# The Risk Function of Breast and Ovarian Cancers in the Avrami–Dobrzyński Cellular Phase-Transition Model

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## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Patients Tested for BRCA1/BRCA2

^{2}equal to 0.99 for both datasets. The results are presented in Figure 1.

_{0}was introduced through the modification of Equation (2) ($P\left(t\right)=1-exp\left[-\alpha {(t-{t}_{0})}^{k}\right]$). A new function (with fixed α and k parameters) was fitted to both sets of data, with and without BRCA1/BRCA2 mutations, and parameter t

_{0}was determined.

_{0}values for both subpopulations and was equal to (2.29 ± 0.20) years. This shift value, in our understanding, represents the difference in the time of cancer occurrence; therefore, this means that patients with BRCA1/BRCA2 mutations are diagnosed with breast or ovary cancer approximately two years earlier than patients without such mutations. This seems to be quite a natural situation, which can be found in the literature. For instance, in the Latvian study, this period varied from ~9 to even ~16 years [5], while in the Belarussian study, it was equal to 4–5 years [4]. Our results are therefore much closer to those of Belarus and suggest that a new oncogenic mutation can be obtained in breast/ovary somatic cells every two years of human life.

#### 2.2. Results for Cancer Patients without Genetic Tests

**Table 1.**Results of the application of the Avrami–Dobrzyński model to breast and ovary cancer clinical data received from the MSCI, divided into three datasets. All uncertainties represent one standard deviation.

No. of the Dataset | No. of Patients | Cancer Type | α Parameter (y^{−1}) | k Parameter | Description |
---|---|---|---|---|---|

1 | 459 | Breast, ovary, or both | (3.6 ± 1.6) 10^{−11} | 6.32 ± 0.12 | Patients without BRCA1/BRCA2 (no-BRCA group); see Figure 1 |

52 | (5.0 ± 6.0) 10^{−10} | 5.69 ± 0.33 | Patients with BRCA1/BRCA2 (BRCA group); see Figure 1 | ||

2 | 20,802 | Breast | (4.16 ± 0.21) 10^{−11} | 5.775 ± 0.013 | Patients diagnosed with breast cancer (C50 group); see Figure 3 |

3 | 9106 | Ovary | (6.58 ± 0.43) 10^{−9} | 4.570 ± 0.016 | Patients diagnosed with ovarian cancer (C56 group); see Figure 3 |

#### 2.3. Protection Curve

^{2}= 0.76) using the following function (Figure 4):

_{1}= (0.883 ± 0.050) y

^{−2}, p

_{2}= (−40.6 ± 2.5) y

^{−1}, p

_{3}= 482 ± 41, q

_{1}= (−45.3 ± 3.5) y

^{−1}, and q

_{2}= 518 ± 42. Other functions, like log-normal, Gaussian, or Weibull distributions, which were also tested as a best fit to the data points from Figure 4, had lower statistical likelihood than the one from Equation (3). One has to note that the shape of the function given by Equation (3) and presented in Figure 4 is a typical function of the adaptation of the organism to the external stressor [13].

#### 2.4. Fractality

## 3. Discussion

- The geometric structure of DNA represents a fractal character; indeed, the DNA globule has many limited fractal elements, such as self-similarity and limited scale-free or power-law distribution of some DNA elements [25,26,27,28]. However, the process in which this fractality is functioning during cancer transformation is still unknown, especially from a dynamic point of view.
- DNA creates a complex multidimensional protein and metabolic network [29,30,31]; the proposed dimensionality problem should not be thought of as a spatial dimension related to the geometry of the DNA globule but rather the effective dimension of the network of protein interactions that are encoded by DNA. If we draw all proteins in the cell into a large network and connect all the nodes of proteins participating in the same processes (they have high affinity, regulate each other, etc.), we will obtain a network with a complicated topology. Such a complex network of connections can be associated with a dimension characterising the structure of this network (e.g., how many proteins are connected by one edge, how many by two edges, three, etc., and how it grows with the number of edges), not the space in which we draw it. In that way, the number of neighbours acts as a mathematical concept of dimension, and this feature does not have to be limited to 3D but can reach any number.

## 4. Materials and Methods

#### 4.1. Clinical Data Collection

#### 4.2. Statistical Analysis

_{0}) function (with additional time shift t

_{0}parameter).

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ATM | Ataxia–telangiectasia mutated |

BOC | Breast and ovarian cancer |

BRCA | Breast Cancer gene |

CDH1 | Cadherin1 |

CHEK2 | Checkpoint kinase 2 |

DNA | Deoxyribonucleic acid |

ICD | International Classification of Diseases |

JMAK | Johnson–Mehl–Avrami–Kolmogorov |

MSCI | Maria Skłodowska-Curie National Research Institute of Oncology |

MSD | MedStream Designer |

NBN | Nijmegen breakage syndrome (nibrin) |

PALB2 | Partner and localizer of BRCA2 |

PTEN | Phosphatase and tensin homolog |

TP53 | Tumor protein p53 |

XRCC2 | X-ray repair cross-complementating 2 |

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**Figure 1.**Cumulative probability of cancer diagnosis as a function of patient’s age at the time of diagnosis for the first set of clinical data (511 cancer patients with or without the BRCA1/BRCA2 mutation) calculated with the Avrami–Dobrzyński model, where α = (3.6 ± 1.6) × 10

^{−11}y

^{−1}and k = 6.32 ± 0.12 for patients without mutations (black line), and α = (5.0 ± 6.0) × 10

^{−10}y

^{−1}and k = 5.69 ± 0.33 for patients with BRCA1/BRCA2 mutations (grey line). All uncertainties represent one standard deviation.

**Figure 2.**Cumulative probability of cancer diagnosis as a function of patient age at the time of diagnosis calculated with the Avrami–Dobrzyński model for the C50 group (

**A**) and the C56 group (

**B**). Grey points represent clinical data, while black lines represent model fit.

**Figure 3.**Ratio values between theoretical and experimental data as a function of patient age calculated for each clinical dataset (see Table 1). These functions represent potentially better immune protection for young women.

**Figure 4.**The average protection curve, f(t), represents the hypothetically better immune barrier for young women. Data points represent averaged values from Figure 3, while the black line is the best fit using the empirical function from Equation (3).

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**MDPI and ACS Style**

Zawadzka, A.; Brzozowska, B.; Matyjanka, A.; Mikula, M.; Reszczyńska, J.; Tartas, A.; Fornalski, K.W.
The Risk Function of Breast and Ovarian Cancers in the Avrami–Dobrzyński Cellular Phase-Transition Model. *Int. J. Mol. Sci.* **2024**, *25*, 1352.
https://doi.org/10.3390/ijms25021352

**AMA Style**

Zawadzka A, Brzozowska B, Matyjanka A, Mikula M, Reszczyńska J, Tartas A, Fornalski KW.
The Risk Function of Breast and Ovarian Cancers in the Avrami–Dobrzyński Cellular Phase-Transition Model. *International Journal of Molecular Sciences*. 2024; 25(2):1352.
https://doi.org/10.3390/ijms25021352

**Chicago/Turabian Style**

Zawadzka, Anna, Beata Brzozowska, Anna Matyjanka, Michał Mikula, Joanna Reszczyńska, Adrianna Tartas, and Krzysztof W. Fornalski.
2024. "The Risk Function of Breast and Ovarian Cancers in the Avrami–Dobrzyński Cellular Phase-Transition Model" *International Journal of Molecular Sciences* 25, no. 2: 1352.
https://doi.org/10.3390/ijms25021352