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Arterial blood pressure analysis based on scattering
transform I
Taous-Meriem Laleg, Emmanuelle Crépeau, Yves Papelier, Michel Sorine
To cite this version:
Taous-Meriem Laleg, Emmanuelle Crépeau, Yves Papelier, Michel Sorine. Arterial blood pressure
analysis based on scattering transform I. EMBC Sciences and Technologies for Health, Aug 2007,
Lyon, France. �inria-00139527v2�
Arterial blood pressure analysis based on scattering transform I
Taous-Meriem Laleg, Emmanuelle Crépeau, Yves Papelier and Michel Sorine
Abstract— This article presents a new method for analyzing
arterial blood pressure waves. The technique is based on
the scattering transform and consists in solving the spectral
problem associated to a one-dimensional Schrödinger operator
with a potential depending linearly upon the pressure. This
potential is then expressed with the discrete spectrum which
includes negative eigenvalues and corresponds to the interacting
components of an N-soliton. The approach is analogous to the
Fourier transform where the solitons play the role of sinus
and cosinus components. The proposed method seems to have
interesting clinical applications. It can be used for example to
separate the fast and slow parts of the blood pressure that
correspond to the systolic (pulse transit time) and diastolic
phases (low velocity flow) respectively.
I. INTRODUCTION
The cardiovascular system, composed of the heart and a
complex vascular network, provides oxygen and nutrients to
all the body. Pressure and flow waves are created by the
beating heart and propagate through the aorta and the major
arteries to the periphery. The analysis of the Arterial Blood
Pressure (ABP) in daily clinical practice is often restricted
to the maximal and the minimal values of the pressure wave
called systolic and diastolic pressures respectively. However,
in this case, none information about the instantaneous varia-
bility of the pressure is provided. Many studies were devoted
to model and analyse the ABP waveform and many models
were proposed from the well-known windkessel model [22]
to the more complex three dimensional models [13].
A standard description of the blood pressure and flow
waves uses the linear Fourier analysis where the waves
are decomposed into sinus and cosinus components. The
Fourier decomposition is applicable if two assumptions are
satisfied. First, the arterial system is supposed to be linear so
that the superposition principle applies. Then, the pressure
and flow must be measured in steady state conditions at
constant heart rate [21]. About 12 to 15 harmonics are needed
for a good reconstruction of the ABP waves. To explain
the waves contour and interpret the changes in the pulse
pressure when it propagates along the arterial tree like the
increase in the amplitude and the decrease in the width
called "Peaking" and "Steepening" phenomena respectively,
this approach supposes the existence of backward waves.
Each harmonic consists then of an incident wave propagating
away from the heart and a reflected wave travelling towards
the heart. Many studies were carried out in order to separate
The authors are with INRIA-Rocquencourt, B.P. 105, 78153 Le Chesnay
cedex, France, taous-meriem.laleg@inria.fr
emmanuelle.crepeau@inria.fr,
yves.papelier@inria.fr,
michel.sorine@inria.fr.
the ABP into its forward and backward components as in the
pioneering work of Westerhof et al [23] followed by many
others [16], [17], [18].
Instead of the usual decomposition of the ABP into a linear
superposition of sinus and cosinus, in this article, we propose
to use a nonlinear superposition of solitary waves or solitons.
The concept of soliton refers in fact to a solitary wave
emerging unchanged in shape and speed from the collision
with other solitary waves [19]. They fascinate the scientists
with their very interesting coherent-structure characteristics
and are used in many fields and to model natural phenomena.
Solitons are solutions of nonlinear dispersive equations like
the Korteweg-de Vries (KdV) equation arising in a variety
of physical problems, for example to describe wave motion
in shallow water canals. This third order nonlinear partial
derivative equation (NPDE) includes both nonlinear and
dispersive effects and solitons result here from a stable
equilibrium between these effects [20], [25].
