Abstract
Transdermal alcohol biosensors detect ethanol excreted through the skin via perspiration, collecting transdermal alcohol concentration (TAC) data. Compared to traditional detection methods, this approach offers enhanced accuracy, convenience, and real-time monitoring. Recent advancements have expanded their applications, including intravenously infused alcohol studies aimed at maintaining blood alcohol concentration (BAC) at a specified level for an extended period (clamping) or following a pre-specified intoxication trajectory (tracking), as well as blind deconvolution of BAC or breath alcohol concentration (BrAC) from TAC data. To model ethanol transport for the detection of electrochemical biosensors, a dynamic control system is established. A primary challenge is the inconsistency in TAC-to-BAC/BrAC mapping under varying conditions. To address this, system parameters are treated as random variables within a robust framework. A discrete-time linear quadratic control and tracking framework, coupled with specific linear quadratic Gaussian compensators for random abstract parabolic systems, along with a real-time deconvolution scheme, has been developed. The establishment of this advanced mathematical model and feedback control strategies improves the reliability and accuracy of TAC-to-BAC/BrAC conversion, improving the applicability of transdermal alcohol biosensors in alcohol research and clinical communities.
Keywords
- transdermal alcohol biosensor
- intravenously infused alcohol therapy
- linear quadratic Gaussian compensator
- discrete-time finite-dimensional approximation and convergence
- random abstract parabolic system
1. Introduction
The recent development of the transdermal alcohol biosensor technique enables the measurement of alcohol excreted through the skin
Biosensors are typically designed to look like a digital watch, a portable fitness monitor, or an ankle bracelet. Two examples are shown in Figure 1. The Secure Continuous Remote Alcohol Monitor (SCRAM) [4] by Alcohol Monitoring Systems, worn on the ankle, was developed decades ago and remains the most widely utilized TAC device in both clinical research and the criminal justice system for monitoring abstinence. Another TAC device, the WrisTAS [5] by Giner, Inc., is worn on the wrist and was primarily developed for medical applications. However, it has not yet been adopted in legal settings due to the lack of standardized protocols for detecting tampering. The latest generation of smartwatch-style TAC device, such as Skyn by BACtrack [6], measures alcohol levels through insensible sweat and has been commercially available since 2015. Another emerging device, ION [7] by Milo Sensors, is a tattoo-like patch with embedded electrodes that transmit results

