus interspecific processes or a species' roles of invader versus resident). For example, when a pathogen is initially introduced into a disease-free community, disease prevalence in both hosts is more sensitive to the burst size of the first host than the second host. In comparison, disease prevalence in each host is more sensitive to its own infection rate than the infection rate of the other host species. In total, this study illustrates that global sensitivity analysis can provide useful insight into how ecological and epidemiological processes shape disease dynamics and how those effects vary across time and system conditions. Our results show that sensitivity analysis can provide quantification and direction when exploring biological hypotheses.

Mathematical Model of Triple-Negative Breast Cancer in Response to Combination Chemotherapies.

Triple-negative breast cancer (TNBC) is a heterogenous disease that is defined by its lack of targetable receptors, thus limiting treatment options and resulting in higher rates of metastasis and recurrence. Combination chemotherapy treatments, which inhibit tumor cell proliferation and regeneration, are a major component of standard-of-care treatment of TNBC. In this manuscript, we build a coupled ordinary differential equation model of TNBC with compartments that represent tumor proliferation, necrosis, apoptosis, and immune response to computationally describe the biological tumor affect to a combination of chemotherapies, doxorubicin (DRB) and paclitaxel (PTX). This model is parameterized using longitudinal [18F]-fluorothymidine positron emission tomography (FLT-PET) imaging data which allows for a noninvasive molecular imaging approach to quantify the tumor proliferation and tumor volume measurements for two murine models of TNBC. Animal models include a human cell line xenograft model, MDA-MB-231, and a syngeneic 4T1 mammary carcinoma model. The mathematical models are parameterized and the percent necrosis at the end time point is predicted and validated using histological hematoxylin and eosin (H&E) data. Global Sobol' sensitivity analysis is conducted to further understand the role each parameter plays in the model's goodness of fit to the data. In both the MDA-MB-231 and the 4T1 tumor models, the designed mathematical model can accurately describe both tumor volume changes and final necrosis volume. This can give insight into the ordering, dosing, and timing of DRB and PTX treatment. More importantly, this model can also give insight into future novel combinations of therapies and how the immune system plays a role in therapeutic response to TNBC, due to its calibration to two types of TNBC murine models.

Physiological insights into electrodiffusive maintenance of gastric mucus through sensitivity analysis and simulations.

It is generally accepted that the gastric mucosa and adjacent mucus layer are critical in the maintenance of a pH gradient from stomach lumen to stomach wall, protecting the mucosa from the acidic environment of the lumen and preventing auto-digestion of the epithelial layer. No conclusive study has shown precisely which physical, chemical, and regulatory mechanisms are responsible for maintaining this gradient. However, experimental work and modeling efforts have suggested that concentration dependent ion-exchange at the epithelial wall, together with hydrogen ion/mucus network binding, may produce the enormous pH gradients seen in vivo. As of yet, there has been no exhaustive study of how sensitive these modeling results are with respect to variation in model parameters, nor how sensitive such a regulatory mechanism may be to variation in physical/biological parameters. In this work, we perform sensitivity analysis (using Sobol' Indices) on a previously reported model of gastric pH gradient maintenance. We quantify the sensitivity of mucosal wall pH (as a proxy for epithelial health) to variations in biologically relevant parameters and illustrate how variations in these parameters affects the distribution of the measured pH values. In all parameter regimes, we see that the rate of cation/hydrogen exchange at the epithelial wall is the dominant parameter/effect with regards to variation in mucosal pH. By careful sensitivity analysis, we also investigate two different regimes representing high and low hydrogen secretion with different physiological interpretations. By complementing mechanistic modeling and biological hypotheses testing with parametric sensitivity analysis we are able to conclude which biological processes must be tightly regulated in order to robustly maintain the pH values necessary for healthy function of the stomach.

Data assimilation of synthetic data as a novel strategy for predicting disease progression in alopecia areata.

