aplicacion de IPFM en la HRV

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The Integral Pulse Frequency Modulation Modelwith Time-Varying Threshold: Application to

Heart Rate Variability Analysis during ExerciseStress Testing.

Raquel Bailón∗, Ghailen Laouini, César Grao, Michele Orini, Pablo Laguna, Senior Member, IEEE, and OlivierMeste, Member, IEEE

Abstract—In this paper an approach for heart rate variability(HRV) analysis during exercise stress testing is proposed basedon the integral pulse frequency modulation (IPFM) model wherea time-varying threshold is included to account for the non-stationary mean heart rate. The proposed technique allows theestimation of the autonomic nervous system (ANS) modulatingsignal using the methods derived for the IPFM model withconstant threshold plus a correction which is shown to beneeded to take into account the time-varying mean heart rate.On simulations, this technique allows the estimation of theANS modulation on the heart from the beat occurrence timeseries with lower errors than the IPFM model with constantthreshold (1.1%±1.3% versus 15.0%±14.9%). On an exercisestress testing database, the ANS modulation estimated by theproposed technique is closer to physiology than that obtainedfrom the IPFM model with constant threshold, which tends tooverestimate the ANS modulation during the recovery and tounderestimate it during the initial rest.

Index Terms—autonomic nervous system modulation, exercisestress testing, heart rate variability, integral pulse frequencymodulation model, respiratory sinus arrhythmia, time-varyingmean heart rate, time-varying threshold.

I. INTRODUCTION

The integral pulse frequency modulation (IPFM) modelhas been proposed for studying the properties of biomedicalsignals, such as nerve spike trains [1], heart rate variability(HRV) [2] and for modeling the information coding and signaltransmission through nervous fibers [3]. In the IPFM modela spike is generated when the integral of a modulating signalreaches a threshold. Each time the threshold is reached a spikeis generated and the integral is reseted.

Several works have used the IPFM model to explain theregulation of the heart rate by the autonomic nervous system(ANS) [1], [4]–[11], where the ANS regulation is represented

R. Bailón, C. Grao, M. Orini and P. Laguna are with the CommunicationsTechnology Group (GTC) at the Aragón Institute of Engineering Research(I3A), University of Zaragoza, María de Luna 1, 50018 Zaragoza, Spain,and CIBER de Bioingeniería, Biomateriales y Nanomedicina (CIBER-BBN),Spain, (e-mail:{rbailon, morini, laguna}@unizar.es)

G. Laouini and O. Meste are with the Laboratorie I3S, Universityof Nice and CNRS, Sophia Antipolis 06903, France, (e-mail:{meste,laouini}@i3s.unice.fr)

This study was supported by Ministerio de Ciencia e Innovación, Spain,under Project TEC2010-21703-C03-02 and TRA2009-0127, in part by theDiputación General de Aragón (DGA), Spain, through Grupos ConsolidadosGTC ref:T30, by ISCIII, Spain, through CIBER CB06/01/0062, and by CAI,Spain, through Programa Europa XXI.

by the modulating signal and each spike generation representsa beat occurrence. Thus, the IPFM model can be used toestimate the ANS modulation on the sinoatrial (SA) node fromthe beat occurrence times, which is of interest in many physi-ological and pathological situations in which the ANS activitymay be altered, unbalanced or damaged. Stress testing [12], tilttable testing [13], and experiments of induced emotions [14]are some examples of these physiological situations, whilemyocardial infarction, diabetic neuropathy [15], and cardiacischemia [16], [17] are examples of the pathological situations.In order to retrieve the ANS modulation on the SA node fromthe beat occurrence time series, different representations ofHRV have been proposed, among which the heart timing signalhas been demonstrated to provide an unbiased estimation ofthe ANS modulation, even in the presence of isolated ectopicbeats [18], [19].

However, the IPFM model assumes a constant threshold,which in HRV analysis represents the mean heart period, notbeing appropriate in certain situations in which the mean heartperiod is time-varying, such as in exercise stress testing. Infact, the necessity of taking into account the time-varying heartperiod in the analysis of HRV during exercise stress testingwas pointed out in [20] and [21], where a different approach,named the pulse frequency modulation model was applied toobtain the modulating signal. The analysis of the IPFM modelwith time-varying threshold has been previously addressedin [22], where the spectrum of the spike train output of theIPFM model is computed, in [23], where an IPFM model withperiodically-varied threshold is studied from the viewpoint ofnonlinear dynamics, in [24], where a parametric approach isproposed to estimate the modulating signal as well as the time-varying threshold of the IPFM model, and in [25], where theeffect of a time-varying threshold on the heart period is studiedfor a sinusoidal modulating signal.

In this paper we propose an approach to the analysis ofthe time-varying threshold IPFM (TVIPFM) model appliedto the analysis of HRV during exercise stress testing, whichallows the estimation of the modulating signal using themethods derived for IPFM model with constant thresholdplus a correction which takes into account the time-varyingthreshold.

The paper is organized as follows. The IPFM model withconstant and time-varying threshold are presented in Sec-



tion II, where it is also described the simulation study designedto evaluate the performance of our approach, which simulatesthe beat occurrence time series during stress testing, as wellas an exercise stress testing database, which is analyzed.Section III presents the results and Section IV a discussionof the proposed approach.

