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 of growth or more), we find that nearly all such tumors grow more and more slowly as the tumor gets larger, with no appreciable period of growth 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. Fig. 1 and 2 show, respectively, a semi-log and a cube root plot of the growth of the Ehrlich ascites tumor, from the data of Klein and Revesz (1953). If the ascites cells had multiplied exponentially, the experimental points should fall on a straight line in Fig. 1; instead they describe a smooth convex curve, no part of which is linear for long enough to justify an interpretation of exponential growth. This fact indicates that the specific growth rate of such tumors decreases with time (Klein and Revesz, 1953), i.e., that the second derivative of the growth function is negative, in contrast to semi-log growth in which the specific growth rate remains constant. ANALYSIS OF TUMOR GROWTH

RECENTLY we have shown (Laird, 1964) that the growth of a variety of tumors of the mouse, rat and rabbit, whether transplanted or primary, is well described by a Gompertzian equation. Such growth may be regarded as an exponential process limited by an exponential retardation, and tumor growth was therefore interpreted as being due mainly, if not entirely, to an exponential proliferation of tumor cells whose successive mean generation times increase according to an exponential equation.For the present study, corresponding points on the growth curves of different tumors have been defined, and the growth rates of the tumors compared at these points. In addition, the growth curves have been extrapolated back to a tumor size of one cell, and the time at which the tumor would have existed as a single cell, the initial rate of tumor growth, and the range of tumor growth from a single cell to the theoretical limiting size have been computed for each tumor. From this analysis several constant relations have emerged, which provide further insight into the general nature of tumor growth. ANALYSIS OF TUMOR GROWTHFor the present study the same tumor data were used as in the first paper of this series (Laird, 1964). The tumors are 19 examples of 12 different tumors of the mouse, rat, and rabbit. The following analysis is based on the same Gompertzian growth equation* as used in the original paper:W -Woe(la)( 1-e-at) (1) where WO is the tumor size at time zero, W is the tumor size at time t, and A and a are constants. Fig. 1 shows the computed Gompertz curve fitted to the E14 tumor, high dose. This example illustrates several pertinent properties of the Gompertz curve : In its early stages the curve is concave upward, it then passes through an inflection point which occurs at about 37 % of the final limiting tumor size, and the curve is then concave downward as it approaches the asymptote. Because of the mathematical nature of the Gompertz function, any Gompertz curve can be considered to begin as a simple exponential process which then is retarded exponentially as * The special form of the Gompertz function used here was developed several years ago by S. A.Tyler of Argonne National Laboratory, for representing our model of exponential growth retarded by an exponential decay of the specific growth rate.

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