Bulletin of Mathematical Biology

, Volume 80, Issue 3, pp 670–686 | Cite as

Asymptotic Relative Risk Results from a Simplified Armitage and Doll Model of Carcinogenesis

Original Article


We examine basic asymptotic properties of relative risk for two families of generalized Erlang processes (where each one is based off of a simplified Armitage and Doll multistage model) in order to predict relative risk data from cancer. The main theorems that we are able to prove are all corroborated by large clinical studies involving relative risk for former smokers and transplant recipients. We then show that at least some of these theorems do not extend to other Armitage and Doll multistage models. We conclude with suggestions for lifelong increased cancer screening for both former smoker and transplant recipient subpopulations of individuals and possible future directions of research.


Relative risk Multistage model Erlang distribution Cancer Smoking Transplant 


  1. Armitage P (1985) Multistage models of carcinogenesis. Environ Health Perspect 63:195–201CrossRefGoogle Scholar
  2. Armitage P, Doll R (1954) The age distribution of cancer and a multi-stage theory of carcinogenesis. Br J Cancer 8:1–12CrossRefGoogle Scholar
  3. Baker SG (2012) Paradoxes in carcinogenesis should spur new avenues of research: an historical perspective. Disrupt Sci Technol 1(2):100–107CrossRefGoogle Scholar
  4. Cannataro Vincent L, McKinley Scott A, St Mary CM (2016) The implications of small stem cell niche sizes and the distribution of fitness effects of new mutations in aging and tumorigenesis. Evolut Appl.  https://doi.org/10.1111/eva.12361 Google Scholar
  5. Corthay A (2014) Does the immune system naturally protect against cancer? Front Immunol.  https://doi.org/10.3389/fimmu.2014.00197 Google Scholar
  6. Ebbert J, Yang P, Vachon C, Vierkant R, Cerhan J, Folsom A, Sellers T (2003) Lung cancer risk reduction after smoking cessation: observations from a prospective cohort of women. J Clin Oncol 21:921–926CrossRefGoogle Scholar
  7. Gsteiger S, Morgenthaler S (2008) Heterogeneity in multistage carcinogenesis and mixture modeling. Theor Biol Med Model 5(13):1–12Google Scholar
  8. Harding C, Pompei F, Lee E, Wilson R (2008) Cancer suppression at old age. Cancer Res 68(11):4465–4478CrossRefGoogle Scholar
  9. Hiller J, Vallejo C, Betthauser L, Keesling J (2017) Characteristic patterns of cancer incidence: epidemiological data, biological theories, and multistage models. Prog Biophys Mol Biol 124:41–48CrossRefGoogle Scholar
  10. Kirschner D, Panetta JC (1998) Modeling immunotherapy of the tumor immune interaction. J Math Biol 37:235–252CrossRefMATHGoogle Scholar
  11. Knudson AG Jr (1971) Mutation and cancer: statistical study of retinoblastoma. PNAS 68(4):820–823CrossRefGoogle Scholar
  12. Moolgavkar S, Venzon D (1979) Two-event models for carcinogenesis: incidence curves for childhood and adult tumors. Math Biosci 47:55–77CrossRefMATHGoogle Scholar
  13. Moolgavkar SH (2004) Commentary: fifty years of the multistage model: remarks on a landmark paper. Int J Epidemiol 33(6):1182–1183CrossRefGoogle Scholar
  14. Nording CO (1951) A new theory on the cancer-inducing mechanism. Br J Cancer 7(1):68–72CrossRefGoogle Scholar
  15. Pompei F, Wilson R (2001) Age distribution of cancer: the incidence turnover at old age. Human Ecol Risk Assess 7(6):1619–1650CrossRefGoogle Scholar
  16. Zhenyi S, Yang Z, Yongqing X, Chen Y, Qiang Y (2015) Apoptosis, autophagy, necroptosis, and cancer metastasis. Mol Cancer 14(48):1–14Google Scholar
  17. Wilkie KP (2013) A review of mathematical models of cancer-immune interactions in the context of tumor dormancy. Adv Exp Med Biol 734:201–234CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceAdelphi UniversityGarden CityUSA
  2. 2.Department of MathematicsUniversity of FloridaGainesvilleUSA

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