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Applications of Transdermal Alcohol Biosensors

Written By

Mengsha Yao and Gary Rosen

Submitted: 30 October 2024 Reviewed: 07 April 2025 Published: 09 June 2025

DOI: 10.5772/intechopen.1010428

Current Developments in Biosensors and Emerging Smart Technologies IntechOpen
Current Developments in Biosensors and Emerging Smart Technologie... Edited by Selcan Karakuş

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Current Developments in Biosensors and Emerging Smart Technologies [Working Title]

Selcan Karakuş

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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 via perspiration [1, 2, 3]. After alcohol is consumed, it enters the bloodstream and circulates, with the majority metabolized in the liver (approximately 90%), some filtered into urine by the kidneys, and a fraction evaporating through the lungs into breath. Eventually, a small amount is excreted through the skin as ethanol molecules, which can be detected by biosensors. Technically speaking, these highly sensitive devices collect ethanol in its gaseous state within a reservoir. Upon contact, the biosensors, functioning as fuel cells, initiate an oxidation-reduction reaction, generating a fixed number of electrons per ethanol molecule. Thus, this generates an electrical current proportional to the ethanol concentration, which is then converted into a measurable signal representing the transdermal alcohol concentration (TAC).

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 via Bluetooth to the wearer’s mobile device. Currently, ION is undergoing laboratory testing and awaiting further development for commercial release. A comprehensive review of various TAC biosensors can be found in Ref. [8].

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 via perspiration, and its eventual detection by the TAC biosensor on the surface of the skin. For specific applications, we formulate various control or tracking problems, adjusting model variables to represent different quantities. In intravenous infused alcohol clamping or tracking studies, we implement a closed loop output feedback system using linear quadratic Gaussian (LQG) control or tracking compensators, where the input variable is the infused alcohol flow rate, and the state variable is BAC. Here, the model incorporates a Michaelis-Menten nonlinear reaction term to describe alcohol enzyme-catalyzed metabolism via alcohol dehydrogenase in the liver. Additionally, we reformulate the estimation or deconvolution of BAC from TAC measurements as an LQG tracking problem, treating the same model as the plant, with measured TAC data as the reference signal. In this case, BAC serves as the input rather than the state variable, eliminating the need to model liver metabolism, and thus, the Michaelis-Menten nonlinear reaction term is omitted from the dynamical model.

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.

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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

x˜ttη=α2x˜η2tη,t>0,η01,dw˜dtt=βx˜ηt0γw˜t,t>0,dv˜dtt=δx˜ηt1Kv˜tM+v˜tϵv˜t+bu˜t,t>0,E1

with boundary conditions and initial conditions in (2):

x˜t0=w˜t,x˜t1=v˜t,t>0,x˜0η=φ0η,η01,w˜0=θ0,v˜0=χ0,E2

where x˜tη be ethanol concentration at time t>0 and depth η01 in the epidermal layer, with the thickness of the layer normalized to one. The two well-mixed compartments in Dirichlet boundary conditions are w˜t, the concentration of ethanol in the transdermal alcohol biosensor, and v˜t, the concentration of ethanol in either the dermal layer with an active blood supply (intravenous alcohol studies) or the blood (deconvolution). The input u˜t is either the infused alcohol flow rate (intravenous alcohol studies) or BAC (deconvolution). φ0, θ0, and χ0 denote the initial conditions for x˜, w˜, and v˜, respectively.

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 K and M for v˜ in system (1) represents the enzyme-catalyzed metabolism of ethanol by alcohol dehydrogenase (ADH) in the liver, converting ethanol into aldehydes and ketones for further processing or excretion via urine or breath. This term exhibits first-order kinetics at lower concentrations (i.e., v˜<<M) and zero-order kinetics at higher concentrations (i.e., M<<v˜). In deconvolution, as liver metabolism is not involved, the Michaelis-Menten term is omitted by setting K=0.

To formulate control or tracking problems, we define the controlled variable z˜ and the observation variable y˜ as

z˜t=v˜torw˜t,y˜t=w˜t,t>0,E3

where v˜ is regulated in intravenously studies, while w˜ is controlled in deconvolution. The observation variable, or system output, is the TAC measurement.

