# Coupling between Blood Pressure and Subarachnoid Space Width Oscillations during Slow Breathing

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## Abstract

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## 1. Introduction

## 2. Results

## 3. Discussion

## 4. Materials and Methods

#### 4.1. Subjects

#### 4.2. Experimental Design

#### 4.3. Measurements

#### 4.4. Mathematical Analysis

#### 4.4.1. Wavelet Analysis

#### 4.4.2. Decomposition and Preprocessing

#### 4.4.3. Modelling of Coupling Functions

#### 4.4.4. Statistical Analysis and Surrogates

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Time evolution of (

**a**) BP and (

**c**) ${\mathrm{SAS}}_{\mathrm{LEFT}}$ width (left hemisphere) signals. The four letters (A–D) correspond to stages of experimental protocol: (A) baseline, ventilation with normal respiration rate; (B) 6 breaths/min ventilation; (C) 6 breaths/min ventilation with resistance; and (D) recovery, ventilation with normal respiration rate. (

**b**,

**d**) The amplitude of wavelet transforms for (

**b**) BP and (

**d**) ${\mathrm{SAS}}_{\mathrm{LEFT}}$ width signal estimated for whole recording (40 min). The signals were measured for one subject.

**Figure 2.**Time evolution of group-averaged a directionality index $D\left(t\right)$ estimated for the phases of the respiratory and cardiac signals components for BP, SAS${}_{\mathrm{LEFT}}$ and SAS${}_{\mathrm{RIGHT}}$. To estimate $D\left(t\right)$, Equation (5) was used. The four letters (A–D) correspond to stages of experimental protocol: (A) baseline, ventilation with normal respiration rate; (B) 6 breaths/min ventilation; (C) 6 breaths/min ventilation with resistance; and (D) recovery, ventilation with normal respiration rate. Negative value of $D\left(t\right)$ means that respiration component of signal drives the cardiac component of signal for whole period of experiment.

**Figure 3.**Group-averaged coupling functions between the phases of respiratory ${\varphi}_{Resp}$ and cardiac ${\varphi}_{Card}$ oscillations estimated for the same signals. Columns correspond to different stages of the experiment: baseline, 6 breaths/min, 6 breaths/min, + resistance and recovery. Additionally, to find insignificant values of coupling functions, a surrogate threshold was estimated. Rows show the coupling functions ${q}^{r,c}$ between the phases of the respiratory and cardiac signals for BP, SAS${}_{\mathrm{LEFT}}$ and SAS${}_{\mathrm{RIGHT}}$, respectively. Note that, when coupling functions lose sinusoidal shape along ${\varphi}_{Resp}$ axis (baseline and recovery stages), weaker interaction between respiration and cardiac components of considered signal is observed.

**Figure 4.**Time evolution of group-averaged a directionality index $D\left(t\right)$ (see Equation (5)) estimated for the phases of the respiratory and cardiac signals for different combination of measured signals. The four letters (A–D) correspond to stages of experimental protocol: (A) baseline, ventilation with normal respiration rate; (B) 6 breaths/min ventilation; (C) 6 breaths/min ventilation with resistance; and (D) recovery, ventilation with normal respiration rate. Negative value of $D\left(t\right)$ means that respiration component drives the cardiac mode of considered signal for whole period of experiment.

**Figure 5.**Group-averaged coupling functions between the phases of respiratory ${\varphi}_{Resp}$ and cardiac ${\varphi}_{Card}$ oscillations estimated for different combination of components of measured signals. Columns correspond to: baseline, 6 breaths/min, 6 breaths/min + resistance and recovery. Additionally, to find insignificant values of coupling functions, a surrogate threshold was estimated. Rows show the coupling functions ${q}^{r,c}$ between the respiratory and the cardiac components for different combinations of measured signals (BP-SAS, SAS-BP and SAS-SAS). Note that a sinusoidal shape of coupling functions along ${\varphi}_{Resp}$ axis is connected with stronger interaction between respiration and cardiac oscillations of considered signals.

**Figure 6.**Statistics for coupling strength. (

**a**) Median values for the coupling strength $\sigma $ for four stages of protocol ((A) baseline, ventilation with normal respiration rate; (B) 6 breaths/min ventilation; (C) 6 breaths/min ventilation with resistance; and (D) recovery, ventilation with normal respiration rate) and the coupling surrogate threshold. In red (black) median values significantly (not significantly) different from mean surrogate plus two standard deviations. (

**b**–

**j**) Box-plots illustrating the distribution within each stage of the protocol of the strength $\sigma $ of the coupling between the respiratory and cardiac components. The Friedman test was used to estimate the p value. ‘A and B’ means that Stage A (baseline) differs significantly from Stage B (6 breaths/min), and similarly for ‘B and D’, etc. Red crosses correspond to outliers.

**Figure 7.**Statistics for coupling similarity. (

**a**) Median values for the coupling similarity for three stages of protocol (B–D) compared to baseline stage (A) ((A) baseline, ventilation with normal respiration rate; (B) 6 breaths/min ventilation; (C) 6 breaths/min ventilation with resistance; and (D) recovery, ventilation with normal respiration rate). (

**b**–

**j**) Box-plots illustrating the distribution of the coupling similarity within each stage of protocol for respiratory and cardiac components of the measured signals. The Friedman test was used to estimate the p value. ‘B and D’ and similar symbols are the same as in the previous figure. Red crosses correspond to outliers.

**Figure 8.**Time evolution (30 s segment) of respiratory component (

**blue line**) and heart rate (

**red line**) extracted from the BP signal from one of the volunteers.

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**MDPI and ACS Style**

Gruszecka, A.; Nuckowska, M.K.; Waskow, M.; Kot, J.; Winklewski, P.J.; Guminski, W.; Frydrychowski, A.F.; Wtorek, J.; Bujnowski, A.; Lass, P.;
et al. Coupling between Blood Pressure and Subarachnoid Space Width Oscillations during Slow Breathing. *Entropy* **2021**, *23*, 113.
https://doi.org/10.3390/e23010113

**AMA Style**

Gruszecka A, Nuckowska MK, Waskow M, Kot J, Winklewski PJ, Guminski W, Frydrychowski AF, Wtorek J, Bujnowski A, Lass P,
et al. Coupling between Blood Pressure and Subarachnoid Space Width Oscillations during Slow Breathing. *Entropy*. 2021; 23(1):113.
https://doi.org/10.3390/e23010113

**Chicago/Turabian Style**

Gruszecka, Agnieszka, Magdalena K. Nuckowska, Monika Waskow, Jacek Kot, Pawel J. Winklewski, Wojciech Guminski, Andrzej F. Frydrychowski, Jerzy Wtorek, Adam Bujnowski, Piotr Lass,
and et al. 2021. "Coupling between Blood Pressure and Subarachnoid Space Width Oscillations during Slow Breathing" *Entropy* 23, no. 1: 113.
https://doi.org/10.3390/e23010113