The circulation stage of the metastatic cascade: A mathematical description and its clinical implications

Abstract

Metastatic cascade is a multi-stage process that starts with separation of a cancer cell from the primary tumor and ends with the emergence of a detectable metastasis. In the process the initiator cancer cell enters the circulatory system (intravasates), flows with the blood, and exits the circulation (extravasates) into an organ or tissue. The time period between intravasation and extravasation constitutes the circulation stage of the metastatic cascade. This stage is unique in that it lends itself naturally to various non-invasive observations and measurements in an individual cancer patient. This creates an opportunity for gaining insight into metastasis, its mathematical modeling, and designing diagnostic/prognostic tools and new cancer therapies. Although mechanisms of intravasation, survival and extravasation of circulating tumor cells (CTCs) are very complex and largely unknown, mathematical modeling of the circulation stage of the metastatic cascade is facilitated by two inter-related factors: a relative simplicity of the circulatory network and the cyclic nature of blood flow. The article presents a single-subject stochastic model of CTC dynamics that leads to simple formulas, applicable to any homogeneous CTC population, for organ-specific extravasation probabilities, the distribution and expected value of the number, X, of circulation cycles completed by a CTC prior to extravasation, and the average circulation time. In particular, we found that the distribution of random variable X is geometric G(x), where parameter x is measurable, at least in principle, in an individual subject. We also discuss implications of our results for cancer research and treatment.

Keywords: Circulating tumor cell; Circulation time; Extravasation probability; Geometric distribution; Half-life; Intravasation.