title: A simple model of breast carcinoma growth may provide explanations for observations of apparently complex phenomena.
abstract: BACKGROUND
  There has been great debate regarding the importance of ductal carcinoma in situ (DCIS) in the breast. Autopsy results that demonstrate a much greater number of these lesions compared with the number of invasive carcinomas, and the numbers of deaths from breast carcinoma each year have been cited as evidence that DCIS rarely leads to invasion and death. These analyses have overlooked the fact that, to sustain a rate of detection each year, there would have to be a reservoir of undetected breast carcinomas growing in the population. The authors developed a simple model that makes this clear. In addition, complex phenomena have been suggested to explain why invasive breast disease may grow more rapidly among very young women and more slowly among the very old. A simple model provides some insight that may simplify the explanation of these observations.
  
  
  METHODS
  The simple model of breast carcinoma growth assumes that there are three types of breast carcinoma that begin each year in a cohort of women. It assumes that all breast carcinomas begin as DCIS and take 9 years to go from a single cell to an invasive lesion for the slowest growing lesions, 6 years for intermediate growing DCIS lesions, and 3 years for fast-growing DCIS lesions. Furthermore, once an invasive clone forms, the model assumes that it will double in 60 days for fast-growing lesions, 120 days for intermediate growing lesions, and 180 days for slow-growing lesions. Three new tumors begin to grow in each successive year (one of each type). The model uses simple vectors that are defined by the size of the tumors and the time since tumor initiation, and it assumes that all tumors are detected when they reach 2 cm in greatest dimension. The model can be used to show graphically how many undetected tumors (DCIS as well as invasive carcinomas) there may be in the population to sustain the detection of three invasive tumors each year.
  
  
  RESULTS
  Using the assumptions described above, the model showed that, by the time the first slow-growing breast carcinoma reaches 2 cm in greatest dimension, there will be 29 other slow-growing tumors that have not reached that size (9 DCIS and 20 smaller invasive carcinomas), 19 moderately growing tumors (6 DCIS and 13 smaller invasive carcinomas), and 9 fast-growing tumors (3 DCIS and 6 smaller invasive carcinomas). This means that, for every three breast tumors that reach 2 cm, the model predicted that there would be another 57 tumors (39 smaller invasive carcinomas and 18 DCIS) that would be undetected "below the surface". The model showed clearly that faster growing tumors would be expected to predominate among the youngest women, because they are the first to "reach the surface"; and, if the number of newly initiated tumors decreases with age, then there will be more of the slowest growing tumors that are left to reach the surface among the oldest women in the population.
  
  
  CONCLUSIONS
  Even if the authors' assumptions are incorrect, their model made it clear that, to diagnose several breast carcinomas per 1000 women each year means that there have to be many more undetected carcinomas in the population to sustain the rate of detection. Although the model did not prove that DCIS may become potentially invasive and lethal, it did demonstrate that, even if all of these in situ lesions become invasive and lethal, many more DCIS lesions would have to be expected in the population than the number of invasive carcinomas detected each year and the number of deaths from breast carcinoma each year. Furthermore, the model provided a simple, purely mechanical illustration that may explain the preponderance of faster growing breast carcinomas among very young women and the preponderance of slower growing tumors among elderly women.
authors.affiliation: Department of Radiology, Massachusetts General Hospital and Harvard Medical School, Boston, Massachusetts, USA. kopans.daniel@mgh.harvard.edu
authors.forename: Daniel B
authors.initials: DB
authors.lastname: Kopans
authors.forename: Elizabeth
authors.initials: E
authors.lastname: Rafferty
authors.forename: Diane
authors.initials: D
authors.lastname: Georgian-Smith
authors.forename: Eren
authors.initials: E
authors.lastname: Yeh
authors.forename: Helen
authors.initials: H
authors.lastname: D'Alessandro
authors.forename: Richard
authors.initials: R
authors.lastname: Moore
authors.forename: Kevin
authors.initials: K
authors.lastname: Hughes
authors.forename: Elkan
authors.initials: E
authors.lastname: Halpern
mesh_terms: D000367:Age Factors; D001943:Breast Neoplasms / Q000175:diagnosis / Q000473:pathology*; D002285:Carcinoma, Intraductal, Noninfiltrating / Q000175:diagnosis / Q000473:pathology*; D002455:Cell Division; D005260:Female; D006801:Humans; D008433:Mathematics; D008954:Models, Biological; D013997:Time Factors