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A novel algorithm for detecting human circadian rhythms using a thoracic temperature sensor

Aly Chkeir*1, Farah Mourad-Chehade1, Jacques Beau4, Monique Maurice4, Sandra Komarzynski4, Francis Levi3, David J.

Hewson2, Jacques Duchêne1,

1Institut Charles Delaunay UMR CNRS 6281, Rosas, University of Technology of Troyes (UTT) 10000 Troyes, France

2University of Bedfordshire, Luton, United Kingdom

3Warwick University, United Kingdom

4INSERM, University of Paris, France

A R T I C L E I N F O

A B S T R A C T

Article history:

Received: 07 May, 2017

Accepted: 29 June, 2017

Online: 19 July, 2017

Circadian rhythms undergo high perturbations due to cancer progression and

worsening of metabolic diseases. This paper proposes an original method for detecting

such perturbations using a novel thoracic temperature sensor. Such an infrared sensor

records the skin temperature every five minutes, although some data might be missing.

In this pilot study, five control subjects were evaluated over four days of recordings. In

order to overcome the problem of missing data, first four different interpolation methods

were compared. Using interpolation helps covering the gaps and extending the

recordings frequency, subsequently prolonging sensor battery life. Afterwards, a

Cosinor model was proposed to characterize circadian rhythms, and extract relevant

parameters, with their confidence limits. A divergence study is then performed to detect

changes in these parameters. The results are promising, supporting the enlargement of

the sample size and warranting further assessment in cancer patients.

Keywords:

Human circadian rhythm

Interpolation

Cosinor

Divergence

Temperature Sensor

1. Introduction

Most biological processes follow circadian rhythms, with

the standard period of about 24 hours. These 24-hour rhythms

are driven by circadian clocks, which have been observed in

plants, fungi, cyanobacteria, and humans [1, 2]. In mammals,

each cell contains molecular clocks, whose coordination is

ensured by the suprachiasmatic nuclei in the hypothalamus

through the generation of rhythmic physiology. Such a

Circadian Timing System (CTS) regulates rhythmically cellular

metabolism and proliferation over the 24 hours.

The body temperature follows circadian rhythms, that are

tightly related to cancer development[3] [4]. Indeed, they play

an important role in the coordination of molecular circadian

clocks in various peripheral organs, such as lung, liver, kidney,

and intestine, and could have an effect on tumors through their

regulatory effects on Heat Shock Factor (HSF), Heat Shock

Proteins (HSPs), and Cold-Induced Proteins [4, 5]. Moreover,

it has been demonstrated that giving chemotherapy at an

accurate circadian timing improves tolerance up to fivefold and

to almost doubles antitumor efficacy, compared to constant

rates or wrongly timed administrations, in both rodent models

and cancer patients[6] [7]. In the same way, the destruction of

the suprachiasmatic nuclei suppressed any circadian rhythm in

body temperature in mice, which causes a 2-3 fold acceleration

of experimental cancer progression [8]. On the other hand, the

temperature circadian rhythms could be disrupted by anticancer

medications, and molecular clocks are impaired, as a function

of dose and circadian timing in mice [9, 10]. Therefore, the

circadian rhythm of body temperature is a useful biomarker of

CTS function[11] [12].

The relevance of monitoring core body temperature for CTS

assessment has been approved in previous works, however they

all suffer from a lack of non-invasive screening tools, which has

limited testing in cancer patients [13, 14]. To overcome such a

problem, one could use the skin temperature, which is

correlated to core temperature. However, changes in skin

temperature display an opposite pattern compared to that in core

body temperature, but with a similar circadian rhythm.

