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 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

The growth rate of human tumours.

“…Whilst in the published literature on the growth of human tumours extrapolation has always been by an assumed exponential curve, the availability of more complete data on experimental tumours enables one in principle to make a more sophisticated extrapolation. This has been done by Laird (1964Laird ( , 1965 The Contribution from Studie8 of Cell Proliferation in Tumours Many of the problems in our understanding of the overall growth of tumours arise out of ignorance of the basic characteristics ofthe proliferating cell population, despite the fact that techniques of investigation of cell population kinetics have been developing rapidly over the past ten years (Wimber, 1963). The application of these techniques to experimental tumours has been shown to be feasible and data are now available on a variety of tumour types (Mendelsohn, Dohan and Moore, 1960;Mendelsohn, 1960Mendelsohn, , 1962Bertalanffy and Lau, 1962;Bertalanffy, 1963;Edwards et al, 1960).…”

Dynamics of Tumour Growth: Comparison of Growth Rates and Extrapolation of Growth Curve to One Cell

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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.

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Dose-response curves for agents that impair cell reproductive integrity. The relation between dose-response curves and the design of selective regimens in cancer chemotherapy.

“…The second case is that of a population growing according to the Gompertz equation, its growth rate decreasing continuously from the start until the population reaches a plateau. This case is typified by a large number of solid and ascitic tumours (Laird 1964(Laird , 1965. The third case is that of a population maintained at a steady-state level by homoeostasis.…”

“…This principle is illustrated in Fig. 1 Laird (1964Laird ( , 1965 This principle is illustrated in Fig. 2 where it is shown that cells growing at the fastest rate permitted by equation (6) (7), (9) and (10) describe the fates of the three types of population with reference to the fractional reduction F caused by each dose and the interval between doses t. In order to determine the effects of particular regimens we now substitute for F in these equations the expressions in equations (1), (2) Table I gives the equations so derived.…”

“…The value of k will be taken as 0 5 as in the previous example. There are no adequate available data from which to derive the parameters of the dose-response curves for solid tumours; we shall here assume that the particular tumour cells in question are highly sensitive and give an exponential dose-response curve with a shoulder when treated with the agent, the value of ac being 0.7 and of fi, 5. Typical values for the Gompertz equation parameters a and b are 0-2 and 0-02 (Laird, 1964 (12) and (13) , has a simple exponential form (Bruce, Meeker and Valeriote, 1966), so equation G in Table I is used; ax = 0-63, k = 0-5 day-'. It is assumed that the dose-response curve for tumour cells (population A) has a shoulder and that the relevant values (equation E, Table I) are a = 0-7, # = 5, a = 0-2, b = 0-02.…”