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
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.
[1] | R. G. Lister, C. Gorenstein, D. Risher-Flowers, H. J. Weingartner, M. J. Eckardt, Dissociation of the acute effects of alcohol on implicit and explicit memory processes, Neuropsychologia, 29 (1991), 1205–1212. doi: 10.1016/0028-3932(91)90034-6 |
[2] | S. W. Brusilow, E. H. Gordes, The Permeability of the Sweat Gland to Nonelectrolytes, Am. J. Dis. Child., 112 (1966), 328–333. |
[3] | M. Gamella, S. Campuzano, J. Manso, G. G. de Rivera, F. López-Colino, A. Reviejo, et al., A novel non-invasive electrochemical biosensing device for in situ determination of the alcohol content in blood by monitoring ethanol in sweat, Anal. Chim. Acta, 806 (2014), 1–7. doi: 10.1016/j.aca.2013.09.020 |
[4] | E. Nyman, A. Palmlöv, The elimination of ethyl alcohol in sweat1, Skandinavisches Archiv Für Physiologie, 74 (1936), 155–159. doi: 10.1111/j.1748-1716.1936.tb01150.x |
[5] | G. Pawan, Physical exercise and alcohol metabolism in man, Nature, 218 (1968), 966–967. doi: 10.1038/218966a0 |
[6] | T. Wade, N. Pai, J. Eisenberg, J. Colford, Do u.s. environmental protection agency water quality guidelines for recreational waters prevent gastrointestinal illness? a systematic review and meta-analysis, Environ. Health Perspect., 111 (2003), 1102–9. doi: 10.1289/ehp.6241 |
[7] | R. M. Swift, Transdermal measurement of alcohol consumption, Addiction, 88 (1993), 1037–1039. doi: 10.1111/j.1360-0443.1993.tb02122.x |
[8] | R. M. Swift, Transdermal alcohol measurement for estimation of blood alcohol concentration, Alcohol. Clin. Exp. Res., 24 (2000), 422–423. doi: 10.1111/j.1530-0277.2000.tb02006.x |
[9] | R. M. Swift, C. S. Martin, L. Swette, A. LaConti, N. Kackley, Studies on a wearable, electronic, transdermal alcohol sensor, Alcohol. Clin. Exp. Res., 16 (1992), 721–725. |
[10] | P. Kriikku, L. Wilhelm, S. Jenckel, J. Rintatalo, J. Hurme, J. Kramer, et al., Comparison of breath-alcohol screening test results with venous blood alcohol concentration in suspected drunken drivers, Forensic Sci. Int., 239 (2014), 57–61. doi: 10.1016/j.forsciint.2014.03.019 |
[11] | E. Schechtman, D. Shinar, An analysis of alcohol breath tests results with portable and desktop breath testers as surrogates of blood alcohol levels, Accid. Anal. Prev., 43 (2011), 2188–2194. doi: 10.1016/j.aap.2011.06.013 |
[12] | A. Jones, L. Andersson, Comparison of ethanol concentrations in venous blood and end-expired breath during a controlled drinking study, Forensic Sci. Int., 132 (2003), 18–25. doi: 10.1016/S0379-0738(02)00417-6 |
[13] | D. A. Labianca, The chemical basis of the breathalyzer: A critical analysis, J. Chem. Educ., 67 (1990), 259–261. doi: 10.1021/ed067p259 |
[14] | A. W. Jones, Determination of liquid/air partition coefficients for dilute solutions of ethanol in water, whole blood, and plasma, J. Anal. Toxicol., 7 (1983), 193–197. doi: 10.1093/jat/7.4.193 |
[15] | Z. Dai, I. G. Rosen, C. Wang, N. P. Barnett, S. E. Luczak, Using drinking data and pharmacokinetic modeling to calibrate transport model and blind deconvolution based data analysis software for transdermal alcohol biosensors, Math. Biosci. Eng., 13 (2016), 911–934. doi: 10.3934/mbe.2016023 |
[16] | D. M. Dougherty, N. E. Charles, A. Acheson, S. John, R. M. Furr, N. Hill-Kapturczak, Comparing the detection of transdermal and breath alcohol concentrations during periods of alcohol consumption ranging from moderate drinking to binge drinking, Exp. Clin. Psychopharmacol., 20 (2012), 373. doi: 10.1037/a0029021 |
