On the relation between size of necrosis and diameter of tumor spheroids.

In many previous experimental studies on multicellular tumor spheroids, the spheroid diameter at which central necrosis develops has been determined to be twice the thickness of the viable cell rim measured at a later stage of spheroid growth. This procedure tacitly assumes that there is a linear relation between the diameter of necrosis and that of the whole spheroid over the entire range of emergence and growth of necrosis. However, some experimental investigations have demonstrated that necroses do not grow gradually with spheroid diameter, but show a rapid initial increase, once a few cells have died. The present article offers an explanation for this phenomenon, which is derived from basic diffusion theory. A theoretical relation between sizes of spheroids and of their central necroses is developed, which is based on the assumption that formation of necrosis is caused by depletion of substrates or accumulation of metabolic waste products. In a second part, the theoretical model is fitted to experimental data from the literature, and oxygen consumption rate as a function of spheroid size is determined. It turns out that the model closely mimics the experimentally observed behavior described above. These experimental results, therefore, do not furnish any evidence for assuming other hypotheses of necrosis formation. Resulting O2 consumption rates are well in the range of previously published data. In all cases, approximations to the measured data are better than the corresponding linear squares fits. At least in some tumor cell lines, depletion of substrates or accumulation of waste products can explain formation of necrosis without the assumption of any additional mechanisms. Moreover, the model presented in this article offers an alternative way of determining the turnover rate of a substrate or metabolic waste product provided that depletion/accumulation of this substance represents the cause for necrosis development.