17-03-2012, 11:21 AM
Individualized Short-Term Core Temperature Prediction in Humans Using Biomathematical Models
Abstract
This study compares and contrasts the ability of three
different mathematical modeling techniques to predict individualspecific
body core temperature variations during physical activity.
The techniques include a first-principles, physiology-based (SCENARIO)
model, a purely data-driven model, and a hybrid model
that combines first-principles and data-driven components to provide
an early, short-term (20–30 min ahead) warning of an impending
heat injury. Their performance is investigated using two
distinct datasets, a Field study and a Laboratory study. The results
indicate that, for up to a 30 min prediction horizon, the purely
data-driven model is the most accurate technique, followed by
the hybrid. For this prediction horizon, the first-principles SCENARIO
model produces root mean square prediction errors that
are twice as large as those obtained with the other two techniques.
Another important finding is that, if properly regularized and developed
with representative data, data-driven and hybrid models
can be made “portable” from individual to individual and across
studies, thus significantly reducing the need for collecting developmental
data and constructing and tuning individual-specific
models.
Index Terms—Core temperature prediction, data-driven model,
first-principles model, heat injury, hybrid model, regularization,
time-series analysis.
I. INTRODUCTION
HEAT injury is the third leading cause of death of student
athletes at U.S. schools [1]. Heat injury is also a problem
for the armed forces, especially during deployments to localities
with very hot climates. Despite thorough prevention programs
developed by the U.S. Army Research Institute of Environmental
Medicine (USARIEM), from 2003 through 2005, there
Manuscript receivedMarch 27, 2007; revised September 11, 2007. This work
was supported in part by the Combat Casualty Care and the Military Operational
Medicine Research Programs of the U.S. Army Medical Research and
Materiel Command, Fort Detrick, MD and in part by the Natick Soldier System
Center. Asterisk indicates the corresponding author.
A. V. Gribok is with the Bioinformatics Cell, Telemedicine and Advanced
Technology Research Center (TATRC), U.S. Army Medical Research and Materiel
Command (USAMRMC), Fort Detrick, MD 21702 USA. He is also with
the Nuclear Engineering Department, University of Tennessee, Knoxville, TN
37996 USA (e-mail: agribok[at]bioanalysis.org).
M. J. Buller is with the U.S. Army Research Institute of Environmental
Medicine (USARIEM), Natick, MA 01760 USA. He is also with Brown University,
Providence, RI 02912 USA (e-mail: mark.j.buller[at]us.army.mil).
∗J. Reifman is with the Biotechnology High Performance Computing
Software Applications Institute, Telemedicine and Advanced Technology Research
Center (TATRC), U.S. Army Medical Research and Materiel Command
(USAMRMC), Fort Detrick, MD 21702 USA (e-mail: jaques.reifman@us.
army.mil).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TBME.2007.913990
were over 4401 heat injuries in the armed forces, of which 784
were heat strokes and 3617 were heat exhaustions [2]. There
were an additional 17 heat-related fatalities during this time period.
Although heat injuries are considered to be preventable,
a previously published study showed that humans lack warning
mechanisms to signal an impending serious heat injury [3];
hence, in certain situations, a reliable system for real-time continuous
monitoring and prediction of body core temperature
would be highly desirable. Such a prediction system, coupled
with the known clinical limit of 40 ◦C [4], could potentially
prevent heat-related injuries.
Recent advances in the ability to monitor physiology variables
have resulted from the development of new biosensors
and information-processing capabilities. These capabilities have
a direct impact on how closely a person’s state can be monitored
during civilian activities or during military operations,
including the possibility of predicting changes in many vital
physiological variables, such as body core temperature, heart
and respiratory rates, and even such subtleties as level of alertness
and performance. The technological breakthroughs in the
development of hardware and firmware were also accompanied
by an equally profound and significant progress in such
fields as data mining and machine learning. New technologies
to collect and store relatively large amounts of physiological
data in the field allow researchers to explore new opportunities
in data-driven methods to forecast physiological variables and
status.
For example, the Warfighter Physiological Status Monitoring
(WPSM) program at the U.S. Army Medical Research
and Materiel Command seeks to develop a soldier-wearable,
computer-based system for providing commanders and medics
with critical physiological status information about dismounted
war fighters [5], [6]. The WPSM system has two primary aims:
the first is to prevent nonbattle injuries, such as heat stroke
and dehydration, and the second is to optimize casualty management
through improved casualty detection, diagnostics, and
triage. These aims require an array of sensors, a personal area
network, and data management software as well as a variety of
decision-support algorithms formonitoring and predicting a soldier’s
physiological status. In this paper, we focus on mathematical
modeling techniques that can be used to prevent impending
nonbattle heat injuries, such as heat exhaustion and heat stroke.
