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Determining blood alcohol concentration from transdermal alcohol data: calibrating a mathematical model using a drinking diary

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Determining blood alcohol concentration from transdermal alcohol data: calibrating a mathematical model using a drinking diary

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DETERMINING BLOOD ALCOHOL CONCENTRATION FROM TRANSDERMAL
ALCOHOL DATA:
CALIBRATING A MATHEMATICAL MODEL USING A DRINKING DIARY

by

Katherine Coste

A Thesis Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
MASTER OF SCIENCE
(APPLIED MATHEMATICS)

August 2013

Copyright 2013 Katherine Coste
ii

Acknowledgments
I would like to express my gratitude to my thesis advisor Professor Gary Rosen
for the continuous support of my work and for his motivation, enthusiasm, and immense
knowledge. Thank you for giving me the opportunity to be a part of the math research
community and for broadening my background. Also, thanks to Susan Luczak for her
research participation and collaboration.

iii

Table of Contents
Acknowledgements ii
List of Tables iv
List of Figures v
Abstract vi
Introduction 1
Deconvolution 5
Calibration Using Drink Diary 11
Numerical Results 16
References 29

iv

List of Tables
Table 1: Summary of Drink Diary 17
Table 2: Calibration results for episode 1 22
Table 3: Calibration results for episode 2 22
Table 4: Calibration results for episode 3 23
Table 5: Calibration results for episode 4 23
Table 6: Calibration results for episode 5 23
Table 7: Calibration results for episode 6 24
Table 8: Calibration results for episode 7 24
Table 9: Calibration results for episode 8 24
Table 10: Calibration results for episode 9 25
Table 11: Calibration results for episode 10 25
Table 12: Calibration results for episode 11 25
Table 13: Mean ± Standard Deviation of calibration results
across all episodes 26

v

List of Figures
Figure 1: Complete TAC and BrAC data for Subject 5122 16
Figure 2: Complete TAC and BrAC data for Subject 5123 17
Figure 3: Widmark Estimated BrAC with TAC and BrAC data
for episode 1 18

Figure 4: Matthews and Miller Estimated BrAC with TAC and
BrAC data for episode 1 19

Figure 5: Forrest Estimated BrAC with TAC and BrAC data
for episode 1 19

Figure 6: Lewis Estimated BrAC with TAC and BrAC data
for episode 1 20

Figure 7: NHTSA Estimated BrAC with TAC and BrAC data
for episode 1 20

Figure 8: Watson et al. Estimated BrAC with TAC and BrAC data
for episode 1 21

Figure 9: Estimated BrAC curve from Matthews and Miller calibrated
model for episode 1 26

Figure 10: Impulse response functions of each calibrated model for
episode 1 27
vi

Abstract
This research improves the BrAC Estimator software, a MATLAB program
designed to produce estimates of blood or breath alcohol concentration (BAC/BrAC)
from a sensor measuring transdermal alcohol concentration (TAC). Specifically, we aim
to eliminate the need for an initial laboratory alcohol challenge session to calibrate the
device that collects the TAC data and instead calibrate the device using a detailed drink
diary of alcohol consumed. We present the mathematics of the model in the BrAC
Estimator software and explain how the model is mathematically calibrated. We then
show how the calibration process can be improved using linear models to estimate BAC
from reported alcohol consumed. Actual subject data was taken over a period of 18 days
and analyzed using the software. For this data, the model was calibrated using
breathalyzer measurements taken during the initial laboratory session and calibrated
using the estimates of BAC from each of the six linear estimation equations. We compare
the results of calibration from each method to show that calibrating using a drink diary
corresponds to a similar output as calibrating using BrAC measurements.

1

Introduction
Alcohol consumed by humans enters the bloodstream and dissolves in the water
of the blood. Blood alcohol concentration (BAC) is a measure of the concentration of
alcohol in blood, which is usually reported as a percentage by volume. For example, a
BAC of 0.08 would mean the blood is 0.08% alcohol by volume. After alcohol is
absorbed into the bloodstream it leaves the body through different systems, such as
exhaling and diffusion through the skin. Breath alcohol concentration (BrAC) is a
measure of the concentration of alcohol exhaled from the lungs. This is measured using a
breathalyzer, which functions by chemically converting the ethanol in the breath into an
electric current. The output of the breathalyzer is an approximation to BAC. In this paper
we do not distinguish between BAC and BrAC, and use them interchangeably.
Transdermal alcohol concentration (TAC) is a measure of the concentration of alcohol in
the vapor at the surface of the skin, usually reported in milligrams per deciliter.
Approximately 1% of the alcohol consumed diffuses through the interstitial fluid
surrounding the cells in the dermal and epidermal layers of the skin and vaporizes at the
surface where it can be measured with a transdermal alcohol sensor (Rosen & Luczak,
2012).
Several devices have been manufactured to measure TACs to produce a running
TAC signal for a few weeks while the device is worn. These devices include the
WrisTAS
™
, developed by Giner, Inc. in Newton, MA, and SCRAM
©
, developed by
Alcohol Monitoring Systems, Inc. in Denver, CO. Data from these TAC sensors correlate
with BrAC, but have previously only been used as abstinence monitors. The WrisTAS
™

