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An Improved Approximation to the Exact Solution of the Two‐Stage Clonal Growth Model of Cancer

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  • Harvey J. Clewell III
  • Dennis W. Quinn
  • Melvin E. Andersen
  • Rory B. Conolly

Abstract

Multistage clonal growth models are of interest for cancer risk assessment because they can explicitly incorporate data on cell replication. Both approximate and exact formulations of the two stage growth model have been described. The exact solution considers the conditional probability of tumors arising in previously tumor‐free animals; the approximate solution estimates total probability of tumor formation. The exact solution is much more computationally intensive when time‐dependent cell growth parameters are included. The approximate solution deviates from the exact solution at high incidences and probabilities of tumor. This report describes a computationally tractable,‘improved approximation’to the exact solution. Our improved approximation includes a correction term to adjust the unconditional expectation of intermediate cells based on the time history of formation of intermediate cells by mutation of normal cells (recruitment) or by cell division in the intermediate cell population (expansion). The improved approximation provided a much better match to the exact solution than the approximate solution for a wide range of parameter values. The correction term also appears to provide insight into the biological factors that contribute to the variance of the expectation for the number of intermediate cells over time.

Suggested Citation

  • Harvey J. Clewell III & Dennis W. Quinn & Melvin E. Andersen & Rory B. Conolly, 1995. "An Improved Approximation to the Exact Solution of the Two‐Stage Clonal Growth Model of Cancer," Risk Analysis, John Wiley & Sons, vol. 15(4), pages 467-473, August.
    • Handle: RePEc:wly:riskan:v:15:y:1995:i:4:p:467-473
    DOI: 10.1111/j.1539-6924.1995.tb00339.x
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    References listed on IDEAS

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    1. Dennis W. Quinn, 1989. "Calculating the Hazard Function and Probability of Tumor for Cancer Risk Assessment When the Parameters Are Time‐Dependent," Risk Analysis, John Wiley & Sons, vol. 9(3), pages 407-413, September.
    2. Todd W. Thorslund & Charles C. Brown & Gail Charnley, 1987. "Biologically Motivated Cancer Risk Models," Risk Analysis, John Wiley & Sons, vol. 7(1), pages 109-119, March.
    3. Annette Kopp‐Schneider & Christopher J. Portier & Claire D. Sherman, 1994. "The Exact Formula for Tumor Incidence in the Two‐Stage Model," Risk Analysis, John Wiley & Sons, vol. 14(6), pages 1079-1080, December.
    4. Suresh H. Moolgavkar & Georg Luebeck, 1990. "Two‐Event Model for Carcinogenesis: Biological, Mathematical, and Statistical Considerations," Risk Analysis, John Wiley & Sons, vol. 10(2), pages 323-341, June.
    5. Qi Zheng, 1994. "On the Exact Hazard and Survival Functions of the MVK Stochastic Carcinogenesis Model," Risk Analysis, John Wiley & Sons, vol. 14(6), pages 1081-1084, December.
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