Separation of arterial pressure into a nonlinear superposition of solitary waves and a windkessel flow

Biomedical Signal Processing and Control 2 (2007) 163–170 www.elsevier.com/locate/bspc

Separation of arterial pressure into a nonlinear superposition of solitary waves and a windkessel flow Taous-Meriem Laleg *, Emmanuelle Cre´peau, Michel Sorine INRIA Rocquencourt, 78153 Le Chesnay Cedex, France Received 8 February 2007; received in revised form 3 May 2007; accepted 21 May 2007 Available online 3 July 2007

Abstract A simplified model of arterial blood pressure intended for use in model-based signal processing applications is presented. The main idea is to decompose the pressure into two components: a travelling wave which describes the fast propagation phenomena predominating during the systolic phase and a windkessel flow that represents the slow phenomena during the diastolic phase. Instead of decomposing the blood pressure pulse into a linear superposition of forward and backward harmonic waves, as in the linear wave theory, a nonlinear superposition of travelling waves matched to a reduced physical model of the pressure, is proposed. Very satisfactory experimental results are obtained by using forward waves, the N-soliton solutions of a Korteweg–de Vries equation in conjunction with a two-element windkessel model. The parameter identifiability in the practically important 3-soliton case is also studied. The proposed approach is briefly compared with the linear one and its possible clinical relevance is discussed. # 2007 Elsevier Ltd. All rights reserved. Keywords: Arterial blood pressure; Solitons; Windkessel model; Identifiability

1. Introduction Nowadays, the cardiovascular diseases constitute one of the most important causes of mortality in the world. Among the parameters that play an important role in the circulatory system is the arterial blood pressure (ABP). Its importance has drawn the attention of the scientific community which try to find mathematical models in order to understand and assess the behavior of this complex system both in normal and pathological cases. Despite the different proposed models, the interpretation of the ABP in daily clinical practice is still too often restricted to the maximal and the minimal values of the pressure called systolic and diastolic pressures respectively and measured using a sphygmomanometry device wrapped around the upper arm. In this case, none information about the instantaneous variability of the pressure is provided. Two approaches can be distinguished for modelling the ABP in large arteries: lumped and distributed models. Lumped * Corresponding author. Tel.: +33 139635758; fax: +33 139635786. E-mail addresses: [email protected] (T.M. Laleg), [email protected] (E. Cre´peau), [email protected] (M. Sorine). 1746-8094/$ – see front matter # 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.bspc.2007.05.004

models, or 0D, models (n D stands for n space variables), like the popular windkessel models, are based on an analogy with simple RLC electrical circuits, the pressure being represented by a voltage and the blood flow-rate by a current [1]. In order to enlarge its frequency domain, the basic two-element windkessel model, a resistance in parallel with a capacitor, has been extended to three and four elements by adding in series a resistance, alone or with an inductance in parallel [2]. Windkessel models successfully explain the diastolic phase but, with their low order, they cannot explain propagation phenomena like the transit delay of the pressure pulse. The 3D distributed models are based on computational fluid dynamic principles and can explain observed phenomena [3]. Too complex for some applications, they can be reduced to 2D or 1D models [4]. Then 0D models can be deduced from 1D models [5,6] to be used as boundary conditions for 1D models [7,8] or in signal analysis as is done here. During the propagation of the pulse pressure (PP) along the arterial tree, it undergoes some changes like an increase in the amplitude with rapid development of a steep front noted as Peaking and Steepening [3]. These phenomena have been explained by the linear superposition of direct and reflected waves, the reflected waves being created when the forward