In our work, a non-invasive wearable sensor, Movisens®

GmbH, Karlsruhe, Germany, has been used. This sensor

monitors skin temperature for up to 200 hours, with the sensor

able to be worn at different positions such as the hip, wrist, or

chest. The temperature signals are collected at a relatively low

ASTESJ

ISSN: 2415-6698

*Corresponding Author: Aly Chkeir, Institut Charles Delaunay UMR CNRS

6281, Rosas, University of Technology of Troyes (UTT) 10000 Troyes, France

Email: aly.chkeir@utt.fr

Advances in Science, Technology and Engineering Systems Journal Vol. 2, No. 4, 105-110 (2017)

www.astesj.com

A. Chkeir et al. / Advances in Science, Technology and Engineering Systems Journal Vol. 2, No. 4, 105-110 (2017)

www.astesj.com 106

sampling frequency with this sensor, that is, one point every

five minutes. Using such a low frequency is possible since

temperature varies slowly over time. Also, a minimal memory

is required and the battery life is extended, due to reduced

energy consumption for the measurement device. However,

having a low signal frequency produces more difficult

complications. Moreover, sensor malfunction or a subject not

wearing it for a period of time causes missing data in the

signals.

This paper proposes an original algorithm to analyze the

circadian rhythms in skin temperature signals. It first develops

an interpolation algorithm to set the signals to a higher

frequency such as one sample per minute, and to fill any gap in

the signals [15]. For this reason, a kernel-based machine

learning algorithm is presented and compared to other classic

interpolation methods. Afterwards, a rhythmometric modeling,

using Cosinor models, is proposed. It aims to extract parameters

such as the MESOR, the amplitude, the orthophase and the

bathyphase from interpolated signals. The paper then proposes

a divergence study over these features to detect changes caused

by chemotherapy. Five healthy subjects’ temperature signals

are involved in this study, as a prerequisite for subsequent

investigations involving a larger number in healthy controls and

cancer patients.

2. Subjects and Methods

2.1. Subjects and materiel

In this study, three female and two male control subjects,

aged 45.2 ±13.6 years, are considered. They were given a

detailed description of the objectives and requirements of the

study before the experiment, and they read and signed an

informed consent prior to testing. The infrared Movisens

(GmbH - move III) sensor was positioned on the thorax of the

subjects to monitor their skin temperature for four days. It

measures 5.0 x 3.6 x1.7 cm3, and weighs 32 g. The sensor is

composed of a tri-axial acceleration sensor (adxl345, Analog

Devices; range: ±8 g; sampling rate: 64 Hz; resolution: 12 bit)

embedded with a temperature sensor (MLX90615 high

resolution 16bit ADC; resolution of 0.02°C). The recorded data

is saved on a memory chip inside the sensor and transmitted to

a server via GPRS. A hypoallergenic patch has been used to

maintain the sensor in the upper right anterior thoracic area of

the subjects.

2.2. Interpolation of temperature signals

The temperature signals have a small frequency, with

occasional missing data. Let denote the

temperature samples collected for a certain subject using the

sensor, with in minutes. The aim of interpolation is to

estimate a function

, that computes the temperature at any

time , and which verifies the following:

(1)

Different algorithms for interpolation such as linear,

polynomial or cubic splines techniques exist in the literature

[15, 16]. In the linear interpolation, temperature is represented

by a straight line between any two consecutive collected

measurements. Having, for ,

(2)

with and

.

This technique is easy to implement, but it yields several

functions, one per interval between two consecutive

measurements. Moreover, it is disadvantageous for large time-

intervals due to the non-specification of the linear estimation.

In the polynomial interpolation, temperature is represented by

a single polynomial function, which fits the measured data. By

using Lagrange polynomials, one obtains the following

function:

(3)

where is the degree of the obtained polynomial and

. By taking close to , the obtained function

is specific, but with highly complex computations. Cubic

splines is the most commonly used interpolation technique. It

computes a set of piece-wise polynomial functions that

maximize the smoothness of the whole curve. The -th splines

function defined over the interval, for ,

is given as follows:

(4)

where:

where

is the derivative of the temperature model

. The

values of the coefficients are then computed iteratively as

shown in [2]. This technique is efficient, but proposes piece-

wise functions, that need iterative computations.

