### Mathematical Biosciences and Engineering

2021, Issue 5: 6739-6770. doi: 10.3934/mbe.2021335
Research article

# Deconvolving breath alcohol concentration from biosensor measured transdermal alcohol level under uncertainty: a Bayesian approach

• Received: 22 May 2021 Accepted: 03 August 2021 Published: 10 August 2021
• The posterior distribution (PD) of random parameters in a distributed parameter-based population model for biosensor measured transdermal alcohol is estimated. The output of the model is transdermal alcohol concentration (TAC), which, via linear semigroup theory can be expressed as the convolution of blood or breath alcohol concentration (BAC or BrAC) with a filter that depends on the individual participant or subject, the biosensor hardware itself, and environmental conditions, all of which can be considered to be random under the presented framework. The distribution of the input to the model, the BAC or BrAC, is also sequentially estimated. A Bayesian approach is used to estimate the PD of the parameters conditioned on the population sample's measured BrAC and TAC. We then use the PD for the parameters together with a weak form of the forward random diffusion model to deconvolve an individual subject's BrAC conditioned on their measured TAC. Priors for the model are obtained from simultaneous temporal population observations of BrAC and TAC via deterministic or statistical methods. The requisite computations require finite dimensional approximation of the underlying state equation, which is achieved through standard finite element (i.e., Galerkin) techniques. The posteriors yield credible regions, which remove the need to calibrate the model to every individual, every sensor, and various environmental conditions. Consistency of the Bayesian estimators and convergence in distribution of the PDs computed based on the finite element model to those based on the underlying infinite dimensional model are established. Results of human subject data-based numerical studies demonstrating the efficacy of the approach are presented and discussed.

Citation: Keenan Hawekotte, Susan E. Luczak, I. G. Rosen. Deconvolving breath alcohol concentration from biosensor measured transdermal alcohol level under uncertainty: a Bayesian approach[J]. Mathematical Biosciences and Engineering, 2021, 18(5): 6739-6770. doi: 10.3934/mbe.2021335

### Related Papers:

• The posterior distribution (PD) of random parameters in a distributed parameter-based population model for biosensor measured transdermal alcohol is estimated. The output of the model is transdermal alcohol concentration (TAC), which, via linear semigroup theory can be expressed as the convolution of blood or breath alcohol concentration (BAC or BrAC) with a filter that depends on the individual participant or subject, the biosensor hardware itself, and environmental conditions, all of which can be considered to be random under the presented framework. The distribution of the input to the model, the BAC or BrAC, is also sequentially estimated. A Bayesian approach is used to estimate the PD of the parameters conditioned on the population sample's measured BrAC and TAC. We then use the PD for the parameters together with a weak form of the forward random diffusion model to deconvolve an individual subject's BrAC conditioned on their measured TAC. Priors for the model are obtained from simultaneous temporal population observations of BrAC and TAC via deterministic or statistical methods. The requisite computations require finite dimensional approximation of the underlying state equation, which is achieved through standard finite element (i.e., Galerkin) techniques. The posteriors yield credible regions, which remove the need to calibrate the model to every individual, every sensor, and various environmental conditions. Consistency of the Bayesian estimators and convergence in distribution of the PDs computed based on the finite element model to those based on the underlying infinite dimensional model are established. Results of human subject data-based numerical studies demonstrating the efficacy of the approach are presented and discussed.

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