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waves, from the heart to the periphery, encounter discontinuities in the arterial properties like a bifurcation or a stenosis. This model has been well known for decades and many studies have been carried out in order to separate the PP into its forward and backward components, as in the pioneering work of Westerhof [9], followed by many others [10–16]. The main idea in the present work is to decompose the pressure into a travelling wave representing the fast propagation phenomena during the systolic phase and a windkessel term representing slow phenomena during the diastolic phase. A decomposition of this kind has been studied in Ref. [17] but the novelty here consists in choosing a forward solitary wave which already captures essential properties of the PP, like the Peaking and the Steepening phenomena. Solitons resulting from a balance between shock wave creation and wave dispersion are proposed. They possess an analytical expression which leads to a reduced ABP model that is easy to identify, an important advantage of the approach. So, instead of the usual decomposition of the ABP as a linear superposition of forward and backward harmonic waves, the suggestion is to use a nonlinear superposition of forward solitary waves completed by a windkessel flow. The use of solitons for analyzing the ABP has been already proposed in Refs. [18–22]. This work is based on the results of [19] where a quasi-1D Navier–Stokes equation is chosen and reduced to a Korteweg–de Vries equation (KdVE) with solitons as particular solutions. It has been shown that only 2 or 3 interacting solitons are sufficient for a good description of the systolic phase [19,23,24]. In the next section, we recall the linear theory based on the decomposition of the ABP into forward and backward waves. Then, in Section 3, we introduce a reduced model: solitons + a two-element windkessel flow. The windkessel term appears here as a correction-term decomposition, needed in the diastolic phase. In Section 4, the identifiability of the 3-soliton’s parameters and of the two-element windkessel model is studied. In Section 5 experimental results are presented, followed by a discussion in Section 6. A conclusion summarizes the different results. 2. ABP as a linear superposition of forward and backward waves The cardiovascular system, composed of the heart and a complex vascular network, feeds out of oxygen and nutrients 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 with a certain speed called a wave speed. The wave contour has been explained by the existence of reflected waves which continuously modify their magnitude and morphology [3,25]. So the pressure wave is composed of incident components and backward components propagating, respectively, away from and towards the heart. In practice, it is impossible to decompose the pressure into all its incident and reflected components. Therefore, this complex wave pattern is often simplified so that the arterial system is considered as a single visco-elastic tube in which the pressure wave is seen as the superposition of an equivalent forward component and an

equivalent backward component as it has been introduced in Ref. [9]. In what follows, we recall briefly this technique. If we note the position on the arterial tree z and the time variable t, then a standard description of the blood pressure Pðz; tÞ and flow Qðz; tÞ consists of a linear superposition of forward and backward waves as described in Ref. [9]: Pðz; tÞ ¼ Pf ðz  c0 tÞ þ Pb ðz þ c0 tÞ;

(1)

Qðz; tÞ ¼ Qf ðz  c0 tÞ þ Qb ðz þ c0 tÞ;

(2)

where subscripts f and b refer, respectively, to the forward and backward components. The parameter c0 represents the Moens– Korteweg sound waves velocity in an elastic vessel filled with blood of density r. It is given by the following relation: sffiffiffiffiffiffiffiffiffiffi Eh0 c0 ¼ ; (3) 2rR0 where E is the wall elasticity coefficient. R0 and h0 are, respectively, the equilibrium radius and the thickness of the vessel. The problem consists then in determining the forward and reflected components of blood flow and pressure. The method proposed in Ref. [9] suggests to use the input impedance concept. The vascular input impedance refers in fact to the relation between the blood pressure and flow with the assumption of linearity of the model. In the frequency domain, this relation Zðz; vÞ at position z and frequency v is given by the ratio of the time Fourier transforms of the pressure and flow at this position: Zðz; vÞ ¼

ˆ vÞ Pˆ f ðz; vÞ þ Pˆ b ðz; vÞ Pðz; : ¼ ˆ b ðz; vÞ ˆ vÞ Q ˆ f ðz; vÞ þ Q Qðz;

(4)

Then, the global reflection coefficient G is defined by [9,12]: G ðz; vÞ ¼

Zðz; vÞ  Z c ; Zðz; vÞ þ Z c

(5)

where Z c denotes the characteristic impedance that can be estimated by several methods (in the case of a reflectionless tube Z c ¼ Z) [9,10,26,27]. Knowing Z and Z c it is possible to estimate G . The forward and backward components of the pressure and flow are then given by [9,25]: Pˆ f ¼

Pˆ ; 1þG

Pˆ b ¼ G Pˆ f ;