TABLE I. TYPICAL REPRODUCING KERNELS

Kernel type

General expression

Gaussian

Polynomial

Exponential

This paper proposes a kernel-based regression approach that

generates a single function [17, 18]. A training database is first

constructed using the collected temperature measurements

. Then a model is computed using this

database, taking time as input and yielding temperature as

output. This model is defined using the kernel-based ridge

regression technique. The model is afterwards applied to other

times, where temperature values are unknown, for

interpolation. Consider a reproducing kernel defined from

to and denote its reproducing kernel Hilbert space.

Some commonly used reproducing kernels are given in Table I.

where the kernel parameters and are positive and is a

positive integer. Then the temperature model is defined by

minimizing the regularized mean quadratic error between the

model’s outputs and the measured data of the learning database:

(5)

where is a regularization parameter that controls the tradeoff

between the training error and the complexity of the solution

and

is the norm in the reproducing kernel Hilbert

space[19]. According to the Representer theorem, that all

A. Chkeir et al. / Advances in Science, Technology and Engineering Systems Journal Vol. 2, No. 4, 105-110 (2017)

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machine learning algorithms share, the minimization problem

could be reduced to a more computation-friendly problem.

Hence, the temperature model could be written as follows:

(6)

where the coefficients are to be determined. Let denote

the column coefficients vector whose -th entry is . By

injecting the model expression in the minimization problem,

one obtains a dual optimization problem whose solution is

given by:

(7)

where is a -by- matrix whose -th entry is given by

, is the -by- identity matrix and is the

temperature column vector, whose -th entry is given by

. Now that the model is defined for each temperature

signal, the temperature value at a given time is estimated by

The main advantage of the kernel-based approach remains in

the fact that a single-function model is obtained, unlike the

linear and cubic spline interpolations, where a piece-wise

expression is obtained. In the following, and for simplicity, the

notation is used for the estimated signal, having a value at

each minute, obtained after interpolation.

2.3. Detection of rhythmicity

The detection of rhythmicity is usually performed in the

frequency domain . The spectral analysis using the “Fourier

transform” is a well-known study to do this . In this

analysis, any signal, regardless of its shape and properties, can

be represented by a complex function of frequency that

highlights the frequencies that make it up. By applying the

inverse Fourier transform, the signal is then decomposed into

an infinite sum of sine and cosine functions of infinite

frequencies [22]. The signals could be deterministic such as

periodic/non-periodic or random such as stationary/non-

stationary. A similar analysis for periodic signals is the Fourier

series analysis, which represents a function as a sum of sine and

cosine functions of different frequencies.

In this paper, an algorithm based on Fourier analysis is first

proposed for frequency and harmonic detection. This algorithm

starts by estimating the fundamental frequency, the

fundamental amplitude and phase, and the harmonic amplitudes

and phases, to evaluate the periodogram. The term

periodogram was introduced by Schuster in December 1934

when Fourier analysis was used to estimate periodicity in

meteorological phenomena . The technique was evaluated

for the first time when inspecting circadian rhythms in the

1950s to measure circadian rhythms of mice after blinding .

The periodogram showed that periodic signals have a frequency

spectrum consisting of harmonics. For instance, if the time

domain repeats at f, the frequency spectrum will contain a first

harmonic at f, a second harmonic at 2f, a third harmonic at 3f,

and so forth. The first harmonic, which is the frequency at

which the time domain repeats itself, is called the fundamental

frequency, and has the highest amplitude. Periodograms and

spectral density were originally used in chronobiology in the

1960s [25].

In order to set the periodogram, a suitable window must be

applied to the signal, to reduce side-lobes. For the proposed

algorithm, a normalized Hanning window has been chosen [26]

since this window does not disturb the position of spectral peaks

in the spectral density, although the amplitude is decreased and

the peak is larger. Having the periodogram and thus the

fundamental and harmonic frequencies, the temperature signal

is then modeled, using the Fourier series as follows:

(8)

where is the angular frequency i.e.

, i.e. is the

fundamental period (duration of one cycle) and is the number

of the considered harmonics with the fundamental frequency.

