Approximation methods for hybrid diffusion systems with state-dependent switching processes : numerical algorithms and existence and uniqueness of solutions

By focusing on hybrid diffusions in which continuous dynamics and discrete events coexist, this work is concerned with approximation of solutions for hybrid stochastic differential equations with a state-dependent switching process. Iterative algorithms are developed. The continuous-state dependent switching process presents added difficulties in analyzing the numerical procedures. Weak convergence of the algorithms is established by a martingale problem formulation first. This weak convergence result is then used as a bridge to obtain strong convergence. In this process, the existence and uniqueness of the solution of the switching diffusions with continuous-state-dependent switching are obtained. Different from the existing results of solutions of stochastic differential equations in which the Picard iterations are utilized, Euler's numerical schemes are considered here. Moreover, decreasing stepsize algorithms together with their weak convergence are given. Numerical experiments are also provided for demonstration

Comparison of Image Registration Based Measures of Regional Lung Ventilation from Dynamic Spiral CT with Xe-CT

Purpose: Regional lung volume change as a function of lung inflation serves as an index of parenchymal and airway status as well as an index of regional ventilation and can be used to detect pathologic changes over time. In this article, we propose a new regional measure of lung mechanics --- the specific air volume change by corrected Jacobian. Methods: 4DCT and Xe-CT data sets from four adult sheep are used in this study. Nonlinear, 3D image registration is applied to register an image acquired near end inspiration to an image acquired near end expiration. Approximately 200 annotated anatomical points are used as landmarks to evaluate registration accuracy. Three different registration-based measures of regional lung mechanics are derived and compared: the specific air volume change calculated from the Jacobian (SAJ); the specific air volume change calculated by the corrected Jacobian (SACJ); and the specific air volume change by intensity change (SAI). Results: After registration, the mean registration error is on the order of 1 mm. For cubical ROIs in cubes with size 20 mm $\times$ 20 mm $\times$ 20 mm, the SAJ and SACJ measures show significantly higher correlation (linear regression, average $r^2=0.75$ and $r^2=0.82$) with the Xe-CT based measure of specific ventilation (sV) than the SAI measure. For ROIs in slabs along the ventral-dorsal vertical direction with size of 150 mm $\times$ 8 mm $\times$ 40 mm, the SAJ, SACJ, and SAI all show high correlation (linear regression, average $r^2=0.88$, $r^2=0.92$ and $r^2=0.87$) with the Xe-CT based sV without significant differences when comparing between the three methods. Conclusion: Given a deformation field by an image registration algorithm, significant differences between the SAJ, SACJ, and SAI measures were found at a regional level compared to the Xe-CT sV in four sheep that were studied