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 v˜t=v˜0. Based on this, the steady state or the equilibrium solution is given by x˜0η=γv˜0γ+βη+βv˜0γ+β, w˜0=βv˜0γ+β, u˜0=δγv˜0bγ+β+Kv˜0bM+v˜0+ϵv˜0b, y˜0=βv˜0β+γ, z˜0=v˜0. We then linearize around x˜0, w˜0, v˜0, u˜0, y˜0, and z˜0 by writing x˜=x˜0+x, w˜=w˜0+w, v˜=v˜0+v, u˜=u˜0+u, y˜=y˜0+y and z˜=z˜0+z and obtain the linearized system for x, w, v, u, y, z given by:

xttη=q12xη2tη,t>0,η01,dwdtt=q3xηt0q4wt+ω1t,t>0dvdtt=q5xηt1qLvt+q2ut+ω2t,t>0,E4

with boundary conditions, controlled variable, observation, and initial value:

xt0=wt,xt1=vt,t>0,zt=vtorwt,yt=wt+ζt,t>0,x0η=φ0ηx˜0η=x0η,t>0,E5

respectively, where in (4) and (5), the parameters q1=α, q2=b, q3=β, q4=γ, q5=δ, and qL=ϵ+KMM+v˜02 with q6=ϵ, q7=K, q8=M are all positive. Additionally, random noise associated with the state variables w and v and the observation variable y is introduced as uncorrelated, zero-mean, stationary Gaussian white noise processes ω1, ω2, and ζ, with variances σ12, σ22, and σ2, respectively.

The state of the system (4) and (5) is given by the triple wvx, with the controlled or tracked quantity as z, the output as y, and the input as u. The goal of the intravenous alcohol studies is to design an output feedback law for the input variable u as a function of observation y=y˜y˜0 such that, when the system (1) is disturbed, the input u˜=u˜0+u drives v to zero and maintains it, ensuring the safety of the test participant (clamping), or enables v to follow a pre-specified BAC trajectory adjusted by subtracting v˜0 (tracking). The challenge lies in achieving this when some model parameters, q=q1q2q3q4q5q6q7q8, are known only up to their distribution in a target population, rather than their precise values.

By setting K=0 to adapt (1)(3) for deconvolution, the linearized models (4) and (5) with q7=q8=0 reduces to the plant (1)(3). Therefore, the linearized models (4) and (5) is applicable to all related applications and will be the primary system discussed in the remainder of this chapter.

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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 Q be a compact subset of the positive orthant of R8, let H=R2×L201 be endowed with the standard inner product and norm. For each qQ, let Hq=R2×L201 with the inner product θξφθ¯ξ¯φ¯q=q1q3θθ¯+q1q5ξξ¯+01φηφ¯η. It is clear that for any φ̂,ψ̂H fixed, the map qφ̂ψ̂q is continuous from Q into R.

Let V=θξφH:φH101θ=φ0ξ=φ1 be the Hilbert space with inner product φ0φ1φφ¯0φ¯1φ¯V=φ0φ¯0+φ1φ¯1+φφ¯L201, where L201 denotes the standard inner product on L201. Standard arguments [33] imply the dense and continuous embeddings VHqV, with these being uniformly bounded with respect to qQ.

Define a bilinear form aq: V×VR by

aqφ0φ1φφ¯0φ¯1φ¯=q1q4q3φ0φ¯0+q1qLq5 φ1φ¯1+q101φηφ¯η.E6

By applying Cauchy-Schwarz and Morrey’s inequalities, it can be verified [17, 21, 36] that the bilinear form aq as defined in (6) is uniformly bounded, continuous with respect to qQ, and coercive, given the compactness of Q. With these properties, the bilinear form aq defines a bounded linear operator Aq:DomAqHH by Aqφ̂ψ̂V,V=aqφ̂ψ̂, for any φ̂DomAq, ψ̂V, where DomAq=φ̂=(φ0φ1φ)V:φH2(01) is independent of qQ. Specifically, for any φ̂=φ0φ1φDomAq, we have Aqφ̂=q3φ0q4φ0q5φ1qLφ1q1φ. Additionally, the operator Aq is densely defined on Hq, regularly dissipative and self-adjoint. Consequently, Aq restricted to its domain is the infinitesimal generator of an analytic semigroup of bounded linear operators on Hq, Tqt:t0 [33, 34].

