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## Revista Brasileira de Anestesiologia

##
*Print version* ISSN 0034-7094*On-line version* ISSN 1806-907X

### Rev. Bras. Anestesiol. vol.54 no.3 Campinas May/June 2004

#### http://dx.doi.org/10.1590/S0034-70942004000300013

**MISCELLANEOUS**

**Spectral entropy: a new method for anesthetic
adequacy ^{*}**

**Entropia espectral: un nuevo método para
adecuación anestésica**

**Rogean Rodrigues Nunes, TSA, M.D. ^{I};
Murilo Pereira de Almeida, M.D.^{II}; James Wallace Sleigh, M.D.^{III}**

^{I}Diretor Clínico e Chefe do Serviço
de Anestesiologia do Hospital São Lucas, de Cirurgia & Anestesia; Mestre
em Cirurgia, Área de Concentração: Anestesiologia pela UFC; Membro
da Sociedade Brasileira de Engenharia Biomédica; Graduando em Engenharia
Eletrônica pela Universidade de Fortaleza

^{II}Professor Adjunto IV do departamento de Física da UFC; Ph.D.
em Matemática Aplicada pela Brown University (Providence, RI, USA); M.Sc.
em Engenharia Civil pela PUC/RJ; Engenheiro Civil pela UFC

^{III}Professor of Anaesthesiology and Intensive Care, Waikato Clinical
School, Medical and Health Sciences, University of Auckland, New Zealand; MBChB
(Cape Town ) 1979, FFARCS (UK) 1985, FANZCA 1995

**SUMMARY**

**BACKGROUND AND OBJECTIVES:** Though universally
employed, clinical signs to evaluate anesthetic adequacy are not reliable. Over
the past years several pieces of equipment have been devised to improve intraoperative
handling of anesthetic drugs, some of them directly measuring cerebral cortical
activity (hypnosis). None of them, however, has offered the possibility of directly
evaluating sub-cortical activity (motor response).

**CONTENTS:** Spectral entropy measures irregularity, complexity or amount
of EEG disorders and has been proposed as indicator of anesthetic depth. Signal
is collected from the fronto-temporal region and processed according to Shannon's
equation (H = - Sp_{k} log p_{k},
where p_{k} represents the probability of a discrete k event), resulting
in two types of analyses: 1) state entropy (SE), which evaluates cerebral cortex
electrical activity (0.8 - 32Hz) and 2) response entropy (RE), containing both
subcortical electromyographic and cortical electroence- phalographic components
and analyzes frequencies in the range 0.8 - 47Hz.

**CONCLUSIONS:** Frontal muscles activation may indicate inadequacy of the
subcortical component (nociception). Such activation appears as a gap between
SE and RE. This, it is possible to directly evaluate both cortical (SE) and
subcortical (RE) components providing better anesthetic adequacy.

**Key Words:** MONITORING, bispectral index,
state entropy, response entropy

**RESUMEN**

**JUSTIFICATIVA Y OBJETIVOS:** El uso de señales
clínicos para evaluar la adecuación de la anestesia, aun cuando utilizada
universalmente, no son confiables. Varios equipamientos surgieron objetivando
el mejor manoseo intra-operatorio de las drogas anestésicas, algunos de
ellos mensurando directamente la actividad cortical cerebral (hipnosis). Entretanto,
ninguno de ellos presenta características directas de evaluación de
la actividad sub-cortical (respuesta motora).

**CONTENIDO:** La entropia espectral mensura la irregularidad, complejidad
o la cantidad de desorden del electroencefalograma y ha sido sugerida como un
indicador del estado anestésico. El señal es colectado en la región
fronto-temporal y tratado a través de la ecuación de Shannon (H
= - Sp_{k} log p_{k}, donde
p_{k} son las probabilidades de un evento discreto k), resultando en
dos tipos de análisis: 1. Entropia de estado (SE), que consiste en la
evaluación de la actividad eléctrica cortical cerebral (0,8-32Hz)
y 2. Entropia de respuesta (RE), que analiza las frecuencias de 0,8-47Hz (contiene
componentes electroence- falográficos-cortical y electromiográficos-sub-cortical).

