# Aberrant Structure MRI in Parkinson’s Disease and Comorbidity with Depression Based on Multinomial Tensor Regression Analysis

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Participants and Clinical Evaluation

#### 2.2. MRI Acquisition and Preprocessing

#### 2.3. Multinomial Tensor Regression

- ${Y}_{i1}=1$ if the ith patient is healthy and 0 otherwise;
- ${Y}_{i2}=1$ if the ith patient has DPD and 0 otherwise;
- ${Y}_{i3}=1$ if the ith patient has NDPD and 0 otherwise.

#### 2.4. Estimation

Algorithm 1. Block relaxation algorithm for maximizing (4). |

Initialize ${\left\{{\beta}_{k}^{{j}^{\left(0\right)}}\right\}}_{j=1,2,3;k=1,2}$ with random values |

repeat $(t\ge 1)$ |

for $j=1,2,3$ and $k=1,2$ do |

${\beta}_{k}^{{j}^{(t+1)}}={\mathrm{argmax}}_{{\beta}_{k}^{j}}l\left(\right)open="("\; close=")">{\beta}_{1}^{{1}^{(t+1)}},{\beta}_{1}^{{2}^{(t+1)}},\dots ,{\beta}_{k}^{j},\dots ,{\beta}_{3}^{{2}^{\left(t\right)}}$ |

end for |

until $|l\left(\right)open="("\; close=")">{\Theta}^{(t+1)})-l({\Theta}^{\left(t\right)}$ |

## 3. Results

#### 3.1. Clinical and Demographic Data

#### 3.2. Quantitative Performance

#### 3.3. Aberrant Structural Brain Regions

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Original MRI images (

**a**–

**c**) and normalized MRI images (

**d**–

**f**). The original images of size (512, 512, 128) were normalized using Statistical Parametric Mapping. Individual images for all subjects were mapped to a common reference space with size (79, 95, 79) to reduce the complexity.

**Figure 2.**Three-dimensional sMRI images for NDPD (

**a**–

**c**), HC (

**d**–

**f**), and the heatmaps for coefficient matrices corresponding to three different surfaces respectively (

**g**–

**i**). Voxels with yellow and dark blue colors correspond to regions with aberrant structural changes for NDPD compared with HC.

**Figure 3.**Three-dimensional sMRI images for DPD (

**a**–

**c**), NDPD (

**d**–

**f**), and the heatmaps for coefficient matrices corresponding to three different surfaces respectively (

**g**–

**i**). Voxels with yellow and dark blue colors correspond to regions with aberrant structural changes for DPD compared with NDPD.

**Table 1.**Clinical and demographic data evaluation of NDPD, DPD, and HC.

^{a}. The p value for gender distribution by Fisher’s exact test.

^{b}. The p value for age by multivariate analysis of variance (MANOVA).

^{c}. The p value for education by MANOVA.

^{d}. The F test statistic and the p value for MMSE scores by MANOVA.

^{e}–

^{g}. The p values for HAMD scores by Paired-Samples t test with Bonferroni correction for further comparison between three groups.

^{h}. The F test statistic and the p value for UPDRS-III by analysis of variance (ANOVA).

^{i}. The F test statistic and the p value for H & Y by ANOVA.

Characteristics | DPD (n = 84) | NDPD (n = 192) | HC (n = 200) | Test Statistic | p Value |
---|---|---|---|---|---|

Sex (M/F) | 36/48 | 104/88 | 96/104 | 0.409 | >0.05 ${}^{a}$ |

Age (year) | $58.1\pm 7.5$ | $57.8\pm 7.0$ | $57.8\pm 5.5$ | 0.021 | >0.05 ${}^{b}$ |

Education (year) | $11.0\pm 3.1$ | $11.8\pm 3.3$ | $11.7\pm 4.8$ | 0.689 | >0.05 ${}^{c}$ |

MMSE | $28.7\pm 1.1$ | $28.6\pm 1.7$ | $29.0\pm 2.3$ | 0.585 | >0.05 ${}^{d}$ |

HAMD | $20.2\pm 4.6$ | $6.9\pm 3.1$ | $2.2\pm 2.3$ | 243.2 ($p<0.05$) | <0.016 ${}^{e}<0.016$ ${}^{f}<0.016$ ${}^{g}$ |

UPDRS-III | $28.3\pm 13.2$ | $26.4\pm 13.3$ | N/A | 0.295 | >0.05 ${}^{h}$ |

H & Y | $1.4\pm 0.6$ | $1.8\pm 0.7$ | N/A | 5.37 | <0.05 ${}^{i}$ |

Model | RI | PA | MAUC |
---|---|---|---|

Multinomial Tensor | 1 | 1 | 1 |

Multinomial Logistic ($d=1000$) | 0.59 | 0.61 | 0.69 |

Multinomial Logistic ($d=3000$) | 0.6 | 0.63 | 0.64 |

Multinomial Logistic ($d=\mathrm{10,000}$) | 0.66 | 0.68 | 0.73 |

3D CNN | 1 | 1 | 1 |

Model | RI | PA | MAUC |
---|---|---|---|

Multinomial Tensor | 0.89 | 0.94 | 0.98 |

Multinomial Logistic ($d=1000$) | 0.49 | 0.44 | 0.55 |

Multinomial Logistic ($d=3000$) | 0.56 | 0.56 | 0.69 |

Multinomial Logistic ($d=\mathrm{10,000}$) | 0.58 | 0.63 | 0.70 |

3D CNN | 0.55 | 0.31 | 0.53 |

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**MDPI and ACS Style**

Cao, X.; Yang, F.; Zheng, J.; Wang, X.; Huang, Q.
Aberrant Structure MRI in Parkinson’s Disease and Comorbidity with Depression Based on Multinomial Tensor Regression Analysis. *J. Pers. Med.* **2022**, *12*, 89.
https://doi.org/10.3390/jpm12010089

**AMA Style**

Cao X, Yang F, Zheng J, Wang X, Huang Q.
Aberrant Structure MRI in Parkinson’s Disease and Comorbidity with Depression Based on Multinomial Tensor Regression Analysis. *Journal of Personalized Medicine*. 2022; 12(1):89.
https://doi.org/10.3390/jpm12010089

**Chicago/Turabian Style**

Cao, Xuan, Fang Yang, Jingyi Zheng, Xiao Wang, and Qingling Huang.
2022. "Aberrant Structure MRI in Parkinson’s Disease and Comorbidity with Depression Based on Multinomial Tensor Regression Analysis" *Journal of Personalized Medicine* 12, no. 1: 89.
https://doi.org/10.3390/jpm12010089