Wavelet Analysis of Blood Pressure Waves

in Vasovagal Syncope

A. Marrone, A.D. Polosa, G. Scioscia, and S. Stramaglia

Dipartimento di Fisica Universit�a di Bari and Sezione INFN di Bari,

Via Amendola 173, I-70126 Bari, Italy

A. Zenzola

Dipartimento di Scienze Neurologiche e Psichiatriche Universit�a di Bari,

Piazza Giulio Cesare 11, I-70100 Bari, Italy

We describe the multiresolution wavelet analysis of blood pres-

sure waves in vasovagal-syncope a�ected patients compared with

healthy people one. We argue that there exist subtle discriminat-

ing criteria which allow us to isolate particular features, common

to syncope-a�ected patients sample, indicating an alternative di-

agnosis method for this syndrome. The approach is similar to that

followed by Thurner et al. but on a di�erent temporal data series.

Key words: Medical Physics; Biological Physics; Data Analysis

PACS: 87.90.+y

Preprint submitted to Elsevier Preprint BARI-TH/305-98

1 Introduction

In recent years wavelet techniques have been successfully applied to a wide

area of problems ranging from the image data analysis to the study of human

biological rhythms [1,2]. In the following we will make use of the so called

Discrete-Wavelet-Transform (DWT) based on basis sets which are orthogonal

and complete so o�ering an alternative to Fourier bases (circular functions) for

the functional decomposition of physiological signal, with particular attention

to possible characteristic patterns connected with vasovagal syncope.

Vasovagal syncope is a sudden, rapid and reversing loss of consciousness, due to

a reduction of cerebral blood ow [3] attributable neither to cardiac structural

or functional pathology, nor to neurological structural alterations, but due to

a bad functioning of the cardiovascular control, induced by that part of the

Autonomic Nervous System (ANS) that regulates the arterial pressure [3,4].

In normal conditions the arterial pressure is maintained at a constant level

by the existence of a negative feed-back mechanism localized in some nervous

centers of the brainstem. As a consequence of a blood pressure variation de-

tected by the baroreceptors, the ANS is able to restore the haemodynamic

situation acting on heart and vases, by means of two e�erent pathways, the

vasovagal and sympathetic one, the former acting in the sense of a reduction

of the arterial pressure, the latter in the opposite sense [5]. Vasovagal syn-

cope consists of an abrupt fall of blood pressure corresponding to an acute

haemodynamic reaction produced by a sudden change in the activity of the

ANS (an excessive enhancement of vasovagal out ow or a sudden decrease of

simpathetic activity) [3].

Vasovagal syncope is a quite common clinical problem and in the 50% of

patients it is non diagnosed, being labeled as syncope of unknown origin [4,6].

Although the clinical features of this condition are fairly evident, diagnostic

certainty is not always achieved [7] or it takes a long observation period.

Anyway, a speci�c diagnosis of vasovagal syncope is practicable [6,8] with

good sensitivity and speci�city with the help of the head-up tilt (HUT) [9],

that o�ers optimal hints also for the pathophysiology hypotheses. During this

test the patient, positioned on a self-moving table, after an initial rest period

in horizontal (basal) position, is suddenly brought in vertical (hortostatic)

position. In such a way the ANS registers a sudden stimulus of reduction of

arterial pressure due to the shift of blood volume to inferior limbs. A wrong

response to this stimulus can induce syncope behavior.

According to some authors, the positivity of HUT means an individual predis-

position toward vasovagal syncope [10]. This statement does not �nd a general

agreement because of the low reproducibility of the test [11] in the same pa-

tient and the extreme variability of the sensitivity in most of the clinical stud-

2

ies [8]. In last years a large piece of work has been devoted to the investigation

of signal patterns that could characterize syncope-a�ected patients. This has

been done especially by means of mathematical analyses of arterial pressure

and heart rate. The spectral analysis (by means of Fast Fourier Transform

algorithms and similar approaches) is a powerful tool to study the frequency

composition of biological signals, but it shows some limits and does not always

permit to achieve clear conclusions and interpretations [12]. One of the main

results obtained in studies on humans and animals adopting these techniques,

is that two particular frequency peaks are always distinguishable and to both

of them has been attributed a speci�c physiological meaning in connection

with sympathetic and vagal nervous control. Actually, at this stage, there is

not a wide agreement even about the physiological origin of such peaks, and

di�erent attempts to characterize typical patterns to discriminate between

patients fainting to the HUT and healthy people, have failed [13].