We define the input operator BqLRHq by Bqu=0q2u0, uR, qQ, and the random noise influence operator B1LR2Hq by B1ω=ω1ω20 for ω=ω1ω2TR2. The observation or output operator CLHqR is given by Cθξφ=θ. The controlled variable operator DLHqR is given by either Dθξφ=νξ (intravenous alcohol studies) or Dθξφ=νθ (deconvolution), for some weight ν>0.

Therefore, (4) and (5) can be rewritten in state space form as the individual model

ẋt=Aqxt+Bqut+B1ωt,yt=Cxt+ζt,x0=x0,E7

where x=wvx is the state of the system (4) and (5), ωt=ω1tω2tT, and the initial value x0=w0v0x0H can be easily computed by the initial values in Eq. (2) and the equilibrium solution given in Section 2.2.

3.2 Random parameters as auxiliary spatial variables and the population model

The parameter vector qQ is uncertain due to variability across individuals, sensors, and environmental conditions. Following the framework in Refs. [17, 21, 24, 25], we formulate a population model by assuming q=q is a random vector with support Q. The distribution of q can be computed independently using population data via various estimation methods, assuming the population is sufficiently large to capture variations across different conditions and that all functions involving q are π-measurable. One approach is parametric estimation, which fits the generalized beta distribution via the method of moments. A non-parametric kernel density approach yields similar results. Refs. [23, 37] discuss these methods and their statistical validation in greater detail. After obtaining the estimated distribution, the state equation is expressed in a weak form, with q treated as additional spatial variables.

Define the Bochner spaces V=Lπ2QV with its dual space V, and H=Lπ2QHq. Then, the assumption on the spaces V, H and identifying H with its dual guarantee that the spaces V, H, and V form the Gelfand triple VHV. Define the π average bilinear form a on V×V by: aφψ=Eπaqφqψq=Qaqφqψqq, for φ,ψV. The boundedness, coercivity, and continuity of the bilinear form aq guarantee that this integral is well defined with respect to q and that a inherits these properties. This bilinear form defines a self-adjoint operator ALVV by AφψV,V=aφψ, φ,ψV. Similarly, the operator A can be restricted to DomA=φV:AφH as the infinitesimal generator of an analytic semigroup Tt:t0 of bounded, self-adjoint operators on H.

Define the other operators BLRH, B1LR2H, and C,DLHR by Buf=BufH, B1ωf=B1ωfH, fH, uR, ωR2, Cφ̂=EπCφ̂, and Dφ̂=EπDφ̂, φ̂H, respectively. Then, we can formulate a population model with observation, corresponding to the linearized system (4) in state space form as

x.t=Axt+But+B1ωt,yt=Cxt+ζt,x0=x0=x0.E8

As demonstrated in Refs. [24, 25], the solution to the population form (8) agrees almost surely or πa.e. qQ with individual form (7). This approach treats random parameters as additional spatial variables while notably avoiding derivatives with respect to these variables. Now, the dynamic system (8) is effectively deterministic and abstract parabolic. From the linear semigroup theory [33], the mild solution of (8) is unique and given by:

xt=Ttx0+0tTtsBus+B1wsds,t0.E9

Since biosensors collect TAC data at discrete intervals, it is necessary to analyze the discrete-time dynamic system. Let τ represent the sampling interval length and consider zero-order hold inputs of the form ut=uk, for tk+1τ, k=0,1,2,. Define xk=x, yk=y, k=0,1,2,. According to (9), define Â=TτLHH, B̂=0τTsBdsLRH, and B̂1=0τTsB1dsLR2H. Since a is coercive with λ0=0, A is invertible. So B̂=A1ÂIB, B̂1=A1ÂIB1, where I is an identity operator in H. In addition, define Ĉ=CLHR, and D̂=DLHR. Therefore, the discrete-time dynamic system corresponding to (8) is given by

xk+1=Âxk+B̂uk+B̂1ω,yk=Ĉxk+ζ,x0=x0.E10
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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 via biosensors.