Figure 1.
(Left) Giner, Inc. (Newton, Massachusetts) WristTAS™ 7 and (Right) SCRAM Systems (Alcohol Monitoring Systems, Littleton Colorado) transdermal continuous alcohol monitoring devices.
Compared to traditional alcohol monitoring methods, wearable biosensors offer superior accuracy, convenience, and real-time data collection. Self-reports are often unreliable due to inaccuracies in estimating alcohol intake and individual differences in metabolism, making it difficult to derive consistent indicators. While blood tests provide the most accurate measurements, they are invasive, expensive, and unsuitable for continuous real-time monitoring. Breath analyzers, though more convenient, can be inaccurate if not used properly, such as when users fail to adequately rinse mouths before taking a reading [9], fail to take deep lung breaths, or do not allow enough time for the device chamber to clear, among other issues. In contrast, wearable biosensors enable passive, unobtrusive, and continuous data collection in real-world settings, overcoming traditional method limitations.
However, the practical utility of transdermal alcohol biosensors presents challenges, as they measure TAC rather than blood or breath alcohol concentration (BAC/BrAC), which are widely recognized in research, medicine, and legal contexts as standard indicators of intoxication and can be easily converted into one another using Henry’s law with a single parameter. Unlike breath analyzers, which provide consistent measurements across individuals and conditions [10], TAC data vary significantly due to physiological factors (e.g., skin thickness, blood flow, perspiration), environmental conditions (e.g., temperature, humidity), and sensor variability. In other words, TAC does not consistently map onto BAC or BrAC across individuals, devices, or settings. To establish TAC biosensors as reliable real-time indicators of alcohol levels, particularly for monitoring in alcohol studies, a reliable and standardized conversion method is essential.
Various TAC-to-BAC/BrAC conversion models have been proposed, examining their relationships from various perspectives. These include traditional linear regression [11, 12], tree-based machine learning [13], physics-informed hidden Markov models [14], and neural networks [15, 16]. However, existing models exhibit certain limitations: Some are overly simplistic and fail to capture the complexity of TAC-BAC/BrAC dynamics observed in empirical data; others rely heavily on individual-specific data and testing conditions, limiting their generalizability; and while some employ advanced machine learning techniques, their formula are often generated based on limited subject data, restricting their applicability across diverse populations and environments.
The following discussion explores the conversion of TAC into BAC or BrAC through practical applications of transdermal alcohol biosensors. These applications include regulating intravenous infused alcohol therapy (clamping and tracking studies) [17] and estimating BAC or BrAC from TAC signals [18]. We propose leveraging innovative transdermal alcohol biosensor technology alongside advanced mathematical models and strategies to develop feedback control systems that fully automate alcohol clamping and tracking studies. These studies aim to either maintain a subject’s BAC at a specified level of intoxication for an extended period or ensure the participant’s BAC follows a pre-specified target trajectory. Additionally, we have identified an efficient method for estimating or deconvolving BAC/BrAC from TAC data, significantly enhancing the practicality of wearable transdermal alcohol sensors for clinicians, researchers, and consumers.
We have developed a semi-linear reaction-diffusion population model that comprehensively represents alcohol metabolism in the liver, the transport of ethanol from the bloodstream through the dermal (actively supplied with blood) and epidermal (lacking active blood supply) layers of skin, its excretion
Given that model parameters—represented as coefficients—can vary due to individual differences, sensor variability, and environmental factors such as ambient temperature and humidity, it is crucial to account for this variability in parameter fitting. Rather than calibrating model parameters solely based on data from a single drinking episode that simultaneously collects BrAC and TAC measurements (e.g., [19, 20]), we adopt a more robust approach by treating these model parameters as random variables. These variables are only known by their distribution, which was derived from BrAC/TAC data across multiple drinking episodes and diverse cohorts under varying ambient conditions. To formalize our population model [21, 22, 23] with random parameters in state space, we adopt a recent framework that formulates abstract random parabolic systems in a weak form within appropriately designed Bochner spaces, incorporating random parameters as additional spatial variables [24, 25].
After reformulating the population model as a random abstract parabolic system within a Gelfand triple framework, we can design control and tracking compensators for the clamping, tracking, and estimation problems. This reformulation enables us to establish standard LQG control and tracking problems (see, for example, [26, 27, 28, 29, 30, 31]) in Hilbert space. By applying linear semigroup theory [32, 33] and LQG control and tracking theory in Hilbert space, we can leverage associated finite-dimensional Galerkin-based approximation, facilitating a more straightforward convergence analysis.
The primary contributions of this chapter are: (1) the development of a physics-based diffusion population model that rigorously characterizes ethanol transport from the body to a TAC biosensor, with or without incorporating hepatic alcohol metabolism; (2) the treatment of model parameters as random variables and their formulation as additional spatial variables within Bochner spaces, enabling the model to accommodate diverse individual differences and environmental conditions; (3) the reformulation of intravenously infused alcohol studies as LQG control and tracking problems, utilizing linear semigroup and LQG control and tracking theories to regulate alcohol infusion and maintain or follow target BAC levels; (4) the transformation of the deconvolution problem into an LQG tracking framework, enabling real-time estimation of BAC or BrAC from TAC data; and (5) the enhancement of transdermal alcohol biosensor utility through a more precise and consistent TAC-to-BAC/BrAC conversion method, facilitating their application in alcohol therapy, research, and clinical monitoring.
2. Modeling the metabolism and transdermal transport of alcohol
We present a dynamical model of alcohol metabolism and transdermal transport, detailing its movement from blood through the skin to a biosensor, accounting for both liver metabolism and dermal transport, with state and input variables defined accordingly. Since the dermal layer (or blood) and the TAC biosensor are modeled as well-mixed compartments linked to the diffusion equation at the epidermal boundaries, we are able to eliminate the need for unbounded [28, 34, 35] input and output operators, which greatly simplifies the mathematical design. Figure 2 provides a flow diagram illustrating the transport process, both with and without liver metabolism.

