Estimating the Distribution of Random Parameters in a Diffusion Equation Forward Model for a Transdermal Alcohol Biosensor

Abstract

We estimate the distribution of random parameters in a distributed parameter model with unbounded input and output for the transdermal transport of ethanol in humans. The model takes the form of a diffusion equation with the input being the blood alcohol concentration and the output being the transdermal alcohol concentration. Our approach is based on the idea of reformulating the underlying dynamical system in such a way that the random parameters are now treated as additional space variables. When the distribution to be estimated is assumed to be defined in terms of a joint density, estimating the distribution is equivalent to estimating the diffusivity in a multi-dimensional diffusion equation and thus well-established finite dimensional approximation schemes, functional analytic based convergence arguments, optimization techniques, and computational methods may all be employed. We use our technique to estimate a bivariate normal distribution based on data for multiple drinking episodes from a single subject.

Keywords: Biosensor data; Blood alcohol concentration; Distributed parameter systems; Distribution estimation; Random parameters; Transdermal alcohol concentration.

Figures

Figure 1.1.

(Left Panel) The WRISTAS™ 7 transdermal alcohol biosensor, (Right Panel) The Alcohol Monitoring System (AMS) Secure Continuous Alcohol Monitoring (SCRAM) system.

Figure 2.1.

Row-wise from the top left: Panel (a) BrAC and TAC data for the 11 drinking episodes, Panel (b) the optimal pdf, Panels (c), (d), (f), (g), and (i) are the plots of training BrAC and corresponding TAC, resulting fit population model estimated TAC, and 75% credible band; Panels (e) and (h) are the cross validation results.