2

is worn on the wrist and SCRAM
©
is worn on the ankle. They both record a TAC signal
continuously while worn. Transdermal alcohol sensing is a useful method for tracking
BAC over a period of time when the devices are interpreted correctly (Rosen & Luczak,
2012).
The goal of this study is to accurately estimate BrAC from the TAC signals. This
can only be done if we know the relationship between the TAC data and the
corresponding BrAC data. In this study, we used the WrisTAS
™
to collect TAC data. We
also use BrAC Estimator software developed on MATLAB to analyze our data. For
estimating BrAC we use a model developed from the properties of ethanol transport from
the blood through the skin to the TAC sensor. There are parameters of this model that
vary from person to person so it must be calibrated individually so that we can accurately
estimate BrAC. These parameters describe the rate of ethanol diffusion through the skin
and the vapor being processed by the TAC sensor. We present the details of the
mathematical model, how it is calibrated, and how it is used to deconvolve BrAC
estimates from TAC signals (Rosen & Luczak, 2012).
There are several phases of this mathematical data analysis system that we follow
to extract BrAC estimates from a TAC signal. First, there is a calibration phase to adjust
the internal parameters of the model to relate the specific wrist device to the individual
wearing it. We need data from the wrist device to compare to a known alcohol standard,
so the calibration phase begins with a laboratory alcohol challenge session. Simultaneous
BrAC measurements and TAC data are obtained and used to fit the necessary parameters.
Then there is a data collection phase where the individual wears the wrist device for the
3

duration of the testing period. This data is analyzed using the calibrated model to produce
estimates of the BrAC data during the testing period (Rosen & Luczak, 2012).
The purpose of this paper is to show that we can complete the calibration phase of
our method without requiring a laboratory alcohol challenge session. For a more practical
solution, we attempt to estimate BAC from a drinking diary and use those data points for
calibration instead of BrAC measurements. The motivation for this change in our method
is that the laboratory alcohol challenge session requires a lengthy participation from the
subject and the researcher to acquire a set of BrAC measurements and TAC data for
calibration. This is a burden we aim to eliminate. Instead, we would ask that the subject
record the amount of alcohol consumed during the first drinking episode. We call this the
drink diary, where we have the data for exactly how much and when all the alcohol is
consumed. We intend to calibrate the wrist device using the drink diary of the first
drinking episode and the simultaneous TAC data.
For a particular drinking episode, given the amount of alcohol an individual
consumed and the time spent consuming alcohol we can estimate BAC during that
episode and afterward until BAC returns to zero. There are estimation equations that take
into account individual differences, such as gender, age, height, and weight, which affect
how the alcohol is processed in the body (Hustad & Carey, 2005). We apply one of these
equations to the drink diary for the first episode and get a set of estimated BACs that can
be used in place of the BrAC measurements that would have been taken during the
original laboratory session. Then the calibration phase continues as before, using the
4

simultaneous estimated BACs and TAC data to fit the model’s parameters. The rest of the
method remains the same.
We present a comparison of the different ways to calibrate the model. We discuss
the challenges of requiring a laboratory session and offer an alternative that uses
estimation equations. The results of calibrating the model using BrAC measurements are
compared to the results of calibrating the model using BAC estimates from each equation
for estimating BAC from a drink diary. We are able to show how well this alternative
method of calibrating the model produces the estimate of BrAC from TAC signals.

5

Deconvolution
In this section, we present the details of the mathematical model discussed in
Rosen and Luczak. We view the system of blood, skin, and TAC sensor as an
input/output model. The model uses BAC at time as the input, , and the TAC data
as the output, . The ethanol in the blood diffuses through the skin and gets read in
through the wrist device, and this is described by a filter kernel function, . The model
is a convolution integral which is of the form

