Karyn L. Sutton

An Inverse Problem Statistical Methodology Summary

We discuss statistical and computational aspects of inverse or parameter estimation problems for deterministic dynamical systems based on Ordinary Least Squares and Generalized Least Squares with appropriate corresponding data noise assumptions of constant variance and nonconstant variance (relative error), respectively. Among the topics included here are mathematical model, statistical model and data assumptions, and some techniques (residual plots, sensitivity analysis, model comparison tests) for verifying these. The ideas are illustrated throughout with the popular logistic growth model of Verhulst and Pearl as well as with a recently developed population level model of pneumococcal disease spread.

A NEW MODEL FOR THE ESTIMATION OF CELL PROLIFERATION DYNAMICS USING CFSE DATA

CFSE analysis of a proliferating cell population is a popular tool for the study of cell division and divisionlinked changes in cell behavior. Recently Banks et al. (2011), Luzyanina et al. (2009), Luzyanina et al. (2007), a partial differential equation (PDE) model to describe lymphocyte dynamics in a CFSE proliferation assay was proposed. We present a significant revision of this model which improves the physiological understanding of several parameters. Namely, the parameter used previously as a heuristic explanation for the dilution of CFSE dye by cell division is replaced with a more physical component, cellular autofluorescence. The rate at which label decays is also quantified using a Gompertz decay process. We then demonstrate a revised method of fitting the model to the commonly used histogram representation of the data. It is shown that these improvements result in a model with a strong physiological basis which is fully capable of replicating the behavior observed in the data.

Validation of flow cytometric competitive binding protocols and characterization of fluorescently labeled ligands

BACKGROUND Fluorescently labeled ligands and flow cytometric methods allow quantification of receptor-ligand binding. Such methods require calibration of the fluorescence of bound ligands. Moreover, binding of unlabeled ligands can be calculated based on their abilities to compete with a labeled ligand. In this study, calibration parameters were determined for six fluorescently labeled N-formyl peptides that bind to receptors on neutrophils. Two of these ligands were then used to develop and validate competitive binding protocols for determining binding constants of unlabeled ligands. METHODS Spectrofluorometric and flow cytometric methods for converting relative flow cytometric intensities to number of bound ligand/cell were extended to include peptides labeled with fluorescein, Bodipy, and tetramethylrhodamine. The validity of flow cytometric competitive binding protocols was tested using two ligands with different fluorescent properties that allowed determination of rate constants both directly and competitively for one ligand, CHO-NLFNYK-tetramethylrhodamine. RESULTS Calibration parameters were determined for six fluorescently-labeled N-formyl peptides. Equilibrium dissociation constants for these ligands varied over two orders of magnitude and depended upon the peptide sequence and the molecular structure of the fluorescent tag. Kinetic rate constants for CHO-NLFNYK-tetramethylrhodamine determined directly or in competition with CHO-NLFNYK-fluorescein were statistically identical. CONCLUSIONS Combination of spectrofluorometric and flow cytometric methods allows convenient calculation of calibration parameters for a series of fluorescent ligands that bind to the same receptor site. Competitive binding protocols have been independently validated.

Label Structured Cell Proliferation Models.

We present a general class of cell population models that can be used to track the proliferation of cells which have been labeled with a fluorescent dye. The mathematical models employ fluorescence intensity as a structure variable to describe the evolution in time of the population density of proliferating cells. While cell division is a major component of changes in cellular fluorescence intensity, models developed here also address overall label degradation.

Dynamic social network models incorporating stochasticity and delays

Networks are typically studied via computational models, and often investigations are restricted to the static case. Here we extend the work in Banks, Karr, Nguyen and Samuels (2008), which demonstrated a simple dynamical system framework in which to study social network behavior, to include a discrete delay. This delay represents the time lag that is likely required for an agent to change his/her own characteristics (e.g., opinions, viewpoints or behavior) after interacting with an agent possessing different characteristics. Thus this modification adds significantly to the relevance of the model in many potential applications. We have shown that the delays can be incorporated into a stochastic differential equations (SDE) framework in an efficient and computationally tractable way. Through numerical studies, we see novel outcomes when stochasticity, delay, or both are considered, demonstrating the need to include these features should they be present in the network application.

Public vaccination policy using an age-structured model of pneumococcal infection dynamics

Public health professionals are charged with the task of designing prevention programs for the effective control of biologically intricate infectious diseases at a population level. The effective vaccination of a population for pneumococcal diseases (infections caused by Streptococcus pneumoniae) remains a relevant question in the scientific community. It is complicated by heterogeneity in individuals’ responses to exposure to the bacterium and their responses to vaccination. Due to these complexities, most modelling efforts in this area have been on the cellular/bacteria level. Here, we introduce an age-structured SEIS-type model of pneumococcal diseases and their vaccination. We discuss the use of this framework in predicting the impact of vaccine strategies, with pneumococcal diseases as an example. Using parameter values reasonable for a developed country, we discuss the effects of targeting the colonization and/or infection stages on the age profiles of morbidity in a population.

Dynamic modeling of behavior change

We consider a conceptual and quantitative modeling approach for investigating dynamic behavior change. While the approach is applicable to behavior change in eating disorders, smoking, substance abuse and other behavioral disorders, here we present our novel dynamical systems modeling approach to understand the processes governing an individual’s behavior in the context of problem drinking. Recent advances in technology have resulted in large intensive longitudinal data sets which are particularly well suited for study within such frameworks. However, the lack of previous work in this area (specifically, on the interand intra-personal factors governing drinking behavior of individuals) renders this a daunting and unique challenge. As a result, issues which are typically routine in mathematical modeling require considerable effort such as the determination of key quantities of interest, and the timescale on which to represent them. We discuss the construction of an initial mathematical model for two starkly distinct individuals and make a case for the potential for such efforts to help in understanding the underlying mechanisms responsible for behavior change in problem drinkers. AMS Subject Classification: 91E10, 34K29, 62G10.

Generalized Sensitivity Analysis for Delay Differential Equations

We present theoretical foundations for traditional sensitivity and generalized sensitivity functions for a general class of nonlinear delay differential equations. Included are theoretical results for sensitivity with respect to the delays. A brief summary of previous results along with several illustrative computational examples are also given.

Using inverse problem methods with surveillance data in pneumococcal vaccination

The design and evaluation of epidemiological control strategies is central to public health policy. While inverse problem methods are routinely used in many applications, this remains an area in which their use is relatively rare, although their potential impact is great. We describe methods particularly relevant to epidemiological modeling at the population level. These methods are then applied to the study of pneumococcal vaccination strategies as a relevant example which poses many challenges common to other infectious diseases. We demonstrate that relevant yet typically unknown parameters may be estimated, and show that a calibrated model may used to assess implemented vaccine policies through the estimation of parameters if vaccine history is recorded along with infection and colonization information. Finally, we show how one might determine an appropriate level of refinement or aggregation in the age-structured model given age-stratified observations. These results illustrate ways in which the collection and analysis of surveillance data can be improved using inverse problem methods.