With respect to circadian rhythms, the rhythm persists in

constant conditions with a period of around 24-hours, i.e.

minutes. takes values from to infinity. The

higher is, the better the model

fits the observed value

. The parameters , and could be computed using

the computations of Fourier series. The main advantage of this

technique is that one is able to determine the exact fundamental

frequency, with its following ones, by analyzing the

periodogram, obtained with the Fourier transform. However, it

needs the data to be equidistant and to cover more than a single

cycle, otherwise the analysis would be erroneous.

One interesting method used for analyzing unequidistant

and time-limited observations is the single Cosinor procedure.

This study was developed to evaluate rhythmicity of un-

equidistant data series[27] [28], and is frequently used in the

analysis of biologic time series that have expected rhythms.

Cosinor uses the least squares method to fit a sum of sine

functions to a time series, as least squares procedures do not

have an equidistant data limitation. Practically, it considers the

Fourier series model of (8) with a precise fundamental

frequency, which corresponds to 24-hours, and number of

harmonics and then computes the parameters, and so

as to minimize the error between the signal and the model

. Let be the residual corresponding to the value ,

that is,

and consider the modeling error as the sum of the squared

residuals () for all the data, that is,

The parameters , and are then obtained using least

squares by setting the derivatives of the over each

parameter equal to zero. Since the temperature follows a

circardian rhythm, then the fundamental frequency corresponds

to 24h. Figure 1 shows an example of a Cosinor model obtained

with. The obtained model will subsequently be used to

compute significant rhythmometric parameters, as shown in the

following paragraph.

Once the temperature signals are modeled, using either

Fourier series or Cosinor, some features are extracted from their

sinusoidal representation, as will be shown in the following

paragraph.

A. Chkeir et al. / Advances in Science, Technology and Engineering Systems Journal Vol. 2, No. 4, 105-110 (2017)

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Figure 1 Circadian Rhythm Model

2.4. Features selection and divergence study

Once the temperature signal is decomposed into cumulative

sine functions, using Fourier series or Cosinor, the objective

here is to determine the sinusoidality of the data. This requires

the extraction of some features from the models by inspecting

their graph plotted against time, as shown in Figure 1. When the

model is composed of more than one trigonometric function,

that is, the fundamental period and some harmonics, four

features could be extracted [29]:

the MESOR (M), for “Midline Estimating Statistic Of

Rhythm”, that is the mean of the model,

the amplitude (A) that is defined as the half of the difference

between the maximum and the minimum of the model in

one fundamental period,

and, finally, the phases of the maximum and the minimum

of the composite model including harmonic terms, which

are called the orthophase ΦO and bathyphase ΦB

respectively.

Figure 1 illustrates these features. For a control subject,

these rhythmometric parameters vary slightly over time;

whereas chemotherapy could produce significant modification

in the circadian rhythm, which yields a divergence of one or

more rhythmometric parameters. In order to detect this

divergence, a sliding window algorithm is considered over the

temperature signal of a period of several days.

Then, for each window, the signal is modeled using the

cumulative sine functions, and the four rhythmometric

parameters are extracted. A statistical test, such as an exact

Fisher test, Wilcoxon test or another, is then applied to check

whether the values through the window diverge from their

previous values. This study is motivated by the perturbation of

the circadian rhythm due to chemotherapy, which induce a

divergence of the statistical distributions of the extracted

rhythmometric parameters.

3. Results

This section starts with the illustration of the effectiveness of

the interpolation techniques. To this end, the collected

temperature signals of two subjects out of the five are

considered. These two have a frequency of one sample per

minute over four days, whereas the remaining are measured with

a rate of one sample every five minutes. An example of a skin

temperature signal while wearing the IR sensor for four days, for

a control pattern with an expected pattern is shown in Figure 2.