Calibrating the model uses BrAC at time t as the input, as this can be measured or
estimated, and deconvolving the model to determine the parameters of . This is a
blind deconvolution because is not a known function, so it must be parameterized
and estimated, based on what we know about the transport of ethanol through the skin as
a diffusion process. We only have BrAC and TAC at discrete time data points, where the
time between points is . This gives us a discrete time convolution, which is of the form

where

,

, and

is the discrete filter to be estimated.
The flow of ethanol from the blood, through the skin, and detection by the TAC
sensor can be described with a diffusion process model. The alcohol moves through the
interstitial fluid surrounding the skin cells and is released as vapor on the surface of the
skin. Let be the concentration of ethanol in the interstitial fluid, in units of
6

mol/cm
2
, at time in seconds and depth in cm. Let be the thickness of the skin in cm,
and let be the diffusivity in units of cm
2
/sec. Then the diffusion model is

The boundaries of the skin are the outer surface, where , and the bottom of all
layers of skin, where . The boundary conditions are determined by setting the flux
at the boundaries of the skin to be proportional to the difference in concentration of
ethanol on either side of the skin, so we have

where is a constant of proportionality in units of cm/sec, is a parameter that
represents the flow of ethanol from the blood into the interstitial fluid of the skin in units
of mol/(cm
2
× BAC or BrAC units), and is the concentration of ethanol in the blood in
either BAC or BrAC units. At time , there should be no alcohol in the skin, so we
have the initial condition

The TAC sensor processes the ethanol vapor at the surface of the skin, which we assume
is a linear relation from the concentration of ethanol at the surface before it evaporates.
So we have

where is the constant of proportionality in units of TAC units × cm
2
/mol.
7

We have the diffusion model from equations (2.3) – (2.7), which has five
parameters it depends on: , , , , and . They are not all independent, so we can
rewrite the model to depend on only two parameters,

and

:

This model can be written in weak form in order to get a programmable model so that we
can use the computer to find the solution. We use

together with the standard
inner product

to construct the weak form of the diffusion
model. Equation (2.8) becomes

We integrate by parts and apply the boundary conditions so the model simplifies to

where

, with the
initial condition, equation (2.11).
Equation (2.14) is true for all

, but the only way we can numerically
evaluate this model to estimate the parameters

and

is when there is a finite basis for
8

the set of functions, . Therefore, we created a new space of functions

spa

, where each

is a “pup te t” fu ctio , which has support on

and takes on values from to in the shape of a triangle. Then any
function

in

can be written as a linear combination of the

, so we have

, where the

are the coefficients of the basis vectors. It
is much easier to represent a function in this space, as one only needs a vector of
numbers in , which will also be an approximation to functions not in this space.
Equation (2.14) can be represented in matrix form if we let it be true for

so that
every is a linear combination of the basis functions,

. So letting

we
have

for each and putting it in matrix form we get

with the initial condition

where

,

, and

.
Equation (2.16) can also be written with

as

9

where

and

. From this we can get a discrete
time formula for

as in equation (2.2) in convolution form by

and the impulse response function, also from equation (2.2), is given by

where

and

is the vector representation of

in

.
In order to calibrate the model, we construct a least squares problem to solve for
the best fitting parameters,

and

. We look for the parameters that minimize

This optimization problem is solved using an iterative constrained gradient based search.
The model is calibrated using data from one drinking episode, typically the first one
recorded. However it may be calibrated on any episode where both TAC and BrAC data
are available simultaneously, though the parameters will vary depending on which
episode is chosen. Then the episode can be deconvolved with the estimated parameters to
give an approximation of BAC for the entire length of time the wrist device is worn. The
deconvolution signal is obtained by a linear least squares fit to the TAC data with a
constraint that the fit must be positive. This is to ensure that the estimate for BAC is
10

mathematically and physically reasonable. Additionally, when one performs a linear least
squares fit to data, it is possible to over-fit the data which results in oscillations and
excessive magnitudes that are not likely to occur physically. Two nonnegative
regularization parameters,

and

, are introduced into the model so that the fit is
smoother and more natural. These parameters are optimally chosen by an additional
calibration phase using the TAC and BrAC data of the chosen episode. Though one can
deconvolve the entire signal, it is observed that the best approximation of BAC occurs on
the time interval of the drinking episode chosen for calibration. Therefore, to get the most
accurate estimation for BAC it is best to deconvolve each episode separately.