The goal of patient-specific treatment of diseases requires a connection between clinical observations with models that are able to accurately predict the disease progression. Even when realistic models are available, it is very difficult to parameterize them and often parameter estimates that are made using early time course data prove to be highly inaccurate. Inaccuracies can cause different predictions, especially when the progression depends sensitively on the parameters. In this study, we apply a Bayesian data assimilation method, where the data are incorporated sequentially, to a model of the autoimmune disease alopecia areata that is characterized by distinct spatial patterns of hair loss. Using synthetic data as simulated clinical observations, we show that our method is relatively robust with respect to variations in parameter estimates. Moreover, we compare convergence rates for parameters with different sensitivities, varying observational times and varying levels of noise. We find that this method works better for sparse observations, sensitive parameters and noisy observations. Taken together, we find that our data assimilation, in conjunction with our biologically inspired model, provides directions for individualized diagnosis and treatments.

Toward Predicting the Spatio-Temporal Dynamics of Alopecia Areata Lesions Using Partial Differential Equation Analysis.

Hair loss in the autoimmune disease, alopecia areata (AA), is characterized by the appearance of circularly spreading alopecic lesions in seemingly healthy skin. The distinct spatial patterns of AA lesions form because the immune system attacks hair follicle cells that are in the process of producing hair shaft, catapults the mini-organs that produce hair from a state of growth (anagen) into an apoptosis-driven regression state (catagen), and causes major hair follicle dystrophy along with rapid hair shaft shedding. In this paper, we develop a model of partial differential equations (PDEs) to describe the spatio-temporal dynamics of immune system components that clinical and experimental studies show are primarily involved in the disease development. Global linear stability analysis reveals there is a most unstable mode giving rise to a pattern. The most unstable mode indicates a spatial scale consistent with results of the humanized AA mouse model of Gilhar et al. (Autoimmun Rev 15(7):726-735, 2016) for experimentally induced AA lesions. Numerical simulations of the PDE system confirm our analytic findings and illustrate the formation of a pattern that is characteristic of the spatio-temporal AA dynamics. We apply marginal linear stability analysis to examine and predict the pattern propagation.

A framework for model analysis across multiple experiment regimes: Investigating effects of zinc on Xylella fastidiosa as a case study.

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.

Analysing the dynamics of a model for alopecia areata as an autoimmune disorder of hair follicle cycling.

Alopecia areata (AA) is a CD8$^{+}$ T cell-dependent autoimmune disease that disrupts the constantly repeating cyclic transformations of hair follicles (HFs). Among the three main HF cycle stages-growth (anagen), regression (catagen) and relative quiescence (telogen)-only anagen HFs are attacked and thereby forced to prematurely enter into catagen, thus shortening active hair growth substantially. After having previously modelled the dynamics of immune system components critically involved in the disease development (Dobreva et al., 2015), we here present a mathematical model for AA which incorporates HF cycling and illustrates the anagen phase interruption in AA resulting from an inflammatory autoimmune response against HFs. The model couples a system describing the dynamics of autoreactive immune cells with equations modelling the hair cycle. We illustrate states of health, disease and treatment as well as transitions between them. In addition, we perform parameter sensitivity analysis to assess how different processes, such as proliferation, apoptosis and input from stem cells, impact anagen duration in healthy versus AA-affected HFs. The proposed model may help in evaluating the effectiveness of existing treatments and identifying new potential therapeutic targets.

Short-Term Antiretroviral Treatment Recommendations Based on Sensitivity Analysis of a Mathematical Model for HIV Infection of CD4âºΤ Cells.

HIV infection is one of the most difficult infections to control and manage. The most recent recommendations to control this infection vary according to the guidelines used (US, European, WHO) and are not patient-specific. Unfortunately, no two individuals respond to infection and treatment quite the same way. The purpose of this paper is to make use of the uncertainty and sensitivity analysis to investigate possible short-term treatment options that are patient-specific. We are able to identify the most significant parameters that are responsible for ART outcome and to formulate some insights into the ART success.

