# Mathematical Modeling for Neuropathic Pain: Bayesian Linear Regression and Self-Organizing Maps Applied to Carpal Tunnel Syndrome

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Participants

#### 2.2. Pain and Related Disability Outcomes

#### 2.3. Pressure and Thermal Pain Threshold Assessment

#### 2.4. Pinch Tip Grip Force Assessment

#### 2.5. Psychological Assessment

#### 2.6. Data Overview and Preprocessing

#### 2.7. Bayesian Linear Regression (BLR)

#### 2.7.1. Method Overview

#### 2.7.2. Bayesian vs. Frequentist Statistics

#### 2.7.3. Bayesian Linear Models LR vs. Nonlinear Models

#### 2.8. Self-Organizing Maps (SOM)

## 3. Results

#### 3.1. Participants

#### 3.2. Bayesian Linear Regression

#### 3.3. Self-Organizing Maps

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Credible intervals for all the parameters in the Bayesian linear regression (BLR) model for pain intensity prediction.

**Figure 3.**Credible intervals for all the parameters in the Bayesian linear regression (BLR) model for function prediction.

**Figure 4.**Credible intervals for all the parameters in the Bayesian linear regression (BLR) model for symptom severity prediction.

**Figure 6.**Self-organizing maps of the clinical, neuro-physiological, psychological, and motor features in women with carpal tunnel syndrome.

**Table 1.**Mean correlation coefficient from 10 training and cross-validation (CV) folds for all output features. Despite the performance differences in training, all algorithms seem to be similarly capable in validation. BLR, Bayesian linear regression; NN, neural network.

Output Feature | Mean Train Correlation | Mean CV Correlation | ||||
---|---|---|---|---|---|---|

BLR | xgboost | NN | BLR | xgboost | NN | |

Function | 0.724 | 0.981 | 0.697 | 0.596 | 0.604 | 0.607 |

Symptoms’ Severity | 0.723 | 0.980 | 0.694 | 0.628 | 0.633 | 0.632 |

Pain Intensity | 0.622 | 0.978 | 0.764 | 0.457 | 0.597 | 0.499 |

Mean | SD | Min | Max | |
---|---|---|---|---|

Age | 45.5 | 9.1 | 21.0 | 64.00 |

Years with pain | 3.5 | 3.0 | 0.5 | 17.00 |

Right side affected * | 0.9 | 0.3 | 0.00 | 1.00 |

Left side affected * | 0.75 | 0.45 | 0.00 | 1.00 |

EMG minimal affectation # | 0.3 | 0.45 | 0.00 | 1.00 |

EMG severe affectation # | 0.4 | 0.5 | 0.00 | 1.00 |

Pain intensity | 5.8 | 2.1 | 0.00 | 10.00 |

Symptom severity | 2.75 | 0.7 | 1.25 | 5.00 |

Function | 2.4 | 0.75 | 1.0 | 4.62 |

Depression (BDI-II) | 4.6 | 2.9 | 0.0 | 15.0 |

CPT carpal tunnel | 19.4 | 6.7 | 5.00 | 30.2 |

CPT hand | 19.2 | 6.45 | 5.00 | 29.75 |

HPT carpal tunnel | 39.9 | 2.6 | 35.2 | 48.45 |

HPT hand | 40.1 | 2.85 | 32.1 | 48.2 |

PPT median nerve | 192.55 | 50.7 | 57.65 | 365.5 |

PPT ulnar nerve | 293.7 | 73.6 | 115.5 | 465.5 |

PPT radial nerve | 225.25 | 61.9 | 109.5 | 433.5 |

PPT cervical spine | 171.1 | 53.75 | 57.0 | 499.5 |

PPT carpal tunnel | 346.05 | 95.4 | 130.5 | 731.0 |

PPT tibialis anterior | 322.85 | 85.5 | 110.5 | 652.5 |

Thumb–index finger pinch tip | 4.15 | 1.7 | 0.5 | 8.5 |

Thumb–little finger pinch tip | 1.1 | 0.8 | 0.0 | 5.5 |

Thumb–middle finger pinch tip | 4.0 | 1.9 | 0.0 | 9.5 |

Thumb–ring finger pinch tip | 2.45 | 1.4 | 0.0 | 6.35 |

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**MDPI and ACS Style**

Pellicer-Valero, O.J.; Martín-Guerrero, J.D.; Cigarán-Méndez, M.I.; Écija-Gallardo, C.; Fernández-de-las-Peñas, C.; Navarro-Pardo, E.
Mathematical Modeling for Neuropathic Pain: Bayesian Linear Regression and Self-Organizing Maps Applied to Carpal Tunnel Syndrome. *Symmetry* **2020**, *12*, 1581.
https://doi.org/10.3390/sym12101581

**AMA Style**

Pellicer-Valero OJ, Martín-Guerrero JD, Cigarán-Méndez MI, Écija-Gallardo C, Fernández-de-las-Peñas C, Navarro-Pardo E.
Mathematical Modeling for Neuropathic Pain: Bayesian Linear Regression and Self-Organizing Maps Applied to Carpal Tunnel Syndrome. *Symmetry*. 2020; 12(10):1581.
https://doi.org/10.3390/sym12101581

**Chicago/Turabian Style**

Pellicer-Valero, Oscar J., José D. Martín-Guerrero, Margarita I. Cigarán-Méndez, Carmen Écija-Gallardo, César Fernández-de-las-Peñas, and Esperanza Navarro-Pardo.
2020. "Mathematical Modeling for Neuropathic Pain: Bayesian Linear Regression and Self-Organizing Maps Applied to Carpal Tunnel Syndrome" *Symmetry* 12, no. 10: 1581.
https://doi.org/10.3390/sym12101581