Computation of nonparametric, mixed effects, maximum likelihood, biosensor data based-estimators for the distributions of random parameters in an abstract parabolic model for the transdermal transport of alcohol

Lernik Asserian; Susan E. Luczak; I. G. Rosen; Lernik Asserian; Susan E. Luczak; I. G. Rosen

doi:10.3934/mbe.2023900

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 11: 20345-20377. doi: 10.3934/mbe.2023900

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Computation of nonparametric, mixed effects, maximum likelihood, biosensor data based-estimators for the distributions of random parameters in an abstract parabolic model for the transdermal transport of alcohol

1.
Department of Mathematics, Stanford University, Stanford, CA 94305, USA
2.
Department of Psychology, University of Southern California, Los Angeles, CA 90089, USA
3.
Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA

Received: 30 June 2023 Revised: 13 October 2023 Accepted: 22 October 2023 Published: 09 November 2023

The existence and consistency of a maximum likelihood estimator for the joint probability distribution of random parameters in discrete-time abstract parabolic systems was established by taking a nonparametric approach in the context of a mixed effects statistical model using a Prohorov metric framework on a set of feasible measures. A theoretical convergence result for a finite dimensional approximation scheme for computing the maximum likelihood estimator was also established and the efficacy of the approach was demonstrated by applying the scheme to the transdermal transport of alcohol modeled by a random parabolic partial differential equation (PDE). Numerical studies included show that the maximum likelihood estimator is statistically consistent, demonstrated by the convergence of the estimated distribution to the "true" distribution in an example involving simulated data. The algorithm developed was then applied to two datasets collected using two different transdermal alcohol biosensors. Using the leave-one-out cross-validation (LOOCV) method, we found an estimate for the distribution of the random parameters based on a training set. The input from a test drinking episode was then used to quantify the uncertainty propagated from the random parameters to the output of the model in the form of a $ 95% $ error band surrounding the estimated output signal.
Citation: Lernik Asserian, Susan E. Luczak, I. G. Rosen. Computation of nonparametric, mixed effects, maximum likelihood, biosensor data based-estimators for the distributions of random parameters in an abstract parabolic model for the transdermal transport of alcohol[J]. Mathematical Biosciences and Engineering, 2023, 20(11): 20345-20377. doi: 10.3934/mbe.2023900

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Abstract

The existence and consistency of a maximum likelihood estimator for the joint probability distribution of random parameters in discrete-time abstract parabolic systems was established by taking a nonparametric approach in the context of a mixed effects statistical model using a Prohorov metric framework on a set of feasible measures. A theoretical convergence result for a finite dimensional approximation scheme for computing the maximum likelihood estimator was also established and the efficacy of the approach was demonstrated by applying the scheme to the transdermal transport of alcohol modeled by a random parabolic partial differential equation (PDE). Numerical studies included show that the maximum likelihood estimator is statistically consistent, demonstrated by the convergence of the estimated distribution to the "true" distribution in an example involving simulated data. The algorithm developed was then applied to two datasets collected using two different transdermal alcohol biosensors. Using the leave-one-out cross-validation (LOOCV) method, we found an estimate for the distribution of the random parameters based on a training set. The input from a test drinking episode was then used to quantify the uncertainty propagated from the random parameters to the output of the model in the form of a $ 95% $ error band surrounding the estimated output signal.

