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.

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

“…It is possible to convert the system (2.1)-(2.5) into an equivalent one that has only two dimensionless parameters instead of the five parameters, D, L, α, β, and γ. These new parameters will be denoted by the vector q = [q 1 , q 2 ] (see [27]) and the system is stated as…”

“…(The λ 0 -shifted form a(q; ·, ·) + λ 0 | · | 2 coercive for some λ 0 ∈ R is fine as well.) We also assume that for each ϕ, ψ ∈ V , the function q → a(q; ϕ, ψ) is measurable with respect to all measures π(θ), in some family of measures parameterized by a vector of parameters θ, where θ ∈ Θ ⊂ R r for some positive integer r. This family of measures and its parameterization will be made more precise below (see also [27]).…”

“…The fact that the input u and output y in the system (2.6)-(2.10) are on the boundary of the domain, requires that some care be exercised in the definitions of the input and output operators in our formulation of the problem (see [27]). Let b(q), c(q) denote elements in V * with the respective maps q →< b(q), ψ > V * ,V and q →< c(q), ψ > V * ,V measurable on Q with respect to all measures π(θ) for ψ ∈ V , where < ·, · > V * ,V again denotes the duality pairing between V and V * .…”

“…In a series of recent studies (e.g. [27,28,29]), we have investigated the construction of a population model, which uses our first principles models to describe the dynamics common to the entire population (i.e., all individuals, devices, and environmental conditions) and then to attribute all un-modeled sources of uncertainty observed in individual data (e.g., variations in human physiology, biosensor hardware, environmental conditions) to random effects. We assume that there is a single underlying mathematical framework that describes the system dynamics that are common to all individuals, environments, and devices in the population (e.g., the physics-based model for the transport of alcohol from the blood, through the skin, and measurement by the sensor), but that individuals in the population exhibit variation in the model parameters (e.g., the rate at which the alcohol is transported, evaporates, etc.).…”

“…In [29] we developed the abstract approximation and convergence theory for fitting or training the population model. In [27] we applied the theory developed in [29] to the alcohol biosensor problem discussed above. In this paper, we are concerned with using the fit population model to deconvolve or an estimate for the input to the model, i.e.…”