11

Calibration Using Drink Diary
To calibrate the transdermal alcohol sensing model, we need simultaneous TAC
and BrAC data to which we fit the two parameters in the model. Therefore, we need
frequent BrAC readings taken during the first drinking episode while wearing the TAC
sensor. There are a few problems with this method of acquiring the calibration data, and
we will present another method that attempts to solve these problems. Currently, to
calibrate the TAC sensor to a subject, that subject must wear the device during a
laboratory alcohol challenge session. Before the alcohol challenge session begins, the
subject ensures that there is no alcohol in their system, by not drinking alcohol for 24
hours and not eating for 4 hours. Also, the subject agrees to refrain from using any other
drugs for 48 hours before, and throughout the two weeks of participation. The TAC
sensor is worn by the subject for 20 minutes before any alcohol is consumed, and then the
subject drinks a room temperature mixture of ethanol and non-caffeinated sugar-free soda
evenly over a 15 minute period. After consuming the alcohol, the subject rinses out their
mouth, in order to remove the possibility of mouth alcohol affecting BrAC readings, and
takes a BrAC reading 5 minutes after finishing the drink. Readings are taken every 15
minutes until BrAC returns to 0.000. Using this data, we are able to calibrate the TAC
sensor so that we can estimate the BAC for the subject given the TAC data during the
two week period of wearing the wrist device (Rosen & Luczak, 2012).
The method described above for collecting initial calibration data worked for the
research participation already conducted; however, if we intend to expand the usage of
this device, we will need to take into consideration the convenience and ethics of this
12

laboratory session. Having a subject come in to the laboratory without drinking alcohol or
eating beforehand and then made to drink alcohol and sit until BrAC is back to zero is a
large inconvenience for the subject. After drinking, it will likely take a few hours for the
BrAC to reach zero again, which is a considerable amount of time to commit to this
session. Additionally, using this device on certain subjects, such as alcoholics trying to
quit drinking and pregnant women, it would raise ethical concerns to bring a subject in to
a laboratory to administer alcohol to them. It would be more ethical and practical to ask
the subject to keep a drink diary of the alcohol they consumed during the first episode of
drinking while wearing the wrist device. We are then able to estimate the BrAC using
several different linear models that need only the number of standard drinks and time
since the last drink as input. Other parameters needed are properties of the subject, such
as gender and weight.
For this paper, we have data from one female subject who wore a wrist device on
each wrist simultaneously during a two week period. Each wrist is called subject 5122
and 5123. She kept a complete, detailed drink diary in addition to frequent BrAC
measurements for each drinking episode. The program needs only the data from the first
episode to calibrate the model, but it is beneficial to have the rest of the data to compare
to the estimated BAC from the model. The model could be calibrated on any actual
drinking episode where we have the complete data set of TAC data and BrAC
measurements. The user of the program inputs the start and end times of the episode so
that we are only using data from one episode. We would like to calibrate the model using
only the drink diary and TAC data, since we can estimate BrAC from the drink diary. We
13

would still input the start and end times of the episode, but we would use the drink diary
information recorded for that period of time to estimate the BrAC instead of using
frequent BrAC measurements during the drinking episode.
From Hustad and Carey, six equations for estimating BrAC are determined to be
reasonably accurate when BrAC is less than 0.08. These equations reconstruct BAC,
which BrAC measures, using the amount of alcohol consumed and time spent drinking.
The equations also require certain information about the subject, such as height, weight,
age, and gender, with the metabolism rate of a subject set at a constant population
average. A widely used formula is Widmark’s Eq uation, which has been adjusted to
create more accurate equations by Matthews and Miller in 1979, Forrest in 1986, Lewis
in 1986, National Highway Traffic Safety Administration (NHTSA) in 1994, and Watson
et al. in 1981. The BrAC estimation equations are: The Widmark formula:

where for men, and for women the Matthews and Miller formula:

the Forrest formula:

where

, percentage body fat for men is

,
percentage body fat for women is

, and

, the Lewis formula:
14

where for men,

and for women

the NHTSA formula:

where for men, and for women and the Watson et al.
formula:

where total body water for men is
total body water for women is , and

. In these models blood alcohol concentration in g/dl, number of
standard drinks consumed, where a standard drink is any drink containing about 0.6 oz of
pure alcohol, which is equivalent to a 12 oz beer, a 5 oz glass of wine, or 1.5 oz of 80-
proof hard liquor, gender constant (7.5 for men, 9.0 for women), weight in
pounds,

metabolism rate of alcohol per hour (0.017 g/dl), time in hours since
first sip of alcohol to time of computation, total volume in ml of drinks consumed
multiplied by the percent of alcohol in the drink multiplied by the density of alcohol (0.79
g/ml) divided by 10, weight in kg, height in meters, age in years, and
height in cm. The basis of these formulas is the evidence that alcohol is absorbed into the
water of the bloodstream and is then released from the body at a constant rate regardless
of the amount that has been consumed. The total amount of water in the body affects how
much alcohol is necessary to reach a certain BAC, and this is influenced by parameters
15