In this paper we perform, by means of DWT, a new, detailed analysis of blood

pressure waves of healthy people (controls) and syncope a�ected patients (pos-

itives) with the main intent to highlight all possible di�erences. The positivity

of examined patients has been clinically established after a long and careful

observation and also results in consequence of repeated HUT tests.

We have succeeded in this task so providing a potential diagnostic method for

an early detection of this syndrome.

The most striking di�erence between Fourier and DWT decomposition is that

the last allows for a projection on modes simultaneously localized in both time

and frequency space, up to the limit of classical uncertainty relations. Unlike

the Fourier bases, which are delocalized for de�nition, the DWT bases have

limited spatial support so being particularly useful for the study of signals

which are known only inside a limited temporal window. For our study, DWT

has been shown to be a powerful tool for putting in evidence some \hidden"

features of the signal we examine that are very di�cult to be detected with a

direct study of Fourier power spectra. So, in a sense, in our investigation, DWT

acts as a microscope allowing us to magnify signal's characteristics otherwise

not easily visible.

The plan of the paper is as follows. In Section 2 we will give a short mathe-

matical introduction to Haar wavelet analysis with the aim of writing down

the formulas that are used in the subsequent sections. In Section 3 the inves-

tigation methods and the results obtained studying blood pressure waves are

fully explained in order to �nd some vasovagal syncope distinctive features.

In Section 4 we develop further data manipulations and �nally we draw our

conclusions in Section 5.

3

2 The Haar wavelet analysis

As told above, in the present paper we will not give a comprehensive introduc-

tion to wavelet mathematics, reporting only a brief account of its aspects that

are relevant for our objectives and referring the reader, interested in further

understanding, to a vast literature on the subject [14].

The Haar wavelet is historically the �rst basis that has been introduced for

wavelet analysis and for many practical purposes it is the simplest to be used

in applications.

Let us consider a function f(x), de�ned in [0; L], representing some data (below

we will consider the systolic/diastolic pressure height taken at di�erent times).

This function is typically known with some �nite resolution �x and it can be

represented as an histogram having 2m bins in such a way that:

L

2m� �x: (1)

Each bin is labeled by an integer n running from 0 to 2m � 1. We can now

de�ne:

fm;n(x) = f(x); x 2 [L

2mn;

L

2m(n+ 1)]: (2)

Obviously holds the relation:

f(x) =2m�1Xn=0

fm;n(x)�(2m

Lx� n) = f (m)(x); (3)

where the function � is de�ned in such a way that:

�(s) =

8><>:1 if 0 � s � 1

0 elsewhere: (4)

By writing f (m)(x) we mean \f looked at scale m". Let us consider now a

roughening of f (m):

f (m�1)(x) =2m�1

�1Xn=0

fm�1;n(x)�(2m�1

Lx� n); (5)

4

where, as it is very easy to check, we have:

fm�1;n(x) =1

2(fm;2n(x) + fm;2n+1(x)): (6)

It is clear that f (m�1)(x) contains less information than f (m)(x), so, in order to

recover the information that has been lost, we should be able to calculate the

di�erence function f (m)(x) � f (m�1)(x). We will call this di�erence function

W (m�1)(x). It can be shown (see [14]) that:

W (m�1)(x) =2m�1

�1Xn=0

Wm�1;n(x) (2m�1

Lx� n); (7)

where:

Wm�1;n(x) =1

2(Wm;2n(x)�Wm;2n+1(x)); (8)

and:

(s) =

8><>:1 if 0 � s < 0:5

�1 if 0:5 � s � 1; (9)

being (s) zero outside of the indicated intervals.

This is properly the main ingredient of the Haar wavelet decomposition. � and

are respectively known as mother and father functions and the coe�cients

fm;n, Wm;n are the mother and father (or wavelet) coe�cients. Mother and

father functions, taken together, generate a compactly supported orthogonal

basis .

According to our de�nition of di�erence function, it is straightforward to ob-

serve that:

f (m)(x) = f (0)(x) +W (0)(x) + :::::: +W (m�1)(x): (10)

Analyzing a signal which oscillates around an average value we realize that,

for all practical purposes, f (0)(x) = 0, so we immediately argue that:

f(x) =m�1Xj=0

2j�1Xn=0

Wj;n(x) (2j

Lx� n); (11)

where summing on all m's corresponds to looking at function f at all possible

scales. This is the wavelet representation of f(x) provided by the Haar basis.

