FORMATION OF NECROTIC CORES IN THE GROWTH OF TUMORS: ANALYTIC RESULTS*

In this article, the author studies the mechanism of formation of necrotic cores in the growth of tumors by using rigorous analysis of a mathematical model. The model modifies a corresponding tumor growth model proposed by Byrne and Chaplain in 1996, in the case where no inhibitors exist. The modification is made such that both necrotic tumors and nonnecrotic tumors can be considered in a joint way. It is proved that if the nutrient supply is below a threshold value, then there is not dormant tumor, and all evolutionary tumors will finally vanish. If instead the nutrient supply is above this threshold value then there is a unique dormant tumor which can either be necrotic or nonnecrotic, depending on the level of the nutrient supply and the level of dead-cell dissolution rate, and all evolutionary tumors will converge to this dormant tumor. It is also proved that, in the second case, if the dormant tumor is necrotic then an evolutionary tumor will form a necrotic core at a finite time, and if the dormant tumor is nonnecrotic then an evolutionary tumor will also be nonnecrotic from a finite time.