ˆf ¼ Q

ˆ Q ; 1G

ˆ b ¼ G Q ˆ f: Q

(6) (7)

This method has the advantage of requiring pressure and flow measurements at only one location but as we have said, it assumes a linear model of the flow. Many studies have followed, aiming to take the nonlinearities into account when separating the PP into forward and backward components. In Refs. [11,13], quasi-nonlinear methods for 1D flow models are used (method of characteristics and split coefficient matrix for the Euler equation) under the linearizing assumption that intersecting forward and backward waves are additive. In Ref. [12] the linear superposition of multiple forward and reflected waves is considered. In Refs. [14,16], a nonlinear separation is proposed using Riemann invariants which takes

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into account nonlinearities in the area–pressure relationship and in the convective term. Other studies have proposed a model based on the propagation of nonlinear waves in fluid filled thin tubes like in Ref. [22] and in the recent work [19] where it has been shown how to derive a KdVE from a quasi 1D Navier–Stokes equation using a singular perturbation technique by taking into account the radial acceleration. A KdVE is in fact a third order derivative equation, nonlinear and dispersive that is integrable and admits as particular solutions nonlinear waves called solitons [28,29]. This situation leads to a simple model of the ABP as it will be presented in the next section. 3. Reduced arterial blood pressure model In this section, we introduce a reduced model of the ABP which consists of the sum of a nonlinear superposition of forward solitary waves and a two-element windkessel model. Before presenting this model, let us recall briefly the definition of a soliton. A solitary wave consists in a nonlinear wave which propagates without changing its shape and velocity. A soliton is a solitary wave that emerges from the collision with other solitary waves, with the same shape and velocity with which it entered. This particle-like property has been observed by Zabusky and Kruskal [30] who gave the name solitons to these waves; the suffix ‘on’ describing a particle-like property. Solitons with their fascinating characteristics have been used in many fields: hydrodynamics, optics, plasma, electrical networks, . . ., etc. [29,28,31]. Solitons have been introduced for modelling ABP in Refs. [19,22] as solutions of a KdVE and in Ref. [21] as solutions of a Boussinesq equation. Perhaps what makes solitons interesting for describing blood flow is their capacity to explain the PP wave contour and the observed phenomena like the Peaking and the Steepening thanks to their properties. More explanation about these remarks will be given in Section 6. Fig. 1 shows three interacting solitons at two positions z1 and z2 of their propagation with z1 < z2 while Fig. 2 represents measured pressures at the aorta and the finger, respectively. One can notice the great ressemblance between the 3 interacting solitons and the measured pressure. In what follows, we are interested in solitons solutions of the full KdVE proposed in Ref. [19] and given by: @Ps @Ps @ 3 Ps þ ðd0 þ d 1 Ps Þ þ d2 3 ¼ 0; @z @t @t

(8)

where Ps ðz; tÞ is the ABP and

d0 ¼

1 ; c0

with rw Remark and the solitons.

d1 ¼ 

3 1 ; 2 A0 c20

d2 ¼ 

Fig. 1. A 3-soliton at two positions: (a) z1 and (b) z2 (z1 < z2 ).

equilibrium between the nonlinearity and the dispersion. An N-soliton solution of a KdVE consists of N individual solitons with different amplitudes and velocities while interacting. With the following new variables: j ¼ t  d0 z;

t ¼ d2 z and

y¼

d1 Ps ; 6d 2

(9)

Eq. (8) becomes a normalized KdVE: @y @y @3 y þ 6y þ 3 ¼ 0; @t @j @j

yðj; 0Þ ¼ y0 ðjÞ:

(10)

The general analytical expression of an N-soliton solution of (10) can be found in Ref. [29]. It is given by: rw h0 R0 ; 2rc30

the wall density and A0 the equilibrium section. in (8) the pressure dependent velocity d 0 þ d1 Ps dispersion term @3 Ps =@t3 that are at the origin of The KdVE solitons result in fact from a stable

yðj; tÞ ¼ 2

@2 ðln det ðMÞÞ ; @j2

(11)

