# Radiobiological Meta-Analysis of the Response of Prostate Cancer to Different Fractionations: Evaluation of the Linear–Quadratic Response at Large Doses and the Effect of Risk and ADT

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## Abstract

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## Simple Summary

## Abstract

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Clinical Dataset

#### 2.2. Radiobiological Modelling: Dose–Response

#### 2.3. Statistical Methods

#### 2.4. Radiobiological Modelling: $\alpha $ and Number of Clonogens

## 3. Results

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Best fits to prostate carcinoma dose–response data obtained with the linear–quadratic model. Results are presented separately for low risk (

**left**panel), intermediate risk (

**central**panel), and high risk (

**right**panel).

**Figure 2.**Best fits to intermediate- and high-risk prostate carcinoma dose–response data obtained with the linear–quadratic model. Results are presented separately for cohorts that used androgen deprivation therapy (ADT) and cohorts that did not use ADT in addition to radiotherapy.

**Figure 3.**Modelled dose per fraction versus number of fractions to achieve 90% control for HR patients treated with radiotherapy and ADT (dashed line). The circles represent the experimental fractionations included in the dataset.

Risk | Number of Schedules | Number of Patients (Range) | Dose per Fraction (Range) | Total Dose (Range) | Treatment Time (Range) | ADT (Fraction of Schedules) | Control at 5 Years (Range) |
---|---|---|---|---|---|---|---|

LR | 35 | 3–550 | 1.8–10 Gy | 33.5–81 Gy | 3–62 days | 3/35 | 0.59–1.00 |

IR | 32 | 7–839 | 1.8–10 Gy | 34–81 Gy | 3–62 days | 9/32 | 0.38–1.00 |

HR | 20 | 12–812 | 1.8–8.5 Gy | 34–81 Gy | 3–62 days | 15/20 | 0.28–0.908 |

**Table 2.**Best fits obtained with the LQ, LQL, and LQmod models to prostate carcinoma dose–response data, separated by risk (low, intermediate, and high risk). For intermediate risk, results are also presented separately for schedules not including ADT. For high risk, results are also presented separately for schedules including ADT. The table shows the best-fitting parameters, maximum likelihood, and $AI{C}_{c}$ values. Improvements on the performance of the LQ model ($\Delta AI{C}_{c}$ > 0) are highlighted in bold and italics. The symbol * indicates that the best-fitting parameter reached the edge of the constraint window.

Risk | Model | Parameters | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\alpha}/\mathit{\beta}$ | ${\mathit{\lambda}}^{\prime}$ | ${\mathit{T}}_{\mathit{k}}$ | $\mathit{\delta}$ | $\mathit{b}$ | ${\mathit{D}}_{\mathit{50}}$ | ${\mathit{\gamma}}_{\mathit{50}}$ | $-\mathit{log}\mathit{L}$ | ${\mathit{AIC}}_{\mathit{c}}$ | $\Delta {\mathit{AIC}}_{\mathit{c}}$ | ||

(Gy) | (Gy day${}^{-\mathbf{1}}$) | (day${}^{-\mathbf{1}}$) | (Gy${}^{-\mathbf{1}}$) | (Gy${}^{-\mathbf{1}/\mathbf{2}}$) | (Gy) | ||||||

LR | LQ | 2.0 | 0.00 | - | - | - | 56.2 | 2.17 | 89.4 | 190.8 | - |

LQL | 2.0 | 0.00 | - | 0.00 | - | 56.2 | 2.17 | 89.4 | 193.7 | −2.9 | |

LQ${}_{\mathrm{mod}}$ | 2.6 | 0.00 | - | - | 0.07 | 55.8 | 2.11 | 89.2 | 193.4 | −2.6 | |

IR | LQ | 3.4 | 0.41 | 24.0 | - | - | 56.9 | 2.14 | 220.9 | 454.2 | - |

LQL | 0.4 | 0.00 | - | 0.28 | - | 62.8 | 2.18 | 201.3 | 418.0 | 36.2 | |

LQ${}_{\mathrm{mod}}$ | 3.5 | 0.41 | 23.9 | - | 0.00 | 56.8 | 2.15 | 220.9 | 457.2 | −3.0 | |

HR | LQ | 2.8 | 0.35 | 21.0 * | - | - | 59.8 | 1.45 | 105.0 | 224.4 | - |

LQL | 2.8 | 0.35 | 21.0 * | 0.00 | - | 59.8 | 1.45 | 105.0 | 228.5 | −4.2 | |

LQ${}_{\mathrm{mod}}$ | 11.2 | 0.34 | 21.0 * | - | 0.75 | 58.7 | 1.47 | 103.9 | 226.2 | −1.8 | |

