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, 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.

Keywords: Deconvolution; Distributed parameter systems; Linear semigroups of operators; Population model; Random abstract parabolic systems; System identification; Transdermal alcohol biosensor.

Figure 7.1:

Alcohol Biosensor Devices: Left panel: The Giner WrisTAS. Right panel: The AMS SCRAM.

Figure 7.2:

Left panel: BrAC and TAC measurements for Example 7.1. Right panel: Optimal pdf obtained using the data from drinking episodes 1, 2, 6, 7, and 11 as the population or training dataset.

Figure 7.3:

Example 7.1 Results. Top row: BrAC estimates for the training datasets assuming the input depends on both time and the random parameters, Bottom row: BrAC estimates for the training datasets assuming the input depends only on time.

Figure 7.4:

Example 7.1 Results. Left Panel: Top row: BrAC estimates for the cross-validation datasets assuming the input, $\mathcal{u}$, depends on both time, t, and the random parameters, $\mathcal{q}$; Bottom row: BrAC estimates for the training datasets assuming the input, u, depends only on time, t. Right Panel: Upper Left: Expected value of the impulse response function or convolution kernel together with 75% credible intervals for population model in Example 7.1, Upper Right: Impulse response function or convolution kernel for population model in Example 7.1 as a function of q = (q₁,q₂), Lower Left: Estimated BrAC, or input signal, $\mathcal{u}$ at time when the expected value is at its peak as a function of q = (q₁,q₂) for drinking episode 1 in Example 7.1, Lower Right: Probability density function for the estimated BrAC, or input signal, $\mathcal{u}$ at time when the expected value is at its peak for drinking episode 1 in Example 7.1. The points marked with red dots in the q₁,q₂ plane in the plots in the upper right and lower left corners of the right panel are the samples from the bivariate normal distribution that were used to compute the 75% credible bands.