M is an N  N matrix with coefficients given by: M mk ¼ dmk þ

2am f ; am þ ak m

m; k ¼ 1 . . . N;

(12)

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Fig. 3. A two-element windkessel model.

solitary waves. Pwk ðtÞ is the output of a two-element windkessel model described in Fig. 3 and given by the following differential Eq. [2,17]: dPwk Pwk ðtÞ P1 QðtÞ ¼ ; ðtÞ þ þ RC C dt RC

(14)

where R denotes the peripheral resistance, C ¼ dV wk =dPwk the compliance of the arteries and QðtÞ the inflow. The inflow can be taken proportional to Ps as proposed in Ref. [17]. So it is written Q ¼ Ps =Rs . 4. Identifiability of the model’s parameters 4.1. Identifiability of a 3-soliton’s parameters

Fig. 2. Measured ABP looks as interacting 3-solitons. (a) Pressure at aorta; (b) Pressure at finger.

In this section we are interested in the identifiability of the 3soliton’s parameters. The identifiability of a 2-soliton’s parameters can be deduced easily from the 3-soliton case. We consider a 3-soliton solution of the KdVE (10), given by (11). For studying identifiability, the definition given in Ref. [32] is used. So, two solutions of (10) in the form of a 3-soliton are assumed such that: j Mmk ¼ dmk þ

where dmk is the Kronecker symbol and f m ðj; tÞ ¼ exp ½am ðj  sm  a2m tÞ;

ðam ; sm Þ 2 Rþ  R:

The parameters d 0 , d1 and d2 characterize the KdVE and depend on the characteristics of the vascular compartment while the parameters a j , j ¼ 1;    ; N characterize the solitons. One can notice from real data that the PP can be approximated by 2 or 3-solitons. Each soliton refers in fact to a well-distinguished peak of the PP. However, as it is noticed in Ref. [19] solitons lead to a good estimation of the systolic phase but need some corrections in the diastolic phase. So this work proposes to describe slow phenomena with a two-element windkessel model. Then, the ABP can be written in the following form: n

Pðz; tÞ ¼  Ps j ðz; tÞ þ Pwk ðtÞ;

(13)

2amj amj

fmj ;

þ akj

(15)

where j ¼ 1; 2, m ¼ 1; 2; 3 and k ¼ 1; 2; 3. The determinant of M j , j ¼ 1; 2 is given by: j

det ðM Þ ¼ 1 þ  þ 

f1j a1j a1j

þ

f2j



a3j a3j

þ

þ 1þ 

þ

2

f3j

 þ

f1j f3j

a1j  a2j a1j

 þ

þ

2

a2j

a2j  a3j

f1j f2j

2

a2j þ a3j

16a1j a2j a3j ða1j

4a1j a3j ða1j þ a3j Þ

þ

a2j Þða1j

 2

þ

a3j Þða2j 

4a2j a3j 2

ða2j þ a3j Þ

þ

a3j Þ

f2j f3j 

4a1j a2j ða1j þ a2j Þ

2

f1j f2j f3j :

For t ¼ 0, fi j can be rewritten as follows:

j¼1

where nj¼1 Ps j ðz; tÞ is a wave term given by a nonlinear superposition of solitary waves Ps j ðz; tÞ. For example a 2soliton can be written as a nonlinear superposition of two

fi j ðj; 0Þ ¼ exp ðaij jÞ exp ðaij sij Þ;

(16)

where j ¼ 1; 2 and i ¼ 1; 2; 3. The aij are positive and can be ordered: a1j > a2j > a3j . Then when j ! þ 1, the asymptotic

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167

 For the zero at infinity :

behaviour is: det ðM j Þ ¼ k j exp ðða1j þ a2j þ a3j ÞjÞ; 1

j ¼ 1; 2:

(17)

P10 þ

2

Now, the question is: does y ðj; 0Þ ¼ y ðj; 0Þ imply unicity of the 3-soliton (i.e M 1 = M 2 )? When y1 ðj; tÞ ¼ y2 ðj; tÞ we have:

1

lim Ps ðsÞ ¼ P20 þ

R1s C 1 s ! 1

1

lim Ps ðsÞ:

R2s C 2 s ! 1

As Ps ð0Þ ¼ lim sPs ðsÞ < þ 1;

@2 ðln det ðM 1 ÞÞ @2 ðln det ðM 2 ÞÞ ¼ : @j2 @j2

s!1

(18)

it follows:

Remark that Eq. (11) with the boundary condition (17) is a Poisson equation. Therefore, from the Dirichlet theorem, its solution is unique. So,

P10 ¼ P20 and R1s C1 ¼ R2s C2 :  For the finite pole: R1 C1 ¼ R2 C 2 .

det ðM 1 Þ ¼ det ðM 2 Þ:

It can be noticed that the parameters R, C and Rs are not identifiable. However T ¼ RC, T s ¼ Rs C and P1 are identifiable, so that it is better to write Eq. (14) describing the winkessel part of the model as follows:

(19)

Now, does the unicity of the determinant of such a matrix (15) imply the unicity of the matrix and therefore the unicity of the 3-soliton? From the asymptotic behaviour (17) we have if

1

2

det ðM Þ ¼ det ðM Þ;

then

By isolating the different exponential terms in the determinant expression, different cases have to be studied. For example, a21 þ a22 is equal to either a11 þ a12 or a12 þ a13 or a11 þ a13. So the solutions a2j consist of a permutation of the a1j . A similar result is found for the s2j . So, there is a finite number of matrices M such that det ðM 2 Þ ¼ det ðM 1 Þ. But, it is important to notice that the greatest of the a2j coincides with the greatest of a1j and so on. Therefore, if a21 > a22 > a23 then the following unique solution is found: s2i ¼ s1i ;

(24)

The parameters to identify are now T, P1 and T s .

a11 þ a12 þ a13 ¼ a21 þ a22 þ a23 :

a2i ¼ a1i ;

dPwk Pwk ðtÞ P1 Ps ðtÞ ¼ ðtÞ þ þ : T Ts dt T

i ¼ 1; 2; 3:

5. Experimental results In this section we present some comparisons of measured and computed ABP signals. Instantaneous ABP has been measured at the finger with a FINAPRES devise [33] and at the aorta with a catheter. The parameters of the two parts of our model, the 2 or 3solitons and the two-element windkessel part, were estimated from real data. In the case of a 2-soliton, eight parameters have to be identified:

(20)

In other terms, if det ðM 1 Þ ¼ det ðM 2 Þ then the 3-solitons defined by M 1 and M 2 are the same.

a1 ;

a2 ;

s1 ;

s2 ;

A¼

6d2 ; d1

T;

Ts

and

P1

In the case of a 3-soliton, we have 10 parameters to identify. The parameters have been identified using a nonlinear optimization based on the Nelder–Mead algorithm. Fig. 4 shows a

4.2. Identifiability of the two-element windkessel model Assuming that Ps ðtÞ is known, the parameters to identify j are R, C, P1 and Rs in Eq. (14). Let Pwk ðtÞ, j ¼ 1; 2 be such that P1wk ðtÞ ¼ P2wk ðtÞ:

(21) j

j

j P1

and The equality of the corresponding parameters, R , C , Rsj , j ¼ 1; 2 has to be shown. Applying the Laplace transform to Eq. (14) gives:   1 1 P1 Ps ðsÞ þ ; (22) sþ Pwk ðsÞ  P0 ¼ RC s RC Rs C j then, for j ¼ 1; 2, Pwk is given by: j ðsÞ ¼ Pwk

j sR j C j P0j þ P1 R j =Rsj Ps ðsÞ þ : 1 þ sR j C j sð1 þ sR j C j Þ

(23)

From Eqs. (21) and (23) it can be deduced:  For the pole 0: P11 ¼ P21 .

Fig. 4. Measured and estimated ABP at the finger: one beat.

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Fig. 5. The estimation error with a 3-soliton alone is well approximated by a two-element windkessel model.

