David M. Bortz

Incorporation of variability into the modeling of viral delays in HIV infection dynamics

We consider classes of functional differential equation models which arise in attempts to describe temporal delays in HIV pathogenesis. In particular, we develop methods for incorporating arbitrary variability (i.e., general probability distributions) for these delays into systems that cannot readily be reduced to a finite number of coupled ordinary differential equations (as is done in the method of stages). We discuss modeling from first principles, introduce several classes of non-linear models (including discrete and distributed delays) and present a discussion of theoretical and computational approaches. We then use the resulting methodology to carry out simulations and perform parameter estimation calculations, fitting the models to a set of experimental data. Results obtained confirm the statistical significance of the presence of delays and the importance of including delays in validating mathematical models with experimental data. We also show that the models are quite sensitive to the mean of the distribution which describes the delay in viral production, whereas the variance of this distribution has relatively little impact.

The Simplex Gradient and Noisy Optimization Problems

Many classes of methods for noisy optimization problems are based on function information computed on sequences of simplices. The Nelder-Mead, multidirectional search, and implicit filtering methods are three such methods. The performance of these methods can be explained in terms of the difference approximation of the gradient implicit in the function evaluations. Insight can be gained into choice of termination criteria, detection of failure, and design of new methods.

COPS: Large-scale nonlinearly constrained optimization problems

The authors have started the development of COPS, a collection of large-scale nonlinearly Constrained Optimization Problems. The primary purpose of this collection is to provide difficult test cases for optimization software. Problems in the current version of the collection come from fluid dynamics, population dynamics, optimal design, and optimal control. For each problem they provide a short description of the problem, notes on the formulation of the problem, and results of computational experiments with general optimization solvers. They currently have results for DONLP2, LANCELOT, MINOS, SNOPT, and LOQO.

Inverse Problems for a Class of Measure Dependent Dynamical Systems

Abstract : We consider a class of probability measure dependent dynamical systems which arise in the study of multiscale phenomena in diverse fields such as immunological population dynamics, viscoelasticity of polymers and rubber and polarization in dielectric materials. We develop an inverse problem framework for studying systems with distributed temporal delays. In particular, we establish conditions for existence and uniqueness of the forward problem and well-posedness (including method stability under numerical approximations) for the inverse problem of estimating the probability measures. We show that a motivating class of models of HIV infection dynamics satisfies all the conditions of our framework, thereby providing a theoretical foundation for inverse problem computations with these models.

Optimizing control of open bay acoustics

∗ Associate Fellow AIAA; President, Innovative Technology Applications Company, Chesterfield, MO † Post Doctoral Researcher, presently residing in Argentina ‡ Ph. D. Candidate, Dept. of Mathematics § Director of the Center for Research in Scientific Computation and University Professor and Drexel Professor of Mathematics ¶ Associate Professor of Mathematics and Associate Director of the Center for Research in Scientific Computation Copyright 2000 The American Institute of Aeronautics and Astronautics Inc. All rights reserved. Abstract

Dynamical system analysis of Staphylococcus epidermidis bloodstream infection.

Unlike many localized infections, the development and resolution of bacteremia involves physical and immunological interactions between many anatomic sites. In an effort to better understand these interactions, we developed a computational model of bacteremia as a dynamical system fashioned after multicompartmental pharmacodynamic models, incorporating bacterial proliferation and clearance in the blood, liver, spleen, and lungs, and the transport of pathogens between these sites. A system of four first-order homogeneous ODEs was developed. Blood and organ bacterial burdens were measured at various time points from 3 to 48 h postinoculation using an LD25 murine model of Staphylococcus epidermidis bacteremia. Using these empiric data, solutions to the mathematical model were recovered. A bootstrap resampling method was used to generate 95% confidence intervals around the solved parameters. The validity of the model was examined in parallel experiments using animals acutely immunocompromised with cyclophosphamide; the model captured abnormalities in bacterial partitioning previously described with this antineoplastic agent. Lastly, the approach was used to explore possible benefits to clinically observed hyperdynamic blood flow during sepsis: in simulation, normal mice, but not those treated with cyclophosphamide, enjoyed significantly more rapid bacterial clearance from the bloodstream under hyperdynamic conditions.

Modeling keratinocyte wound healing dynamics: Cell-cell adhesion promotes sustained collective migration.

