Cancer cell population growth kinetics at low densities deviate from the exponential growth model and suggest an Allee effect
Authors: Johnson Kaitlyn E. , Howard Grant , Mo William , Strasser Michael K. , Lima Ernesto A. B. F. , Huang Sui , Brock Amy 
Published: March 2019

Models of cancer cell population expansion assume exponential growth kinetics at low cell densities, with deviations from exponential growth only at higher densities due to limited resources such as space and nutrients. However, recent pre-clinical and clinical observations of tumor initiation or recurrence indicate the presence of tumor growth kinetics in which growth rates scale with cell numbers. These observations are analogous to the cooperative behavior of species in an ecosystem described by the ecological principle of the Allee effect. In preclinical and clinical models however, tumor growth data is limited by the lower limit of detection (i.e. a measurable lesion) and confounding variables, such as tumor microenvironment and immune responses may cause and mask deviations from exponential growth models. In this work, we present alternative growth models to investigate the presence of an Allee effect in cancer cells seeded at low cell densities in a controlledin vitrosetting. We propose a stochastic modeling framework to consider the small number of cells in this low-density regime and use the moment approach for stochastic parameter estimation to calibrate the stochastic growth trajectories. We validate the framework on simulated data and apply this approach to longitudinal cell proliferation data of BT-474 luminal B breast cancer cells. We find that cell population growth kinetics are best described by a model structure that considers the Allee effect, in that the birth rate of tumor cells depends on cell number. This indicates a potentially critical role of cooperative behavior among tumor cells at low cell densities with relevance to early stage growth patterns of emerging tumors and relapse.

Author Summary

The growth kinetics of cancer cells at very low cell densities are of utmost clinical importance as the ability of a small number of newly transformed or surviving cells to grow exponentially and thus, to “take off” underlies tumor formation and relapse after treatment. Mathematical models of stochastic tumor cell growth typically assume a stochastic birth-death process of cells impacted by limited nutrients and space when cells reach high density, resulting in the widely accepted logistic growth model. Here we present an in-depth investigation of alternate growth models adopted from ecology to describe potential deviations from a simple cell autonomous birth-death model at low cell densities. We show that our stochastic modeling framework is robust and can be used to identify the underlying structure of stochastic growth trajectories from both simulated and experimental data taken from a controlledin vitrosetting in which we can capture data from the relevant low cell density regime. This work suggests that the assumption of cell autonomous proliferation via a constant exponential growth rate at low cell densities may not be appropriate for all cancer cell growth dynamics. Consideration of cooperative behavior amongst tumor cells in this regime is critical for elucidating strategies for controlling tumor cell growth.