References

[1]	D. A. Labianca, The chemical basis of the Breathalyzer: A critical analysis, J. Chem. Educ., 67 (1990), 259–261. https://doi.org/10.1021/ed067p259 doi: 10.1021/ed067p259
[2]	J. T. Sakai, S. K. Mikulich-Gilbertson, R. J. Long, T. J. Crowley, Validity of transdermal alcohol monitoring: fixed and self-regulated dosing, Alcohol.: Clin. Exp. Res., 30 (2006), 26–33. https://doi.org/10.1111/j.1530-0277.2006.00004.x doi: 10.1111/j.1530-0277.2006.00004.x
[3]	R. M. Swift, Transdermal alcohol measurement for estimation of blood alcohol concentration, Alcohol.: Clin. Exp. Res., 24 (2000), 422–423.
[4]	P. R. Marques, A. S. McKnight, Field and laboratory alcohol detection with 2 types of transdermal devices, Alcohol.: Clin. Exp. Res., 33 (2009), 703–711. https://doi.org/10.1111/j.1530-0277.2008.00887.x doi: 10.1111/j.1530-0277.2008.00887.x
[5]	H. T. Banks, K. Ito, Approximation in LQR problems for infinite dimensional systems with unbounded input operators, J. Math. Syst. Estim. Control, 7 (1997), 1–34.
[6]	H. T. Banks, K. Kunisch, Estimation Techniques for Distributed Parameter Systems, Birkhauser, Boston, (1989).
[7]	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. https://doi.org/10.3934/mbe.2016023 doi: 10.3934/mbe.2016023
[8]	M. A. 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.
[9]	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.
[10]	W. F. Smith, J. Hashemi, F. Presuel-Moreno, Foundations of Materials Science and Engineering, 3$^{rd}$ edition, McGraw-Hill, New York, (2004).
[11]	M. Allayioti, C. Oszkinat, E. Saldich, L. Goldstein, S. E. Luczak, C. Wang, et al., Parametric and non-parametric estimation of a random diffusion equation-based population model for deconvolving blood/breath alcohol concentration from transdermal alcohol biosensor data with uncertainty quantification, in American Control Conference (ACC), (2023). https://doi.org/10.23919/ACC55779.2023.10156287
[12]	K. Hawekotte, S. E. Luczak, I. G. Rosen, A Bayesian approach to quantifying uncertainty in transport model parameters for, and breath alcohol concentration deconvolved from, biosensor measured transdermal alcohol level, Math. Biosci. Eng., 18 (2021), 6739–6770.
[13]	H. Liu, L. Goldstein, S. E. Luczak, I. G. Rosen, Confidence bands for evolution systems described by parameter-dependent analytic semigroups, in SIAM Conference on Control and its Applications, (2023). https://doi.org/10.1137/1.9781611977745.17
[14]	H. Liu, L. Goldstein, S. E. Luczak, I. G. Rosen, Delta-method induced confidence bands for a parameter-dependent evolution system with application to transdermal alcohol concentration monitoring, in Conference on Decision and Control, (2023).
[15]	M. Sirlanci, S. E. Luczak, C. E. Fairbairn, D. Kang, R. Pan, X. Yu, et al., Estimating the distribution of random parameters in a diffusion equation forward model for a transdermal alcohol biosensor, Automatica, 106 (2019), 101–109. https://doi.org/10.1016/j.automatica.2019.04.026 doi: 10.1016/j.automatica.2019.04.026
[16]	M. Sirlanci, S. E. Luczak, I. G. Rosen, Approximation and convergence in the estimation of random parameters in linear holomorphic semigroups generated by regularly dissipative operators, in American Control Conference (ACC), (2017), 3171–3176. https://doi.org/10.23919/ACC.2017.7963435
[17]	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), 287–329. https://doi.org/10.12732/caa.v23i2.4 doi: 10.12732/caa.v23i2.4
[18]	C. Oszkinat, T. Shao, C. Wang, I. G. Rosen, A. D. Rosen, E. Saldich, et al., Estimation and uncertainty quantification via forward and inverse filtering for a covariate-dependent, physics-informed, hidden Markov model, Inverse Probl., 38 (2022). https://doi.org/10.1088/1361-6420/ac5ac7 doi: 10.1088/1361-6420/ac5ac7
[19]	C. Oszkinat, S. E. Luczak, I. G. Rosen, Uncertainty quantification in estimating blood alcohol concentration from transdermal alcohol level with physics-informed neural networks, IEEE Trans. Neural Networks Learn. Syst., 34 (2023), 8094–8101. https://doi.org/10.1109/tnnls.2022.3140726 doi: 10.1109/tnnls.2022.3140726
[20]	C. Oszkinat, S. E. Luczak, I. G. Rosen, An abstract parabolic system-based physics-informed long short-term memory network for estimating breath alcohol concentration from transdermal alcohol biosensor data, Neural Comput. Appl., 34 (2022), 1–19. https://doi.org/10.1007/s00521-022-07505-w doi: 10.1007/s00521-022-07505-w
[21]	H. T. Banks, W. C. Thompson, Least Squares Estimation of Probability Measures in the Prohorov Metric Framework, Technical report, (2012).