The use of solitons to describe the ABP was already
introduced in [24] and in [15] where a KdV equation and a
Boussinesq equation were respectively proposed as a blood
flow model. Recently, in [2], [9] an interesting reduced model
of the ABP was introduced. The latter consists of a sum of
a 2 or 3-solitons, solution of a KdV equation, describing
fast phenomena during the systolic phase and a 2-element
windkessel model describing slow phenomena during the
diastolic phase. We recall that the systolic phase corresponds
to the contraction of the heart, driving blood out of the left
ventricle while the diastolic phase corresponds to the period
of relaxation of the heart. We point out that the introduction
of solitons in the ABP model explains the peaking and the
steepening phenomena [9].
The decomposition of the ABP into a nonlinear super-
position of solitons introduced in this article is based on
an elegant mathematical transform : the scattering transform
for a one-dimensional Schrödinger equation [1], [5], [6].
This Scattering-Based Signal Analysis (SBSA) method was
introduced in [10]. The main idea in the SBSA consists in
solving the spectral problem of a one-dimensional Schrödin-
ger operator with a potential depending linearly upon the
pressure wave. This potential is then expressed with the
discrete spectrum which includes negative eigenvalues and
corresponds to the interacting components of an N-soliton.
In the next section, we recall the basis of the SBSA method
that is used to reconstruct and analyse the ABP waves.
Section III illustrates a good agreement between real and
reconstructed pressures using the SBSA. We also present an
interesting clinical application which consists in separating
the fast and slow components of the ABP which are related
to the systolic and diastolic phases respectively. Finally a
conclusion summarizes the different results.
II. A SCATTERING-BASED SIGNAL ANALYSIS
METHOD
In this section, we introduce an original method for
reconstructing signals based on the scattering transform. We
start by briefly recalling the basis of the Direct and Inverse
Scattering Transforms (DST & IST) and then present the
main idea in the SBSA technique.
A. Scattering transform for a Schrödinger equation
The stationary one-dimensional Schrödinger equation is
given by :
−
∂
2
ψ
∂
x
2
+V
ψ
=
λ ψ
, −∞ < x < +∞. (1)
where x is the space variable,
λ
and
ψ
are respectively
the eigenvalues and the associated eigenfunctions. V is the
potential of the Schrödinger operator L(V ) = −
∂
2
∂
x
2
+V.
In this study we suppose that the function V belongs to
the Schwartz space S (R) of regular and rapidly decreasing
functions on R.
The DST of the potential V is the solution of the spectral
problem for L(V ). The spectrum of this operator has two
components : a continuous spectrum including positive ei-
genvalues and a discrete spectrum with negative eigenvalues
[1], [4], [5], [6], [12]. Denoting the positive eigenvalues
by
λ
= k
2
, the continuous spectrum is characterized by
the following asymptotic boundary conditions where T (k)
and R(k) are respectively the transmission and the reflection
coefficients associated to V :
ψ
(x, k) → T (k)exp (−ikx), x → −∞, (2)
ψ
(x, k) → exp(−ikx)+R(k)exp(ikx), x → +∞. (3)
Conservation of energy leads to |T (k)|
2
+ |R(k)|
2
= 1.
We note the negative eigenvalues
λ
n
= −
κ
2
n
, n = 1, · ·· , N
with N their number. The discrete eigenfunctions
ψ
n
are L
2
-
normalized such that :
Z
+∞
−∞
ψ
n
(x)
2
dx = 1. (4)
and behave as
ψ
n
(x) ∼ c
n
exp(−
κ
n
x) in the limit of large x
where the coefficients c
n
are defined by :
c
n
= lim
x→+∞
exp(
κ
n
x)
ψ
n
(x), n = 1, ·· · , N. (5)
So, the DST S of V is the collection of data S(V ) called
scattering data and defined by :
S(V ) = (R,
κ
n
, c
n
, n = 1, ·· · , N). (6)
The inverse problem or IST consists in reconstructing the
Schrödinger operator’s potential from its spectral data. The
IST is based on the Gel’fand-Levitan-Marchenko (GLM)
integral equation [1], [5]. In this study we are only interested
in a special situation which corresponds to a reflectionless
potential.
