Inverse Modeling of Diffusive Processes Using Instantaneous Measurements and Stochastic Differential Equations

Abstract

<p>Modeling the .dispersion of diffusive sources and signal processing algorithms for its prediction is an important issue in many applications such as cardiac activation, drug delivery, and environmental monitoring. This work focuses on the development of computationally efficient algorithms for modeling diffusion processes and estimation of their different properties.</p> <p>First, we implement the well known Fick's law of diffusion for localizing and estimating the properties of diffusive sources. Moreover, we propose a new model for the cardiac activation using inhomogeneous reaction-diffusion equations in the presence of diffusivity disorders. We also derive corresponding statistical signal processing algorithms for estimating (localizing) parameters describing these anomalies using ECG/MCG sensor arrays.</p> <p>However, in some applications, such as drug delivery and capillary exchange process, where low-intensity diffusive sources are considered, random effects such as Brownian motion should be accounted for. Hence, we propose a computationally efficient framework for localizing low-intensity diffusive sources using stochastic differential equations. To achieve computational efficiency, we model the dispersion using the Fokker-Planck equation and derive corresponding inverse model and maximum likelihood estimator of source intensity, location and release time. Also, we expand our stochastic model to account for drift and propose an algorithm for the estimation of boundary properties.</p> <p>Finally, we present a novel technique for modeling the exchange process and particle clearance in capillary networks using coupled stochastic- differential and Navier-Stokes equations. Numerical examples are used to demonstrate the applicability of our models.</p>