[17] | D. M. Dougherty, N. Hill-Kapturczak, Y. Liang, T. E. Karns, S. E. Cates, S. L. Lake, et al., Use of continuous transdermal alcohol monitoring during a contingency management procedure to reduce excessive alcohol use, Drug Alcohol Depend., 142 (2014), 301–306. doi: 10.1016/j.drugalcdep.2014.06.039 |
[18] | D. M. Dougherty, T. E. Karns, J. Mullen, Y. Liang, S. L. Lake, J. D. Roache, et al., Transdermal alcohol concentration data collected during a contingency management program to reduce at-risk drinking, Drug Alcohol Depend., 148 (2015), 77–84. doi: 10.1016/j.drugalcdep.2014.12.021 |
[19] | M. Dumett, I. G. Rosen, J. Sabat, A. Shaman, L. Tempelman, C. Wang, et al., Deconvolving an estimate of breath measured blood alcohol concentration from biosensor collected transdermal ethanol data, Appl. Math. Comput., 196 (2008), 724–743. |
[20] | I. G. Rosen, S. E. Luczak, W. W. Hu, M. Hankin, Discrete-time blind deconvolution for distributed parameter systems with dirichlet boundary input and unbounded output with application to a transdermal alcohol biosensor, in 2013 Proceedings of the Conference on Control and Its Applications, Society for Industrial and Applied Mathematics, 2013,160–167. |
[21] | I. G. Rosen, S. E. Luczak, J. Weiss, Blind deconvolution for distributed parameter systems with unbounded input and output and determining blood alcohol concentration from transdermal biosensor data, Appl. Math. Comput., 231 (2014), 357–376. |
[22] | G. D. Webster, H. C. Gabler, Feasibility of transdermal ethanol sensing for the detection of intoxicated drivers, Annual Proceedings / Association for the Advancement of Automotive Medicine, 51 (2007), 449–464. |
[23] | G. D. Webster, H. C. Gabler, Modeling of transdermal transport of alcohol effect of body mass and gender, Biomed. Sci. Instrum., 44 (2008), 361–366. |
[24] | M. Sirlanci, S. E. Luczak, I. G. Rosen, Estimation of the distribution of random parameters in discrete time abstract parabolic systems with unbounded input and output: Approximation and convergence, Commun. Appl. Anal., 23 (2019), 44. |
[25] | M. Sirlanci, I. G. Rosen, S. E. Luczak, C. E. Fairbairn, K. Bresin, D. Kang, Deconvolving the input to random abstract parabolic systems: a population model-based approach to estimating blood/breath alcohol concentration from transdermal alcohol biosensor data, Inverse Probl., 34 (2018), 125006. doi: 10.1088/1361-6420/aae791 |
[26] | N. Hill-Kapturczak, J. D. Roache, Y. Liang, T. E. Karns, S. E. Cates, D. M. Dougherty, Accounting for sex-related differences in the estimation of breath alcohol concentrations using transdermal alcohol monitoring, Psychopharmacology, 232 (2014), 115–123. |
[27] | T. E. Karns-Wright, J. D. Roache, N. Hill-Kapturczak, Y. Liang, J. Mullen, D. M. Dougherty, Time delays in transdermal alcohol concentrations relative to breath alcohol concentrations, Alcohol Alcohol., 52 (2017), 35–41. doi: 10.1093/alcalc/agw058 |
[28] | M. Yao, S. E. Luczak, I. G. Rosen, Linear quadratic Gaussian control of random abstract parabolic systems, IEEE Contr. Syst. Lett., submitted for publication. |
[29] | R. Adams, J. Fournier, Sobolev Spaces, ISSN, Elsevier Science, 2003. |
[30] | H. Tanabe, Equations of evolution, Monographs and Studies in Mathematics. |
[31] | A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer, 1983. |
