title: The time-evolution of DCIS size distributions with applications to breast cancer growth and progression.
abstract: Ductal carcinoma in situ (DCIS) lesions are non-invasive tumours of the breast that are thought to precede most invasive breast cancers (IBCs). As individual DCIS lesions are initiated, grow and invade (i.e. become IBC), the size distribution of the DCIS lesions present in a given human population will evolve. We derive a differential equation governing this evolution and show, for given assumptions about growth and invasion, that there is a unique distribution which does not vary with time. Further, we show that any initial distribution converges to this stationary distribution exponentially quickly. Therefore, it is reasonable to assume that the stationary distribution governs the size of DCIS lesions in human populations which are relatively stable with respect to the determinants of breast cancer. Based on this assumption and the size data of 110 DCIS lesions detected in a mammographic screening programme between 1993 and 2000, we produce maximum likelihood estimates for certain growth and invasion parameters. Assuming that DCIS size is proportional to a positive power p of the time since tumour initiation, we estimate p to be 0.50 with a 95% confidence interval of (0.35, 0.71). Therefore, we estimate that DCIS lesions follow a square-root growth law and hence that they grow rapidly when small and relatively slowly when large. Our approach and results should be useful for other mathematical studies of cancer, especially those investigating biological mechanisms of invasion.
authors.affiliation: Melbourne School of Population and Global Health, University of Melbourne, Level 3/207 Bouverie St., Carlton 3053, Australia jdowty@unimelb.edu.au.
authors.forename: James G
authors.initials: JG
authors.lastname: Dowty
authors.affiliation: International Agency for Research on Cancer, 150 Cours Albert Thomas, 69372 Lyon cedex 08, France.
authors.forename: Graham B
authors.initials: GB
authors.lastname: Byrnes
authors.affiliation: Victorian Cytology Service, 752 Swanston Street, Carlton 3053, Australia and Melbourne School of Population and Global Health, University of Melbourne, Level 3/207 Bouverie St., Carlton 3053, Australia.
authors.forename: Dorota M
authors.initials: DM
authors.lastname: Gertig
mesh_terms: D000368:Aged; D001943:Breast Neoplasms / Q000000981:diagnostic imaging / Q000209:etiology* / Q000473:pathology*; D002285:Carcinoma, Intraductal, Noninfiltrating / Q000000981:diagnostic imaging / Q000209:etiology* / Q000473:pathology*; D016208:Databases, Factual; D018450:Disease Progression; D005260:Female; D006801:Humans; D016013:Likelihood Functions; D008327:Mammography / Q000706:statistics & numerical data; D055641:Mathematical Concepts; D008875:Middle Aged; D008954:Models, Biological; D009361:Neoplasm Invasiveness / Q000473:pathology
keywords: DCIS; breast cancer; cancer progression; maximum likelihood; population dynamics