We compare and contrast the ability of three types of models (a
first-principles model, a purely data-driven model, and a hybrid
model that combines first-principles and data-driven components)
to produce short-term (20–30 min), individual-specific
0018-9294/$25.00 © 2008 IEEE
1478 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 55, NO. 5, MAY 2008
predictions of body core temperature variations during physical
activity.1
Physiological models commonly rely on first-principles
knowledge about various mechanisms in the human body and
their associated dynamics. Although some underlying physiological
phenomena are not well understood and are therefore
unmodeled, the resulting first-principles models may still
be effective in predicting some population-average responses
with certain fidelity. However, unless the model parameters
are constantly adjusted, based, for example, on measurements
from a specific individual, in general, first-principles models
are not capable of representing interindividual variability
[7], [8], leading to inaccurate predictions for specific individuals.
Individuals with similar anthropomorphic characteristics
and subject to the same workload and environmental
conditions may yield very different physiological responses.
Interindividual variation in physiological response is particularly
critical at limiting thresholds of physiological health, such
as at extreme values of core temperature, where small variations
can make a difference between a suitable recovery and
an irreversible pathological condition. The need to represent
interindividual variability can be addressed by developing models
that utilize historic and current data that are specific to the
individual.
One approach to improve the fidelity of first-principles models
and account for interindividual variability is to incorporate
data-driven or “black box” models into the first-principles
model to create a “hybrid” model [9]. In this case, the datadriven
portion of the hybrid model is intended to capture the
dynamics and the physiological idiosyncrasies of each particular
individual, which the first-principles model cannot capture,
by “learning,” during the “training” phase, the residuals
between predictions produced by the first-principles model
and the actual measurements. This allows hybrid models to
account for interindividual variability and also for parts of
the poorly modeled dynamics. The hybrid approach was introduced
to the physiological community in a previous study
[9], where different hybrid schemes were presented and contrasted.
Hybrid models have been widely used in system identification
and control in industrial processes and have proven
to be quite effective [10], [11]. Hybrid modeling of physiological
dynamics holds equal promise in this regard [12],
[13].
Another approach to physiological predictions is to employ a
purely data-driven model. A stand-alone, data-driven model can
be trained on historical data, and subsequently used to predict
future unknown data. The historical data can include independent
variables related to the predicted variable aswell as delayed
instances of the predicted variable itself, that is, previous core
temperature measurements in this case. An inherent limitation
of purely data-driven models is their inability to extrapolate reliably
beyond the distribution of the “training” data. However,
1In collecting the data presented in this manuscript, the investigators adhered
to the policies for protection of human subjects as prescribed in Army Regulation
70–25, and the research was conducted in adherence with the provisions of 45
CFR Part 46. The subjects gave their informed consent for the laboratory study
after being informed of the purpose, risks, and benefits of the study.
linear data-driven models are quite often good extrapolators if
the underlying dependencies can be reasonably modeled by linear
laws. Furthermore, many physiological variables are tightly
bounded by homeostatic limits, thus simplifying the problem
of collecting data that cover all physiologically plausible situations.
These provide an opportunity to properly train linear
data-driven models on representative samples of historical
data and determine their generalization effectiveness, including
their ability to be made “portable” from one individual to
another.
Another general limitation of data-driven modeling is the
possibility of “excessive explanation” of the training data, leading
to an “overfitted” model with poor generalization capabilities.
The problem of overfitting is quite often understated
in the case of linear data-driven models; however, this effect
is as detrimental in linear models as it is in their nonlinear
counterparts. This paper demonstrates that proper regularization
of purely data-driven models and the data-driven
portion of hybrid models is crucial to their generalization capabilities,
since it precludes overfitting and produces models
that capture the underlying data dependencies but not their
idiosyncrasies.
II. METHODS
A. First-Principles SCENARIO Model
The first-principles SCENARIO model [14], [15], developed
at USARIEM,was designed to estimate and predict core temperature,
heart rate, and sweat rate, without requiring prior knowledge
and direct measurement of these physiological variables.
The underlyingmodel for SCENARIO simulates the time course
of core temperature variations, while taking into account different
factors that affect human thermoregulation. The temperature
distribution within the human body is represented by a lumpparameter
model consisting of six concentric cylindrical compartments.