Figure 2.
Signal flow diagram of the transport system describing the entire process with and without metabolism in the liver.
By linearizing the original nonlinear model around a nominal operating regime, a linearized system is prepared for integrating our linear compensators for both control and tracking problems.
2.1 Individual model for the metabolism and transdermal transport of ethanol
The dynamical model in (1) is a hybrid, semi-linear, one-dimensional, PDE/ODE reaction-diffusion equation
with boundary conditions and initial conditions in (2):
where
This model incorporates the fundamental physics and physiology of ethanol transport through the skin within its structure. In the TAC chamber, ethanol inflow is proportional to the flux from the epidermal layer, while linear outflow is proportional to the ethanol concentration in the chamber. Similarly, at the dermal layer or blood, ethanol outflow is proportional to both the flux and concentration at the epidermal-dermal boundary. All parameters in (1) are positive due to their physical and physiology significance (see Ref. [17]). Their values, which vary across conditions, will need to be estimated from input/output data.
The Michaelis-Menten nonlinear reaction term involving
To formulate control or tracking problems, we define the controlled variable
where
2.2 Linearization about steady state
In intravenously infused alcohol studies, the dynamic system (1) involves the Michaelis-Menten nonlinear reaction term. A linear compensator is designed to regulate around the steady state or equilibrium solution to (1) and (2). The desired clamping BAC is set to be
with boundary conditions, controlled variable, observation, and initial value:
respectively, where in (4) and (5), the parameters
The state of the system (4) and (5) is given by the triple
By setting
3. The state space formulation of the individual and population models
We first reformulate the linearized individual model (4) and (5) as an abstract parabolic system in a Gelfand triple of Hilbert spaces, which serves as a generalized form of the alcohol biosensor problem with deterministic parameters. We then define the population model by constructing appropriate Bochner spaces and treating parameters as random variables in additional spatial dimensions. Finally, we derive a corresponding discrete-time dynamic system that aligns with the data collection method of biosensors.
3.1 The formulation of the individual model
Let
Let
Define a bilinear form
By applying Cauchy-Schwarz and Morrey’s inequalities, it can be verified [17, 21, 36] that the bilinear form
We define the input operator
Therefore, (4) and (5) can be rewritten in state space form as the individual model
where
3.2 Random parameters as auxiliary spatial variables and the population model
The parameter vector
Define the Bochner spaces
Define the other operators
As demonstrated in Refs. [24, 25], the solution to the population form (8) agrees almost surely or
Since biosensors collect TAC data at discrete intervals, it is necessary to analyze the discrete-time dynamic system. Let
4. Linear quadratic Gaussian control and tracking problems
We extend the standard linear quadratic Gaussian (LQG) control and tracking theory from finite-dimensional to infinite-dimensional Bochner spaces (see Refs. [27, 28, 38]). The LQG compensators, along with control and tracking theories, are applied to problems involving intravenously infused alcohol studies and the blind deconvolution of BAC/BrAC from TAC measurements obtained
4.1 Problem formulation
We formulate a standard discrete-time LQ control in the Bochner space for the population system (10). Consider the discrete-time LQG control problem in
where
Recalling the definition of controlled variable operator
We also formulate a standard discrete-time LQ tracking in Bochner space for the population system (10). Consider the discrete-time LQG tracking problem in
where
The tracking problem
In discrete-time setting, all involved operators are bounded, making the sampled time LQ theory in Hilbert space on both finite and infinite time horizon entirely analogous to the finite-dimensional case. The primary distinction lies in replacing matrices in the Riccati equations and gain expressions with their operator counterparts. Since the finite horizon results follow similarly, we focus solely on the infinite horizon solutions for both control and tracking problems, assuming full-state observability (i.e., no noise terms
Theorem 5.2 in Refs. [40] implies the uniform exponential stability of the semigroup
and for every initial value
where
For the tracking problem
4.2 LQG compensator
Since the plant (10) is influenced by random noise processes

Figure 3.
Flow diagram of LQG control problem with a compensator.