[32] | K. J. Hawekotte, Obtaining Breath Alcohol Concentration from Transdermal Alcohol Concentration Using Bayesian Approaches, PhD thesis, University of Southern California, 2021. |
[33] | M. Schultz, Spline Analysis, Prentice-Hall Series in Automatic Computation, Pearson Education, Limited, 1972. |
[34] | H. T. Banks, K. v. Bremen-Ito, U. S. R. Association., I. for Computer Applications in Science and Engineering., A unified framework for approximation in inverse problems for distributed parameter systems, National Aeronautics and Space Administration, Langley Research Center, Institute for Computer Applications in Science and Engineering Hampton, Va, 1988. |
[35] | H. T. Banks, K. Kunisch, Estimation Techniques for Distributed Parameter Systems, Birkhauser Boston, Inc., Secaucus, NJ, USA, 1989. |
[36] | T. Kato, Perturbation theory for linear operators, Grundlehren der mathematischen Wissenschaften, Springer Berlin Heidelberg, 2013. |
[37] | M. Dashti, A. M. Stuart, The Bayesian Approach to Inverse Problems, 311–428, Springer International Publishing, Cham, 2017. |
[38] | T. Bui-Thanh, O. Ghattas, J. Martin, G. Stadler, A computational framework for infinite-dimensional bayesian inverse problems part i: The linearized case, with application to global seismic inversion, SIAM J. Sci. Comput., 35 (2013), A2494–A2523. |
[39] | N. Petra, J. Martin, G. Stadler, O. Ghattas, A computational framework for infinite-dimensional bayesian inverse problems, part II: Stochastic newton MCMC with application to ice sheet flow inverse problems, SIAM J. Sci. Comput., 36 (2014), A1525–A1555. |
[40] | A. M. Stuart, Inverse problems: A bayesian perspective, Acta Numer., 19 (2010), 451–559. doi: 10.1017/S0962492910000061 |
[41] | M. J. Schervish, Theory of statistics, Springer Science & Business Media, 2012. |
[42] | T. Choi, R. V. Ramamoorthi, Remarks on consistency of posterior distributions, vol. Volume 3 of Collections, 170–186, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2008. |
[43] | S. Walker, New approaches to bayesian consistency, Ann. Stat., 32 (2004), 2028–2043. |
[44] | T. Choi, M. Schervish, Posterior consistency in nonparametric regression problems under gaussian process priors, 2004. |
[45] | N. Choudhuri, S. Ghosal, A. Roy, Bayesian estimation of the spectral density of a time series, J. Am. Stat. Assoc., 99 (2004), 1050–1059. doi: 10.1198/016214504000000557 |
[46] | S. Seubert, J. Wade, Fréchet differentiability of parameter-dependent analytic semigroups, J. Math. Anal. Appl., 232 (1999), 119–137. doi: 10.1006/jmaa.1998.6249 |
[47] | W. Cheney, Calculus in Banach Spaces, 115–169, Springer New York, New York, NY, 2001. |
[48] | J. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, 1971. |
[49] | C. J. Gittelson, R. Andreev, C. Schwab, Optimality of adaptive galerkin methods for random parabolic partial differential equations, J. Comput. Appl. Math., 263 (2014), 189–201. doi: 10.1016/j.cam.2013.12.031 |
[50] | C. Schwab, C. J. Gittelson, Sparse tensor discretization of high-dimensional parametric and stochastic PDEs, Acta Numer., 20 (2011), 291–467. doi: 10.1017/S0962492911000055 |
[51] | E. B. Saldich, C. Wang, I. G. Rosen, L. Goldstein, J. Bartroff, R. M. Swift, et al., Obtaining high-resolution multi-biosensor data for modeling transdermal alcohol concentration data, Alcohol. Clin. Exp. Res., 44 (2020), 181A. |
[52] | C. Rasmussen, C. Williams, Gaussian Processes for Machine Learning, Adaptive Computation and Machine Learning, MIT Press, Cambridge, MA, USA, 2006. |