# Arterial blood pressure analysis based on scattering transform II

## Summary (2 min read)

### Introduction

- The cardiovascular system, composed of the heart and a complex vascular network, provides oxygen and nutrients to all the body.
- 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.
- Many studies were carried out in order to separate.
- The decomposition of the ABP into a nonlinear superposition 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].

### II. A SCATTERING-BASED SIGNAL ANALYSIS METHOD

- The authors introduce an original method for reconstructing signals based on the scattering transform.
- The authors 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 spectrum of this operator has two components : a continuous spectrum including positive eigenvalues and a discrete spectrum with negative eigenvalues [1], [4], [5], [6], [12].
- Denoting the positive eigenvalues by λ = k2, 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.
- The IST is based on the Gel’fand-Levitan-Marchenko (GLM) integral equation [1], [5].
- This means that there is no contribution from the continuous spectrum.
- 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 (κn,cn) such that 4κ2n gives the speed of the soliton and cn its position.

### B. SBSA principle

- The authors now present the SBSA technique which is essentially inspired from the results established in the case of a reflectionless potential that have been recalled in the previous subsection.
- 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) : ŷ = 4χ−1 N ∑ n=1 κnψ2n + ymin, (9) where −κ2n and ψn, n = 1, · · · ,N are respectively the N negative eigenvalues and the corresponding L2-normalized eigenfunctions for the potential V , determined by the DST.
- 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.

### III. A SOLITON-BASED DECOMPOSITION OF THE ABP

- The SBSA method is applied to the ABP signal.
- An interesting application is also presented which consists in separating systolic and diastolic phases.

### A. Reconstruction of the ABP

- The Schrödinger operator potential V depends linearly upon the ABP signal V (t) = −χP(t).
- Fig. 2 and Fig. 3 compare measured and estimated pressures at the aorta and at the finger levels for different values of the number N(χ) of the solitons’ components.
- In Fig. 4, several beats of measured and reconstructed pressures are considered.

### B. Separation of the systolic and diastolic phases

- Now the authors exploit the SBSA to separate the pressure into its fast and slow parts which correspond respectively to the systolic and diastolic phases.
- Then, the following P̂f (t) = 4χ−1 N f ∑ n=1 κnψ2n , (10) P̂s(t) = 4χ−1 N ∑ n=N f +1 κnψ2n , (11) are good candidates to represent respectively the fast and slow phenomena in the ABP.
- 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.
- Promising results are presented in their second article [11].

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