Dynamics of Tumor Growth

Abstract

IT is commonly believed that tumor growth under ideal conditions is a simple exponential process terminated by the exhaustion of the nutritional support provided by the host. However, a survey of the literature shows that exponential growth of tumors has been observed only rarely and then only for relatively brief periods. When we consider those tumors whose growth has been followed over a sufficiently extensive range (100 to 1000-fold range ofgrowth or more), we find that nearly all such tumors grow more and more slowly as the tumor gets larger, with no appreciable period ofgrowth at a constant specific growth rate as would be expected for simple exponential growth. This continuous deceleration of growth has the consequence in many cases that the diameter (if a solid tumor) or the cube root of total cell number (if an ascites tumor) when plotted against time gives a close approximation to a straight line (Mayneord, 1932; Schrek, 1935; Klein and Revesz, 1953; Patt and Blackford, 1954). Mayneord (1932) has shown that cube root growth could be readily explained in mathematical terms if the active growth of a solid tumor were limited to a thin layer of cells at the surface of the tumor. However, in practice most solid tumors do not grow only at the surface, and in the case of ascites tumors it has been possible to Jabel the DNA of nearly 100 per cent of the tumor cells (Baserga, Kisieleski, and Halvorsen, 1960), indicating that almost all of these cells are viable and proliferating. Hence, although cube root growth has been empirically established for many tumors, it is difficult to relate it mathematically to proliferation of tumor cells. The present study offers a model of tumor cell proliferation that would account for the observed course of tumor growth. Furthermore, it will be shown that there is a distinct difference between the continuous and regular slowing characteristic of tumor growth and the more abrupt cessation of exponential growth observed when bacterial cultures outgrow their nutrient supply. Finally, implications of this new interpretation of tumor growth will be discussed in relation to concepts of hosttumor interaction.