Combining Two Methods of Global Sensitivity Analysis to Investigate MRSA Nasal Carriage Model.

We apply two different sensitivity techniques to a model of bacterial colonization of the anterior nares to better understand the dynamics of Staphylococcus aureus nasal carriage. Specifically, we use partial rank correlation coefficients to investigate sensitivity as a function of time and identify a reduced model with fewer than half of the parameters of the full model. The reduced model is used for the calculation of Sobol' indices to identify interacting parameters by their additional effects indices. Additionally, we found that the model captures an interesting characteristic of the biological phenomenon related to the initial population size of the infection; only two parameters had any significant additional effects, and these parameters have biological evidence suggesting they are connected but not yet completely understood. Sensitivity is often applied to elucidate model robustness, but we show that combining sensitivity measures can lead to synergistic insight into both model and biological structures.

Theoretical and experimental evidence for eliminating persister bacteria by manipulating killing timing.

Formation of a transient sub-population of bacteria, referred to as persisters, is one of the most important and least understood mechanisms that bacteria employ to evade elimination. Persister cells appear to be slow-growing bacteria that are broadly protected from a wide range of antibiotics. Using both theoretical and experimental methods, we show that alternating the application and withdrawal of antibiotics can be an effective treatment-as long as the timing of the protocol is estimated with precision. More specifically, we demonstrate that timing the alternating treatment based on theoretical predictions is confirmed using experimental observations. These results support a large class of theoretical studies that show that, even without complete understanding of the biological mechanisms, these models can provide insight into properties of the system.

Mathematical Model for MRSA Nasal Carriage.

An interesting biological phenomenon that is a factor for the spread of antibiotic-resistant strains, such as MRSA, is human nasal carriage. Here, we evaluate several biological hypotheses for this problem in an effort to better understand and narrow the scope of the dominant factors that allow these bacteria to persist in otherwise healthy individuals. First, we set up and analyze a simple PDE model created to generally mimic the interactions of the microbes and nasal immune response. This includes looking at different types of diffusion and chemotaxis terms as well as different boundary conditions. Then, using sensitivity analysis, we walk through several biological hypotheses and compare to the model's results looking for persistent infection scenarios indicated by the model's bacteria component surviving over time.

Mathematical model for alopecia areata.

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.

Global sensitivity analysis used to interpret biological experimental results.

Modeling host/pathogen interactions provides insight into immune defects that allow bacteria to overwhelm the host, mechanisms that allow vaccine strategies to be successful, and illusive interactions between immune components that govern the immune response to a challenge. However, even simplified models require a fairly high dimensional parameter space to be explored. Here we use global sensitivity analysis for parameters in a simple model for biofilm infections in mice. The results indicate which parameters are insignificant and are 'frozen' to yield a reduced model. The reduced model replicates the full model with high accuracy, using approximately half of the parameter space. We used the sensitivity to investigate the results of the combined biological and mathematical experiments for osteomyelitis. We are able to identify parts of the compartmentalized immune system that were responsible for each of the experimental outcomes. This model is one example for a technique that can be used generally.

Modelling mechanical and chemical treatment of biofilms with two phenotypic resistance mechanisms.

Bacterial biofilms are notoriously difficult to eradicate owing to a number of tolerance mechanisms including physiological, physical, genotypic and phenotypic variations. Recent focus has shifted to phenotypic tolerance which is apparently the main defence mechanism that protects biofilms against long-term disinfection. Previous mathematical models have addressed phenotypic dynamics by considering adaptive response and persister formation separately. The aim of this manuscript is to consider a combined model to understand the interplay between these two defence mechanisms. We find that each mechanism protects the biofilm differently and hence responds differently to antibiotic challenge. We focus on on-off dosing that has been shown to eradicate each subpopulation alone. Our results indicate that the combined resistance exhibits qualitatively similar behavior to persister formation for short dosing times, and similar behavior to adaptive resistance for long dosing times. To further contrast the behavior of the model under different parameter regimes, we explore two classes of combination treatment that include mechanical and chemical treatments. The examples focus on different applications - pipe clearance and dental carrie prevention - and demonstrate the underlying conclusion that adaptive and persister mechanism provide protection for different challenges and are thus not redundant systems and each may require specific treatment plans.

