Gompertz growth equations

The patterns of growth shown in Hg. 15.1 can be described by what are known as Gompertz growth equations, which have the form ... 

Because this class of functions has been used extensively to model cell growth kinetics, these functions can be modified to describe subpopulations of cells. The equations below describe a Gompertz growth model that has been modified to allow cells to oscillate between a therapeutically sensitive state (Rs) and a resistant state (Rr). The same modification can also be made to the simple growth model (28). 

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 ... 

The Gompertz equation is often used for description of tumors growth kinetics ... 

For example for Walker carcinosarcoma the data on growth of tumor volume V are well described by Gompertz equation ... 

The observation that the specific growth rate of many normal tissues and tumors decays approximately exponentially with time led to the extensive use of the Gompertz equation (Gompertz, 1825) ... 

The growth of pellets is not always exponential because of mass transfer limitations (Metz and Kossen, 1977 Pirt, 1975 Whitaker and Long, 1973). Increasing the size of pellets results in a significant decrease in reaction rate, and the apparent Kg value increases (e.g., Kobayashi et al., 1973). To describe the growth of pellets, one can use either the exponential law or the Monod relationship. However, the logistic equation (Kendall, 1949) in the form of Equ. 5.74 or the Gompertz equation (Chiu and Zajic, 1976)... 

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.

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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