Figure 4.
Flow diagram of LQG tracking problem with a compensator.
In the observer or estimator, the state covariance operator and output covariance matrix are given by
where
whose solution exists for the same reason as that of ARE (13). Since
Based on separation principle (see Theorem 5.3 in [38]), the optimal controller for
And the tracking controller for
Where
5. Finite-dimensional approximation and convergence
The infinite-dimensional dynamic system (10) requires a finite-dimensional approximation [27, 28, 41] for practical computation. A Galerkin method [26] based on the weak form of (10) and the corresponding convergence theory for closed-loop LQ control and tracking problems is developed. This involves constructing finite-dimensional subspaces
Let
Following Ref. [26], we construct a Galerkin approximation of
Using a variational corollary of Trotter-Kato theorem (see Theorem 2.1, 2.2, and 2.3 in [36]), we establish the convergence of the approximating semigroups
We set
Consider the infinite time horizon LQG control problems in their finite-dimensional approximations:
Similar to the infinite-dimensional case, applying separation principle, the unique solution to
with
Since
Consider the infinite time horizon LQG tracking problems in their finite-dimensional approximations:
As in the infinite-dimensional case, by separation principle, the optimal control
Analogous to the control problem, the convergence of
The equations in Section 5, though finite-dimensional, are operator equations. To enable computations, they must be converted into equivalent matrix equations. However, due to the non-orthonormal nature of the chosen basis for
6. Numerical studies
We validate our approach through simulations demonstrating how a TAC biosensor, combined with the proposed controller, operates in the three scenarios described in Section 1. First, the parameters
6.1 Human subject validation data
We validated the model (1) using inverse problem techniques, fitting it to human subject data from the Luczak Laboratory at the University of Southern California. Validation involved BrAC and TAC data from 40 participants (50% female, ages 21–33 years, 35% with BMI
6.2 Parameter setting
To reduce variance in the BrAC/TAC training set relative to the full population, the parameter vectors
6.3 Simulating intravenous alcohol infusion in clamping studies
The objective of simulating intravenous alcohol infusion in clamping studies is to maintain a subject’s BAC at a target level of
Using the
We set the total running time to be 15 or 30 hours and assumed a noise level of approximately
We present simulation results for both controlled BAC and alcohol infusion rate across the merged cohort using various compensator designs. In the figures, the target equilibrium BAC and mean target infusion rate are shown in green, while the sample mean controlled BAC and infusion rate are plotted in blue. Black error bars indicate point-wise sample standard deviations, and red error bars represent two standard deviations. The initial BAC was set to 0.04 g/dL, with the control objective of raising it to 0.06 g/dL and maintain that level.
For comparison, the left and center panels of Figure 5 show controlled BAC and intravenous infusion rates for the cohort, where each compensator was designed using its plant’s fitted deterministic parameters. Here, the plant (i.e., the subject-specific parameters) was assumed known, and the compensator was based on the individual model (1) and (2) rather than the population model, with parameters fitted from each subject’s BrAC and TAC data. The plant was simulated with

Figure 5.
Closed-loop BAC (left) and corresponding intravenous ethanol infusion rate (center) using an optimal compensator designed with known plant parameters, as well as the open-loop BAC (right).
Figure 6 presents results for the cohort using a compensator designed based on the population model fitted in Sections 4 and 5. The left panel shows the compensator-controlled BAC, while the center panel displays the corresponding controller. To test robustness, plant parameters were randomly varied by up to 20% during the simulation. The right panel shows controlled BAC for one such case, demonstrating that the compensator’s performance remained largely consistent with the fixed-parameter scenario in the left panel.

