The time-evolution of DCIS size distributions with applications to breast cancer growth and progression

TL;DR

This paper models the evolution of DCIS size distributions over time, deriving a differential equation, and uses data to estimate growth parameters, suggesting DCIS grows roughly as the square root of time.

Contribution

It introduces a differential equation model for DCIS size evolution and estimates growth parameters from empirical data, linking mathematical modeling with breast cancer progression.

Findings

Stationary distribution of DCIS sizes exists and is unique.

DCIS growth follows a square-root law with respect to time.

Estimated growth exponent p is approximately 0.50.

Abstract

Ductal carcinoma {\em in situ} (DCIS) lesions are non-invasive tumours of the breast which are thought to precede most invasive breast cancers (IBC). 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. It is therefore reasonable to assume that the stationary distribution is equal to the true DCIS size distribution, at least for 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 program 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.