Applications of Transdermal Alcohol Biosensors

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 x˜tη be ethanol concentration at time t>0 and depth η∈01 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

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:

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

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.

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 q∈Q, let Hq=R2×L201 with the inner product θξφθ¯ξ¯φ¯q=q1q3θθ¯+q1q5ξξ¯+∫01φηφ¯ηdη. It is clear that for any φ̂,ψ̂∈H fixed, the map q↦φ̂ψ̂q is continuous from Q into R.

Let V=θξφ∈H:φ∈H101θ=φ0ξ=φ1 be the Hilbert space with inner product φ0φ1φφ¯0φ¯1φ¯V=φ0φ¯0+φ1φ¯1+φ′φ¯′L201, where ⋅⋅L201 denotes the standard inner product on L201. Standard arguments [33] imply the dense and continuous embeddings V ↪ Hq ↪ V∗, with these being uniformly bounded with respect to q∈Q.

Define a bilinear form aq⋅⋅: V×V→R by

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 q∈Q, and coercive, given the compactness of Q. With these properties, the bilinear form aq⋅⋅ defines a bounded linear operator Aq:DomAq⊂H→H by Aqφ̂ψ̂V∗,V=−aqφ̂ψ̂, for any φ̂∈DomAq, ψ̂∈V, where DomAq=φ̂=(φ0φ1φ)∈V:φ∈H2(01) is independent of q∈Q. Specifically, for any φ̂=φ0φ1φ∈DomAq, we have Aqφ̂=q3φ′0−q4φ0−q5φ′1−qLφ1q1φ′′. 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:t≥0 [33, 34].

We define the input operator Bq∈LRHq by Bqu=0q2u0, u∈R, q∈Q, and the random noise influence operator B1∈LR2Hq by B1ω=ω1ω20 for ω=ω1ω2T∈R2. The observation or output operator C∈LHqR is given by Cθξφ=θ. The controlled variable operator D∈LHqR 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

where x=wvx is the state of the system (4) and (5), ωt=ω1tω2tT, and the initial value x0=w0v0x0⋅∈H 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 q∈Q 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 V ↪ H ↪ V∗. Define the π− average bilinear form a⋅⋅ on V×V by: aφψ=Eπaqφqψq=∫Qaqφqψqdπq, for φ,ψ∈V. 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 A∈LVV∗ by AφψV∗,V=−aφψ, φ,ψ∈V. Similarly, the operator A can be restricted to DomA=φ∈V:Aφ∈H as the infinitesimal generator of an analytic semigroup Tt:t≥0 of bounded, self-adjoint operators on H.

Define the other operators B∈LRH, B1∈LR2H, and C,D∈LHR by Buf=B⋅ufH, B1ωf=B1ωfH, f∈H, u∈R, ω∈R2, Cφ̂=EπCφ̂, and Dφ̂=EπDφ̂, φ̂∈H, respectively. Then, we can formulate a population model with observation, corresponding to the linearized system (4) in state space form as

As demonstrated in Refs. [24, 25], the solution to the population form (8) agrees almost surely or π−a.e. q∈Q 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:

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 t∈kτk+1τ, k=0,1,2,⋯. Define xk=xkτ, yk=ykτ, k=0,1,2,⋯. According to (9), define Â=Tτ∈LHH, B̂=∫0τTsBds∈LRH, and B̂1=∫0τTsB1ds∈LR2H. Since a⋅⋅ is coercive with λ0=0, A is invertible. So B̂=A−1Â−IB, B̂1=A−1Â−IB1, where I is an identity operator in H. In addition, define Ĉ=C∈LHR, and D̂=D∈LHR. Therefore, the discrete-time dynamic system corresponding to (8) is given by

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-∈l2k0k1−1R for minimizing the quadratic performance index

where xk is generated by dynamic system (10), r̂>0 is a constant, Q̂,Ĝ∈LHH are given by Q̂=D̂∗D̂, and Ĝ=ρ̂D̂∗D̂ 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-∈l2k0k1−1R for minimizing the quadratic performance index

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 D̂xk−ξk2.

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:t≥0 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 Π̂∈LHH to the following algebraic Riccati equation (ARE):

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

where F̂=r̂+B̂∗Π̂B̂−1B̂∗Π̂Â, 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=Â−B̂F̂x-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=Â∗K−KB̂B̂∗KB̂+r̂−1B̂∗KÂ+Q̂ exists and is unique, self-adjoint, and positive semi-definite. The efficacy of the tracking controller (15) is guaranteed, where F̂=B̂∗KB̂+r̂−1B̂∗KÂ∈LHR and Θ=B̂∗KB̂+r̂−1B̂∗θ∈R with θ=Â∗I+r̂−1KB̂B̂∗−1θ+D̂∗ξ.

4.2 LQG compensator

Since the plant (10) is influenced by random noise processes B̂1ωkτ and ζkτ, 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.

In the observer or estimator, the state covariance operator and output covariance matrix are given by Q˜=B̂1ΣB̂1∗∈LHH where Σ=diagσ12σ22∈R2×2 and R˜=σ2>0, respectively. The observer or state estimator for both infinite time horizon problems P1 and P2 takes the form

where φ˜0∈H is arbitrary. The operators that are known as Kalman gains L˜∈LRH for k≥0 are given by L˜=ÂS˜Ĉ∗σ2+ĈS˜Ĉ∗−1, with positive semi-definite, self-adjoint operator S be given by ARE

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

And the tracking controller for P2 is given by

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).

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=q1q2…q8 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 q1∼q1,nq1 with q1,n=0.74 and q1∼Beta3.49,3.60. Similarly, scaled gamma distributions were fit to q2∼q2,nq2 with q2,n=0.71 and q2∼Gamma8.79,0.06, and to q3∼q3,nq3 with q3,n=19.70 and q3∼Gamma2.08,0.16. The remaining parameters were assigned their sample means: q¯4=3.9×10−1, q¯5=1.6×10−4, q¯6=3.6×10−4, q¯7=1.1×10−2, and q¯8=2.1×10−5. 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 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.

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.

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 ξk−v˜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 q1∼Beta1.53,2.54 with support 0.25,0.54, q2∼Beta7.52,17.72 with support 0.18,0.76, q3∼Beta1.38,1.59 with support 3.20,10.91. For the remaining parameters, we used their sample means, yielding q¯4=1.63×10−1, q¯5=2.18×10−4, q¯6=4.88×10−4, q¯7=1.23×10−2, and q¯8=2.26×10−5.

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.

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, yk∈R2, 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=0k−1δuj.

The sampling rate was set at τ=1/60, with parameter distributions specified as: q1∼Beta32 with support 01, q2∼Beta79 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.