Table II. Interpolation results

Interpolation

methods

Interpolation mean error ± SD

0

segment

removed

1

segment

removed

2

segments

removed

3

segments

removed

Linear

0.13±0.7

0.16±0.8

0.18±0.7

0.19±0.8

Cubic splines

0.15±0.7

0.17±0.8

0.19±0.8

0.25±0.9

Polynomial

0.80±1.4

0.80±1.4

0.81±1.4

0.81±1.4

Kernel-based

0.06±0.2

0.08±0.3

0.11±0.3

0.13±0.4

Figure 2 Skin temperature signal

Figure 3 Kernel-based interpolation signal (dashed line) and its true measured

one (straight line) for one subject over five hours, i.e. 300 minutes

In order to compare the interpolation approaches, the signals

of the two subjects were divided into segments of 1440 minutes,

i.e. one-day length, leading to eight segments. Then, these

segments are resampled by taking one point every five minutes,

leading to segments of 288 points. Thereafter, the linear,

polynomial, cubic spline and kernel-based interpolation

techniques were applied over these segments and the results are

compared to the original observed signal. For the kernel-based

interpolation, a Gaussian kernel was used, using a cross-

validation algorithm to select the optimal values of the

bandwidth and the regularization parameter according to the

learning data . The mean errors for the interpolations are

shown in Table II. These were computed by averaging the

absolute difference between the observed and the estimated

signals for the eight segments. In order to simulate the missing

data, a 30-min segment was subsequently removed from the one-

day length segment. This 30-min segment corresponds to six

consecutive points among the 288-point segments. Then, two

and three 30-min segments are removed. These segments were

randomly selected within each one-day length segment. Table II

shows the results for segments with no missing data (0 segment

removed), one 30-min segment removed, two segments

removed, and three segments removed. It is worth noting that

simulations are performed 50 times and errors are averaged over

all results, since the removed segments are selected randomly.

The table shows that the Kernel-based algorithm yields better

results with less estimation error in all cases, which is expected

due its malleability and its adaptation to the curve, even under

non-linear conditions.

The measured temperature signal for a typical subject for a

five-hour period is shown in Figure 3. The figure also shows the

interpolated signal obtained with the kernel-based interpolation.

01000 2000 3000 4000 5000 6000

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Time (min)

Temperature (°C)

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The plot shows that the computed signal is close to the measured

one, with a smooth curve produced.

Then, the objective is to illustrate the rhythmometric modeling

techniques, which use the Fourier series or Cosinor. To this end,

we consider the 5 temperature signals over a 4-day period,

having passed the interpolation phase, i.e. the signals have no

gaps, with a rate of one sample per minute. We start by applying

the Fourier series, taking the fundamental period to be equal to

24 hours, i.e. 1440 minutes, followed by 3 harmonics, i.e. 12

hours, 8 hours and 6 hours. We consider that the modeling error

is the average of the absolute differences between the 4-day

temperature signals and their modeled ones for the 5 signals. In

this case, the modeling error of the Fourier series technique with

a 24-hour period is equal to 0.77. We then apply the Fourier

transform to the signals to obtain their periodograms. These

computations showed that the local maxima of the spectral

energy are not obtained exactly at the frequencies 1/24, 1/12, 1/8

and 1/6, but very close to them.

Figure 4 Periodogram generated by the Fourier analysis of the time series

Figure 4 shows a periodogram obtained for a certain subject,

where the fundamental frequency is equal to 1/23.5. Let be

the fundamental period obtained by taking the maximal point of

the periodogram, i.e. is very close to 24 hours. If we take

and the following three local maxima, and perform the

inverse Fourier transform, the modeling error decreases to 0.73,

which was expected since this way more information is

considered in the modeling. Having the fundamental period not

equal to 24h is related to the fact that only four cycles (4 days)

are considered, which is not enough for Fourier analysis.

Afterwards, by applying Cosinor computations, while taking the

fundamental period equal to 24h, and considering the following

3 harmonics, the modeling error is the least, being equal to 0.53.

This shows the power of such a method with limited-duration

signals.

Figure 5 Temperature modeled signals using the Fourier series with a 24h

period in thin dashed line, the Fourier transform with a 23.5h period in thin

straight line and the Cosinor in thick dashed line, and the original signal in thick

straight line.

