title: Order-of-magnitude estimates of latency (time to appearance) and refill time of a cancer from a single cancer 'stem' cell compared by an exponential and a logistic equation.
abstract: OBJECTIVE
  The time required before a mass of cancer cells considered to have originated from a single malignantly transformed cancer 'stem' cell reaches a certain number has not been studied. Applications might include determination of the time the cell mass reaches a size that can be detected by X-rays or physical examination or modeling growth rates in vitro in order to compare with other models or established data.
  
  
  METHODS
  We employed a simple logarithmic equation and a common logistic equation incorporating 'feedback' for unknown variables of cell birth, growth, division, and death that can be used to model cell proliferation. It can be used in association with free or commercial statistical software.
  
  
  RESULTS
  Results with these two equations, varying the proliferation rate, nominally reduced by generational cell loss, are presented in two tables. The resulting equation, instructions, examples, and necessary mathematical software are available in the online appendix, where several parameters of interest can be modified by the reader www.uic.edu/nursing/publicationsupplements/tobillion_Anderson_Rubenstein_Guinan_Patel1.pdf.
  
  
  CONCLUSIONS
  Reducing the proliferation rate by whatever alterations employed, markedly increases the time to reach 10(9) cells originating from an initial progenitor. In thinking about multistep oncogenesis, it is useful to consider the profound effect that variations in the effective proliferation rate may have during cancer development. This can be approached with the proposed equation, which is easy to use and available to further peer fine-tuning to be used in future modeling of cell growth.
authors.affiliation: Division of Biology, Hektoen Institute, Chicago, IL 60612, USA. Kanderso427@sbcglobal.net
authors.forename: Ken M
authors.initials: KM
authors.lastname: Anderson
authors.forename: Marvin
authors.initials: M
authors.lastname: Rubenstein
authors.forename: Patrick
authors.initials: P
authors.lastname: Guinan
authors.forename: Minu
authors.initials: M
authors.lastname: Patel
mesh_terms: D000465:Algorithms; D002277:Carcinoma / Q000473:pathology; D002452:Cell Count; D002453:Cell Cycle; D002455:Cell Division; D002471:Cell Transformation, Neoplastic / Q000235:genetics; D003043:Cocarcinogenesis; D003198:Computer Simulation; D019337:Hematologic Neoplasms / Q000473:pathology; D006801:Humans; D016015:Logistic Models; D008954:Models, Biological; D009362:Neoplasm Metastasis; D014411:Neoplastic Stem Cells / Q000166:cytology*; D013997:Time Factors; D047368:Tumor Burden