**CONCLUSIONES:** La activación de la musculatura frontal puede indicar
inadecuación del componente sub-cortical (nocicepción). Esta activación
es observada como un "gap" entre SE y RE. De este modo, es posible
evaluar directamente tanto el componente cortical (SE), como el sub-cortical
(RE), posibilitando mejor adecuación de los componentes anestésicos.

**INTRODUCTION**

EEG changes are correlated to anesthetic concentrations
in the effector site, but predominantly describe the hypnotic effect of such
agents. Several methods have been applied to raw EEG signal: 95% spectral edge
frequency (derived from power spectrum), bispectral analysis (incorporates the
degree of phase coupling among sinusoidal components) ^{1}, approximate
entropy (quantifies data regularity in a temporal series) and spectral entropy
(quantifies data regularity in the frequency domain) ^{2,3}. The differential
of devices using spectral entropy analysis is that, in addition to processing
EEG signal (cortical), they also analyze frontal electromyographic activity
(sub-cortical) ^{4}, which may provide important subsidies for better
anesthetic components control.

The Concept of Entropy

It was Claude Shannon ^{5} who, in late
1940s, developed the modern concept of "logic" or "information"
entropy as part of information or uncertainty measurement. Information theory
was dealing with the newly born data communication science. Shannon's entropy
(H) is given by the following equation:

*H = - *S*p _{k}
log p_{k}*, where p

_{k}are the probabilities of a discrete

*k*event.

It is a measure of data dispersion. Data with wide and flat probability distribution will present a high entropy level. Data with narrow and peaked distribution will have a low entropy value. When applied to EEG, entropy is a statistical descriptor of EEG signal variability (comparable to other descriptors, such as spectral edge or low frequency passages during general anesthesia).

There are several concepts and analytical techniques to quantify irregularities of stochastic signals, such as EEG. Entropy is one of them. As a physical concept, entropy is proportional to the logarithm of the number of microstates available to a thermodynamic system, thus being related to the magnitude of system disorder.

However, it is difficult to define "disorder".
Boltzmann ^{6} (Figure
1) has shown that thermodynamic entropy could be accurately described as
the proportionality constant *k* (Boltzmann's) multiplied by the logarithm
of the number of independent microstates (w)
available in the system:

*S = k log (*w*) *

Boltzmann was able to explain changes in macro-parameters (such as temperature) as from changes in kinetic energy of a collection of individual molecules and has become a pioneer in statistical mechanics science. Thermodynamic entropy has a well-established physics basis. It is possible to derive Shannon's entropy equation (or "information" entropy) (H) from Boltzmann's thermodynamic formula (S).

There are important neurophysiological signs
that the usefulness of information entropy estimators as a measure of cortical
function is due to the fact that, as cortex enters unconsciousness, there is
a true neuronal decrease in the logarithm of the number of accessible microstates
(S) ^{7,8}. So, "entropy" could mean more than a mere statistical
measure of EEG pattern and, in a way, could truly reflect intracortical information
flow.

Entropy is the logarithm of the number of modes in which a microstate may rearrange and still produce the same macrostate. The difference between true thermodynamic entropy and other information entropies is that kinetic energy distribution of individual molecules is not necessarily involved with information entropy estimators. In a thermodynamic system, collision events spread kinetic energy through the substance. Entropy is a measure of such dispersion. In the cerebral cortex, synaptic events spread electric charges through the cortex. Assuming that cortical information processing is linked to electrical charges distribution and transfer, EEG entropy measures cortical pyramidal cells activity.

