Deconvolving the input to random abstract parabolic systems: a population model-based approach to estimating blood/breath alcohol concentration from transdermal alcohol biosensor data

Abstract

The distribution of random parameters in, and the input signal to, a distributed parameter model with unbounded input and output operators for the transdermal transport of ethanol are estimated. The model takes the form of a diffusion equation with the input, which is on the boundary of the domain, being the blood or breath alcohol concentration (BAC/BrAC), and the output, also on the boundary, arXiv:1807.05088v1 [math.OC] 13 Jul 2018 being the transdermal alcohol concentration (TAC). Our approach is based on the reformulation of the underlying dynamical system in such a way that the random parameters are treated as additional spatial 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 a functional diffusivity in a multi-dimensional diffusion equation. The resulting system is referred to as a population model, and well-established finite dimensional approximation schemes, functional analytic based convergence arguments, optimization techniques, and computational methods can be used to fit it to population data and to analyze the resulting fit. Once the forward population model has been identified or trained based on a sample from the population, the resulting distribution can then be used to deconvolve the BAC/BrAC input signal from the biosensor observed TAC output signal formulated as either a quadratic programming or linear quadratic tracking problem. In addition, our approach allows for the direct computation of corresponding credible bands without simulation. We use our technique to estimate bivariate normal distributions and deconvolve BAC/BrAC from TAC based on data from a population that consists of multiple drinking episodes from a single subject and a population consisting of single drinking episodes from multiple subjects.

Estimating the distribution of random parameters in a diffusion equation forward model for a transdermal alcohol biosensor

“…Once the distribution of the random parameters has been estimated, this takes the form of a deconvolution problem. In [20] we use the results presented here together with the framework in [9] and [17] to do just that. We obtain an estimate of the input along with error bars or credible bands.…”

“…For a number of reasons, the approach we take here, and in particular our population model as defined in Section 3 below, is especially well suited for our estimation problem as given in (1.6) with the underlying dynamics (1.1), (1.2) being described by a random PDE. Indeed, 1) it does not require repeated simulation, 2) it takes particular advantage of the underlying parabolic structure of the model's state equation, 3) it lends itself extremely well to functional analytic arguments for convergence of the estimators based on finite dimensional approximation, the central focus of this study, 4) based on our working hypothesis concerning our data as stated in the previous paragraph and the statistical model given in (1.5), the output of the population model is precisely what is required to evaluate the naïve pooled data based performance index, ( ; ) given in (1.6), and 5) it is especially well suited for deconvolving an estimate of BrAC from TAC which is our ultimate goal [20].…”

“…In addition, in our treatment here, we are only concerned with the fitting of the random parameters in the forward model. The inverse problem involving the deconvolution of the BAC/BrAC from the TAC signal once the forward model with random elements has been fit is treated elsewhere [20].…”

Effects of stomach content on the breath alcohol concentration‐transdermal alcohol concentration relationship

“…Our research team has created the BrAC Estimator software, which utilises physics‐ and physiological‐based mathematical models we have developed to translate TAC into eBrAC [4,8,14‐20]. We assume the dynamics of the process described by the model are common to every person.…”

“…There are two key parameters in the model— q 1 , the rate at which alcohol diffuses through the skin, and q 2 , the net rate at which alcohol enters and leaves the skin and is processed by the TAC device. The optimal parameter values for each person‐device pairing (denoted q 1 * and q 2 *) can be obtained during an alcohol administration session where an individual wears a particular device and is monitored for both TAC and BrAC [8,14,15] or by using population‐based parameter estimates [4,17–19]. Once obtained, the optimal parameter values can be used to model all drinking episodes for that person‐device pair.…”