Figure 5 shows the modeled signals in a thin dashed line for the

Fourier series using a 24h period, a thin straight line for the

Fourier transform using a 23.5h period and a thick

dashed line for the Cosinor analysis. It also shows the original

signal in thick straight line, for a one day period going from 5

a.m. till 5 am. the following day. The curves show that the

obtained signals follow the original one with a small modeling

error. The rhythmometric parameters, i.e. orthophase and

bathyphase, are then computed. Table III shows their mean

values in degrees with the standard deviations over the 5 signals.

Here the parameters are extracted from the rhythmometric

models obtained for the whole 4-days signals. As expected,

considering the fundamental period from the periodogram,

modeling leads to a difference in the characteristics equivalent

to 30 minutes, whereas both Fourier series and Cosinor lead to

close values. This study shows that the Cosinor computations have

promising results, to be validated with more signals later on.

4. Discussion

This paper proposed a longitudinal evaluation of skin

temperature measurement, collected using a thoracic sensor.

The collected signals have a sampling frequency of one point

every five minutes, with a length of four days for control

subjects. At a first phase, we performed interpolation, using a

kernel-based machine learning technique, to resample the

signals to a higher frequency of one point every minute and to

fill in the gaps in the measured signals. This step is crucial for

the following data processing techniques. A rhythmometric

study is then followed up on resampled signals, using either a

Fourier representation or Cosinor. A Fourier analysis with the

Fourier transform and the Fourier series is first proposed to

model the signals, then a Cosinor model is constructed using

Fourier analysis and least squares computations. The advantage

of Cosinor remains in its robustness against un-equidistant and

low-duration data, which is not the case for Fourier analysis.

Rhythmometric parameters such as MESOR, amplitude,

orthophase and bathyphase, are then extracted using the

obtained model. For a given temperature signal of a given

subject, such computations are performed for each sliding

window of the signal, leading to a set of random variables that

are the rhythmometric parameters. Then a divergence study is

applied to detect any perturbation of the rhythm. In fact,

chemotherapy can alter the circadian rhythm of the patients,

therefore it is expected in that case to observe a divergence of

the distributions of the rhythmometric parameters between the

time span on chemotherapy and that before or after it. To detect

such divergence, we can separate each parameter values into

two sets (on or off chemotherapy), then apply a statistical test

to detect the divergence between their distributions. This study

is promising since chemotherapy-induced perturbations have

been documented for over 15 anticancer drugs in experimental

models, and for several drug combination protocols in cancer

patients [6 14 15].

The kernel-based interpolation phase was validated using

different cases of missing data in the signals, and by comparing

them to well-known interpolation methods such as linear,

polynomial and cubic splines. A comparison between the

rhythmometric modeling techniques is then conducted,

showing that Cosinor analysis leads to fewer modeling errors.

Having only five signals with a 4-day duration is not enough to

draw conclusions, however the results are promising for future

works. Thus a larger study involving more healthy controls and

5 7 9 11 13 15 17 19 21 23 1 3

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35

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37

38

Temperature(°C)

Time (hours)

Table III. Rhythmometric parameters

Mean (±SD)

Fourier

series

(24h)

Fourier

transform

()

Cosinor

model

(24h)

Orthophase

96.9±0.6

104.1±1

97.8±0.5

Bathyphase

27.58±0.8

38.8±0.8

28.4±0.5

www.astesj.com 110

cancer patients is planned to further determine the relevance of

the methodology developed here and the minimum number of

harmonics. It should be possible to evaluate the effect of

harmonics such as 12h, 8h, and 6h, and even to use only specific

harmonics of varying frequency in order to determine the exact

moment when the circadian rhythm is modified. Advanced

processing such as the goodness of fit and the rhythm detection

will also be applied on the signals using the F-test. Moreover,

an evaluation of the IR sensor is also to be done, to verify if the

observed temperature with such a sensor remains a mirror of the

central temperature.

5. Acknowledgments

The authors would like to thank all clinicians, clinical

research associates and technicians for their involvement in the

device design and the protocol definition, set-up and

monitoring. This work was supported by the Champagne-

Ardenne Regional Council; PICADO project (Projet Innovant

pour le Changement d’Ampleur de la Domomédecine).

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