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 H subject to the discrete-time linear dynamic system (10):

P1 choose an input u-l2k0k11R for minimizing the quadratic performance index

Ĵu=k=k0k11Q̂xkxkH+r̂uk2+Ĝxk1xk1H,E11

where xk is generated by dynamic system (10), r̂>0 is a constant, Q̂,ĜLHH are given by Q̂=D̂D̂, and Ĝ=ρ̂D̂D̂ for some nonnegative weight ρ̂. k1 can be either finite or infinite (in the latter case, ρ̂=0).

Recalling the definition of controlled variable operator Dθξφ=νξ, the control problem P1 aligns with the objective of the intravenously infused clamping studies. Specifically, it seeks an optimal feedback law for the infused alcohol flow rate u that drives the disturbance of BAC v to zero (associated with the term Q̂xkxkH in (11)) while maintaining it within the safety bounds for participants (associated with the term r̂uk2 in (11)).

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 H subject to the discrete-time linear dynamic system (10):

P2 Choose an input u-l2k0k11R for minimizing the quadratic performance index

Ĵu=k=k0k11D̂xkξk2+r̂uk2+ρ̂D̂xk1ξk12,E12

where xk is generated by dynamic system (10), ξk is the given reference signal to be tracked, and k1 can be either finite or infinite (in the latter case, ρ̂=0).

The tracking problem P2 is formulated for both intravenously infused tracking studies and blind deconvolution. In the case of intravenous tracking, the reference signal ξk represents a pre-specified target BAC trajectory adjusted by subtracting the equilibrium solution v˜0. The controlled variable operator Dθξφ=νξ directly relates to BAC disturbances, with the objective of determining an optimal feedback law for the infused alcohol flow rate u to ensure BAC follows the target trajectory. For blind deconvolution, the problem is formulated as tracking problem with the objective of designing an output feedback law for the input BAC signal u that forces the population model to track the given TAC signal. Here, the reference signal ξk is the given TAC signal, and the controlled variable operator, Dθξφ=νθ, corresponds to TAC measurement. In both problems, the objective is achieved by minimizing the quadratic performance index (12), which includes a term measuring the deviation D̂xkξk2.

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 ω and ζ in (10)). The general full-state solutions to the discrete-time LQ control and tracking problems for both finite and infinite horizons are given by the LQ theory in Refs. [27, 39].

Theorem 5.2 in Refs. [40] implies the uniform exponential stability of the semigroup Tt:t0 and consequently of Â, along with the positive semi-definite property of Q̂, which guarantees the existence of an admissible control for P1 for any initial state xk0. Hence, there exists a unique positive semi-definite, self-adjoint solution Π̂LHH to the following algebraic Riccati equation (ARE):

Π̂=Â*Π̂Π̂B̂r̂+B̂Π̂B̂1B̂Π̂Â+Q̂,E13

and for every initial value xk0, the optimal closed-loop solution to P1 is unique and generated by the linear control law

u-k=F̂x-k=f^x-kH,k=k0,k0+1,,E14

where F̂=r̂+B̂Π̂B̂1B̂Π̂Â, f^=F̂ is the corresponding functional gain obtained by Riesz Representation Theorem, Ĵu-= Π̂xk0xk0H, and that the optimal trajectory x-kk=k0 is given by x-k+1=ÂB̂F̂x-k, x-k0=xk0.