Shannon was the first to define entropy in the
information theory in 1948. Then, in 1984 ^{9}, Johnson and Shore have
applied it to the power spectrum of a signal. Within this context, entropy describes
irregularity, complexity or level of uncertainty of a signal. Let's look
at a simple example: a signal where sequential values alternate between one
fixed magnitude and other has zero entropy, that is, the signal is totally regular
and predictable. A signal where sequential values are determined by randomized
numbers generator has higher complexity levels and entropy.

The Concept of Uncertainty

Consider an experiment A, for example, throwing
a coin in the air, with two possible results A_{1} and A_{2},
with associated probabilities p_{1} and p_{2}, being p_{1}
+ p_{2} = 1. This representation may be illustrated as follows:

In this case, probability in relation to results may be represented as follows:

Now, consider an experiment B associated to the following probability scheme:

It is reasonable to think that there are different uncertainty levels for experiments A and B, based on a psychological sense. This suggests, as a principle for the task of probabilities, that uncertainty resulting from the probability scheme of an experiment should be maximally subject to restrictions of any other available information related to the experiment. To create this quantitative concept, a mathematical expression for uncertainty and a probability scheme are necessary, that is, Shannon's Entropy:

, where:

p_{i} represents the probability related
to the experiment and n the number of probabilities.

In applying this equation to the experiment above, we obtain:

For the scheme A, a value of E = 0.69 and for
the scheme B, a value of E = 0.056, showing that the level of uncertainty for
scheme B is much lower as compared to scheme A ^{10}. This may also
be applied to power distribution analysis of an EEG signal, determining the
level of uncertainty, for example, of a patient being unconscious based on the
power distribution analysis in the frequency domain.

Figure 2 shows power spectrum of an anesthetized patient without spectral entropy (Shannon's) analysis.

Entropy is an intuitive parameter in the sense that it is possible to visually distinguish a regular from an irregular signal. In addition, entropy is independent of absolute scales, such as signal amplitude or frequency: a simple sine wave is perfectly regular, regardless being fast or slow. For EEG applications, this is a significant property since, as we know, there are absolute individual frequencies variation on EEG rhythms.

There are many ways to calculate the entropy
of a signal. In the time domain, one may consider, for example, approximate
entropy (Pincus ^{11}, Bruhn ^{12}) or Shannon's entropy
(Shannon ^{5}. Bruhn ^{13}). In the frequency domain, one may
calculate spectral entropy (Shannon ^{5}, Johnson ^{9}). To
optimize the speed in which information is derived from the signal, one may
combine time and frequency domains. Spectral entropy algorithm has the specific
advantage of being able to explicitly separate contributions to entropy coming
from any frequency range. For an optimal response time, one may develop calculations
in a way that time window length for each frequency is individually selected.
This leads to a concept that could be named "Spectral entropy with balanced
time and frequency".

What Does This Mean to EEG?

Since EEG allows, to a certain extent, the visualization of cortical processes, changes in EEG entropy should measure changes within the cerebral cortex itself. Considering that the main conscious cortex function is information processing and generation, it would be fair to assume that some kind of "information measurement" would be very useful. The problem is that the word "information", similarly to "disorder"; has several meanings and connotations, and should be carefully and scientifically defined to be applied within this context. Perhaps the simplest and most practical description of EEG entropies would be: "measures of the extent to which limitations (in our case, general anesthesia) decrease the number of accessible status available to the cortex". It follows that, while one could expect that, a higher number of microstates would be associated to a more "complex" system, entropy per se does not necessarily provide a direct measure of the "complexity" of a system, which has other variability implications in response to inputs, etc.

If entropy can be defined as the logarithm of
the number of more commonly accessible cortical microstates, the question raised
is: What are cortical microstates? There are increasingly more neurophysiological
evidences that cognitive activity involves the transient formation and dissolution
of interconnecting neuronal cortical structures ("activation" and
"acquiescence") (John et al., 1997 ^{14}).