The in vitro migration of keratinocyte cell sheets displays behavioral and biochemical similarities to the in vivo wound healing response of keratinocytes in animal model systems. In both cases, ligand-dependent Epidermal Growth Factor Receptor (EGFR) activation is sufficient to elicit collective cell migration into the wound. Previous mathematical modeling studies of in vitro wound healing assays assume that physical connections between cells have a hindering effect on cell migration, but biological literature suggests a more complicated story. By combining mathematical modeling and experimental observations of collectively migrating sheets of keratinocytes, we investigate the role of cell-cell adhesion during in vitro keratinocyte wound healing assays. We develop and compare two nonlinear diffusion models of the wound healing process in which cell-cell adhesion either hinders or promotes migration. Both models can accurately fit the leading edge propagation of cell sheets during wound healing when using a time-dependent rate of cell-cell adhesion strength. The model that assumes a positive role of cell-cell adhesion on migration, however, is robust to changes in the leading edge definition and yields a qualitatively accurate density profile. Using RNAi for the critical adherens junction protein, α-catenin, we demonstrate that cell sheets with wild type cell-cell adhesion expression maintain migration into the wound longer than cell sheets with decreased cell-cell adhesion expression, which fails to exhibit collective migration. Our modeling and experimental data thus suggest that cell-cell adhesion promotes sustained migration as cells pull neighboring cells into the wound during wound healing.

Multicellularity and Antibiotic Resistance in Klebsiella pneumoniae Grown Under Bloodstream-Mimicking Fluid Dynamic Conditions

BACKGROUND While the importance of fluid dynamical conditions is well recognized in the growth of biofilms, their role during bacteremia is unknown. We examined the impact of physiological fluid shear forces on the development of multicellular aggregates of Klebsiella pneumoniae. METHODS Wild-type and O-antigen or capsular mutants of K. pneumoniae were grown as broth culture in a Taylor-Couette flow cell configured to provide continuous shear forces comparable to those encountered in the human arterial circulation (ie, on the order of 1.0 Pa). The size distribution and antibiotic resistance of aggregates formed in this apparatus were determined, as was their ability to persist in the bloodstream of mice following intravenous injection. RESULTS Unlike growth in shaking flasks, bacteria grown in the test apparatus readily formed aggregates, a phenotype largely absent in capsular mutants and to a lesser degree in O-antigen mutants. Aggregates were found to persist in the bloodstream of mice. Importantly, organisms grown under physiological shear were found to have an antibiotic resistance phenotype intermediate between that of fully planktonic and biofilm states. CONCLUSIONS When grown under intravascular-magnitude fluid dynamic conditions, K. pneumoniae spontaneously develops into multicellular aggregates that are capable of persisting in the circulation and exhibit increased antibiotic resistance.

Determination of personalized diabetes treatment plans using a two-delay model

Diabetes cases worldwide have risen steadily over the past few decades, lending urgency to the search for more efficient, effective, and personalized ways to treat the disease. Current treatment strategies, however, may fail to maintain oscillations in blood glucose concentration that naturally occur multiple times per day, an important element of normal human physiology. Building upon recent successes in mathematical modeling of the human glucose-insulin system, we show that both food intake and insulin therapy likely demand increasingly precise control over insulin sensitivity if oscillations at a healthy average glucose concentration are to be maintained. We then model and describe personalized treatment options for patients with diabetes that maintain these oscillations. We predict that for a person with type II diabetes, both blood glucose levels can be controlled and healthy oscillations maintained when the patient gets an hour of daily exercise and is placed on a combination of Metformin and sulfonylurea drugs. We note that insulin therapy and an additional hour of exercise will reduce the patient׳s need for sulfonylureas. Results of a modeling analysis suggest that, with constant nutrition and controlled exercise, the blood glucose levels of a person with type I diabetes can be properly controlled with insulin infusion between 0.45 and 0.7μU/mlmin. Lastly, we note that all suggested strategies rely on existing clinical techniques and established treatment measures, and so could potentially be of immediate use in the design of an artificial pancreas.

Physics of biofilms: the initial stages of biofilm formation and dynamics

One of the physiological responses of bacteria to external stress is to assemble into a biofilm. The formation of a biofilm greatly increases a bacterial populations resistance to a hostile environment by shielding cells, for example, from antibiotics. In this paper, we describe the conditions necessary for the emergence of biofilms in natural environments and relate them to the emergence of biofilm formation inside microfluidic devices. We show that competing species of Escherichia coli bacteria form biofilms to spatially segregate themselves in response to starvation stress, and use in situ methods to characterize the physical properties of the biofilms. Finally, we develop a microfluidic platform to study the inter-species interactions and show how biofilm-mediated genetic interactions can improve a species? resistance to external stress.