Arterial blood pressure analysis based on scattering transform II
Summary (1 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.
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.
- 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.
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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Cites background from "Arterial blood pressure analysis ba..."
...Flow and pressure in the carotid artery are naturally aperiodic due to heart rate variability, breathing and other physiological factors (Holdsworth et al. 1999; Laleg et al. 2007)....
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References
193 citations
"Arterial blood pressure analysis ba..." refers methods in this paper
...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....
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190 citations
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"Arterial blood pressure analysis ba..." refers background or methods in this paper
...squared differences between adjacent beat-to-beat intervals in overtraining subjects was described in RR but not in the ABP [ 1 ]....
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...The analysis of mean values and beat-to-beat variability of cardiovascular (CV) time series has been widely used as a non invasive approach to study the control of the autonomic nervous system (ANS) on the CV function [ 1 ], [8]....
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...The aim of this kind of studies is to assess the quality of training, to avoid overtraining and to improve CV adjustment to exercise [ 1 ], [12]....
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142 citations
"Arterial blood pressure analysis ba..." refers methods in this paper
...For a convenient use, we solve the SBSA by replacing the space variable x by the time variable t....
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71 citations
"Arterial blood pressure analysis ba..." refers background or methods in this paper
...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]....
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...The IST is based on the Gel’fand-Levitan-Marchenko (GLM) integral equation [1], [5]....
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...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]....
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Frequently Asked Questions (7)
Q2. What is the spectrum of the Schrödinger operator?
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].
Q3. what is the main idea of the SBSA technique?
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.
Q4. What is the spectral function of the Schrödinger operator?
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.
Q5. What is the main idea of the SBSA technique?
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.
Q6. What is the DST of the potential V?
In this study the authors 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 ).
Q7. What is the method used to analyze the ABP?
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.