such as gender, age, height, weight, and percentage of body fat. In addition, the liver can
only process so much alcohol per hour, so that is why a population average is used for the
rate of alcohol metabolism per hour. To keep the formulas from producing negative
BACs, we added a nonlinear saturation condition where we set the estimated BAC to
zero when the result of the estimation equation would have been negative.
The more accurate the drink diary is, the more accurate these equations will be for
estimating BrAC. This means keeping notes frequently and observing precise amounts of
alcohol consumed, which may be difficult when under the influence of alcohol, and
outside a laboratory setting this is a heavy burden on the subject. The drink diary is
merged with the TAC data from the wrist device so that the time of the drinking episode
is in sync with the time of the drink diary. The program lets the user identify the start and
end times of the drinking episode on which to calibrate the TAC sensor, and selects the
drink diary information for that episode. That information is then converted into BrAC
using one of the five BrAC estimation equations and runs the rest of the program as if the
input were actual BrAC measurements. This estimated BrAC acts in place of BrAC
during the calibration phase of the program. The performance of the five linear models
can be compared to each other, as well as to the actual BrAC measurements for this data
set because we have this information recorded for every drinking episode of subject 5122
and 5123. We are ultimately looking for an accurate way to calibrate the transdermal
alcohol sensing model without requiring BrAC measurements.

16

Numerical Results
We have 18 days of data during which the subject wore a wrist device on each
wrist and recorded 11 drinking episodes for the drink diary. Frequent BrAC
measurements were also taken and plotted on the same graph as the raw TAC data for
comparison. The entire datasets of alcohol level over time for both subjects 5122 and
5123 are shown in figures 1 and 2, respectively, which shows that the wrist device is
consistent since the two datasets are very similar. In these figures, the horizontal axis
reports the number of the TAC data point, which have been recorded every 5 minutes.
The left vertical axis shows the TAC data in blue, measured in milligrams per deciliter.
The right vertical axis shows the BrAC data in red, measured in percent alcohol.
Figure 1: Complete TAC and BrAC data for Subject 5122

500 1000 1500 2000 2500 3000 3500 4000 4500
0
50
Field Data for Patient: 5122
Data Point
Alcohol Level (TAC) (mg/dl)

500 1000 1500 2000 2500 3000 3500 4000 4500
0
0.05
0.1
Alcohol Level (BrAC) (% alcohol)
TAC Data
BrAC Data
17

Figure 2: Complete TAC and BrAC data for Subject 5123

Table 1 summarizes the drink diary, giving the number of standard drinks consumed and
the total duration of time spent drinking. A more detailed drink diary is recorded and
synced with the TAC data for calibration purposes.
Table 1: Summary of Drink Diary
Drinking Episode Standard Drinks Duration of Drinking (min)
1 2 15
2 1 45
3 2 75
4 3.5 150
5 3 300
6 1 30
7 1.5 90
8 1 30
9 2.5 120
10 1 60
11 3 120

500 1000 1500 2000 2500 3000 3500 4000 4500
0
50
Field Data for Patient: 5123
Data Point
Alcohol Level (TAC) (mg/dl)

500 1000 1500 2000 2500 3000 3500 4000 4500
0
0.05
0.1
Alcohol Level (BrAC) (% alcohol)
TAC Data
BrAC Data
18

We can convert this drink diary into estimated BrAC so that we can calibrate the model
on any drinking episode using the same method as though we had actual BrAC
measurements. It is most typical to calibrate the model based on the first drinking
episode, so we plot the TAC data from the corresponding data points and estimate BrAC
from the first episode of the drink diary by each of the linear estimation equations, (3.1) –
(3.6). We also plot the actual BrAC measurements from the first episode to show that
they are similar to the estimated quantities. Figures 3 – 8 represent this data for each
estimation method using the data of the first drinking episode of subject 5122. Both the
BrAC data and estimated BrAC are reported in percent alcohol and correspond to the
right hand vertical axis. The TAC data is as above in figures 1 and 2 on the left hand
vertical axis. The horizontal axis is now reported in hours.
Figure 3: Widmark Estimated BrAC with TAC and BrAC data for episode 1

0 2 4 6 8 10 12 14 16
0
20
40
Adjusted Challenge Data for Patient: 5122
Time (hours)
Alcohol Level (TAC) (mg/dl)

0 2 4 6 8 10 12 14 16
0
0.05
0.1
Alcohol Level (BrAC) (% alcohol)
TAC Data
BrAC Data
Estimated BrAC
19

Figure 4: Matthews and Miller Estimated BrAC with TAC and BrAC data for episode 1

Figure 5: Forrest Estimated BrAC with TAC and BrAC data for episode 1

0 2 4 6 8 10 12 14 16
0
20
40
Adjusted Challenge Data for Patient: 5122
Time (hours)
Alcohol Level (TAC) (mg/dl)