II. METHODS AND MATERIALS

A. The integral pulse frequency modulation model

In order to relate and derive the ANS influence on the beatoccurrence time series, tk, which is the available information,we rely on the IPFM model. The IPFM model is based onthe hypothesis that the ANS influence on the SA node can berepresented by the modulating signal M(t), and a beat triggerimpulse is generated when the integral of 1 +M(t) reaches athreshold T , which represents the mean heart period, resettingthe integrator [5], see Fig. 1(a). The modulating signal M(t)is assumed to be causal, band-limited and M(t) < 1.

1 +M(t) ∫+

−T

· · ·tk

Reset(a)

1 +M(t) ∫+

−T (t)

· · ·tk

Reset(b)

1 +M(t)+MT(t) ∫

+

−TDC

· · ·tk

Reset(c)

Figure 1. Block diagram of the different IPFM models: (a) the classicalIPFM model, (b) the TVIPFM model, and (c) the alternative TVIPFM model.

Based on the IPFM model and assuming that the first beatoccurs at time t0 = 0, the beat occurrence time series can begenerated as solution of

k =

∫ tk0

1 +M(t)

Tdt, (1)

where k and tk represent the beat order and occurrence timeof this kth beat [8], respectively. The term

dHR(t) =1 +M(t)

T(2)

represents the instantaneous heart rate in Hz, T is the meanRR interval (heart period) in seconds in the analyzed epoch,1/T the mean heart rate, and dHRV(t) =

M(t)T

represents theHRV in Hz. Note that dHRV(t) represents the time-varying partof the instantaneous heart rate dHR(t).

The IPFM model can be generalized to continuous timerewriting (1) as [18]

κ(t) =

∫ t0

1 +M(τ )

Tdτ, (3)

where κ(t) is the continuous beat order function such thatκ(tk) = k.

Note that dHR(t) can be obtained differentiating κ(t) withrespect to t, dHR(t) = κ

′(t), without any knowledge of T .Then, the modulating signal M(t) can be obtained by

rewriting from (2)

M(t) = T · dHR(t)− 1. (4)

In practice, κ̂(t) is estimated by a spline interpolation of thepairs (tk, k), then analytically derived to obtain d̂HR(t) [18], T̂is estimated as the mean heart period in the analyzed interval,and, finally

M̂(t) = T̂ · d̂HR(t)− 1. (5)

In situations in which the mean heart period T is time-varying, i.e. T (t), such as during stress testing, the estimateM̂(t) in (5) also contains the variations of T (t), which usuallyare of much lower frequency. For that reason, and to overpassthis limitation, M̂(t) is usually high-pass filtered in order toremove the variations non-related to the modulating signal,assuming that these variations are of lower frequency and donot overlap with those related toM(t). This estimate is denotedM̂0(t). As it will be shown in Section III, the amplitude of thevariations of M̂0(t) are still affected by the time-varying heartperiod T (t), making M̂0(t) a bias estimator.

B. The integral pulse frequency modulation model with time-varying threshold

In the TVIPFM model the integral of 1+M(t) is comparedto a time-varying threshold T (t), representing the time-varyingmean heart period, which can be decomposed as a constantcomponent TDC and a time-varying component TAC(t), T (t) =TDC + TAC(t) [24] (see Fig. 1(b)).

Assuming, as in Section II-A, that M(t) is causal, band-limited, M(t) < 1 and that T (t) is constant between twosuccessive beats, the beat occurrence time series can beapproximated by

k ≈∫ tk0

1 +M(t)

T (t)dt, (6)

being the instantaneous heart rate

dHR(t) =1 +M(t)

T (t). (7)

Assuming that the variations of the term 1T (t) are slower

than those of the term M(t)T (t)

, and that their spectral componentsdo not overlap, a time-varying mean heart rate, dHRM(t), canbe defined

dHRM(t) =1

T (t). (8)



Then, the HRV signal dHRV(t) can be computed as

dHRV(t) = dHR(t) − dHRM(t) =M(t)

T (t), (9)

From (9) it is evident that the modulating signal can beobtained by correcting the HRV signal dHRV(t) by the time-varying mean heart rate dHRM(t),

M(t) = dHRV(t)T (t) =dHRV(t)

dHRM(t). (10)

Based on equation (7), different approaches can be consid-ered for the estimation of the modulating signal M(t).

1) Approach A: Equations (7), (8), and (9), can be seenfrom an alternative IPFM model with constant threshold, inwhich the effect on the output of the variations of the meanheart period, TAC(t), can be attributed to an extra modulat-ing signal MT(t), additional to M(t), causal, band-limited,MT(t) < 1, and whose spectral components are lower than,and do not overlap with, those of M(t). The block diagramof this alternative TVIPFM model can be seen in Fig. 1(c).If the variations of the time-varying threshold, T AC(t), aresmall compared to its mean value, TDC, the TVIPFM modelof Fig. 1(b) and the alternative model of Fig. 1(c) can beshown to be approximately equivalent.