For the tracking problem P2, we assume that the reference signal ξk converges to a fixed reference value ξ. By theorem 2.4 in Ref. [27], Â open loop stable implies the solution to the ARE K=ÂKKB̂B̂KB̂+r̂1B̂KÂ+Q̂ exists and is unique, self-adjoint, and positive semi-definite. The efficacy of the tracking controller (15) is guaranteed, where F̂=B̂KB̂+r̂1B̂KÂLHR and Θ=B̂KB̂+r̂1B̂θR with θ=ÂI+r̂1KB̂B̂1θ+D̂ξ.

u-k=F̂x-k+Θ=B̂KB̂+r̂1B̂KÂxk+θE15

4.2 LQG compensator

Since the plant (10) is influenced by random noise processes B̂1ω and ζ, and the full state is not directly measurable, the feedback controllers in (14) for P1 and (15) for P2 cannot be directly implemented. To address this limitation, we introduce the state observers or estimators in the form of a Kalman filter into the feedback loop, enabling a transition from state feedback to output feedback. The combined feedback control law and observer form an LQG compensator, applicable to both control and tracking problems. Flow diagrams illustrating the closed-loop system for the LQG control and tracking problems with a compensator are given in Figure 3 [17] and Figure 4 [37], respectively.

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 Q˜=B̂1ΣB̂1LHH where Σ=diagσ12σ22R2×2 and R˜=σ2>0, respectively. The observer or state estimator for both infinite time horizon problems P1 and P2 takes the form

x˜k+1=Âx˜k+B̂uk+L˜ykĈx˜k,x˜k0=φ˜0,E16

where φ˜0H is arbitrary. The operators that are known as Kalman gains L˜LRH for k0 are given by L˜=ÂS˜Ĉσ2+ĈS˜Ĉ1, with positive semi-definite, self-adjoint operator S be given by ARE

S˜=ÂS˜S˜Ĉσ2+ĈS˜Ĉ1ĈS˜Â+Q˜,E17

whose solution exists for the same reason as that of ARE (13). Since L˜ is in LRH, it follows that L˜=l˜H. The element l˜ in H is the optimal functional observer gain.

Based on separation principle (see Theorem 5.3 in [38]), the optimal controller for P1 is given by

u-k=F̂x˜k=f^xkH,k=k0,k0+1,k0+2,E18

And the tracking controller for P2 is given by

u-k=F̂x˜k+Θ=B̂KB̂+r̂1B̂KÂx˜k+θ,E19

Where x˜k is generated by (16), the feedback operators F̂, and functional control gains f^ in (18) are the same as those in (14), and the feedback operators F̂, Θ, and θ in (19) are the same as those in (15).

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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 VN, with their orthogonal projections converging strongly to the identity in V, and defining converging sequences of approximating operators to Â, B̂, B̂1, Ĉ, and D̂.

Let N be the multi-index N=nm1m2.m8 where N implies both n and mi, i=1,2,,8. Assume the random parameter qi has support aibi, forming the compact subset Q=×i=18aibi. Partition aibi into mi equal subintervals, and let χjmi be the characteristic function of the j-th subinterval for j=1,2,,mi. For n=1,2,, let φjnj=0n be the standard linear B-splines on 01 with respect to the uniform mesh and define φ̂jn=φjn0φjn1φjnV. Let J=j0j1j8 where j00,1,2n and ji12mi. Set ΦJN=φ̂j0nΠi=18χjimi and VN=spanJΦJN. The orthogonal projection PN of H onto VN converges strongly to the identity in both H and V by standard spline theory and piecewise constant approximation in L2.

Following Ref. [26], we construct a Galerkin approximation of A. Define the operators AN on VN by restricting the bilinear form a to VN×VN. Specifically, for φN,ψNVN, ANφNψNVN,VN=aφNψN= QaqφNqψNqq. Since AN is linear in a finite-dimensional space, it serves as the infinitesimal generator of a uniformly continuous semigroup TNt=eANt for t0. Consequently, we define ÂNLVNVN as ÂN=TNτ=eANτ.

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 TNt. For each xH, TNtPNxTtx in the V norm (so also in the H norm) for t>0, uniformly in t on compact sub intervals. Given the definitions Â=Tτ and ÂN=TNτ, it follows that ÂNPNÂ strongly in V (so also in H) as N. Furthermore, since both Tt and TNt are self-adjoint in H, we have ÂNPNÂ strongly in H as N.

