# Development of “Mathematical Technology for Cytopathology,” an Image Analysis Algorithm for Pancreatic Cancer

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Design

#### 2.2. Procedure of EUS-FNA/EUS-FNB and Diagnosis

#### 2.3. Automatic Diagnosis Assistance System

#### 2.4. The Mathematical Method

#### 2.4.1. The Reaction–Diffusion System

_{a}is the diffusion coefficient. The value a is a threshold for running the FitzHugh–Nagumo (FHN) equations, i.e., Equations (3) and (4) [19,21]. The initial value a

_{0}of a is determined by the following:

_{max}and I

_{min}are the maximum and minimum brightness values of the pixels, respectively.

_{u}and D

_{v}are the diffusion coefficients for the variables u and v, respectively. The parameter $\epsilon $ is a positive small constant (0 < $\epsilon $ < 1). The parameter b is a positive constant and is spatially homogeneous. The initial value of u is defined as follows:

#### 2.4.2. Numerical Computations

_{a}) = 2500.0. Next, Equations (1), (3) and (4) were calculated with t in the interval [0.005, 0.02]. The parameter values were ε = 0.0002 and b = 20.0. Our numerical computation stopped after approximately 4–5 s, using an ordinary laptop computer. The final result of u was translated to the binary images.

#### 2.5. Calculating the Quantitative Index

_{u}and D

_{v}) can be regarded as reflecting the state of the tissue staining. In this study, we selected three parameters for D (50, 100, and 150). Important indices for detecting adenocarcinoma cells were identified by calculating the accuracy, sensitivity, and specificity of the quantitative index.

#### 2.6. Classifying Tissues as Normal or Adenocarcinoma Tissues

#### 2.7. Evaluating the Classification Accuracy

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The left panels: Diff-Quick stain of adenocarcinoma and benign pancreatic tissue. The right panels: “reaction–diffusion method” of adenocarcinoma and benign pancreatic tissue.

**Figure 2.**If the value of $u$ is in interval (

**i**), then the reaction term is negative. If ${u}_{t}$ is considered negative, then the value of $u$ decreases. Conversely, if $u$ is in interval (

**ii**), then the value of $u$ increases. Here, let interval (

**i**) be (a, 0) and interval (

**ii**) be (0, b). Therefore, the value of $u$ finally converges to that of a (black) or b (white).

**Figure 3.**The left panels show the real image of capillaries at the base of the fingernails and the image after processing using the reaction–diffusion method. The right panels show the real image of silicon–boron compounds and the image after processing using the reaction–diffusion method.

**Figure 4.**The reaction–diffusion images with different D values: As the D value decreased, the unnecessary parts of the edges became visible, instead of the core content. In contrast, as the D value increased, the unnecessary parts of the edges disappeared, while the content showed a tendency to almost disappear.

Quantitative Index | Accuracy (%) | Sensitivity (%) | Specificity (%) | |
---|---|---|---|---|

Univariate analysis | Number of pixels | 71 | 75 | 65 |

Area | 39 | 47 | 28 | |

Interquartile area range | 67 | 71 | 61 | |

Area/pixel | 68 | 72 | 63 | |

Average perimeter of the connected components | 57 | 64 | 48 | |

Average circularity of the connected components | 46 | 53 | 36 | |

Interquartile circularity range of the connected components | 43 | 50 | 33 | |

Multivariate analysis | Number of pixels + interquartile area range + average perimeter of the connected components | 75 | 78 | 70 |

Quantitative Index | Accuracy (%) | Sensitivity (%) | Specificity (%) | |
---|---|---|---|---|

Univariate analysis | Number of pixels | 68 | 72 | 62 |

Area | 42 | 49 | 31 | |

Interquartile area range | 67 | 71 | 61 | |

Area/pixel | 66 | 71 | 60 | |

Average perimeter of the connected components | 40 | 49 | 27 | |

Average circularity of the connected components | 24 | 34 | 10 | |

Interquartile circularity range of the connected components | 69 | 74 | 64 | |

Multivariate analysis | Number of pixels + interquartile area range | 70 | 74 | 65 |

Quantitative Index | Accuracy (%) | Sensitivity (%) | Specificity (%) | |
---|---|---|---|---|

Univariate analysis | Number of pixels | 56 | 62 | 48 |

Area | 52 | 58 | 44 | |

Interquartile area range | 69 | 73 | 64 | |

Area/pixel | 55 | 61 | 48 | |

Average perimeter of the connected components | 46 | 54 | 34 | |

Average circularity of the connected components | 23 | 33 | 9 | |

Interquartile circularity range of the connected components | 74 | 78 | 69 | |

Multivariate analysis | Area/pixel + interquartile circularity range of the connected components | 74 | 77 | 70 |

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**MDPI and ACS Style**

Yamada, R.; Nakane, K.; Kadoya, N.; Matsuda, C.; Imai, H.; Tsuboi, J.; Hamada, Y.; Tanaka, K.; Tawara, I.; Nakagawa, H.
Development of “Mathematical Technology for Cytopathology,” an Image Analysis Algorithm for Pancreatic Cancer. *Diagnostics* **2022**, *12*, 1149.
https://doi.org/10.3390/diagnostics12051149

**AMA Style**

Yamada R, Nakane K, Kadoya N, Matsuda C, Imai H, Tsuboi J, Hamada Y, Tanaka K, Tawara I, Nakagawa H.
Development of “Mathematical Technology for Cytopathology,” an Image Analysis Algorithm for Pancreatic Cancer. *Diagnostics*. 2022; 12(5):1149.
https://doi.org/10.3390/diagnostics12051149

**Chicago/Turabian Style**

Yamada, Reiko, Kazuaki Nakane, Noriyuki Kadoya, Chise Matsuda, Hiroshi Imai, Junya Tsuboi, Yasuhiko Hamada, Kyosuke Tanaka, Isao Tawara, and Hayato Nakagawa.
2022. "Development of “Mathematical Technology for Cytopathology,” an Image Analysis Algorithm for Pancreatic Cancer" *Diagnostics* 12, no. 5: 1149.
https://doi.org/10.3390/diagnostics12051149