Let’s start by rewriting (7)

dHR(t) =1 +M(t)

T (t)=1 +M(t)

TDC + TAC(t)=

1 +M(t)

TDC

(1 + TAC(t)

TDC

)≈ 1 +M(t)

TDC

(1− TAC(t)

TDC

), (11)

that, neglecting second order terms, becomes

dHR(t) ≈1 +M(t) − TAC(t)

TDC

TDC

, (12)

and, identifying terms in (2), MT(t) = −TAC(t)TDC

.In this case, the instantaneous heart rate dHR(t) can be

obtained differentiating κ(t) with respect to t, just as inSection II-A,

dHR(t) = κ′(t) ≈ 1 +M(t) +MT(t)

TDC

. (13)

The time-varying mean heart rate dHRM(t) is defined as

dHRM(t) ≈ 1 +MT(t)

TDC

, (14)

and the HRV signal dHRV(t) is computed as

dHRV(t) ≈ dHR(t) − dHRM(t) ≈ M(t)

TDC

. (15)

Assuming

dHRM(t) ≈ 1 +MT(t)

TDC

≈ 1

TDC

, (16)

the expression in (10) is still valid. In order to estimate M(t),the term d̂HR(t) is first estimated from (13), where κ̂(t) isobtained by spline interpolation of the pairs (t k, k). Then,the term d̂HRM(t) is estimated by low-pass filtering d̂HR(t) and

d̂HRV(t) is estimated from the middle term in (15). Finally, theestimate of M(t), denoted M̂A(t), is obtained from (10) as

M̂A(t) =d̂HRV(t)

d̂HRM(t). (17)

This approach allows the interpretation of the effect of atime-varying threshold on the IPFM model as due to an extramodulating signal MT(t), different from M(t), and to use therobust methods derived for the analysis of HRV based on theclassical IPFM model with constant threshold [18], [19], whichtake into account the presence of ectopic beats.

2) Approach B: An alternative approach to the estimationof the modulating signal M(t) which combines the resultsfrom the TVIPFM model with those of the IPFM model withconstant threshold is the following.

Assuming that the quantity dHR(t) is constant over two suc-cessive beat times (tk−1 < t < tk), where no more informationis available, the integration between two successive pulses maybe computed by

1 =

∫ tktk−1dHR(t)dt ≈ dHR(tk−1)(tk − tk−1)

≈ dHR(tk)(tk − tk−1) ≈ dHR (tkc) (tk − tk−1), (18)

where tkc =tk−1+tk2

represents the in–between beat occur-rence time.

Then, replacing dHR (tkc) by its value in (7), we get

M (tkc) ≈T (tkc)

(tk − tk−1) − 1 =T (tkc)− (tk − tk−1)

(tk − tk−1) . (19)

The term T (t) is estimated from d̂HRM(t), as in approachA, by T̂ (t) = 1

d̂HRM(t), and then evaluated at t = tkc . The

term T̂ (tkc) is substituted in (19) to obtain what is denotedM̂B (tkc), and the continuous signal M̂B(t) is estimated byspline interpolation of the pairs (tkc , M̂B (tkc)). An alternativeapproach for the estimation of T̂ (tkc) is described in Ap-pendix A.

3) Approach C: Yet another approach for the estimation ofthe modulating signal M(t) based on the TVIPFM model canbe considered [25] from (19)

M (tkc) ≈T (tkc)− (tk − tk−1)

T (tkc). (20)

The estimate in (20) is denoted M̂C (tkc); the continuoussignal M̂C (t) can be estimated by spline interpolation of thepairs (tkc , M̂C (tkc)).

C. Materials

1) Simulation study: Due to the unavailability of a “true”reference modulating signal, representing the ANS regulationon the heart, a simulation study has been designed to evaluatethe performance of the different approaches presented in thispaper.

The ANS modulation during stress testing may be modeledby two sinusoids representing the LF and HF components,respectively. Thus, the analytic signal of the modulating signalM(t) can be modeled as [12]

aM(t) = ALF(t)ejΦLF(t) + AHF(t)e

jΦHF(t). (21)



The LF and HF components are defined by the amplitudesALF(t) and AHF(t), and the instantaneous frequencies FLF(t) =12πdΦLF(t)dt

and FHF(t) =12πdΦHF(t)dt

, respectively. The very lowfrequency (VLF) component is not considered in this modelsince it cannot be estimated during stress testing. The FLF(t) isassumed to be constant during the stress test and set to 0.1 Hz,while the FHF(t) is set to be the respiratory frequency measuredduring a stress test recording using an airflow thermistor [26].The frequency FHF(t) and amplitudes ALF(t) and AHF(t) of thesimulated signals are displayed in Fig. 2(a).

200 400 600 8000.2

0.3

0.4

0.5

0.6

0.02

0.04

0.06

0.08

0.1

Freq

uenc

y(H

z)

Time (s)

Am

plitu

de(A

U)

200 400 600 8000.3

0.5

0.7

0.9

1.1

Time (s)

T(t)

(s)

(a) (b)

Figure 2. Simulated signal: (a) FHF(t) (dashed-dotted line), ALF(t) (dashedline) and AHF(t) (solid line), (b) T (t)

A time-varying mean heart period T (t), displayed inFig. 2(b), is simulated as the inverse of a time-varying meanheart rate which increases linearly [27]–[29] from 60 bpm (1Hz) at the beginning of the stress test to 180 bpm (3 Hz) atpeak stress, and decreases linearly during recovery until theinitial value of 60 bpm.

Given the modulating signal M(t) and the time-varyingthreshold T (t), the beat occurrence time series tk is gener-ated based on the TVIPFM model of Fig. 1(b). Then, thetime series tk are contaminated with additive white Gaussiannoise (AWGN) in order to simulate the jitter in the QRSfiducial point of the estimated beat occurrence time series,i.e. t̂k = tk+vk, where vk is a series of AWGN with varianceσ2. Different values of σ2 (0.04·10−5, 2.5·10−5 and 10·10−5s2 ) have been considered representing different jitter in theQRS fiducial point (2, 5 and 10 ms, respectively).