A reflectionless potential is a potential for which the
reflection coefficient R(k) is zero. This means that there is
no contribution from the continuous spectrum. This situation
leads to an interesting representation of the potential with
the discrete spectrum only [6] as it is given in the following
theorem :
Theorem : If V is a reflectionless potential of the Schrö-
dinger equation (1) then :
V (x) = −4
N
∑
n=1
κ
n
ψ
2
n
(x), x ∈ R. (7)
An interesting relation between solitons, solutions of a
KdV equation and the discret spectrum of the Schrödinger
operator was introduced in [6]. Indeed, from a reflectionless
potential of (1) that evolves, in time and space, according
to a KdV equation, there emerge, for t → +∞, N solitons,
each one characterized by a pair (
κ
n
, c
n
) such that 4
κ
2
n
gives
the speed of the soliton and c
n
its position. Therefore each
component −4
κ
n
ψ
2
n
of the previous sum refers to a single
soliton.
B. SBSA principle
We now present the SBSA technique which is essentially
inspired from the results established in the case of a re-
flectionless potential that have been recalled in the previous
subsection.
We note y the signal that we want to reconstruct and ana-
lyse and we suppose that the potential V of the Schrödinger
operator L depends linearly upon y :
V (y) = −
χ
(y − y
min
), (8)
where y
min
is a constant such that y − y
min
> 0 and
χ
a
positive parameter to determine.
The main idea is then to find the parameter
χ
such that
the signal y is well approximated by the reflectionless part of
the potential which can be written then using equation (7) :
ˆy = 4
χ
−1
N
∑
n=1
κ
n
ψ
2
n
+ y
min
, (9)
where −
κ
2
n
and
ψ
n
, n = 1, · ·· , N are respectively the N
negative eigenvalues and the corresponding L
2
-normalized
eigenfunctions for the potential V , determined by the DST.
This situation corresponds in fact to the decomposition of
the signal y into a nonlinear superposition of solitons [10].
It is important to recall the role of the parameter
χ
and
its influence on the number of negative eigenvalues. The
potential V is a well of variable depth which is determined
by
χ
. The number of the negative eigenvalues N(
χ
) is a
nondecreasing function of
χ
and there is an infinite unboun-
ded sequence of values of
χ
at which N(
χ
) is incremented
by one, the new eigenvalue being born from the continuous
spectrum [7], [10], [14].
Determining the parameter
χ
determines the number N(
χ
)
of negative eigenvalues and hence the number of solitons
components required for a satisfying approximation of the
signal y. Fig. 1 summarizes the SBSA method.
Fig. 1. Signals analysis with the SBSA method
III. A SOLITON-BASED DECOMPOSITION OF THE
ABP
In this section, the SBSA method is applied to the ABP
signal. An interesting application is also presented which
consists in separating systolic and diastolic phases. For a
convenient use, we solve the SBSA by replacing the space
variable x by the time variable t.
A. Reconstruction of the ABP
Let P(t) be the ABP signal. The Schrödinger operator
potential V depends linearly upon the ABP signal V (t) =
−
χ
P(t).
Fig. 2 and Fig. 3 compare measured and estimated pres-
sures at the aorta and at the finger levels for different
values of the number N(
χ
) of the solitons’ components.
We notice that only 5 to 10 components are sufficient for
a good approximation of the ABP. In Fig. 4, several beats of
measured and reconstructed pressures are considered.
