A BIOLOGICAL-PROCESS-BASED MODELING APPROACH FOR ESTIMATING THE MEAN SOJOURN TIME OF INVASIVE BREAST TUMORS

Purpose: Mammography aims to detect breast tumors (BTs) prior to becoming clinically symptomatic. Sojourn time (ST), the length of time when cancer is screen-detectable, is an important measure for determining the optimal screening interval. Several authors have estimated mean sojourn times (MSTs) by fitting parametric distributions to screening trial data. These estimates vary considerably across studies. Our study introduces a novel method for estimating MST by using a stochastic tumor growth model.

Method(s):  We adopted a biological-process-based modeling approach to estimate the MST by using a birth-death process (BDP) of BT cells in an oblate spherical tumor. The forward Kolmogorov equation (FKE) for the BDP was formulated and solved analytically.  The solution to the FKE gives the probability density function (pdf) of the number of tumor cells at any given time following tumor initiation. By defining two threshold sizes of 10^4 and 10^7 for screen-detected and clinically-detected tumors, respectively, we estimated the pdf for ST. We derived a simple analytical expression for calculating MST by using the estimated pdf. We also solved the FKE by using the tau-leap simulation method to validate the results from the analytical method. A tumor doubling time of 130 days was used to parameterize the BDP, based on a literature review.

Result(s): The analytical and tau-leap simulation methods yielded MST estimates of 734 and 743 days, respectively. Our method gave an estimate comparable to the lower MST estimate of 767 days reported by a study using data from the Swedish two-county study and to the result from a simulation-based estimation method using piecewise pdf for the ST and data from the HIP trial (730 days). Our estimate was significantly lower than those based on the Nijmegen trial data (1131 days). In contrast, our estimate was higher compared to the MST estimates from a statistical model using exponential pdf for the ST to fit the HIP trial data (621 days). 

Conclusion(s):   The incorporation of a simple biological process to estimate MST may be valuable for reducing the uncertainty in the estimates based on parametric assumptions. Moreover, the modeling approach exemplifies the potential linkage between modeling at the cellular level and patient or clinical intervention level.