# Machine Learning Models and Statistical Complexity to Analyze the Effects of Posture on Cerebral Hemodynamics

^{1}

^{2}

^{*}

## Abstract

**:**

_{2}reactivity and neurovascular-coupling) that affect cerebral blood flow (BF) are included. In this work, we analyzed postural influences using non-linear machine learning models of dCA and studied characteristics of cerebral hemodynamics under statistical complexity using eighteen young adult subjects, aged 27 ± 6.29 years, who took the systemic or arterial blood pressure (BP) and cerebral blood flow velocity (BFV) for five minutes in three different postures: stand, sit, and lay. With models of a Support Vector Machine (SVM) through time, we used an AutoRegulatory Index (ARI) to compare the dCA in different postures. Using wavelet entropy, we estimated the statistical complexity of BFV for three postures. Repeated measures ANOVA showed that only the complexity of lay-sit had significant differences.

## 1. Introduction

_{2}and cerebral neurovascular-coupling (NVC).

## 2. Materials and Methods

#### 2.1. Subjects and Measurement

#### 2.2. Methods

#### 2.2.1. Machine Learning Models of dCA

^{^}(n)) Equations (4) and (5) are used, where n, n

_{p}is the delay in the BP signal, n

_{v}corresponds to BFV recurrences, and f represents a non-linear function. Estimates for kernel radial basis function of SVM shown in Equation (3).

^{^}(n) = f(p(n), p(n − 1), …, p(n − n

_{p}))

^{^}(n) = f(v

^{^}(n − 1), v

^{^}(n − 2), …v

^{^}(n − n

_{v}), p(n), p(n − 1), …, p(n − n

_{p}))

_{p}) and the BFV recurrences (n

_{v}) was conducted empirically. The hyper-parameters of non-linear ν-SVR (i.e., C, v, ε and σ) were bounded by grid search.

^{^}) estimated by the model. The best model for each subject is the one with the highest CC in the validation segment. However, high correlation is not enough to guarantee that a model’s response has physiological plausibility. To solve this problem, we implemented a computational routine based on the indications suggested by Ramos et al. [35], where the physiological quality is evaluated according to the response of the BFV to negative BP step, using three simple criteria: (i) the response should be reduced to at least 40% of the original level from the BFV signal’s mean level; (ii) the response return must be between the minimum and average value plus 10%; and (iii) the BFV return must be between 3 and 6 s after the pressure drop. That automatically discards models that generate non-physiological responses. Training and validating subroutines were implemented using the R environment [36] and libsvm [37] in package e1071 [38].

#### 2.2.2. Statistic Complexity of Hemodynamics

**Wavelet distribution.**This discrete transform represents the signal v(n) in its coefficients ${C}_{j}\left(k\right)$. Since the $\psi $ functions turn out to be an orthogonal basis, the energy of each scale j, the sum of the squared coefficients, at the different frequency levels j = 1…N results in the energy for each resolution level j, Equation (7).

**Entropy and complexity.**The total entropy of the signal will correspond to the classical definition given by Shannon et al. [41], which is obtained in this case from the relative wavelet energy. A measure of the information contained in the signal, Rosso et al. [40] Equation (8):

**Blood flow velocity analysis.**Many times, the value of the complexity represented by the different states is enough to apply the statistical tests directly to the complexities of the analysis. However, Equation (11) shows a compound characteristic that considers entropy, but its dependence is not direct since the imbalance is added as a product.

#### 2.3. Statistical Analysis

## 3. Results

#### 3.1. SVM Models

#### 3.2. Statistic Entropy and Complexity

## 4. Discussion

_{2}variation.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Classic Dynamic Autoregulation Index

_{1}and x

_{2}, as shown in the equations in Equation (A1).

T | D | K | ARI |
---|---|---|---|

2.00 | 1.70 | 0.00 | 0 |

2.00 | 1.60 | 0.20 | 1 |

2.00 | 1.50 | 0.40 | 2 |

2.00 | 1.15 | 0.60 | 3 |

2.00 | 0.90 | 0.80 | 4 |

1.90 | 0.75 | 0.90 | 5 |

1.60 | 0.65 | 0.94 | 6 |

1.20 | 0.55 | 0.96 | 7 |

0.87 | 0.52 | 0.97 | 8 |

0.65 | 0.50 | 0.98 | 9 |

^{^}(t) is predicted from the model, which can be compared with the real speed v(t), and whose highest correlation represents the model’s quality (lowest error).