It would be fair to assume that such coherent
structures are effectively functional cortical microstates. EEG perceives the
activity on the scalp as a broader range white noise spectrum (Stam et al.,
1993 ^{15}, Thomeer et al., 1994 ^{16}).

If (1) the awaken state requires efficient generation
of several cortical microstates, and (2) EEG signal microstates reflect, in
a way, cortical microstates, then decreased EEG entropy (as seen in general
anesthesia) indicates a decreased number of available cortical microstates (Weiss,
1992 ^{17}).

The cortex-consciousness paradox, if we define entropy as "disorder", is that higher EEG entropy levels of the awaken cortex imply that cortex is more disorganized awaken as compared to unconsciousness. The paradox draws attention to problems inherent to equations involving entropy and disorder. We still have no means to identify high dimension signals generated by the conscious cortex during cognition and, for such, to call them "noise"! This is why we prefer to define entropy in terms of available microstates and not of "order". It might be more illustrative to describe entropy as "freedom of choice". Conscious cortex is free to move among a huge number of available microstates.

Spectral Entropy

There are several ways to estimate changes in the amplitude of EEG power spectrum. These methods use the amplitude component of the power spectrum as "probabilities" in entropy calculation. With the use of space-frequency, we are defining microstates in terms of change indices. A wide range of accessed frequencies (that is, a flat spectrum) implies several added change indices of pyramidal cell membrane potentials.

The prototype of this group is Spectral Entropy
(S) (Inoye et al., 1991 ^{18}, Fell et al., 1996 ^{19}). S corresponds
to Shannon's formula duly normalized and applied to power spectrum density
of EEG signal. So:

*S = *S*p _{k}
log p_{k}/log(N)*, where:

*p _{k}* are spectral amplitudes
of

*k*frequencies region.

S*p _{k}
* = 1, and N = number of frequencies.

So, the starting point for calculations is signal
spectrum. Several spectral transformations are needed to obtain the spectrum.
Here, however, we will hold to Fourier's transformed discontinuing ^{20},
which allows for the transformation of a set of signal values x(t_{i})
sampled during moment t_{i} within a signal's sample into a set
of complex numbers X(f_{i}), of frequencies f_{i}.

where:

being: F_{n}(t_{i})
the n^{th} component of a total of N components, and a_{n
}the coefficient associated to the n^{th} component.

Spectral components X(f_{i}) may be evaluated
by an effective calculation technique called "Fourier's fast transformed"
(FFT). Spectral entropy concept comes from an information measurement called
"Shannon's entropy". When applied to signal power spectrum, spectral
entropy is obtained ^{9}. The following steps are necessary to calculate
spectral entropy for a certain signal interval (epoch) within a given frequency
range [f_{1}, f_{2}].

From Fourier's transformed X(f_{i})
of signal x(t_{i}), power spectrum P(f_{i}) is calculated by
multiplying by themselves Fourier's transformed amplitudes of each element
X(f_{i}):

(1)

where X^(f_{i}) is Fourier's complex
conjugated X(f_{i}) and the star (*) indicates multiplication.

Power spectrum is then normalized. Normalized
power spectrum P_{n}(f_{i}) is calculated by the establishment
of a normalization constant C_{n}, so that the sum of normalized power
spectrum on the selected frequency region [f_{1}, f_{2}]
equals to 1:

(2)

This now represents a probabilities diagram.