A sampling frequency of 1000 Hz is used for the simulatedM(t) and T (t).

2) Stress testing database: A database of stress testingrecordings belonging to subjects with different training statusis analyzed. All subjects were non-smokers and none wastaking any medication. Physical activity and alcohol andcaffeine beverages consumption were prohibited 24 hoursbefore any exercise testing session. Following a 5-minuteresting period during which subjects stayed seated on thebicycle, all subjects performed a maximal graded exercise testwith constant pedalling frequency, completing the exercisetest without any clinical abnormality or discomfort. Initialworkload and subsequent 2-minute step workload incrementswere set to ensure a 12- to 15-minute maximal exercise test.Subjects characteristics are shown in Table I, more informationcan be found in [30].

During the exercise test and the preceding 5 minutes (rest),a three-lead ECG was recorded and digitized on-line by a 12-

Table ISTUDY POPULATION CHARACTERISTICS.

Group: # of Pedalling Pmax V̇O2peak Agesubjects frequency (rpm) (W) (ml·min−1·kg−1) (yr)

A: 2 70, 75 225± 10 37±2 33 ± 3B: 9 (5)70, (3)75, 80 282±30 46±4 29±5C: 3 (3)80 340±20 61±2 30±5D: 9 (3)75, (3)85, (3)90 430±30 70±5 25±2A: sedentary men; B: less than 10 hour training per week; C: more than 10hour training per week, and D: high level athletes; P max: maximal poweroutput; V̇O2peak: peak O2 consumption.

bit analog-to-digital converter at a sampling rate of 1000 Hzon a personal computer.

Detection of the beat occurrence time series t̂k is performedon one of the ECG leads, placed collinearly to the standardDII derivation directly on the chest in order to avoid limbsmotion artifacts, after removing baseline with a high-pass finiteimpulse response filter.

D. Performance measurements

In order to evaluate the different approaches, differentperformance measurements are considered. First, the estimatedmodulating signal M̂(n) is compared to the simulated modu-lating signal M(n) (signals M̂(n) and M(n) represent M̂(t)and M(t), respectively, sampled at a sampling frequency ofFs Hz). Then, clinical HRV parameters, namely the power ofthe low frequency (LF) and high frequency (HF) components,derived from M̂(n) (P̂LF(n) and P̂HF(n), respectively) arecompared with those derived from M(n) (PLF(n) and PHF(n),respectively). Parameters PLF(n) and PHF(n) are computed ateach time instant n as

PLF(n) =1

2

1

2K − 1m2∑m=m1

PM(n,m),

PHF(n) =1

2

1

2K − 1m4∑m=m3

PM(n,m), (22)

where PM(n,m) is the discrete Smoothed Pseudo Wigner-Ville Distribution (SPWVD) of M(n), defined as

PM(n,m) = 2

K−1∑k=−K+1

|h(k)|2⎡⎣

N−1∑n′=−N+1

g(n′)aM(n+ n′ + k)a∗M(n+ n′ − k)⎤⎦

× e−j2π mM k ;m = −M + 1, . . . ,M, (23)

where n and m are the time and frequency indices, respec-tively, and aM(n) is the analytic signal of M(n). The termsg(n) and |h(k)|2 represent the time and frequency smoothingwindows, respectively, chosen to be

g(n) =

⎧⎨⎩

1

2N − 1 , n = −N + 1, . . . , N − 10, otherwise,

(24)

and

|h(k)|2 ={e−γ|k|, k = −K + 1, . . . , K − 10, otherwise.

(25)

Parameter values used in the estimation of PLF(n) and PHF(n)are given in Table II, where the LF band is defined in its



Table IIPARAMETER VALUES.

Parameter 2M 2K − 1 γ 2N − 1 m1 m2 m3 m4

Value 1024 1023 164

41 � 0.04FsM� � 0.15

FsM� �Fr(n)−0.07

FsM� �Fr(n)+0.07

FsM�

(units) (samples) (samples) (samples−1) (samples) (samples) (samples) (samples) (samples)

standard way from 0.04 to 0.15 Hz, while a time-varying HFband is defined centered on the respiratory frequency, Fr(n)(in Hz), with a bandwidth of 0.14 Hz [31]. Parameters P̂LF(n)and P̂HF(n) are computed from (22) and (23) substitutingM(n)with M̂(n).

For each value of σ2 a total of Q=100 realizations havebeen simulated. A relative error trend, eqθ%(n), is defined foreach realization q and for each parameter θ(n), where θ(n) ∈{M(n), PLF(n), PHF(n)},

eqθ%(n) =

θ̂q(n) − θ(n)ηθ(n)

× 100(%). (26)

The term θ̂q(n) represents the estimate of θ(n) in realization q,and the term ηθ(n) ∈ {bM(n), PLF(n), PHF(n)}, where bM(n)is the envelope of M(n).

An averaged relative error trend, eθ%(n), can be defined as,

eθ%(n) =1

Q

Q∑q=1

eqθ%(n). (27)

For each realization q, the mean and SD of the error trendsare computed and then averaged among the Q realizations,giving the performance measurements μθ% and σ2θ%,

μθ% =1

Q

Q∑q=1

1

Nq

Nq∑n=1

∣∣eqθ%(n)∣∣ ,

σ2θ% =1

Q

Q∑q=1

1

Nq − 1Nq∑n=1

⎛⎝∣∣eq

θ%(n)∣∣ − 1Nq

Nq∑n=1

∣∣eqθ%(n)∣∣⎞⎠2

,

(28)

where Nq is the number of samples of θ̂q(n).