5

What we learn is that the di�erence functions W enable us to project the

function f(x) on a new basis set. Furthermore, the orthogonality properties

of mother and father function allow us to write down the coe�cients of the

Discrete Wavelet Transform. A particular choice of normalization gives us:

Wm;n = 2�m0=2

L�1Xi=0

fi (2�m0

i� n); (12)

where m0 = 10�m being L = 210 the total number of pressure wave maxima

fi constituting the histogram function that we want to study. So it is clear

that, substituting m0 ! m, we have:

W10�m;n = 2�m=2L�1Xi=0

fi (2�mi� n): (13)

Let us call W10�m;n = W 0

m;n in the following. The variability of the wavelet

coe�cients for each pressure wave has been parameterized at the di�erent

scales (di�erent values of m) by means of their standard deviations:

�(m) =

"1

N � 1

N�1Xn=0

(W 0

m;n � hW0

m;ni)2

# 1

2

; (14)

where N is the number of wavelet coe�cients at a given scale m (N = L=2m).

3 Blood pressure waves

In the previous section we introduced the technical tools that we will use

in our analysis of blood pressure waves. Basically, all we need for our pur-

poses is the expression of the uctuation in Eq. (14). Our main results are in

fact obtained examining plots of scale m versus �(m) in which the function

f(x) gives the height of maxima of systolic/diastolic blood pressure waves of

healthy and syncope-a�ected patients recorded in basal position. The record-

ing period, before the tilting, is twenty minutes long. During this time the

following biological signals of the subject are recorded: E.C.G. (e�ected in

D-II), E.E.G., the thoracic breath, the arterial blood pressure (by means of a

system �napres 2300 Ohmeda Co. USA, applied to a �nger of the left hand).

The variable x is a time variable. We assume that the time interval separat-

ing two pressure maxima is, in normal conditions, almost equal to a certain

average value that we could establish case by case.

Of course di�erent individuals have di�erent average values of pressure so their

6

5

10

15

20

25

1 2 3 4 5 6 7 8

Fig. 1. Standard deviations of the wavelet coe�cients (see Eq. (14)) of the Systolic

Basal Pressure in syncope-a�ected patients (positive) and healthy people (control).

Note the quite evident separation among positives and controls at m = 5 marked

by the short solid line.

blood wave signal has a di�erent average height, but a similar shape. What

we have really care of is the variation of height between neighboring pressure

maxima and not their absolute height values. Wavelet analysis looks exactly

where we want to look. In fact it is not di�cult to realize that, in Haar basis,

wavelet coe�cients are rescaled derivatives of the function being analyzed. If

we have two oscillating functions similar in shape but with two di�erent aver-

age height values, we could say that these functions are di�erent by a constant,

so this di�erence, induced by reasons external to what we really want to study,

is soon removed by computing the wavelet coe�cients and uctuations.

In order to understand better the role of the height variations between pres-

sure maxima with respect to the absolute value of each single height we also

perform speci�c analyzes. We will come back on this point in Section 4.

Looking at Fig. 1 it seems that there is a particularly clear distinction between

healthy people and patients having a syncope clinical behavior (established

after a long clinical observation period). This distinction appears traced in

correspondence of m = 5 scale value where we can observe a tendency of �'s

7

related to positives to be lower than �'s related to controls. For smaller values

of m there is no sensitive distinction between controls and positives, while,

for bigger values our analysis begins to be less and less signi�cant due to the

narrowness of our temporal observation window.

If we interpret this analysis as a possible discriminating test, we observe that it

has a sensitivity of 90% and a speci�city of 78%. These empirical observations

have been subject of a further study focused to understanding why m = 5

is the relevant discriminating scale or possibly the onset of a discriminating

region between controls and positives.

The result of this investigation is that m = 5 separation corresponds to the

fact that positives must have in their Fourier power spectrum, relative to

the above-mentioned pressure signal, a \hidden" suppression of a certain low

frequency range (roughly centered around 0:02 � 0:03 Hz, being m = 5 the

resolution window corresponding to about 25 pressure maxima) that is not

easily detectable looking only at Fourier spectra. What we just want to stress

here is how wavelet analysis has helped us to �nd this feature.

One of the main drawback of wavelets with respect to Fourier bases, besides

their greater technical complication, is a less immediate physical meaning.

On the other hand there are well known mathematical expressions connecting

wavelet and Fourier coe�cients of the same function [14], but, again, many

computational di�culties have to be faced if one wants to implement these

formulas. Anyway, with a few straightforward and qualitative arguments, it is

possible to immediately �gure out a frequency spectrum interpretation of the

m = 5 separation.