0 2 4 6 8 10 12 14 16
0
0.05
0.1
Alcohol Level (BrAC) (% alcohol)
TAC Data
BrAC Data
Estimated BrAC
0 2 4 6 8 10 12 14 16
0
20
40
Adjusted Challenge Data for Patient: 5122
Time (hours)
Alcohol Level (TAC) (mg/dl)

0 2 4 6 8 10 12 14 16
0
0.05
0.1
Alcohol Level (BrAC) (% alcohol)
TAC Data
BrAC Data
Estimated BrAC
20

Figure 6: Lewis Estimated BrAC with TAC and BrAC data for episode 1

Figure 7: NHTSA Estimated BrAC with TAC and BrAC data for episode 1

0 2 4 6 8 10 12 14 16
0
20
40
Adjusted Challenge Data for Patient: 5122
Time (hours)
Alcohol Level (TAC) (mg/dl)

0 2 4 6 8 10 12 14 16
0
0.05
0.1
Alcohol Level (BrAC) (% alcohol)
TAC Data
BrAC Data
Estimated BrAC
0 2 4 6 8 10 12 14 16
0
20
40
Adjusted Challenge Data for Patient: 5122
Time (hours)
Alcohol Level (TAC) (mg/dl)

0 2 4 6 8 10 12 14 16
0
0.05
0.1
Alcohol Level (BrAC) (% alcohol)
TAC Data
BrAC Data
Estimated BrAC
21

Figure 8: Watson et al. Estimated BrAC with TAC and BrAC data for episode 1

Tables 2 – 13 present the calibration results for subject 5122 using the different
methods of calibration. We have BrAC measurements and drink diary information for
each episode so each table represents one of the 11 drinking episodes. Table 13
represents an average for each parameter across all episodes. We present the mean plus or
minus the standard deviation of each parameter so we can see if the methods of BrAC
estimation are valid for use in the calibration. The model is calibrated by estimating
BrAC from each equation (3.1) – (3.6) and using the estimation in place of the BrAC
measurements. For comparison, the model is also calibrated using the BrAC
measurements. Without using drink diary information, the calibration parameters vary
widely over all 11 episodes so it is hard to say what the correct parameters should be and
if using drink diary information achieves the same correct parameters. Occasionally, it
0 2 4 6 8 10 12 14 16
0
20
40
Adjusted Challenge Data for Patient: 5122
Time (hours)
Alcohol Level (TAC) (mg/dl)

0 2 4 6 8 10 12 14 16
0
0.05
0.1
Alcohol Level (BrAC) (% alcohol)
TAC Data
BrAC Data
Estimated BrAC
22

seems that the program calibrates the model somewhat unnaturally when it results in one
parameter being much higher than the other, as seen in Matthews and Miller in Table 6.
This may be due to the program trying to fit unnatural data or the model not being
accurate in certain types of drinking episodes. However, comparing the mean and
standard deviation of the calibration parameters obtained from each method shows that
using drink diary information to calibrate the model gives appropriate quantities for these
parameters. We can see this in table 13 that there are values of the calibration parameters
that fit close to every mean value, within a standard deviation. Based on these calibration
results, it is reasonable to say that using drink diary information calibrates the model just
as well as using BrAC measurements.
Table 2: Calibration results for episode 1

BrAC measurements 0.93 1.24 0.10 0.10
Widmark 0.56 1.54 0.10 0.10
Matthews and Miller 0.77 1.14 0.10 0.10
Forrest 0.55 1.56 0.10 0.10
Lewis 0.53 1.64 0.10 0.10
NHTSA 0.67 1.29 0.10 0.10
Watson et al. 0.62 1.37 0.10 0.10

Table 3: Calibration results for episode 2

BrAC measurements 0.47 1.17 0.10 0.10
Widmark 0.52 1.33 0.10 0.14
Matthews and Miller 0.60 0.94 0.11 0.13
Forrest 0.52 1.35 0.10 0.14
Lewis 0.50 1.44 0.10 0.14
NHTSA 0.57 1.08 0.11 0.13
Watson et al. 0.55 1.15 0.12 0.13

23

Table 4: Calibration results for episode 3

BrAC measurements 1.55 1.77 0.10 0.11
Widmark 1.35 2.20 0.10 0.11
Matthews and Miller 4.22 1.54 0.10 0.11
Forrest 1.31 2.23 0.10 0.11
Lewis 1.18 2.37 0.10 0.11
NHTSA 2.15 1.79 0.10 0.11
Watson et al. 1.75 1.92 0.10 0.11