III. RESULTS

A. Simulation study

From the simulated t̂k series, the modulating signal M(n)is estimated using the approaches M̂0(n), M̂A(n), M̂B(n) andM̂C(n), described in Sections II-A and II-B. A samplingfrequency of Fs= 4 Hz and a 5th order spline interpolationis used in all the approaches. The estimate M̂0(n) is obtainedby high-pass filtering M̂(n) with a cut-off frequency of 0.03Hz, which is the same cut-off frequency used to obtain thetime-varying mean heart rate d̂HRM(n). This cut-off frequencyis below the lower limit of the LF band (0.04–0.15) Hz. Therespiratory frequency, Fr(n), needed for the estimation ofPHF(n), is assumed to be equal to the instantaneous frequencyof the HF component, and known.

Figure 3 displays the SPWVD of the simulated modulatingsignal M(n), as well as for M̂0(n) and M̂A(n). It can beobserved that M̂A(n) and M(n) present similar characteristics,

such as the progressive diminution of both the LF and HFamplitudes during the exercise, which is noticeable as lightergrays approximately from second 400 to 600. This effectcannot be appreciated in M̂0(n), which presents even darkergrays around the stress peak.

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uenc

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y(H

z)

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Figure 3. Smoothed pseudo Wigner-Ville distribution of (a)M(n), (b) M̂0(n)and (c) M̂A(n). In the the gray scale, lighter grays stand for lower values anddarker grays for higher values.

(a)

PM(n,m)

(b)

PM̂0(n,m)

(c)

PM̂A(n,m)

Table III displays the performance measurements obtainedby the approaches proposed in this paper to estimate themodulating signal based on the TVIPFM model.

Regarding the parameters μM%±σM%, it can be appreciatedthat all the three methods based on the TVIPFM model(M̂A(n), M̂B(n), and M̂C(n)) obtained notably lower valuesthan the method based on the IPFM model with constantthreshold (M̂0(n)), for all values of σ2 considered, being thedifferences larger for lower values of σ2. In the absence ofnoise, M̂A(n) obtained the lowest values, while the differencesbetween M̂A(n), M̂B(n), and M̂C(n) became insignificant whennoise was added. Estimates M̂B(n) and M̂C(n) obtained slightlylower values than M̂A(n) for σ=5 ms. Regarding the clinicalHRV parameters (μPLF%

± σPLF%, and μPHF%

± σPHF%), ap-

proaches A, B and C obtained much lower values than the



Table IIIPERFORMANCE MEASUREMENTSμθ% ± σθ% OBTAINED BY M̂0(n),

M̂A(n), M̂B(n) AND M̂C(n), FOR DIFFERENT VALUES OF σ.

σ μM% ± σM% (%)(ms) M̂0(n) M̂A(n) M̂B(n) M̂C(n)

0 15.0 ± 14.9 1.1 ± 1.3 3.6 ± 2.6 5.8 ± 4.92 18.4 ± 17.4 7.8 ± 7.8 7.4 ± 6.3 8.9 ± 7.15 28.1 ± 27.4 19.0 ± 18.9 16.1 ± 15.4 16.9 ± 15.4σ μPLF%

± σPLF%(%)

(ms) M̂0(n) M̂A(n) M̂B(n) M̂C(n)0 48.0± 31.0 0.9 ± 1.9 1.4 ± 1.9 1.2 ± 2.02 48.0 ± 31.1 1.5 ± 2.0 1.8 ± 2.1 1.7 ± 2.25 48.2 ± 32.9 3.9 ± 4.9 3.1 ± 2.7 3.1 ± 2.9σ μPHF%

± σPHF%(%)

(ms) M̂0(n) M̂A(n) M̂B(n) M̂C(n)0 48.7 ± 31.9 0.7 ± 0.5 12.6 ± 5.1 11.5 ± 4.92 49.1 ± 32.9 4.0 ± 3.3 12.6 ± 6.4 11.5 ± 6.35 47.1 ± 33.2 10.4 ± 8.8 14.5 ± 9.1 13.8 ± 8.9

one based on M̂0(n). For μPLF%± σPLF%

, methods A, B andC obtained similar results, which slightly increased when σincreased. For μPHF%

± σPHF%, approach A obtained notably

lower values than approaches B and C.Figure 4 displays the averaged relative error trend eθ%(n)

corresponding to M̂0(n), M̂A(n), M̂B(n), and M̂C(n) for aσ=2 ms. Note that the average relative error trends eM%(n),ePLF%(n), and ePHF%(n), corresponding to the IPFM modelwith constant threshold (related to M̂0(n)) are dependent onthe value of T (t), being minimum when T (t) equals its meanvalue and maximum for the lowest and highest values ofT (t). However, the average error trends corresponding to theTVIPFM model (related to M̂A(n), M̂B(n) and M̂C(n)) do notdepend on the value of T (t), although a peak is observedin the vicinity of stress peak, where there is an abrupt non-physiological change in T (t), which can not be followed byd̂HRM(n).