If one simply tries to calculate, with the help of a computer, Haar wavelet

coe�cients Wm;n of a sinus (or cosinus) function using Eq. (12), he will soon

realize that what is obtained are derivatives of sinus (or cosinus) rescaled on

the x-axis. Passing from an m value to the next m + 1 value, a derivative

is made and x units are scaled by a dilation factor a < 1. This means that,

passing from an m to the next, the sinusoidal function will have a greater

uctuation over the zero average value (the area below the curve must remain

the same after the rescaling of coordinates).

According to what has just been said, a low frequency sinusoidal function,

with a small weight coe�cient, will start uctuating in a considerable way

only going up with m values.

As known from Fourier analysis, each function, under very general conditions,

can be written down as a sum of weighted sinusoidal functions. If we take

two functions di�ering in their Fourier coe�cient values only in a certain low

frequency common range and if we calculate their �(m) for di�erent m's, we

expect that the function having the smaller weight coe�cients in the low fre-

quency domain will also have the smaller � values, for certain particular m

values, with respect to the other. For greater m's, the depressed low frequency

harmonics begin to uctuate in a more sensitive way, so that we can foresee

again a decreasing in the di�erences between �(m) values belonging to the

two functions having di�erent low frequency weights.

8

0

2

4

6

8

10

12

14

1 2 3 4 5 6 7 8

Fig. 2. As in Fig. 1 but for the Diastolic Basal Pressure.

With this simple picture in mind we soon understand that (see Fig. 1), since

positives appear to have smaller �(5) with respect to controls and, since in

the above explained context m = 5 is a su�ciently high value, then positives

are expected to have some low frequency depression in their Fourier spectrum

relative to the blood pressure wave signal.

An analysis similar to that carried out for Systolic Pressure waves has been

performed for Diastolic Pressure waves belonging to the same patients and

healthy subjects. As it is shown in Fig. 2, a clear evidence of separation be-

tween controls and positives is lost even if, at m = 5, controls tend to accu-

mulate towards higher �-values than positives. This loss of sensitivity may be

due to the shorter variability range of the wavelet coe�cients of the Diastolic

Pressures with respect to the Systolic ones, as can be extrapolated looking at

the small values of the �'s in Fig. 2 than those in Fig. 1.

9

0

2

4

6

8

1 2 3 4 5 6 7 8

Fig. 3. Standard deviations of the reconstructed function of the Systolic Basal Pres-

sure after the application of a �lter to the temporal series (see the text).

4 Data manipulations

First of all we want to follow a �lter-technique used in [2] in order to improve

the clearness of our results. As one can see from Fig. 3, we do not succeed

in obtaining a better separation between control and positive data points, so

concluding that the method, seeming very e�ective in [2], probably deserves

a further attention if it is to be used successfully for the manipulation of the

kind of data we are examining.

What is done is the following: the temporal series is reconstructed using the

set of its own wavelet coe�cients after a manipulation consisting in putting

allWm;n with m di�erent from s equal to zero (see Fig. 3). The reconstruction

is obviously obtained through a wavelet antitrasformation. Then we evaluate

�f which is simply the standard deviation of this new reconstructed temporal

series. We neither obtain a very encouraging increasing in the gap between

positives and controls nor observe the opening of a gap region as in [2]. For

this reason, at least in this work, we do not push further this kind of analysis.

10

0

2.5

5

7.5

10

1 2 3 4 5 6 7 8

Fig. 4. Standard deviations of the wavelet coe�cients of the Systolic Basal Pressure

computed after a randomly reordering of the pressure wave maxima.

The second test we perform on our data is aimed to put in evidence the

importance of the temporal evolution of the pressure maxima heights with

respect to their magnitude. If we simply put in a di�erent temporal succession

the same initial data, as it is done in Fig. 4, we lose the somehow discriminating

behavior of Fig. 1, so arguing that the information, able to create a certain gap

between positives and controls, mostly comes from the temporal disposition

of maxima.

5 Conclusions and perspectives

We are aware that the analysis here suggested is far to be used as an operative

diagnosing method of vasovagal syncope but we show that it strongly suggests

(with its 90% sensitivity) a direction to look at. We think that with more

careful and longer-period data recording techniques this task could be easily

achieved.

Moreover we �nd meaningful that also our contribution puts in evidence the

11

capability of wavelet analysis to give new insights going beyond those provided

by Fourier analysis in the context of such problems at frontier between physics

and pathophysiology.

Many considerations of pathophysiologic nature may be done about previous

results, but we think these deserve an appropriate work.

Acknowledgement

The authors are grateful to Doc. M. Osei Bonsu for giving us the possibility

to access at the not elaborated blood pressure data.

A. D. P. acknowledges Prof. R. Gatto for his long and kind hospitality at the

University of Geneva.

12

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