During the sum stage, spectral entropy corresponding
to the frequency range [f_{1}, f_{2}] is calculated
as a sum:

(3)

Then, entropy value is normalized to vary between
1 (maximum irregularity) and 0 (total regularity). Value is divided by factor
log (N[f_{1}, f_{2}]), where N[f_{1}, f_{2}]
equals the total number of frequency components in the range [f_{1},
f_{2}]:

(4)

Figures
3 to 5 show these stages. Figures
3(a), 4(a) and 5(a)
show three signal sections corresponding to different entropy values. In this
simple example, we have considered signal sections with 8 spectral components,
from which 0 frequency component (or the one assumed to be zero) was omitted,
so that N = 7 frequency components are analyzed. Figure
3(a) shows a perfect sinusoidal wave. Figure
4(a) shows a sinusoidal wave with superimposed with white noise, while figure
5(a) illustrates a perfectly randomized white noise signal. Fourier's discontinuous
spectra of such signals, normalized according to equation (2), were plotted
in figures 3(b), 4(b)
and 5(b), respectively.
Then, normalized spectral components P_{n}(f_{i}) were plotted
in figures 3(c), 4(c)
and 5(c) using Shannon's
function to obtain contributions:

Since: log(x)=n(x)/n(10), then we may replace the expression above:

Spectral Entropy with Balanced Time and Frequency

In real time signal analysis, signal values x(t_{i})
are sampled within a finite time window (epoch) with a selected length and a
given sampling frequency. This time window moves step by step to provide updated
spectrum estimates. Window length choice (epoch) is related to the choice of
the frequency range under consideration, since time window has to be long enough
to estimate slower signal variations (lower frequencies).

An EEG signal comprises a wide selection of frequencies, from slow delta frequencies (as from 0.5 Hz) to 50 Hz. A 0.5 Hz frequency would need up to 30 s time window to obtain 15 complete 0.5 Hz variation cycles. A 50 Hz frequency would need just 0.3 s of data to obtain the same number of complete cycles.

Clearly, a single time window with fixed length is not the best choice to get the information in the fastest and most reliable way. To find the optimal point between time and frequency resolution, the already mentioned algorithm uses a set of window lengths selected so that each frequency component is obtained from the most adequate window for that specific frequency. This way, information is extracted from the signal in the fastest possible way. The approach is closely related to the idea of the transformation of small waves, which being packets of finite variable range waves with a more or less constant number of variations, provide a break-even point between time and frequency resolution. The selected technique combines the advantage of small waves analysis with fast Fourier's analysis, as well as the possibility of explicitly considering the contribution of any given frequency range and the effective implementation in software. The basic idea is shown in figure 6.

Surface Electromyography, State Entropy and Response Entropy

Surface electromyography (EMGs) is the algebraic
sum of the electric activity of a population of muscle fibers. There is a direct
relationship between EMGs amplitude and muscle tension during isometric contraction
^{21}. In conscious patients under no anesthetic effect, high tonic
activity observed in electromyography is positively correlated to the level
of stimulation, vigilance or psychological stress. Increased EMGs phase activity
is associated to somatic stress periods, such as pain ^{22}. For unconscious
patients monitoring, EMGs should be performed on frontal muscles, which have
a relatively fixed length, thus decreasing the potentially complicating influence
of fiber length variation (isotonic contraction). These muscles are also preferred
because they are innervated by special visceral efferent fibers of the facial
nerve. It is important to remember that these muscles are derived from branchial
archs which are considered visceral formations. Frontal muscles EMGs provides
a simple and noninvasive measure of an autonomic tone aspect. Voluntary and
involuntary frontal muscles contractions are responses brought by different
pathways innervation, as well as tonic activity (basal) and phase activity (abruptly
increased) may be differently affected by abnormal levels of vigilance or neuromuscular
blockers ^{23-25}.

Decreased vigilance following general anesthesia
is typically associated to dramatic decrease in tonic frontal activity. Some
authors ^{24,26,27} have concluded that increased phase activity in
the presence of EMGs amplitude depressing drugs is an indication of inadequate
anesthesia. In addition, these investigators have observed that phase activity
measured by EMGs may be seen in the presence of neuromuscular blockers.

It has been shown that at least three different types of stimulations - emotion, sounds and ischemia - may evoke surface EMGs amplitude phase increase during low vigilance states.