We set B̂N=AN1ÂNINPNBLRVN. Similarly, we set B̂1N=AN1ÂNINPNB1LR2VN, and set ĈN=ĈPNLVNR, D̂N=D̂PNLVNR. By previous definitions, we set ĜN=PNĜPN=ρ̂D̂N D̂NLVNVN, Q̂N=PNQ̂PN=D̂ND̂NLVNVN, and Q˜N=B̂1NΣB̂1N LVNVN. Using standard arguments [27, 28], the operators B̂N, B̂1N, ĈN, D̂N, Q̂N, ĜN, and Q˜N can be shown to strongly or uniformly converge to their infinite-dimensional counterparts as N.

Consider the infinite time horizon LQG control problems in their finite-dimensional approximations:

P1N For every N, choose an input u-Nl2k0R for minimizing the approximating quadratic performance index ĴNuN=k=k0 Q̂NxkNxkNH+r̂ukN2, where xkN is generated by the approximating discrete-time dynamic system

xk+1N=ÂNxkN+B̂Nuk+B̂1Nω,ykN=ĈNxkN+ζ,x0N=PNx0=PNx0.E20

Similar to the infinite-dimensional case, applying separation principle, the unique solution to P1N is expressed in a closed-loop linear state feedback form u-kN=F̂NxkN=f^Nx^kNH,k=k0,k0+1,, where F̂N=r̂+B̂NΠ̂NB̂N1B̂NΠ̂NÂN, and Π̂N is the unique positive semi-definite, symmetric solution to the approximating ARE

Π̂N=ÂNΠ̂NΠ̂NB̂Nr̂+B̂NΠ̂NB̂N1B̂NΠ̂NÂN+Q̂N;E21

f^N denotes the optimal functional control gains; the approximating state estimator x˜kN is generated by

x˜k+1N=ÂNx˜kN+B̂NukN+L˜NykĈNx˜kN,E22

with x˜0N=PNφ˜0, where the approximating Kalman gains L˜NLRVN is given by L˜N=ÂNS˜NĈNσ2+ĈNS˜NĈN1, with the positive semi-definite, self-adjoint operators S˜N be given by the approximating ARE

S˜N=ÂNS˜NS˜NĈNσ2+ĈNS˜NĈN1ĈNS˜NÂN+Q˜N.E23

Since TNt and TNt are uniformly continuous, exponentially stable, and analytic semigroups on H, the same arguments as in Section 4.1 and 4.2 ensure the existence of the solution Π̂N to the approximating ARE (21) and S˜N to the approximating ARE (23), as well as their uniform boundedness. By Theorem 3.10 in Ref. [27], the sequences Π̂N and S˜N converge strongly in H to the solution Π and S˜ of AREs (13) and (17), respectively. Applying Theorem 3.9 in Ref. [27], the approximating closed-loop controls u-N converge to u- in l2, whose inner product and norm are defined by xyl2=k=k0k1xkykH for any x and y in l2k0k1H. Additionally, F̂NPN converges uniformly to F̂ in LHR; the approximating cost functions ĴNu-N also converge to Ĵu- as N. In practical applications, the implemented control is given by u-kN=f^Nx-kH, k=k0,k0+1,, where x-k denotes the trajectory given by (10) with uk=u-kN, k=k0,k0+1,.

Consider the infinite time horizon LQG tracking problems in their finite-dimensional approximations:

P2N for every N, choose an input u-Nl2k0R for minimizing the approximating quadratic performance index ĴNuN=k=k0D̂NxkNξk2+r̂ukN2, where xkN is generated by the approximating discrete-time dynamic system (20).

As in the infinite-dimensional case, by separation principle, the optimal control u-N for approximating tracking problem P2N is given by u-kN=F̂Nx˜kN+ ΘN=B̂NKNB̂N+r̂1B̂NKNÂNx˜kN+θN, where F̂N=B̂NKNB̂N+r̂1B̂NKNÂN, ΘN=B̂NKNB̂N+r̂1 B̂NθN, with θN=ÂNI+r̂1KNB̂NB̂N1θN+D̂Nξ, KN is the unique, positive, semi-definite, self-adjoint solution to the approximating ARE KN=ÂNKNKNB̂NB̂NKNB̂N+r̂1B̂NKNÂN+Q̂N, and the approximating state estimator x˜kN is generated by (22).