B. Stress testing database

From the t̂k series estimated from the stress testing record-ings described in Section II-C2, the modulating signal isestimated using M̂0(n) and M̂A(n). Estimates M̂0(n) and M̂A(n)are then filtered with a time-varying filter centered on therespiratory frequency [21] with a bandwidth of 0.14 Hz, andthe envelopes of the filtered signals denoted b̂M0, HF(n) andb̂MA, HF(n), respectively. Estimates M̂0(n), M̂A(n), b̂M0, HF(n),b̂MA, HF(n), b̂MA, HF(n) − b̂M0, HF(n) and d̂HRM(n), are displayedin Fig. 5 for one of the subjects (with less than 10 hourtraining per week), who performed the stress test at a pedallingfrequency of 75 rpm. It can be appreciated that differencesbetween b̂M0, HF(n) and b̂MA, HF(n) depend on the time-varyingmean heart rate d̂HRM(n), being larger when d̂HRM(n) departsfrom its mean value. This can lead to erroneous interpretationof the ANS response to exercise when estimating its evolutionfrom the changes observed in the estimated modulating signal.By using M̂0(n) in this example, the return of the vagal activity(measured from b̂M0, HF(n)) observed in the early stage of therecovery (from second 1200 to 1400) reaches similar valuesthan those corresponding to the beginning of the exercise(before second 300). This conclusion is slightly different byusing b̂MA, HF(n), where the two levels are different. Indeed,

the latter conclusion is more likely from a physiological pointof view [32].

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Figure 5. Estimates for one of the subjects of Section II-C2:(a) M̂0(n),(b) M̂A(n), (c) b̂M0, HF(n) (grey) and b̂MA, HF (n) (black), (d) b̂MA, HF (n)-b̂M0, HF(n), (e) d̂HRM(n).

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Figure 4. Average relative error trend (a) eM%(n), (b) ePLF%(n) and (c) ePHF%(n), corresponding, from top to bottom, to M̂0(n), M̂A(n), M̂B(n), andM̂C(n), for a σ=2 ms.

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the example of Fig. 5, i.e. that M̂0(n) tends to overestimatevagal activity during the recovery and to underestimate itduring the initial rest.

Inspection of Fig. 5 reveals an abrupt increase in d̂HRM(n)just after the onset of exercise (second 300), which is visiblein all subjects of the database, and may be due to thecentral command or the exercise pressor reflex in responseto exercise [33]. The plateau observed from second 300 to450, visible in some but not all the subjects, correspondsto the lowest intensity exercise and it may be associated toa transient response of the ANS at the onset of exercise.Note that a similar plateau is also observed in b̂MA, HF(n) atlower values than those during rest, which then decreasesconcomitant with the linear increase in d̂HRM(n), till second1000, when it slightly increases. During the recovery, there isan abrupt increase in b̂MA, HF(n), which gradually decreasesas the heart rate decreases, reaching values close to thoseobserved during exercise. There is evidence that the HF power,as well as other HRV indices, remains reduced with respect toresting values after 5, 10, 15, 30 minutes or even an hour ofthe cessation of exercise [32], depending on exercise intensityand modality [34].

IV. DISCUSSION

A. Methodological aspects

In this paper the TVIPFM model is applied to the analysisof HRV during exercise stress testing.

The classical IPFM model has been previously used in HRVanalysis to estimate the ANS modulation on the heart from thebeat occurrence time series, even in the presence of isolatedectopic beats, when the mean heart period can be consideredconstant [18], [19]. However, in situations in which the meanheart period is time-varying, such as during exercise stresstesting, the estimation of the modulating signal in (1) needsto be modified, as proposed in (10), to account for the meanheart period modulation.

Three approaches have been considered to analyze theTVIPFM model which allow the estimation of the modulatingsignal using the methods derived for the IPFM model withconstant threshold plus a correction which takes into accountthe time-varying threshold. The three approaches are basedon the hypothesis that, in the time-varying threshold case,the instantaneous heart rate, derived as in the classical IPFMmodel, can be written as dHR(t) =

1+M(t)T (t)

. A time-varyingmean heart rate, which is the inverse of the time-varyingmean heart period, is estimated by low-pass filtering dHR(t).



Approach A estimates the modulating signal multiplying theHRV signal with the time-varying mean heart period; approachB, dividing the variability of the heart period signal bythe heart period signal itself; and approach C, dividing thevariability of the heart period signal by the time-varying meanheart period.

Results from the simulation study show that the threeapproaches estimate the modulating signal as well as clinicalHRV parameters with lower error than the classical IPFMmodel with constant threshold, being the differences largerfor lower levels of noise. Approaches A, B and C estimatethe modulating signal with similar errors, being approach Athe one achieving the lowest error in the absence of noise(1.1%±1.3%). In the presence of high levels of noise (σ=5ms) approaches B and C obtained lower estimation errors thanapproach A, which can be due to the low pass-filtering effectassociated to the heart period signal on which approaches Band C are based [18]. It is worth noting that, while the powerof the LF component is estimated with similar errors by thethree approaches for all levels of noise, the power of the HFcomponent is estimated with notably lower error by approachA (0.7%±0.5%) than by approaches B (12.6%±5.1%) andC (11.5%±4.9%). This, again, can be explained by the lowpass-filtering effect when estimating HRV from the heartperiod signal [18], and it is not related to the influence ofsympathetic and parasympathetic stimulation on the sinusnode. An alternative approach for the estimation of approachesB and C, which is not based on dHR(t) but only on the heartperiod dHP(t) signal is considered in Appendix A, yieldingsimilar results (not shown).