[22]	H. T. Banks, K. B. Flores, I. G. Rosen, E. M. Rutter, M. Sirlanci, W. C. Thompson, The Prohorov metric framework and aggregate data inverse problems for random PDEs, Commun. Appl. Anal., 22 (2018), 415–446.
[23]	M. Davidian, D. Giltinan, Nonlinear Models for Repeated Measurement Data, Chapman and Hall, New York, (1995).
[24]	M. Davidian, D. M. Giltinan, Nonlinear models for repeated measurement data: An overview and update, Agric. Biol. Environ. Stat., 8 (2003), 387–419. https://doi.org/10.1198/1085711032697 doi: 10.1198/1085711032697
[25]	E. Demidenko, Mixed Models, Theory and Applications, 2$^{nd}$ edition, John Wiley and Sons, Hoboken, (2013).
[26]	M. Lovern, M. Sargentini-Maier, C. Otoul, J. Watelet, Population pharmacokinetic and pharmacodynamic analysis in allergic diseases, Drug Metab. Rev., 41 (2009), 475–485. https://doi.org/10.1080/10837450902891543 doi: 10.1080/10837450902891543
[27]	R. Tatarinova, M. Neely, J. Bartroff, M. van Guilder, W. Yamada, D. Bayard, et al., Two general methods for population pharmacokinetic modeling: non-parametric adaptive grid and non-parametric Bayesian, J. Pharmacokinet. Pharmacodyn., 40 (2013), 189–199. https://doi.org/10.1007/s10928-013-9302-8 doi: 10.1007/s10928-013-9302-8
[28]	J. Li, S. E. Luczak, I. G. Rosen, Comparing a distributed parameter model-based system identification technique with more conventional methods for inverse problems, J. Inverse Ill-Posed Probl., 27 (2019), 703–717. https://doi.org/10.1515/jiip-2018-0006 doi: 10.1515/jiip-2018-0006
[29]	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. https://doi.org/10.1088/1361-6420/aae791 doi: 10.1088/1361-6420/aae791
[30]	M. Yao, S. E. Luczak, I. G. Rosen, Tracking and blind deconvolution of blood alcohol concentration from transdermal alcohol biosensor data: A population model-based LQG approach in Hilbert space, Automatica, 147 (2023). https://doi.org/10.1016/j.automatica.2022.110699 doi: 10.1016/j.automatica.2022.110699
[31]	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), 373–81. https://doi.org/10.1037/a0029021 doi: 10.1037/a0029021
[32]	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. https://doi.org/10.1016/j.drugalcdep.2014.12.021 doi: 10.1016/j.drugalcdep.2014.12.021
[33]	C. E. Fairbairn, D. Kang, N. Bosch, Using machine learning for real-time BAC estimation from a new-generation transdermal biosensor in the laboratory, Drug Alcohol Depend., 216 (2021), 108205. https://doi.org/10.1016/j.drugalcdep.2020.108205 doi: 10.1016/j.drugalcdep.2020.108205
[34]	B. Lindsay, The geometry of mixture likelihoods: a general theory, Ann. Stat., 11 (1983), 86–94. https://doi.org/10.1214/aos/1176346059 doi: 10.1214/aos/1176346059
[35]	A. Mallet, A maximum likelihood estimation method for random coefficient regression models, Biometrika, 73 (1986), 645–656. https://doi.org/10.2307/2336529 doi: 10.2307/2336529
[36]	J. Kiefer, J. Wolfowitz, Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters, Ann. Math. Stat., 27 (1956), 887–906. https://doi.org/10.1214/aoms/1177728066 doi: 10.1214/aoms/1177728066
[37]	H. Tanabe, Equations of Evolution (Monographs and Studies in Mathematics), Pitman Publishing, (1979).
[38]	J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer Berlin, Heidelberg, (1971).
[39]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, (1983).
[40]	H. T. Banks, K. Kunisch, The linear regulator problem for parabolic systems, SIAM J. Control Optim., 22 (1984), 684–698. https://doi.org/10.1137/0322043 doi: 10.1137/0322043
[41]	H. T. Banks, K. Ito, A Unified Framework for Approximation in Inverse Problems for Distributed Parameter Systems, NASA. Hampton, VA. Technical Reports NASA-CR-181621, (1988).
[42]	R. A. Adams, J. J. F. Fournier, Sobolev Spaces, Elsevier, (2003).
[43]	M. H. Schultz, Spline Analysis, Prentice-Hall, (1973).
[44]	S. E. Luczak, I. G. Rosen, T. L. Wall, Development of a real-time repeated-measures assessment protocol to capture change over the course of drinking episodes, Alcohol Alcohol., 50 (2015), 1–8. https://doi.org/10.1093/alcalc/agu100 doi: 10.1093/alcalc/agu100
[45]	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). https://doi.org/10.1111/acer.14358 doi: 10.1111/acer.14358
[46]	A. Kryshchenko, M. Sirlanci, B. Vader, Nonparametric estimation of blood alcohol concentration from transdermal alcohol measurements using alcohol biosensor devices, Adv. Data Sci. Adapt. Anal., 26 (2021), 329–360. https://doi.org/10.1007/978-3-030-79891-8_13 doi: 10.1007/978-3-030-79891-8_13

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