Analogous to the control problem, the convergence of KN, FN, u-N, and ĴN to their infinite-dimensional counterparts, as N, can be established (see [18, 27], eq. (2.47) in [39]).

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 VN, careful attention is required to preserve the symmetry of the resulting matrix ARE. Further details are provided in Refs. [27, 28].

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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 qR8 in the individual model (1) and (2) are fitted using BrAC and TAC data from drinking episodes collected in the Luczak Laboratory at the University of Southern California [42]. Next, we design and compare various plant and compensator combinations using either individual or population models for simulating intravenous alcohol infusion. Finally, we apply our approach to the blind deconvolution, estimating BAC from TAC while incorporating regularization to address uncertainty in the forward convolution filter or kernel.

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 > 25.0), who consumed the same alcohol dose across multiple drinking sessions on different days. BrAC was measured using a breath analyzer (Intoximeters, Inc., St. Louis, MO), while TAC was recorded via SCRAM-CAM TAC sensors (Alcohol Monitoring Systems, AMS, Littleton, CO) placed on the upper arm. Participants consumed Vodka (40% ethanol) mixed with fruit juice and sugar-free soda, avoiding pure alcohol intake. The final dataset comprised 267 BrAC/TAC measurements, 140 from males and 127 from females. Although the model fit data from a diverse cohort at different times using various TAC sensors, parameter values varied across individuals. A detailed account of data collection and model validation for individual and population models is available in Ref. [17].

6.2 Parameter setting

To reduce variance in the BrAC/TAC training set relative to the full population, the parameter vectors q=q1q2q8 were clustered into sub-populations using a regression model. Predictor variables included patient-, sensor-, and environment-dependent covariates such as age, sex, race/ethnicity, body measurements (e.g., BMI, waist-to-hip ratio, percent body fat), drinking behavior, ambient temperature, humidity, and sensor manufacturer. After clustering, sub-populations were selected to fit specific distributions to parameters qi, i=1,2,3 under different scenarios, employing the method of moments [43]. For simplicity, qi, i=1,2,3 were assumed independent. To reduce the compensator dimension, and since parameters qi, i=4,5,6,7,8 had small magnitudes and variance, their sample means were used in the population model. Additional parameters were also determined, with specific values provided in the following subsections.

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 0.06g/dL by infusing 6%V/V ethanol solution. The plant was simulated using the individual model from (1) and (2), with parameters obtained by fitting that model to the subject’s BrAC and TAC data from the laboratory described in Section 6.1. The compensator was designed using the population model and a state estimator based on the linearized models (4) and (5). This approach is practical since, in real applications, the compensator would be designed from a population model derived from a cohort that includes the subject based on selected covariates, even if the subject was not part of the training dataset.

Using the nth order statistic, we fit a scaled beta distribution to q1q1,nq1 with q1,n=0.74 and q1Beta3.49,3.60. Similarly, scaled gamma distributions were fit to q2q2,nq2 with q2,n=0.71 and q2Gamma8.79,0.06, and to q3q3,nq3 with q3,n=19.70 and q3Gamma2.08,0.16. The remaining parameters were assigned their sample means: q¯4=3.9×101, q¯5=1.6×104, q¯6=3.6×104, q¯7=1.1×102, and q¯8=2.1×105. Linearization was performed around the plant steady state solution, x˜0, w˜0, and u˜0, which were computed based on the parameter vector q and corresponded to the target BAC level v˜0=0.06.

We set the total running time to be 15 or 30 hours and assumed a noise level of approximately 10%, with σ1=σ2=σ=.006 in both the design and simulation phases. In the performance index, we choose ρ̂=ν=5.0 and r̂=0.05. If the control penalty is too low, the alcohol infusion rate may become negative or exceed safe levels for human experiments or treatments. The plant was simulated with n=32, while the compensator was designed with n=4 and mi=4, i=1,2,3, resulting in a compensator dimension of 5×43=320. The sampling rate was 1/60Hz or τ=0.0167 hours (one sample per minute).

