A dynamical model for the growth and size distribution of multiple metastatic tumors

Metastasis is the spread of tumors culminating in the establishment of one or more secondary tumors at remote sites. In deciding the best treatment for cancer therapy, estimations of the colony size of metastatic tumors and predictions of the future spread of colonies are needed. A dynamical model for the colony size distribution of multiple metastatic tumors is presented here. The dynamics is described by equations that incorporate both the colonization by metastasis and the growth of each colony. When the colony growth is subject to the Gompertz function, the explicit solution obtained tends to an asymptotic stable distribution that shows a monotonically decreasing or U-shaped pattern according to the values of clinically significant parameters, such as the colonization coefficient and the fractal dimension of blood vessels. This predicted colony size distribution agrees well with successive data of a clinically observed size distribution of multiple metastatic tumors of liver cancer. The combined analysis of the theoretical colony size distribution and clinical data will give useful information on the diagnosis and the therapy for cancer patients.