Nonlinear Filtering Methodologies for Parameter Estimation and Uncertainty Quantification in Noisy, Complex, Biological Systems

A model is a set of equations constructed to represent the interactions of various variables within a biological or physical process. These mathematical models are used to obtain a more thorough understanding of a system or to gain information not easily obtained through other means. Measurements of system components are frequently collected and are used to validate the model through the solution of the inverse problem. The inverse problem is defined as calculating the optimal parameter values to obtain the best possible fit of the model the data. However, as the systems of interest become more complex, the solution to the inverse problem becomes increasingly difficult.;A common method to solve the inverse problem is to use a nonlinear least squares (NLS) approach which aims to minimize the residual, the difference between the data and the model. However, this methodology presents a certain set of assumptions which may not hold for complicated biological models. An alternate method addressed in this thesis is the use of Kalman filtering. The Kalman filter is a recursive algorithm that optimally combines the uncertainties in the model and data to yield an improved final estimate. Carrying out the inverse problem utilizing this methodology has a number of advantages and has shown favorable results.;One area where these methodologies have proven fruitful is in cardiovascular modeling. The cardiovascular system is a branching network of vessels which transports blood and nutrients throughout the body while removing wastes. At the center of this process is the heart, which is the mechanism that facilitates transport through pumping. The heart and vasculature are controlled through the autonomic nervous system. As the cardiovascular system is so important to homeostasis, obtaining measurements on immediate variables of interest is difficult. Mathematical modeling is one way to gather more understanding. Using a simplified model of the cardiovascular system and the autonomic nervous system, the Kalman filter is used to illustrate their interplay. The advantages are shown over a NLS approach due to the ability to take into account modeling errors.;For many problems, measurements are collected for a multitude of individuals, representing a population. A standard approach is to fit each individual using NLS and then do statistical analysis on the individual parameters. However, this has been known to introduce bias to the final estimates. An improved method is introduced that accounts for inter- and intra-individual variability called nonlinear mixed effects. Using the Kalman filter within this framework allows the estimation of the population parameters, along with model misspecification, and time varying parameters. Using a population pharmacokinetic study, nonlinear mixed effects was carried out utilizing the Kalman filter referred to as stochastic nonlinear mixed effects. The results of this highlight the utility of the stochastic nonlinear mixed effect method through refinement of noisy model components.