Table 5: Calibration results for episode 4

BrAC measurements 0.29 1.20 0.21 0.90
Widmark 0.27 1.12 0.21 0.90
Matthews and Miller 0.28 0.88 0.21 0.90
Forrest 0.27 1.13 0.21 0.90
Lewis 0.26 1.18 0.21 0.90
NHTSA 0.27 0.97 0.21 0.90
Watson et al. 0.27 1.02 0.21 0.90

Table 6: Calibration results for episode 5

BrAC measurements 2.35 1.47 0.16 0.37
Widmark 1.23 1.54 0.15 0.71
Matthews and Miller 12.53 1.05 0.26 1.99
Forrest 1.17 1.58 0.15 0.71
Lewis 1.00 1.69 0.20 0.65
NHTSA 2.92 1.22 0.27 1.39
Watson et al. 2.04 1.32 0.27 1.39

24

Table 7: Calibration results for episode 6

BrAC measurements 0.46 1.04 0.11 0.25
Widmark 0.34 1.90 0.14 0.28
Matthews and Miller 0.39 1.32 0.13 0.32
Forrest 0.33 1.93 0.14 0.28
Lewis 0.32 2.07 0.14 0.28
NHTSA 0.37 1.53 0.14 0.28
Watson et al. 0.36 1.64 0.14 0.28

Table 8: Calibration results for episode 7

BrAC measurements 0.82 0.83 0.11 0.20
Widmark 1.05 1.23 0.10 0.11
Matthews and Miller 2.23 0.72 0.13 0.21
Forrest 1.04 1.26 0.10 0.11
Lewis 1.00 1.38 0.10 0.11
NHTSA 1.59 0.89 0.11 0.20
Watson et al. 1.35 0.99 0.10 0.11

Table 9: Calibration results for episode 8

BrAC measurements 0.53 0.20 0.10 0.10
Widmark 0.49 2.75 0.10 0.10
Matthews and Miller 0.60 1.76 0.10 0.10
Forrest 0.48 2.81 0.10 0.10
Lewis 0.47 3.01 0.10 0.10
NHTSA 0.55 2.10 0.10 0.10
Watson et al. 0.53 2.30 0.10 0.10

25

Table 10: Calibration results for episode 9

BrAC measurements 1.11 1.53 0.19 1.08
Widmark 0.96 1.10 0.15 0.71
Matthews and Miller 2.07 0.74 0.21 0.90
Forrest 0.94 1.12 0.15 0.70
Lewis 0.87 1.20 0.15 0.70
NHTSA 1.45 0.87 0.21 0.90
Watson et al. 1.24 0.94 0.15 0.70

Table 11: Calibration results for episode 10

BrAC measurements 2.27 1.76 0.14 0.28
Widmark 1.18 3.65 0.11 0.25
Matthews and Miller 2.03 2.17 0.11 0.25
Forrest 1.17 3.71 0.11 0.25
Lewis 1.17 3.98 0.11 0.25
NHTSA 1.59 2.67 0.11 0.25
Watson et al. 1.41 2.97 0.11 0.25

Table 12: Calibration results for episode 11

BrAC measurements 0.63 0.97 0.10 0.11
Widmark 2.87 0.40 0.20 1.14
Matthews and Miller 6.65 0.32 0.21 0.90
Forrest 2.79 0.41 0.20 1.14
Lewis 2.52 0.42 0.20 1.14
NHTSA 4.39 0.35 0.20 1.40
Watson et al. 3.75 0.37 0.20 1.14

26

Table 13: Mean ± Standard Deviation of calibration results across all episodes

BrAC measurements 1.04±0.72 1.20±0.45 0.13±0.04 0.33±0.34
Widmark 0.98±0.73 1.71±0.89 0.13±0.04 0.41±0.38
Matthews and Miller 2.94±3.72 1.14±0.53 0.15±0.06 0.54±0.59
Forrest 0.96±0.71 1.74±0.91 0.13±0.04 0.41±0.38
Lewis 0.89±0.63 1.85±0.98 0.14±0.05 0.41±0.38
NHTSA 1.50±1.27 1.34±0.65 0.15±0.06 0.52±0.52
Watson et al. 1.26±1.02 1.45±0.72 0.15±0.06 0.47±0.48

Figure 9 shows the estimated BrAC curve for episode 1 from the calibrated model using
the Matthews and Miller equation for estimating BrAC. This is representative of the
results of determining BrAC from any of the calibrated models. Specifically, the
calibration parameters for this model are

,

,

, and

.
Figure 9: Estimated BrAC curve from Matthews and Miller calibrated model for episode 1

0 1 2 3 4 5 6 7 8
0
0.01
0.02
0.03
0.04
0.05
0.06
Results of Regularized TAC Data Inversion: Patient ID: 5122
Time (hours)
Alcohol Level (BrAC) (% alcohol)