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 n=32 and the compensator with n=4. The right panel of Figure 5 illustrates the uncontrolled (zero control) BAC for the cohort, where the input is set to uk=u˜0. As shown in the plot and confirmed by standard analysis, while the system eventually tends toward the target BAC v˜0, convergence rate is notably slow.

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 6%V/V ethanol solution. The tracking controller is designed using the linearized model (4) and (5) with the v-component tracking ξkv˜0, where ξk represents the target BAC signal. If uk denotes the resulting tracking compensator for the linearized system, then u˜=u˜0+uk is the corresponding tracking controller for the nonlinear plant (1) and (2). Using the approach outlined in earlier sections and the population model based on known parameter distributions, we designed the output-feedback tracking compensator (19).

We fit four-parameter Beta distributions to qi, i=1,2,3, yielding q1Beta1.53,2.54 with support 0.25,0.54, q2Beta7.52,17.72 with support 0.18,0.76, q3Beta1.38,1.59 with support 3.20,10.91. For the remaining parameters, we used their sample means, yielding q¯4=1.63×101, q¯5=2.18×104, q¯6=4.88×104, q¯7=1.23×102, and q¯8=2.26×105.

The total runtime was set to 120 hours, with noise levels of σ1=σ2=σ=.004, corresponding to 5% of the initial alcohol level in the target signal. In the performance index, we set ρ̂=ν=1.0 and r̂=0.5. For approximation, we used n=4 and mi=6, i=1,2,3, resulting in an approximating compensator of dimension 5×63=1080. The simulation executed rapidly even when increasing n (e.g., n=8,12, or 16), with minimal impact on performance. This held true on both desktop and laptop machines, including computations for the control and observer Riccati equations, which are typically computed offline. The sampling rate was 1/60 Hz, with a 1-minute interval (τ=0.0167 hours).

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 (n=64), and a third using incorrect plant parameter values.

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 η=1 represents the dermal layer of the skin, and the input u˜ represents BAC. The nonlinear Michaelis-Menten reaction term for liver metabolism is absent (K = 0 in (1) and (2) or q7=q8=0 in (4) and (5)). The output is TAC, and the plant is modeled using the population model, allowing full-state observation without the need for an observer. Furthermore, given the ill-posed nature of this inverse problem, regularization is applied by penalizing both the magnitude and rate of change of BAC to prevent overfitting and avoid non-physical oscillations.

Regularization is incorporated by augmenting the state with the BAC signal, uk, introducing a new control variable, δuk, and adding a state equation Δuk=δuk with the initial condition u0=0. The state vector is now defined as xk=xkukT, where the magnitude of u is regularized through a quadratic state penalty term, while any unwanted oscillations are mitigated by the control penalty term involving δuk. The output variable to be tracked, ykR2, consists of the TAC signal as the first component and zero as the second. Once the optimal δuk is determined, the estimated BAC, uk, is computed from uk=Σj=0k1δuj.

The sampling rate was set at τ=1/60, with parameter distributions specified as: q1Beta32 with support 01, q2Beta79 with support 03, while fixed values were assigned to the remaining parameters: q3=3.99, q4=0.05, q5=0.086, q6=0.85. We set ρ̂=100, ν=0.8, the penalty weight on uk (now part of the D̂ operator) to be .01, and the control penalty on δuk to r̂=100. The model was approximated with n=8 and m1=m2=16. The dataset used for this analysis consisted of TAC measurements collected by one of the co-authors (S.E.L.) [19, 20] using a WrisTAS 7TM transdermal alcohol biosensor, manufactured by Giner, Inc. of Waltham, MA. Simultaneous BrAC data were also collected, which allowed us to evaluate the effectiveness of the proposed method. Figure 9 presents the target TAC signal, the tracking TAC, and the deconvolved versus true BrAC.

Figure 9.

Tracked TAC and estimated BrAC.

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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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Written By

Mengsha Yao and Gary Rosen

Submitted: 30 October 2024 Reviewed: 07 April 2025 Published: 09 June 2025