Inverting the diffusion-convection equation for gas desorption through an homogeneous membrane by Kalman filtering

This paper presents a dynamic transport model of a gaseous compound such as carbon dioxide based on the diffusion-convection through a three layered media composed of: a liquid medium (blood), a membrane (skin), a gaseous medium (air). The objective is to estimate the signal defined by the time variations in the concentration of the gaseous compound dissolved in the liquid medium based solely on the measurement signal defined by the time variations of the concentration of the gaseous compound in the gaseous medium. This dynamic model makes it possible to formulate the direct transport model in the form of a Markovian model with hidden states in order to generate synthetic data. We propose to implement a Kalman filter to calculate from the noisy observed variables, the hidden variables of the model, and in particular the concentration of the gaseous compound in the liquid medium. The challenge is to model the temporal evolution of a concentration profile as a function of time and depth taking into account the heterogeneity of the diffusion coefficients and the partition coefficients associated with the three media considered. The objective of this time recursive processing is to design an algorithm, which can be carried out on an embedded processor, taking into account the constraints of limited computing capacity. The application we are dealing with concerns the transcutaneous measurement of blood carbon dioxide in the forearm using an autonomous wristband-type worn device, in particular for monitoring respiratory diseases at home[1], [2].