Estimated BrAC
Challenge BrAC Data
Estimated Challenge BrAC
27

Figure 10 plots the calibrated impulse response functions based on calibrating the
model on episode 1 by the BrAC measurements and by each method of estimating BrAC.
Equivalently, they are the convolution kernals, , from equation (2.1) where the input
is an instantaneous impulse. The impulse response functions represent the TAC data if
the BrAC is an instantaneous impulse of 1 mg/dl, or 0.001% alcohol. These curves are all
very similar, which is consistent with how close the corresponding calibration parameters
are for the seven different calibrations.
Figure 10: Impulse response functions of each calibrated model for episode 1

We have shown that BrAC can be sufficiently estimated from an accurate drink
diary and used with simultaneous TAC data to calibrate the skin-TAC sensor system. It
does not matter which of the equations we use to estimate BrAC from the drink diary
because they all produce similar estimations. The calibration tables show that the
0 1 2 3 4 5 6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Time (hours)
Alcohol Level (TAC) (mg/dl)

BrAC measurements
Widmark
Matthews and Miller
Forrest
Lewis
NHTSA
Watson et al.
28

parameters do not vary too much when considering a specific episode. The inconsistency
of the parameters across different episodes is a separate concern, involving adjusting
terms of the model relating alcohol, blood, and skin, which this thesis does not address.
We calibrated this model using drink diary information and compared the results to the
calibrated models using BrAC measurements. We compared the different methods of
calibration by looking at the calibration parameters and the impulse response graphs. The
calibrated models from drink diary information are similar enough to those of the original
calibration method so these numerical results support the method of replacing frequent
BrAC measurements with an accurate drink diary.

29

References
Forrest, A. R. W. (1986). The estimatio of Widmark’s factor. Journal of the Forensic
Science Society, 26, 249-252.
Hustad, J. T. P., & Carey, K. B. (2005). Using calculations to estimate blood alcohol
concentrations for naturally occurring drinking episodes: A validity study.
Journal of Studies on Alcohol, 66, 130-138.
Lewis, M. J. (1986). The individual and the estimation of his blood alcohol concentration
from i take, with particular refere ce to the “hip -flask” dri k. Journal of the
Forensic Science Society, 26, 19-27.
Matthews, D. B., & Miller, W. R. (1979). Estimating blood alcohol concentration: Two
computer programs and their applications in therapy and research. Addictive
Behaviors, 4, 55-60.
National Highway Traffic Safety Administration. (1994). Computing a BAC estimate.
Washington: Department of Transportation.
Rosen, I. G., & Luczak, S. (2012). Mathematical modeling of BrAC from transdermal
alcohol data: The introduction of BrAC estimator software program and its
comparison to alternate real-time data collection methods. Manuscript submitted
for publication, Behavioral Research Methods.
Watson, P. E., Watson, I. D., & Batt, R. D. (1981). Prediction of blood alcohol
concentrations in human subjects: Updating the Widmark equation. Journal of
Studies on Alcohol, 42, 547-556.

Abstract (if available)

Abstract This research improves the BrAC Estimator software, a MATLAB program designed to produce estimates of blood or breath alcohol concentration (BAC/BrAC) from a sensor measuring transdermal alcohol concentration (TAC). Specifically, we aim to eliminate the need for an initial laboratory alcohol challenge session to calibrate the device that collects the TAC data and instead calibrate the device using a detailed drink diary of alcohol consumed. We present the mathematics of the model in the BrAC Estimator software and explain how the model is mathematically calibrated. We then show how the calibration process can be improved using linear models to estimate BAC from reported alcohol consumed. Actual subject data was taken over a period of 18 days and analyzed using the software. For this data, the model was calibrated using breathalyzer measurements taken during the initial laboratory session and calibrated using the estimates of BAC from each of the six linear estimation equations. We compare the results of calibration from each method to show that calibrating using a drink diary corresponds to a similar output as calibrating using BrAC measurements.

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Asset Metadata

Creator Coste, Katherine A. (author)

Core Title Determining blood alcohol concentration from transdermal alcohol data: calibrating a mathematical model using a drinking diary

School College of Letters, Arts and Sciences

Degree Master of Science

Degree Program Applied Mathematics

Publication Date 07/10/2013

Defense Date 06/24/2013

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag BrAC estimation,drinking diary,OAI-PMH Harvest,transdermal alcohol sensor

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Rosen, I. Gary (committee chair), Luczak, Susan (committee member), Wang, Chunming (committee member)

Creator Email katherinecoste@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-285337

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Document Type Thesis

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Rights Coste, Katherine A.

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Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

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