Monte Carlo simulation-based approach to model the size distribution of metastatic tumors

The size distribution of metastatic tumors and its time evolution are traditionally described by integrodifferential equations and stochastic models. Here we develop a simple Monte Carlo approach in which each event of metastasis is treated as a chance event through random-number generation. We demonstrate the accuracy of this approach on a specific growth and metastasis model by showing that it quantitatively reproduces the size distribution and the total number of tumors as a function of time. The approach also yields statistical distribution of patient-to-patient variations, and has the flexibility to incorporate many real-life complexities.

Figure 1

Cumulative size distribution of tumors from our simulations (symbols) averaged over 200 patients for four different times of observation. The solid lines are results from Fig. 4 (upper) of IKS (Ref. [12]) corresponding to 432 days, 559 days, 632 days, and 732 days, respectively. (See text.)

Figure 2

Total numbers of tumors from our simulation averaged over 200 patients as a function of time, compared with the results from Iwata et al. (Ref. [12]).

Figure 3

A breakdown of the number of secondary and tertiary metastasized tumors as a function of time; (a) linear-linear plot; (b) log-log plot. The results displayed are from our numerical simulations averaged over 200 patients. Also shown are the number of corresponding large tumors of sizes >10${}^9$ cells (1 cm${}^3$ or larger).

Figure 4

Statistical distribution of the time at which the first metastasis of the primary and the secondary tumors occur. The results are from our simulations on 5000 patients. Solid curves are best fits using the skewed normal distribution (SND). Both curves are negatively skewed with skewness factors −0.13 and −0.21 for the primary and secondary curves, respectively. (See text and Table ).