In this paper, a time-varying HF band centered on therespiratory frequency [31] has been considered both in thesimulation study and in the stress testing database. When asimultaneously recorded respiration signal is not available,respiratory frequency can be derived from the ECG signalusing, e.g., the method described in [26], which has beenshown to provide reliable respiratory frequency estimatesduring stress testing.

The TVIPFM has been also applied to the analysis of HRVin [24], where the modulating signal as well as the time-varying threshold are decomposed into a series of orthogonalbasis functions. Despite its computational complexity, one ofthe advantages of the method in [24] is that it does notrequire the time-varying threshold to be of lower frequencythan the modulating signal. However, the application of themethod in [24] to the analysis of HRV during exercise stresstesting requires further considerations since the frequency bandcovered by the basis functions depends on the mean heart rate,which is time-varying during exercise stress testing. If the mostrestrictive frequency band, given by the lowest mean heart rate,is chosen for the basis functions, it may not be large enoughto include the LF component as well as the HF component,centered on respiratory frequency, or even other componentssuch as the pedaling component, when the exercise intensityis high. This could be overcome by performing the analysisin short-time intervals, whose length is inversely proportionalto the frequency resolution of the basis functions.

A note of caution should be considered when using different

representations of HRV, since the effect of a time-varyingmean heart rate may be different. For example, if the powerof the LF and HF components are obtained from the heart ratesignal an increase in mean heart rate leads to an overestimationof the modulating signal (see Figs. 3, 4 and 5), while if theheart period signal is used, it leads to an underestimation ofthe modulating signal [20], [21], [25]. See Appendix B for ananalytical example. The later case is, in fact, an interpretationof approaches B and C.

B. Physiological aspects

An important result derived from the simulation studyis that the estimation error of the modulating signal (andof the clinical HRV parameters) obtained by the classicalIPFM model is dependent on the value of the time-varyingthreshold, while it does not depend on it when the TVIPFMmodel is considered. This is of particular importance whenthe ANS evolution is estimated from the changes observedin the estimated modulating signal (or in the clinical HRVparameters), and may lead to erroneous interpretation of theANS response to exercise. One example is the mechanicalstretching modulation of the sinus node due to respiration,which may be overestimated if the time-varying threshold isnot considered. Another example is the return of vagal activityduring recovery, which is relevant in stress testing studies [35],[36], and may be overestimated by the IPFM model withconstant threshold.

The suitability of the TVIPFM model to the analysis ofHRV during exercise stress testing may be justified by theobservation that the mean heart period is not constant duringthe exercise but increases with work load to satisfy theincreasing metabolic demand. But the TVIPFM model may beuseful for the analysis of HRV in a wider context. It is worthymentioning that threshold modulation was first introduced inthe classical IPFM model in order to study the possibilityof transmitting two informations, with potentially differentsources, along the same channel [22]. The TVIPFM has beenapplied to HRV analysis in [24] to discriminate between theautonomic nervous system modulation of the SA node andthe stretch induced effect, since there is evidence that thevariations in sinus rhythm caused by autonomic modulationand stretch-induced variations are mediated through differentmechanisms: the rate of depolarization for the former andthe threshold level for the later. Moreover, the autonomicmodulation itself has been shown to affect the sinus rhythmthrough different mechanisms. For example, the stimulation ofthe sympathetic nerves releases the hormone norepinephrine,which is believed to increase the permeability of the fibermembrane to sodium and calcium ions, which in turn, causesa more positive resting potential, and also causes increasedrate of upward drift of the membrane potential during dias-tolic depolarization [37]. Parasympathetic stimulation releasesacetylcholine, which may cause a decrease in the slope ofdiastolic depolarization, an increase (in absolute value) inmaximum diastolic potential and an increase in the thresholdpotential [38], [39]. In the TVIPFM model the changes in theslope of the diastolic depolarization membrane potential can



be modeled by the modulating signal M(t) while changes inboth threshold potential and maximum diastolic potential maybe modeled by the time-varying threshold T (t).

One of the limitations of the TVIPFM model concerningthe physiological interpretation of HRV is that it cannot distin-guish from different sources of sympathetic or parasympatheticstimulations which share the same frequency band.

V. CONCLUSION

In this work a new technique for the analysis of HRVduring exercise stress testing has been introduced based onthe TVIPFM model. On a simulation study, this techniqueallows the estimation of the ANS modulation on the heart fromthe beat occurrence time series with lower bias and SD thanthe classical IPFM model with constant threshold. Estimationerrors achieved by this technique are independent from thetime-varying mean heart rate, as opposed to the ones obtainedby the classical IPFM model. On an exercise stress testingdatabase, the ANS modulation estimated by the proposedtechnique is closer to physiology than that obtained from theclassical IPFM model with constant threshold. In situationswhere HRV measurements at different mean heart periods arecompared, the proposed correction of HRV measurements withthe time-varying mean heart rate should be considered.

APPENDIX AAn alternative approach for the estimation of T̂ (tkc), which

may be used in (19) and (20), is the following. Note that thanksto (19) and assuming that M (tkc) is small compared to 1, theheart period signal, estimated by the interval function, can bewritten as

d̂HP(tk) = dIF(tk) = tk − tk−1 =T (tkc )

1 +M (tkc )(29)

≈ T (tkc ) [1 −M (tkc )] ≈ T (tkc ) − T (tkc )M (tkc ) .which can be considered to be approximately composed of

a low pass component, T (tkc) , and a high pass component,T (tkc)M (tkc). Assuming that T (t) is low frequency and M(t)high frequency, the separation of low and high frequencycomponents is straightforward. So, T̂ (tkc) is easily computedlow-pass filtering d̂HP(tk), and, consequently, M̂ (tkc) by using(19) or (20).