Fig. 8. Measured and estimated ABP at the finger: several beats.

good agreement between estimated pressure and measured pressure at the finger in the case of a 3-soliton + a two-element windkessel model. Here, the identification of the parameters have been done for one beat. In Fig. 5 the estimation error when using a 3-soliton alone is represented and is well approximated by a two-element windkessel model. Then the procedure has been extended to a sequence of beats. Fig. 6 shows the reconstruction of the pressure measured at the finger with a 2-soliton + a two-element windkessel model while Figs. 7 and 8 illustrate the case of a 3-soliton for measured pressures at aorta and at finger, respectively. 6. Discussion

Fig. 6. Measured and estimated ABP at the finger: several beats.

Fig. 7. Measured and estimated ABP at the aorta: several beats.

The different phenomena observed when the PP propagates along the arterial tree were explained in the linear theory by the existence of backward waves. For example, the Peaking was associated with an increase in the backward wave velocity, this wave going back earlier. The increase in the velocity results from the changes in the vessels’ characteristics (the increase in stiffness and the decrease in section). However, the nonlinear superposition of forward solitons can also explain this phenomenon. Therefore, the Peaking can be explained by an increase in the soliton velocity which leads to an increase in its amplitude (this is one of the interesting characteristics of the solitons). There is always some arbitrariness in choosing a function basis to represent the solution of an evolution equation, unless this basis has some special properties as in the case of eigenfunctions for a linear system. The decomposition of the PP wave into a superposition of harmonic forward and backward waves corresponds to a small amplitude, high frequency approximation of the linearized flow equations which, in this case can be reduced into a linear second order wave equation. It is also a convenient representation of functions due to Fourier analysis and calculus. This linear decomposition of waves necessitates in general between 6 and 12 components. As described before, the proposed approach, due to the choice of forward waves matched to the pressure waveform (in fact

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particular solutions of a reduced model of the flow) can neglect the backward waves and needs only a small number of components: 2- or 3-solitons are sufficient. An important advantage of this approach is that it leads to a reduced PP model with a small number of identifiable parameters, as was proved in the previous section. The linear theory leads to a good local model of the PP [9] but it cannot help to solve the problem of the distal-to-proximal transfer-function estimation. Remark that this interesting problem is still open [34]. It gives a good description of the PP at the measurement point of the arterial tree but it does not allow the description of the PP in a different location because it does not take into account propagation phenomena. The nonlinear superposition of solitons constitutes a global model in time and space and it is a good candidate to tackle this problem of transfer-function estimation. This may lead to the estimation of proximal pressure (at aortic or ventricle levels) from distal one (at finger level for example) using only non-invasive measurements. This problem is the subject of current research. Another important point concerns clinical applications. In the case of the linear decomposition, pressure and flow measurements at a same point are needed. But, it is difficult to have joint measurements of flow-rate and aortic pressure. The latter can be obtained with invasive techniques and the former from image processing. In the proposed approach, only noninvasive pressure measurements are needed to identify the model. This is potentially an important advantage. Also, with this approach, some parameters are identified which have to be interpreted. The arterial tree is represented by the characteristics d i , i ¼ 0; 1; 2 of an equivalent unique vessel and of a windkessel model; the PP by the parameters a j , j ¼ 1;    ; N of an N-soliton with N ¼ 2; 3. All these parameters seem to give new insight on the blood flow. 7. Conclusion In this paper, a reduced blood pressure model is proposed. The theoretical considerations suggest that ABP can be seen as the sum of two terms: an N-soliton (with N ¼ 2 or 3 in the present experiments) and a two-element windekessel model. The former takes into account fast wave phenomena and the latter slow phenomena. The agreement between estimated and real pressure is satisfactory. The introduction of forward solitons explains the Peaking and the Steepening phenomena. Unlike the linear approach, which necessitates simultaneous blood pressure and flow measurements, the proposed model requires only pressure measurements. It depends on a small number of parameters that are easy to identify. It seems that these results on ABP waveform analysis, in particular in the systolic phase, can lead to some interesting clinical applications. Acknowledgments The authors would like to thank Pr. M. Slama and Dr. Y. Papelier (Hospital Be´cle`re, Clamart) for providing us with pressure data.

169

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