Heat flowis then modeled by a set of macroscopic energy
conservation equations based on heat convection between
the central blood compartment and the adjacent core, muscle,
fat, and vascular skin compartments; radial heat conduction between
every pair of adjacent compartments; and air convection,
radiation, and sweat evaporation between the superficial avascular
skin layer and the environment and transition through the
clothing [14], [15]. The energy conservation equations are represented
by a set of six ordinary differential equations that can
be expressed as
dT
dt
= A(t)T(t) + B(t) (1)
where T(t) ∈ R6×1 is a vector representing the bulk temperatures
in each of the six modeled compartments, and
A(t) ∈ R6×6 is a time-varying matrix determined by parameters,
such as the conductance between two adjacent compartments
and blood flow between the compartments. The vector
B(t) ∈ R6×1accounts for the secondary inputs to the system,
and it is primarily governed by the metabolic rate in each of
the compartments, as well as the respiration rate. The various
GRIBOK et al.: INDIVIDUALIZED SHORT-TERM CORE TEMPERATURE PREDICTION IN HUMANS USING BIOMATHEMATICAL MODELS 1479
factors that affect human thermoregulation and used as input to
SCENARIO include:
1) environmental: mean radiant temperature, ambient temperature,
relative humidity, wind speed;
2) activity: walking speed, pack weight (load), terrain factor,
slope/grade, water intake;
3) individual characteristics: age, weight, height, fat percentage;
Being a first-principles model, SCENARIO does not use past
temperature measurements to produce future core temperature
predictions. Another advantage is that, based on the range of
applicability of each underlying model component, the range
of applicability of the overarching model can be determined a
priori. In addition, SCENARIO can predict other physiological
variables, such as heart and sweat rates. However, because SCENARIO
was designed as a mission-planning tool, as opposed to
an early thermal warning system, it is not expected to perform as
well as customized models for short-term temperature predictions,
where core temperatures are highly correlated. Although
SCENARIO’s input parameters are specific to an individual’s
characteristics, internally, it does not represent parameter model
differences to fully account for interindividual variability. Additionally,
since all parameters are estimated on the basis of experimental
data, inherent observation error and limited sample size
may lead to discrepancies that, compounded, could contribute to
model inaccuracy. Furthermore, due to simplifying modeling assumptions
and unmodeled (unknown) physiology, SCENARIO
does not fully represent some of the physiological dynamics.
Hence, SCENARIO is partly used here as a benchmark, and
it is selected among other first-principles models [16]–[18] because
it has been traditionally used by the Army to analyze
the human response to heat stress and was readily available to
the authors.We acknowledge, however, that the reported results
are only applicable to SCENARIO and cannot be generalized
to other first-principles models, which may demonstrate better
performance under similar conditions.
B. Data-Driven Modeling
Data-driven linear models have been used for time-series prediction
since the early 1970s [19]. One of the most widely
used linear models is the autoregressive (AR) model [10],
which allows for the inference of estimates ˆyn , at time n,
n = m + 1, . . . , N, of signal y as a function of previous observations
ˆyn =
m
i=1
biyn−i + εn . (2)
where b represents the vector of AR coefficients to be determined,
εn denotes white noise with unknown variance, N
denotes the number of data samples, and m is the order of
the model, i.e., the number of previous measurements used
to predict the future measurement ˆyn . Interchanging ˆyn for
yn , and defining the (N − m) × (m) design matrix U and the
Fig. 1. Data-driven approach to physiological time-series prediction; y is the
actual core temperature measurement, ˆy is the predicted core temperature, ε
is the residual between the measured core temperature and the predicted core
temperature, and inputs represent exogenous data into to the system, such as
ambient temperature and past measurements of core temperature. The crossing
of the AR box signifies that the AR coefficients are computed during the training
phase.
(N − m) × (1) and (m) × (1) vectors y and b, respectively, as
U =
ym ym−1 · · · y1
ym+1 ym · · · y2
...
...
. . .
... yN −
1
yN −
2
·
·
·
yN −
m
,
y =
ym+1
ym+2
...
yN
, b=
b1
b2
...
bm
(3)
we arrive at an overdetermined system of linear equations. This
system can be solved for b by the least-squares (LS) method,
which seeks b that minimizes
argmin
b
y − Ub2 (4)
provided the design matrix U is well-conditioned. In addition to
the estimation of the coefficients b, the model’s order also needs
to be determined, which can be done by using some analytical
criterion, like the minimum description length approach [20]
and Akaike information criterion [21], or by cross-validation.
Data-driven models are generally used in problems where
obtaining a first-principles model is either impractical or difficult
due to excessive complexity of the underlying phenomena to
be modeled, and it was a motivating factor for this study. A
schematic diagram of the data-driven approach is presented in
Fig. 1. The advantage of the data-driven approach is that the
explicit relationships between the input–output variables in the
modeled phenomenon do not need to be known and can be
“learned” during the “training” phase. The approach, however,
is highly dependent on data availability and on the quality of the
available data. Another difficulty is that learning input–output
dependencies from noisy data constitutes an ill-posed problem,
since several models may explain the training data quite well,
generally due to model overfitting, although not all models will
posses good generalization capability.