IR | LQ | 2.8 | 0.32 | 21.0 * | - | - | 58.1 | 1.85 | 157.6 | 328.7 | - |

(no ADT) | LQL | 0.5 | 0.00 | - | 0.24 | - | 63.6 | 2.01 | 138.7 | 294.6 | 34.1 |

LQ${}_{\mathrm{mod}}$ | 2.8 | 0.32 | 21.0 * | - | 0.00 | 58.1 | 1.85 | 157.6 | 332.4 | −3.7 | |

HR | LQ | 2.1 | 0.00 | - | - | - | 58.5 | 0.95 | 72.6 | 161.8 | - |

(ADT) | LQL | 2.1 | 0.00 | - | 0.00 | - | 58.5 | 0.95 | 72.5 | 167.6 | -5.8 |

LQ${}_{\mathrm{mod}}$ | 18.7 | 0.00 | - | - | 1.99 | 56.8 | 0.88 | 71.2 | 164.9 | −3.1 |

**Table 3.**Best-fitting parameters and 95% confidence intervals (within parentheses) of prostate carcinoma dose–response data obtained with the linear–quadratic model. Results are separated by risk and for intermediate and high risk are also presented separately for schedules that included or did not include ADT. Data for IR with ADT and HR with no ADT are shown only for illustrative purposes, because, due to the low number of schedules, the confidence intervals are very wide.

$\mathit{\alpha}/\mathit{\beta}$ (Gy) | ${\mathit{\lambda}}^{\prime}$ (Gy day${}^{-1}$) | ${\mathit{T}}_{\mathit{k}}$ (day${}^{-1}$) | ${\mathit{D}}_{50}$ (Gy) | ${\mathit{\gamma}}_{50}$ | |
---|---|---|---|---|---|

LR | 2.0 | 0 | - | 56.2 | 2.17 |

(1.7, 2.3) | (0, 0.13) | (54.4, 58.0) | (1.90, 2.47) | ||

IR | 3.4 | 0.41 | 24.0 | 56.9 | 2.14 |

(3.0, 4.0) | (0.31, 0.49) | (21.0, 25.5) | (55.5, 57.9) | (1.92, 2.40) | |

HR | 2.8 | 0.35 | 21.0 | 59.8 | 1.45 |

(1.4, 4.2) | (0, ∞) | (21.0, ∞) | (57.1, 63.9) | (1.07, 1.83) | |

IR | 2.8 | 0.32 | 21.0 | 58.1 | 1.85 |

(no ADT) | (2.1, 3.5) | (0.09, 0.46) | (21.0, 27.3) | (56.5, 60.0) | (1.55, 2.14) |

HR | 2.1 | 0 | - | 58.5 | 0.95 |

(ADT) | (1.5, 3.5) | (0, 0.31) | (54.3, 61.5) | (0.75, 1.25) | |

IR | 0.1 | 0 | - | 8.1 | 0.20 |

(ADT) | (0, ∞) | (0, ∞) | (0.4, 40.5) | (0.11, 0.80) | |

HR | 100.0 | 3.31 | 39.9 | 54.6 | 6.68 |

(no ADT) | (7.1, ∞) | (1.09, ∞) | (21.0, 40.8) | (49.8, 60.5) | (2.70, 10.67) |

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**MDPI and ACS Style**

Pardo-Montero, J.; González-Crespo, I.; Gómez-Caamaño, A.; Gago-Arias, A.
Radiobiological Meta-Analysis of the Response of Prostate Cancer to Different Fractionations: Evaluation of the Linear–Quadratic Response at Large Doses and the Effect of Risk and ADT. *Cancers* **2023**, *15*, 3659.
https://doi.org/10.3390/cancers15143659

**AMA Style**

Pardo-Montero J, González-Crespo I, Gómez-Caamaño A, Gago-Arias A.
Radiobiological Meta-Analysis of the Response of Prostate Cancer to Different Fractionations: Evaluation of the Linear–Quadratic Response at Large Doses and the Effect of Risk and ADT. *Cancers*. 2023; 15(14):3659.
https://doi.org/10.3390/cancers15143659

**Chicago/Turabian Style**

Pardo-Montero, Juan, Isabel González-Crespo, Antonio Gómez-Caamaño, and Araceli Gago-Arias.
2023. "Radiobiological Meta-Analysis of the Response of Prostate Cancer to Different Fractionations: Evaluation of the Linear–Quadratic Response at Large Doses and the Effect of Risk and ADT" *Cancers* 15, no. 14: 3659.
https://doi.org/10.3390/cancers15143659