Kalman filter parameter estimation for a nonlinear diffusion model of epithelial cell migration using stochastic collocation and the Karhunen-Loeve expansion.

Several Kalman filter algorithms are presented for data assimilation and parameter estimation for a nonlinear diffusion model of epithelial cell migration. These include the ensemble Kalman filter with Monte Carlo sampling and a stochastic collocation (SC) Kalman filter with structured sampling. Further, two types of noise are considered -uncorrelated noise resulting in one stochastic dimension for each element of the spatial grid and correlated noise parameterized by the Karhunen-Loeve (KL) expansion resulting in one stochastic dimension for each KL term. The efficiency and accuracy of the four methods are investigated for two cases with synthetic data with and without noise, as well as data from a laboratory experiment. While it is observed that all algorithms perform reasonably well in matching the target solution and estimating the diffusion coefficient and the growth rate, it is illustrated that the algorithms that employ SC and KL expansion are computationally more efficient, as they require fewer ensemble members for comparable accuracy. In the case of SC methods, this is due to improved approximation in stochastic space compared to Monte Carlo sampling. In the case of KL methods, the parameterization of the noise results in a stochastic space of smaller dimension. The most efficient method is the one combining SC and KL expansion.

A three-dimensional mathematical and computational model of necrotizing enterocolitis.

Necrotizing enterocolitis (NEC) is a severe disease that affects the gastrointestinal (GI) tract of premature infants. Different areas of NEC research have often been isolated from one another and progress on the role of the inflammatory response in NEC, on the dynamics of epithelial layer healing, and on the positive effects of breast feeding have not been synthesized to produce a more integrated understanding of the pathogenesis of NEC. We seek to synthesize these areas of research by creating a mathematical model that incorporates the current knowledge on these aspects. Unlike previous models that are based on ordinary differential equations, our mathematical model takes into account not only transient effects but also spatial effects. A system of nonlinear transient partial differential equations is solved numerically using cell-centered finite differences and an explicit Euler method. The model is used to track the evolution of a prescribed initial injured area in the intestinal wall. It is able to produce pathophysiologically realistic results; decreasing the initial severity of the injury in the system and introducing breast feeding to the system both lead to healthier overall simulations, and only a small fraction of epithelial injuries lead to full-blown NEC. In addition, in the model, changing the initial shape of the injured area can significantly alter the overall outcome of a simulation. This finding suggests that taking into account spatial effects may be important in assessing the outcome for a given NEC patient. This model can provide a platform with which to test competing hypotheses regarding pathological mechanisms of inflammation in NEC, suggest experimental approaches by which to clarify pathogenic drivers of NEC, and may be used to derive potential intervention strategies.

Mathematical modeling in necrotizing enterocolitis--a new look at an ongoing problem.

Necrotizing enterocolitis (NEC) is the most common and lethal disease that affects the gastrointestinal (GI) tract of the premature infant. The etiology of NEC remains undefined. The only consistent epidemiological precursors for NEC are prematurity and enteral alimentation. Various inflammatory mediators, including tumor necrosis factor (TNF)-a, interleukin (IL)-1, IL-6, IL-8, IL-10, IL-18, platelet-activating factor (PAF), and nitric oxide (NO) have been implicated in the pathogenesis of NEC, but the kinetics and role of these agents are ill-defined. Currently, there are no biomarker predictors of NEC risk and severity. Sera or tissue from early time points in the development of the disease may help delineate early inflammatory events that predispose an individual to NEC, thus providing an interventional opportunity. We suggest that the lack of diagnostic and therapeutic modalities for NEC are due to the absence of a systems view of the disease, which in turn is hindered by a lack of sensitive physiological measurements that predict perturbations in the intestinal tissue compartment and an inability to reliably test serial samples for the presence of inflammatory mediators in small volumes and in a high-throughput manner. Computational modeling is a useful tool in the study of complex systems such as the inflammatory process. Computation models provide an "existence proof" for a given mechanism, uncover subtle inconsistencies between the underlying hypotheses and quantitative data, and force one to ask how much is known. We suggest that a properly validated and calibrated mathematical model of inflammation and its pathologic consequences in NEC will be useful for predicting the physiologic and biologic response in infants suffering from the disease.