B. Separation of the systolic and diastolic phases
Now we exploit the SBSA to separate the pressure into
its fast and slow parts which correspond respectively to the
systolic and diastolic phases. We suppose that the N
f
= 2 or
3 largest
κ
2
n
describe fast phenomena and the N
s
= N − N
f
smallest
κ
2
n
describe slow phenomena. Then, the following
ˆ
P
f
(t) = 4
χ
−1
N
f
∑
n=1
κ
n
ψ
2
n
, (10)
ˆ
P
s
(t) = 4
χ
−1
N
∑
n=N
f
+1
κ
n
ψ
2
n
, (11)
are good candidates to represent respectively the fast and
slow phenomena in the ABP. This is confirmed by the
1.6 1.8 2 2.2 2.4 2.6 2.8 3
50
55
60
65
70
75
80
85
90
95
Time (s)
Arterial Blood Pressure (mmHg)
Estimated pressure
Real pressure
1.6 1.8 2 2.2 2.4 2.6 2.8 3
50
55
60
65
70
75
80
85
90
95
Time (s)
Arterial Blood Pressure (mmHg)
Estimated pressure
Real pressure
1.6 1.8 2 2.2 2.4 2.6 2.8 3
50
55
60
65
70
75
80
85
90
95
Time (s)
Arterial Blood Pressure (mmHg)
Estimated pressure
Real pressure
1.6 1.8 2 2.2 2.4 2.6 2.8 3
50
55
60
65
70
75
80
85
90
95
Time (s)
Arterial Blood Pressure (mmHg)
Estimated pressure
Real pressure
Fig. 2. Measured and reconstructed pressures at the aorta with N solitons.
From left to right : N = 3, N = 5 (Up). N = 7, N = 9 (Down)
1.6 1.8 2 2.2 2.4 2.6 2.8 3
50
60
70
80
90
100
110
120
130
140
Time (s)
Arterial Blood Pressure (mmHg)
Estimated pressure
Real pressure
1.6 1.8 2 2.2 2.4 2.6 2.8 3
50
60
70
80
90
100
110
120
130
140
Time (s)
Arterial Blood Pressure (mmHg)
Estimated pressure
Real pressure
1.6 1.8 2 2.2 2.4 2.6 2.8 3
50
60
70
80
90
100
110
120
130
140
Time (s)
Arterial Blood Pressure (mmHg)
Estimated pressure
Real pressure
1.6 1.8 2 2.2 2.4 2.6 2.8 3
50
60
70
80
90
100
110
120
130
140
Estimated pressure
Real pressure
Fig. 3. Measured and reconstructed pressures at the finger with N solitons.
From left to right : N = 3, N = 5 (Up). N = 7, N = 9 (Down)
experimental results. For N
f
= 2 and N = 9, Fig. 5 and Fig. 6
show that
ˆ
P
f
and
ˆ
P
s
are respectively localized during the
systole and the diastole, as expected.
This decomposition of the ABP into fast and slow parts
completes the results obtained in [2], [3], [8], [9].
IV. CONCLUSIONS
This article presents a new method for analyzing ABP
waves. This approach is based on the scattering transform
and deals with the solution of the spectral problem of a
perturbed Schrödinger operator for a given potential. The
latter is then expressed in a new base which components are
solitons. It seems through the satisfactory results obtained
that this method can lead to interesting clinical applications,
for instance the separation of the ABP into its systolic and
diastolic phases. Moreover, the SBSA provides interesting
ABP indices that can be analyzed in various clinical and
physiological conditions. Promising results are presented in
0 2 4 6 8 10 12 14
50
55
60
65
70
75
80
85
90
95
100
Time (s)
Arterial Blood Pressure (mmHg)
Estimated pressure
Real pressure
0 2 4 6 8 10 12 14
50
60
70
80
90
100
110
120
130
140
Time (s)
Arterial Blood Pressure (mmHg)
Estimated pressure
Real pressure
Fig. 4. Multi-beat measures and estimates : Aorta (up) and Finger (down)
1.6 1.8 2 2.2 2.4 2.6 2.8 3
0
10
20
30
40
50
60
70
80
90
100
Time (s)
Arterial Blood Pressure (mmHg)
Estimated pressure
Real pressure
Fig. 5.
ˆ
P
f
and fast systolic phenomena
our second article [11].
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