Figure 6.
Population-model-based compensator-controlled BAC (left) and corresponding controller or ethanol infusion rate (center), and the controlled BAC when plant parameters fluctuate by up to 20% at random intervals (right).
In Figure 7, we present results from using a compensator designed based on the individual model with incorrect parameters. For each subject, design parameters were randomly chosen from another subject in the cohort. The left panel shows the controlled BAC for one simulation, with the corresponding controller in the center panel. The right panel displays the controlled BAC for a less extreme version of the same simulation.

Figure 7.
Controlled BAC from individual model-based compensators, where parameters are randomly selected from the cohort instead of matching the plant’s parameters (left) and the corresponding control inputs (center), and a compensator-controlled BAC for another simulation with a different random selection of design parameters from the cohort (right).
6.4 Simulating intravenous alcohol infusion in tracking studies
The objective of simulating intravenous alcohol infusion in tracking studies is to ensure the subject’s BAC follows a pre-specified target BAC trajectory by infusing
We fit four-parameter Beta distributions to
The total runtime was set to 120 hours, with noise levels of
Figure 8 illustrates the trajectory of the output feedback tracking controller alongside the controlled and target BAC. For comparison, three compensator designs are included: one using our proposed scheme (i.e., compensator (19)), another based on the true plant parameters (

Figure 8.
Population model-based compensator-controlled BAC and corresponding tracking controller or ethanol infusion rate.
6.5 Estimating BAC or BrAC from the TAC signal
The objective is to perform blind deconvolution to estimate BAC from TAC with regularization. This can be rephrased as determining an input signal (BAC) that enables the population model to track a given TAC signal. This problem remains strictly linear, where the compartment at
Regularization is incorporated by augmenting the state with the BAC signal,
The sampling rate was set at

Figure 9.
Tracked TAC and estimated BrAC.
7. Discussion and conclusions
The rigorous mathematical theory and numerical simulations have demonstrated the efficiency and convergence of finite-dimensional approximations of LQG compensators designed using our general approach.
In the simulations of intravenously infused alcohol in clamping studies, our approximated LQG compensator performed nearly as well as the optimal infinite-dimensional compensator designed using the actual plant parameters (Figures 5 and 6). In contrast, a compensator that matches the plant structure but is incorrect designed with random selected parameters provides a poor strategy and may lead to instability (see Figure 7). Moreover, the right panel in Figure 6 demonstrates that our compensator design exhibits robustness against random variations in plant parameters and maintains stability to a certain extent.
In the simulations of intravenously infused alcohol tracking studies, the simulated BAC obtained using our approach closely follows the target BAC, and the infused alcohol flow rate (control) from our approach nearly overlaps with the control computed when the plant parameters are known exactly (see Figure 8).
Our population model-based tracking scheme also provides an efficient and stable algorithm for the real-time deconvolution of BAC or BrAC from a TAC signal. As shown in Figure 9, the estimated BrAC and simulated TAC closely track the actual BrAC and target TAC, respectively. In practice, estimating BAC or BrAC from TAC could potentially be performed in real time directly on a mobile device (e.g., an iPhone, Fitbit, or Apple Watch). This represents a significant advancement toward providing reliable BrAC based on TAC in real time or near real time.
As a summary, our research has enabled the replacement of traditional breathalyzers with advanced TAC biosensor technology to involve the BAC detection in certain alcohol study scenarios through a passive, noninvasive approach. We developed and validated a comprehensive approach for leveraging transdermal alcohol biosensors in conjunction with rigorous mathematical modeling and linear quadratic Gaussian control and tracking theories, applied to both intravenous alcohol studies and the blind deconvolution of BAC from TAC data. By fitting different models to experimental data, we demonstrated that our population model-based compensator, with parameter distributions fit to data, performs well in scenarios where the plant system parameters are either fixed but unknown or vary randomly over time. Additionally, we have shown that our tracking scheme yields an efficient and stable algorithm capable of estimating BAC or BrAC from TAC signals in real time. Importantly, the steady-state Riccati operators and control gains can be pre-computed offline, enabling real-time BAC or BrAC estimation on a mobile device through relatively simple calculations.
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