EMGs may be useful for opioids titration: facial
muscles are not only voluntary, but also innervated by brainstem centers related
to emotions/stress. Although the intraoperative electromyographic evaluation
with patient's movement is a quantal phenomenon, minor EMGs changes may
reflect inadequate analgesia (sob-cortical component with inadequate blockade).
Mathews et al. ^{28} have shown that postoperative opioids decrease
EMGs. In addition, EMGs may respond faster than BIS in emergence reactions.
Kem et al. ^{29} have concluded that EMGs shows good correlation with
responses to noxious stimulations applied to volunteers. Shander et al. ^{30}
have also shown that EMGs may predict analgesic requirements, allowing for a
more effective intraoperative control of analgesia (sub-cortical element). Lennon
^{31} has concluded that moderate neuromuscular block levels may be
achieved without impairing electromyographic monitoring of the facial nerve.
Edmonds ^{25} has shown that EMGs in response to stress (pain) may assess
brainstem function, which is independent of consciousness level (cortex). Dutton
^{32} in a recent study has concluded that electromyographic response
may be used to estimate anesthetic depth.

A biopotential captured in the frontal region includes a significant electromyographic component created by muscle activity. The electromyographic signal has a wide spectrum similar to noise and, during anesthesia, it typically predominates in the frequencies above 30 Hz. Electroencephalographic signal component predominates in lower frequencies (up to approximately 30 Hz) contained in biopotentials existing in the electrodes. At higher frequencies, electroence- phalographic power decreases exponentially (Figure 7).

Electroencephalographic signals of sudden appearance often indicates that patient is responding to some external stimulations, such as nociception as a consequence of some surgical event. These responses may be result of inadequate anesthesia. If stimulation continues and no additional analgesic is given, it is possible that the hypnotic level may get more superficial. So, electromyographic signal may indicate emergence imminence. Note that, due to the higher electromyographic signal frequency, sampling time might be significantly shorter than for lower frequency electroence- phalographic signals. This way, electromyographic data may be refreshed more frequently so that the method may rapidly indicate changes in patient's state.

For better clarity, one should consider two entropy indicators, one solely on the dominating EEG frequency range, and the other on the complete frequency range, including electroencephalographic and electromyographic components. "State entropy" (SE) is calculated on the frequency range 0.8 Hz to 32 Hz. It includes the electroence-phalographic-dominating part of the spectrum and, as a consequence, primarily reflects patient's cortical state. SE time windows are optimally chosen for each specific frequency component, varying 60 s to 15 s as previously explained. "Response entropy" (RE) is calculated on the frequency range 0.8 Hz to 47 Hz. It includes the electroencephalo- graphic-dominating part and the electromyographic-dominating part of the spectrum. RE time windows are optimally chosen for each frequency, being the longest equal to 15.36 s and the shortest equal to 1.92 s (applied to frequencies 32 Hz to 47 Hz).

The two entropy parameters should be normalized
so that RE equals SE whenever electromyographic power (the sum of spectral power
between 32 Hz and 47 Hz) equals zero. This way the difference between RE and
SE will be an indicator of electromyographic activity. At this point, frequency
range between 0.8 Hz and 32 Hz will be called 'R_{low}', and
frequency range from 32 Hz to 47 Hz will be called 'R_{high}'.
The combined 0.8 Hz to 47 Hz frequency range will be called 'R_{low+high}'.
It follows from equations 1 to 4, that when spectral components within R_{high
}equal zero, non-normalized entropy values S[R_{low}] and
S[R_{low}+R_{high}] will coincide, while for normalized
entropies there is SN[R_{low}] > SN[R_{low}+R_{high}]
inequality. The normalization stage (4) is then redefined for SE as follows:

(5)

For RE, normalized entropy value is calculated according to equation (4):

(6)

As a consequence, RE varies 0 to 1, while SE
varies 0 to log(N[R_{low}])/log(N[R_{low+high}])
< 1. Both entropy values coincide when P(f_{i}) = 0 for all f_{i}
of the [R_{high}] range. When there is electromyographic
activity, spectral components of the [R_{high}] range
are significantly different from zero and RE is higher than SE.