APPENDIX BUsing as modulating signal M(t) = A cos(2πf1t) into the

IPFM model, equation (1) for successive beats leads to

tk − tk−1 + A

2πf1(sin(2πf1tk) − sin(2πf1tk−1)) = T. (30)

If f1(tk − tk−1) ≈ f1T � 1, the heart period signal,estimated by the interval function, can be written as [25]

d̂HP(tk) ≈ T −AT cos(πf1(2tk − T )), (31)

which represents an estimate of M(t) with a scaling factorproportional to the mean heart period T . If A� 1, the heartrate signal, estimated by the inverse interval function d̂HR(tk) =dIIF(tk) =

1tk−tk−1 , can be written as

d̂HR(tk) ≈1

T+A

Tcos(πf1(2tk − T )), (32)

which, in contrast with the heart period signal, represents anestimate of M(t) with a scaling factor inversely proportionalto T .

ACKNOWLEDGEMENTS

Authors would like to thank Dr. G. Blain and Dr. S. Bermonfor providing the exercise stress data.

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[36] J. Leino, M. Virtanen, M. Kähönen, K. Nikus, T. Lehtimäki, T. Kööbi,R. Lehtinen, V. Turjanmaa, J. Viik, and T. Nieminen, “Exercise-test-related heart rate variability and mortality: The finnish cardiovascularstudy,” Int. J. Cardiol., pp. –, 2009. doi: 10.1016/j.ijcard.2008.12.123.

[37] A. Guyton, Textbook of Medical Physiology. Saunders, 2005.[38] B. Hoffman, “Autonomic control of cardiac rhythm,” Bull. N.Y. Acad.

Med., vol. 43, no. 12, pp. 1087–1096, 1967.[39] A. Bucchi, M. Baruscotti, R. Robinson, and D. DiFrancesco, “Mod-

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Raquel Bailón was born in Zaragoza, Spain, in1978. She received the M.Sc. degree in telecom-munication engineering and the Ph.D. degree inbiomedical engineering from the University ofZaragoza (UZ), Zaragoza, Spain, in 2001 and 2006,respectively. Since 2003, she was an Assistant Pro-fessor with the Department of Electronic Engineer-ing and Communications, UZ, where she is an Asso-ciate Professor since 2010. She is also a Researcherwith the Aragón Institute for Engineering Researchand also with the Biomedical Research Networking

Center in Bioengineering, Biomaterials and Nanomedicine. Her current re-search interests include the biomedical signal processing field, specially inthe analysis of the dynamics and interactions of cardiovascular signals.

César Grao was born in Teruel, Spain, in 1985.He received the M.Sc. degree in telecommunicationsengineering from the University of Zaragoza (UZ) in2009.

Ghailen Laouini received the M.Sc. Degree inapplied mathematics from the University of Pierreet Marie Curie, France in 2008. Currently he isworking toward the Ph.D. Degree in automatic andsignal processing at the BIOMED project at theLaboratory of Informatics, Signals and Systems,University of Nice Sophia Antipolis.

Michele Orini received the M.S. in biomedicalengineering from the Politecnico di Milano, Italy,and the engineering degree from the École CéntraleParis, France. Since 2007 he is enrolled in the PhDprogram in biomedical engineering at the Universityof Zaragoza, Spain, and at the Politecnico di Milano,Italy. His research activity is in the field of non-stationary biomedical signal processing and focuseson the dynamic interactions between cardiovascularsignals.

Pablo Laguna was born in Jaca (Huesca), Spainin 1962. He received the Physics degree (M.S.) andthe Doctor in Physic Science degree (Ph.D.) from theScience Faculty at the University of Zaragoza, Spain,in 1985 and 1990, respectively. The Ph.D. thesis wasdeveloped at the Biomedical Engineering Division ofthe Institute of Cybernetics (U.P.C.-C.S.I.C.). He isFull Professor of Signal Processing and Communi-cations in the Department of Electrical Engineeringat the Engineering School, and a researcher at theAragón Institute for Engineering Research (I3A),

both at University of Zaragoza, Spain. He is also a member of the CIBER-BBN research center of the Instituto de Salud Carlos III, Spanish Ministryof Science and Innovation. From 1992 to 2005 was Associated professor atsame University and From 1987 to 1992 he worked as Assistant professor ofAutomatic control in the Department of Control Engineering at the PolitecnicUniversity of Catalonia (U.P.C.), Spain and as a Researcher at the BiomedicalEngineering Division of the Institute of Cybernetics (U.P.C.-C.S.I.C.). Hisprofessional research interests are in Signal Processing, in particular appliedto biomedical applications. He is, together with L. Sörnmo, the author ofBioelectrical Signal Processing in Cardiac and Neurological Applications(Elsevier, 2005)

Olivier Meste received the M.Sc. in automaticand signal processing and the Ph. D. degree inscientific engineering from the University of Nice-Sophia Antipolis, France, in 1989 and 1992, respec-tively. He is currently working as a Full Professorat the University of Nice-Sophia Antipolis and asa researcher at the I3S Laboratory. His researchinterests are in digital processing, time-frequencyrepresentations and modelling of biological signalsand systems, including ECG, EMG and EEG. He isactually member of the BISP technical committee of

the IEEE Signal Processing Society.

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