^{^}(t) for each produced maneuver at p(t), which can be compared to the real speed v(t), and whose highest correlation represents the model’s quality (lowest error).

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**Figure 1.**BFV signal for subject #12, in three postures, lay (dashed line), stand (solid line) and sit (point line).

**Figure 2.**ROC curves to classify paired comparisons of the three postures. (

**a**) Among the SVM-NARX model postures’ dCA values. (

**b**) Among the BFV postures’ complexities.

**Figure 3.**Complexity–entropy plane for lay–sit comparison. Subjects represented by circles correspond to the sitting posture, and subjects represented by triangles represent the lying posture.

**Figure 4.**Average NARX estimated BFV response to negative BP step (solid black line), lay (dashed line), stand (solid grey line), and sit (pointed line).

Posture | Lay | Stand | Sit | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Signal | BFV | BP | BFV | BP | BFV | BP | ||||||

Side | Right | Left | Mean | - | Right | Left | Mean | - | Right | Left | Mean | - |

Mean | 58.52 | 60.02 | 59.27 | 74.96 | 54.36 | 52.65 | 53.50 | 84.81 | 53.69 | 52.07 | 52.88 | 81.45 |

Std | 13.48 | 14.23 | 13.68 | 9.03 | 11.47 | 10.31 | 10.79 | 10.93 | 13.50 | 11.22 | 12.26 | 11.87 |

CoV | 0.23 | 0.24 | 0.23 | 0.12 | 0.21 | 0.20 | 0.20 | 0.13 | 0.25 | 0.22 | 0.23 | 0.15 |

Model | n_{p} | n_{v} | C | v | σ | CC Lay | CC Stand | CC Sit |
---|---|---|---|---|---|---|---|---|

FIR linear | [1–10] | - | [−2, 14 e^{inf}] | [0, 1–0, 9] | [−1, 5] | 0.611 | 0.721 | 0.626 |

FIR non-linear | [1–10] | - | [−2, 14 e^{inf}] | [0, 1–0, 9] | [−1, 5] | 0.655 | 0.742 | 0.682 |

AR linear | [1–8] | [1–6] | [−2, 14 e^{inf}] | [0, 1–0, 9] | [−1, 5] | 0.553 * | 0.706 | 0.599 |

AR non-linear | [1–8] | [1–6] | [−2, 14 e^{inf}] | [0, 1–0, 9] | [−1, 5] | 0.749 | 0.809 * | 0.761 |

**Table 3.**The results of the application of the repeated measures ANOVA methods on the ARIs’ values, for the four models.

Model | Lay | Sit | Stand | p-Values ANOVA |
---|---|---|---|---|

FIR ARI | 4.69 ± 2.51 | 3.6 ± 2.15 | 4.76 ± 2.23 | 0.2522 |

NFIR ARI | 4.41 ± 1.94 | 4.12 ± 2.58 | 4.97 ± 2.03 | 0.3201 |

ARX ARI | 4.41 ± 2.57 | 4.25 ± 1.72 | 4.42 ± 1.92 | 0.9683 |

NARX ARI | 4.92 ± 2.31 | 3.84 ± 2.28 | 4.51 ± 2.77 | 0.6991 |

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**MDPI and ACS Style**

Chacón, M.; Rojas-Pescio, H.; Peñaloza, S.; Landerretche, J.
Machine Learning Models and Statistical Complexity to Analyze the Effects of Posture on Cerebral Hemodynamics. *Entropy* **2022**, *24*, 428.
https://doi.org/10.3390/e24030428

**AMA Style**

Chacón M, Rojas-Pescio H, Peñaloza S, Landerretche J.
Machine Learning Models and Statistical Complexity to Analyze the Effects of Posture on Cerebral Hemodynamics. *Entropy*. 2022; 24(3):428.
https://doi.org/10.3390/e24030428

**Chicago/Turabian Style**

Chacón, Max, Hector Rojas-Pescio, Sergio Peñaloza, and Jean Landerretche.
2022. "Machine Learning Models and Statistical Complexity to Analyze the Effects of Posture on Cerebral Hemodynamics" *Entropy* 24, no. 3: 428.
https://doi.org/10.3390/e24030428