With these definitions in mind, SE and RE have different information purposes for the anesthesiologist. State entropy is a sufficiently stable quantity to allow the anesthesiologist to have an idea of the cortical state of the patient at any time by checking a single number. SE time windows are selected so that transient fluctuations are removed from data. Response entropy, on the other hand, rapidly reacts to changes. The different roles of these parameters are typically shown during emergence, when RE is first increased together with muscle activation, being followed few seconds later by SE.

Entropy during Burst Suppression

When burst suppression is started, RE and SE values are calculated as they would be during more superficial hypnosis levels. The part of the signal with suppressed EEG is treated as a perfectly regular signal with zero entropy, while burst-associated entropy is calculated as already described.

Usually, burst suppression is quantified by the representation of the relative suppression amount, called "burst suppression ratio" (BSR), within one minute to obtain a sufficiently stable estimate. The one-minute window has a sufficiently large sample of bursts and suppression to provide a stable indication of the relative amount of suppressed EEG; in fact, much shorter windows would result in highly fluctuating BSR values. For the same reason, a one-minute window - and not a varied set of time windows - is applied to all frequency components of SE and RE values whenever some suppressed interval (epoch) is detected during the previous minute of data.

Burst suppression is detected by the technique
described by Särkelä et al. ^{33}. To eliminate baseline fluctuations,
a local mean of each signal sample is subtracted. Then, the signal is divided
in two frequency ranges by elliptic filters. Threshold frequencies of low and
high filters are 20 Hz and 75 Hz, respectively. Low frequency range is used
to detect burst suppression pattern, while high frequency range is used for
detecting artifacts. An energy operator is used to estimate both signal power
ranges of each 0.05 s interval (epoch). Suppression is detected when estimated
signal power is below a prefixed threshold during at least 0.5 s without artifacts.
BSR is the percentage of 0.05 s intervals (epochs) which may be considered as
suppressed during the previous 60 s (Figure
8).

Optimizing the Use of Entropies

RE and SE parameters were designed to be read together with many other important information on the screen of a single monitor. To present more effectively entropy data, some changes have been made on the way these parameters are displayed thus optimizing their applicability. A 2-digit integer number (for example, 39) is easier to be read on a screen than a decimal value (0.39) or a 3-digit value (539). For this reason, original entropy values, which continuously vary 0 to 1, were converted into an integer scale 0 to 100.

A relatively high portion of the original mathematical scale of entropy values occupies a range where the level of hypnosis may be considered too deep. On the other hand, the most adequate hypnosis and emergence range remains between 0.5 and 1. So, a mere division of the original scale in equidistant integers from 0 to 100 would result in a somewhat poor resolution, of the frequency of interest and in an unnecessarily high resolution at deeper levels. So, the changing of the original continuous entropy scale [0…1] into an integer scale [1 …100] was obtained through a non-linear transformation. This transformation is defined by a monotonous "spline" function F(S) able to plot scale [0…1] on the scale [1…100].

"Spline" functions are continuous, as continuous are their derivatives of any order. Hence, a transformation operation defined by monotonous "spline" function is perfectly smoothed, with no discontinuities or fluctuations. The F(S) function used for transformation is shown in figure 9, where its illustrated that the slope reaches its maximum in the range of anesthesia and clinical emergencies, allowing optimal resolution in this range. RE varies 0 to 100 and SE 0 to 91 (Figure 9).