Modelling the interaction between the host immune response, bacterial dynamics and inflammatory damage in comparison with immunomodulation and vaccination experiments.

The immune system is a complex system of chemical and cellular interactions that responds quickly to queues that signal infection and then reverts to a basal level once the challenge is eliminated. Here, we present a general, four-component model of the immune system's response to a Staphylococcal aureus (S. aureus) infection, using ordinary differential equations. To incorporate both the infection and the immune system, we adopt the style of compartmenting the system to include bacterial dynamics, damage and inflammation to the host, and the host response. We incorporate interactions not previously represented including cross-talk between inflammation/damage and the infection and the suppression of the anti-inflammatory pathway in response to inflammation/damage. As a result, the most relevant equilibrium of the system, representing the health state, is an all-positive basal level. The model is able to capture eight different experimental outcomes for mice challenged with intratibial osteomyelitis due to S. aureus, primarily involving immunomodulation and vaccine therapies. For further validation and parameter exploration, we perform a parameter sensitivity analysis which suggests that the model is very stable with respect to variations in parameters, indicates potential immunomodulation strategies and provides a possible explanation for the difference in immune potential for different mouse strains.

Concepts in disinfection of bacterial populations.

Bacteria are responsible for a range of human diseases that are difficult to clear for a variety of reasons. Understanding the multilayered mechanisms that protect pathogens is of primary importance for developing effective treatments. Research over the past several decades has shown that bacteria can seek protected environments passively by avoiding deadly environments or actively by manipulating their phenotypic expression and gathering in structured biofilm communities. This article outlines the main tolerance mechanisms that have been studied and describes many of the mathematical models that indicate directions for new treatments. It is essential to understand that bacterial populations exploit a range of mechanisms to survive challenges and the mechanisms are often intertwined. This provides an opportunity for mathematical modeling to help provide insights into experiments that may uncouple particular mechanisms or optimize treatments.

Pattern formation exhibited by biofilm formation within microfluidic chambers.

This article investigates the dynamics of an important bacterial pathogen, Xylella fastidiosa, within artificial plant xylem. The bacterium is the causative agent of a variety of diseases that strike fruit-bearing plants including Pierce's disease of grapevine. Biofilm colonization within microfluidic chambers was visualized in a laboratory setting, showing robust, regular spatial patterning. We also develop a mathematical model, based on a multiphase approach that is able to capture the spacing of the pattern and points to the role of the exopolymeric substance as the main source of control of the pattern dynamics. We concentrate on estimating the attachment/detachment processes within the chamber because these are two mechanisms that have the potential to be engineered by applying various chemicals to prevent or treat the disease.

Effect of periodic disinfection on persisters in a one-dimensional biofilm model.

It is well known that disinfection methods that successfully kill suspended bacterial populations often fail to eliminate bacterial biofilms. Recent efforts to understand biofilm survival have focused on the existence of small, but very tolerant, subsets of the bacterial population termed persisters. In this investigation, we analyze a mathematical model of disinfection that consists of a susceptible-persister population system embedded within a growing domain. This system is coupled to a reaction-diffusion system governing the antibiotic and nutrient. We analyze the effect of periodic and continuous dosing protocols on persisters in a one-dimensional biofilm model, using both analytic and numerical method. We provide sufficient conditions for the existence of steady-state solutions and show that these solutions may not be unique. Our results also indicate that the dosing ratio (the ratio of dosing time to period) plays an important role. For long periods, large dosing ratios are more effective than similar ratios for short periods. We also compare periodic to continuous dosing and find that the results also depend on the method of distributing the antibiotic within the dosing cycle.