## Relationship between the generalized equivalent uniform dose formulation and the Poisson statistics-based tumor control probability model.

Publication ,  Journal Article
Zhou, S-M; Das, S; Wang, Z; Marks, LB
Published in: Med Phys
September 2004

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.

Med Phys

0094-2405

September 2004

31

9

2606 / 2609

## Location

United States

• Treatment Outcome
• Survival Analysis
• Relative Biological Effectiveness
• Poisson Distribution
• Nuclear Medicine & Medical Imaging
• Neoplasms
• Models, Statistical

### Citation

APA
Chicago
ICMJE
MLA
NLM
Zhou, S.-M., Das, S., Wang, Z., & Marks, L. B. (2004). Relationship between the generalized equivalent uniform dose formulation and the Poisson statistics-based tumor control probability model. Med Phys, 31(9), 2606–2609. https://doi.org/10.1118/1.1783532
Zhou, Su-Min, Shiva Das, Zhiheng Wang, and Lawrence B. Marks. “Relationship between the generalized equivalent uniform dose formulation and the Poisson statistics-based tumor control probability model.Med Phys 31, no. 9 (September 2004): 2606–9. https://doi.org/10.1118/1.1783532.
Zhou, Su-Min, et al. “Relationship between the generalized equivalent uniform dose formulation and the Poisson statistics-based tumor control probability model.Med Phys, vol. 31, no. 9, Sept. 2004, pp. 2606–09. Pubmed, doi:10.1118/1.1783532.

Med Phys

0094-2405

September 2004

31

9

2606 / 2609

United States