If low frequency components amplitudes are especially
high - like during very deep anesthesia - the difference between response and
situation entropy, as they are originally obtained by equations (5) and (6),
may remain below the integer number resolution displayed on the monitor's
screen. To provide a detectable indication of electromyographic activity, the
treatment of frequency components in the range [R_{high}] =
[32 Hz , 47 Hz] is modified. Instead of applying the normalization constant
C_{n} to all frequency components according to equation (2), in this
situation a different normalization constant C_{n}^{high} for
the range [R_{high}] is used. While C_{n} is below a
threshold data C_{n}^{limit}, it is assumed that C_{n}^{high}
will equal C_{n}, but if it goes beyond C_{n}^{limit},
C_{n} is considered equal to C_{n}^{limit}. This modification
assures that electromyographic activity is detectable on screen in any situation.

**TREATMENT OF RAW SIGNAL FOR ARTIFACTS DETECTION
AND REMOVAL**

To analyze artifacts, electromyographic signal is divided in 0.64 s intervals (epochs), including 256 signal values. These intervals (epochs) are inspected to detect and remove the following artifacts:

Electrocauterization artifacts

The device is highly tolerable to electrocauterization, so very seldom data have to be rejected during electrocauterization. To detect these situations, power in 200 Hz to 1000 Hz frequency range is continuously measured. If power goes beyond a pre-established threshold, electromyographic data collected in 66 Hz to 86 Hz frequency range are inspected to check whether electrocauterization has affected the signal. If so, the interval (epoch) will be excluded from next analyses.

ECG and Pacemaker Artifacts

The high sampling frequency of 400 Hz makes it easy to distinguish between acute peaks associated to ECG and Pacemaker and the underlying electroencephalographic signal. These artifacts are then removed, subtracting distortion from the underlying signal, which is useful for entropy calculation.

EMG

As previously discussed, EMG is treated as a component-signal and not as an artifact.

Eye Movements and Blinking, Movement Artifacts

The 0.64 s interval (epoch) is too short for the reliable detection of all intervals (epochs) containing such artifacts. So, they are considered in two stages:

**Stage 1.** A stationary analysis is made
for the signal within a time window of 24 * 0.64 = 15.36 s (including 6,144
signal values). Signal is classified as stationary or non-stationary according
to statistical distribution of signal values throughout and within the 24 intervals
(epochs).

**Stage 2.** For each interval (epoch), 5
signal characteristics in time and frequency domains are calculated. These characteristics
are simultaneously considered in the corresponding five-dimension parametric
space divided in "normal signal" and "artifact-contaminated signal"
regions. Intervals (epochs) are accepted or rejected, depending on the parametric
space they belong to. There are two sets of rejection rules: one set of stricter
rules used when signal section is analyzed in Stage 1 and classified as non-stationary,
and a set of less strict rules for when analyzed signal section is considered
stationary.

Clinical Application

State entropy reading is related to the level of unconsciousness uncertainty (cortical activity), that is, the lower the state entropy the lower the level of unconsciousness uncertainty. Similarly, the lower the level of sub-cortical activity control uncertainty (electromyographic activity), the lower the response entropy value (Table I).

Electrodes Positioning and Usage Directions

1. Clean the skin with alcohol and let it dry;

2. Position circle #1 in the center of the frontal region, approximately 4 cm above the nose, and circle #3 on the anterior temporal region, between internal palpebral angle and the pileous line (Figure 10);

3. Press sensor borders to assure adhesion;

4. Firmly press circles 1, 2 and 3 for 5 seconds to assure adequate contact.

**CONCLUSION**

Frontal muscles activation may indicate inadequate sub-cortical component. This activation may be observed as a gap between state and response entropy. So, there is the possibility of directly evaluating both cortical (SE) and sub-cortical (RE) components, aiming at better anesthetic components adequacy.

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**Correspondence to**

Dr. Rogean Rodrigues Nunes

Rua Gothardo Moraes, 155/1201 Bloco Dunas, Papicu

60190-801 Fortaleza, CE

E-mail: rogean@fortalnet.com.br

Submitted for publication June 12, 2003

Accepted for publication October 7, 2003

* Received from Serviço de Anestesiologia do Hospital São Lucas de Cirurgia & Anestesia, Fortaleza, CE