### Abstract

The generalized equivalent uniform dose (GEUD) model uses a power-law formalism, where the outcome is related to the dose via a power law. We herein investigate the mathematical compatibility between this GEUD model and the Poisson statistics based tumor control probability (TCP) model. The GEUD and TCP formulations are combined and subjected to a compatibility constraint equation. This compatibility constraint equates tumor control probability from the original heterogeneous target dose distribution to that from the homogeneous dose from the GEUD formalism. It is shown that this constraint equation possesses a unique, analytical closed-form solution which relates radiation dose to the tumor cell survival fraction. It is further demonstrated that, when there is no positive threshold or finite critical dose in the tumor response to radiation, this relationship is not bounded within the realistic cell survival limits of 0%-100%. Thus, the GEUD and TCP formalisms are, in general, mathematically inconsistent. However, when a threshold dose or finite critical dose exists in the tumor response to radiation, there is a unique mathematical solution for the tumor cell survival fraction that allows the GEUD and TCP formalisms to coexist, provided that all portions of the tumor are confined within certain specific dose ranges.

Original language | English (US) |
---|---|

Pages (from-to) | 2606-2609 |

Number of pages | 4 |

Journal | Medical physics |

Volume | 31 |

Issue number | 9 |

DOIs | |

State | Published - Sep 2004 |

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### Keywords

- Cell survival fraction
- EUD
- TCP

### ASJC Scopus subject areas

- Biophysics
- Radiology Nuclear Medicine and imaging

### Cite this

*Medical physics*,

*31*(9), 2606-2609. https://doi.org/10.1118/1.1783532

**Relatioship between the generalized equivalent uniform dose formulation and the Poisson statistics-based tumor control probability model.** / Zhou, Su Min; Das, Shiva; Wang, Zhiheng; Marks, Lawrence B.

Research output: Contribution to journal › Article

*Medical physics*, vol. 31, no. 9, pp. 2606-2609. https://doi.org/10.1118/1.1783532

}

TY - JOUR

T1 - Relatioship between the generalized equivalent uniform dose formulation and the Poisson statistics-based tumor control probability model

AU - Zhou, Su Min

AU - Das, Shiva

AU - Wang, Zhiheng

AU - Marks, Lawrence B.

PY - 2004/9

Y1 - 2004/9

N2 - The generalized equivalent uniform dose (GEUD) model uses a power-law formalism, where the outcome is related to the dose via a power law. We herein investigate the mathematical compatibility between this GEUD model and the Poisson statistics based tumor control probability (TCP) model. The GEUD and TCP formulations are combined and subjected to a compatibility constraint equation. This compatibility constraint equates tumor control probability from the original heterogeneous target dose distribution to that from the homogeneous dose from the GEUD formalism. It is shown that this constraint equation possesses a unique, analytical closed-form solution which relates radiation dose to the tumor cell survival fraction. It is further demonstrated that, when there is no positive threshold or finite critical dose in the tumor response to radiation, this relationship is not bounded within the realistic cell survival limits of 0%-100%. Thus, the GEUD and TCP formalisms are, in general, mathematically inconsistent. However, when a threshold dose or finite critical dose exists in the tumor response to radiation, there is a unique mathematical solution for the tumor cell survival fraction that allows the GEUD and TCP formalisms to coexist, provided that all portions of the tumor are confined within certain specific dose ranges.

AB - The generalized equivalent uniform dose (GEUD) model uses a power-law formalism, where the outcome is related to the dose via a power law. We herein investigate the mathematical compatibility between this GEUD model and the Poisson statistics based tumor control probability (TCP) model. The GEUD and TCP formulations are combined and subjected to a compatibility constraint equation. This compatibility constraint equates tumor control probability from the original heterogeneous target dose distribution to that from the homogeneous dose from the GEUD formalism. It is shown that this constraint equation possesses a unique, analytical closed-form solution which relates radiation dose to the tumor cell survival fraction. It is further demonstrated that, when there is no positive threshold or finite critical dose in the tumor response to radiation, this relationship is not bounded within the realistic cell survival limits of 0%-100%. Thus, the GEUD and TCP formalisms are, in general, mathematically inconsistent. However, when a threshold dose or finite critical dose exists in the tumor response to radiation, there is a unique mathematical solution for the tumor cell survival fraction that allows the GEUD and TCP formalisms to coexist, provided that all portions of the tumor are confined within certain specific dose ranges.

KW - Cell survival fraction

KW - EUD

KW - TCP

UR - http://www.scopus.com/inward/record.url?scp=4644260418&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4644260418&partnerID=8YFLogxK

U2 - 10.1118/1.1783532

DO - 10.1118/1.1783532

M3 - Article

C2 - 15487743

AN - SCOPUS:4644260418

VL - 31

SP - 2606

EP - 2609

JO - Medical Physics

JF - Medical Physics

SN - 0094-2405

IS - 9

ER -