Growth curves Gompertz

Figure 3.2 Example of Gompertz growth curve for parameter value 6 = (0.4, 0.04, 0.3). ...

Figure 3.3 Approximations of the Gompertz growth curve based on Taylor expansion for the internal exponential term.

Gompertz growth curves can be used to represent the tendency of many products and industries to grow at a declining rate as they reach maturity. 

If we use the Gompertz growth curve, it s easy to get F2009 = 9.948 hundred... 

Antineoplastic Agents. Figure 2 Growth curve of tumor cells according to Gompertz. 

The different curves obtained by increasing the number of terms in the Taylor expansion are represented in Figure 3.3 on top of the Gompertz curve itself. The exponential growth model can thus be now justified not only because it fits well the data but also because it can be seen as a first approximation to the Gompertz growth model, which is endowed with a mechanistic interpretation, namely, competition between the catabolic and anabolic processes. 

The study of tumor growth forms the foundation for many of the basic principles of modem cancer chemotherapy. The growth of most tumors is illustrated by the Gompertzian tumor growth curve (Fig. 124-5). Gompertz was a German insurance actuary who described the relationship between age and expected death. This mathematical model also approximates tumor-cell proliferation. In the... 

Buchanan, R. L., Whiting, R. C., Damert, W. C. (1997). When is simple good enough a comparison of the Gompertz, barany, and three-phase linear models for fitting bacterial growth curves. Food Microbiology, 14, 313-326. 

According to properties of the function Y = Y(t), Eq. (3.5) has properties of common growth curves. Hence, we attempt to use value K obtained by Gompertz model approximation as the value K for Eq. (3.4). In order to get Gompertz model parameters, methods like Sanwa method, reciprocal summation method can be used to solve the needed parameters. The introduction of Sanwa method is that the entire time series is divided into intervals which are equal to each other, and the logarithms sum of three observation values is used to calculate the parameters. 

The calculated value of (log Y, — log F, i)/(log F, i log F, 2) varies between [0.085941, 1.0444288], approximately tending to constant 0.86148976 shown in Table 3.1. This shows the raw data have a growth curve characteristics. Gompertz curve can be used for approximate calculation of dynamic equation (3.4) parameter K. The following Sanwa method is used to estimate value of K. 

Figure 1.1 Simulation of (a) logistic (symmetrical) and (b) Gompertz (asymmetrical) growth curves. See equations (1.2.5) and (1.2.6), respectively.

Winsor, C. P. (1932). The Gompertz curve as a growth curve. Proceedings of the National Academy of Sciences, USA, 18,1-8. 

Gompertz is a United Kingdom statistician and mathematician who discovered a type of curve used to predict the population growth in 1820. American scholar R. Prescott firstly applied this curve to forecasting market development in 1922 [3, 5, 6]. Here is Gompertz curve equation in general form ... 

For the record, there are constancies in these equations. The Gompertz curve implies a constant rate of decline in the increments to the logarithm of the observed value. Logistic growth connotes a constant rate of dechne in the decrements to the reciprocal of the observed value. These features become evident when the equations are converted to the form of a modified exponential curve. 

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