Data-driven models can also be nonlinear, represented by
artificial neural networks (ANNs), for example. The difference
between AR and ANN models is that AR models can only
capture linear dependencies present in the data, while ANNs
can also accommodate nonlinear relationships. However, due
1480 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 55, NO. 5, MAY 2008
Fig. 2. Hybrid approach to physiological time-series predictions; z is the
SCENARIO core temperature estimates, y is the actual core temperature measurements,
ˆy is the predicted value of the residual ε (i.e., the difference between
the SCENARIO prediction and the core temperature measurement), inputs are
exogenous inputs to the system, and δ is the residual between ˆy and ε.
to the presence of local minima in the cost function, ANNs
may be harder to train. Also, in cases where the process input–
output dependencies are linear, they provide no added benefit.
In a previous core temperature prediction study by our group,
ANNs failed to outperform linear models [13].
C. Hybrid Modeling
Another modeling approach is the hybrid technique, which
tries to capitalize on the best parts of both models—firstprinciples
and data-driven. The general idea of hybrid modeling
is presented in Fig. 2, where the data-driven component is represented
by an AR model. The hybrid approach first attempts to
predict a data value for a physiological variable using the firstprinciples
model. The residual value ε of this prediction (i.e.,
the difference between the first-principles model prediction and
the measured value) is then presented to a data-driven model
as a target signal, and the data-driven model is trained to fit ε
based on its past values and, possibly, exogenous inputs. After
the training is complete, new data are predicted by adding the
predictions of the first-principles model with those of the datadriven
model. In our implementation, only delayed instances of
the residual signal ε are used as inputs to the data-driven portion
of the hybrid.
An important difference from the purely data-driven approach
is that, in hybrid modeling, the data-driven component learns the
residuals between the first-principles predictions and the actual
measurements, while in purely data-driven modeling, the model
learns the actual measurements. Several arguments have been
put forward to justify the use of hybrid modeling for time-series
predictions. For example, it was shown previously [22] that,
provided the first-principles model has the same form as the
true process model, the hybrid is guaranteed to converge to the
true model as the amount of training data increases indefinitely.
Another argument is that the residualsmay be easier to learn than
the actual measurements [23] because the residuals only cover
a subspace of the whole process space. Significant successes
in applying hybrid models have been reported in chemical and
biochemical engineering [23]. However, to produce accurate
predictions, hybrid models require high-fidelity first-principles
models capable of accurately predicting both the training data
and the testing data. If they fail to produce good predictions for
the training data, the target signal for the data-driven part of the
hybrid will not be adequate. Also, if they fail to accuratelymodel
the testing data, the hybrid predictions will not be accurate, since
in this case, the overall prediction error will be dominated by the
error produced by the first-principles component of the model.
D. Regularization of the Data-Driven and Hybrid Models
As mentioned earlier, fitting a data-driven AR model to data
(either as a stand-alone module or as part of a hybrid model) requires
estimation of the AR coefficients as one of the steps. The
coefficients are usually determined by minimizing the LS functional
in (4). Unfortunately, due to the highly correlated nature
of the core temperature signal, the designmatrix U is quite often
ill-conditioned or even numerically rank deficient. This causes
the estimates of the AR coefficients b to be highly unstable, producing
poor-quality predictions, i.e., degraded generalization.
The reason for the degraded performance is that the unconstrained
minimization of (4), when U is ill-conditioned, causes
the solution to be dominated by high-frequency components
that overfit the training data [24]. The practical consequences of
the ill-conditioning of the design matrix U are demonstrated in
Section III.
It is well known that the LS solution to (4) yields an unbiased
estimator with the smallest variance among unbiased estimators
[25]. Although unbiasedness is intuitively desired, in practice, it
could be quite useless due to the potential large variance of the
unbiased estimator. To deal with this problem, a class of biased
estimators known as regularized least squares was proposed
by Tikhonov [24]. In this method, the minimization of (4) is
replaced by the minimization of the augmented functional
argmin
b
y − Ub2 + λ2 Lb2 (5)
where the regularization parameter λ controls the tradeoff between
the smoothness of the solution and its fit to the training
data, and L is a well-conditioned matrix; for example, a discrete
approximation of a second-order derivative operator was used
in this study. The major benefit of the regularized LS estimate is
that it reduces the variance of the solution by introducing a small
bias to generate a much smaller estimation error, defined as the
variance plus the square of the bias between the true (unknown)
parameter and its estimate [26].