Gompertz equation

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 ... 

Recently, Foubert et al. (43) developed a new, empirical model (Foubert model) to predict the kinetics of fat crystallization. Other authors have used a reparameterized Gompertz equation (Gompertz model) to empirically describe crystallization kinetics of fats (44, 45). 

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 experimental curves/j(/) were fitted to the numerical expression given by a reparametrized Gompertz equation ... 

No convenient modifications of the basic logistic and Gompertz equations will yield a linear form, so it was necessary to resort to the relatively new technique of nonlinear least-squares regression to fit these curves to the data. We fitted the Gompertz equation in its usual form ... 

In the case of the Gompertz equation, the exact computations employed here can be found in II, Harnett, 1975, 448-451 for the logistic equation, the computations used are from II, Olinick, 1978,62-63. 

FIGURE 11.17 Symmetrical and asymmetrical dose-response curves, (a) Symmetrical Hill equation with n = 1 and EC5o = 1.0. Filled circle indicates the EC50 (where the abscissa yields a half maximal value for the ordinate). Below this curve is the second derivative of the function (slope). The zero ordinate of this curve indicates the point at which the slope is zero (inflection point of the curve). It can be seen that the true EC50 and the inflection match for a symmetrical curve, (b) Asymmetrical curve (Gompertz function with m = 0.55 and EC50= 1.9). The true EC50 is 1.9, while the point of inflection is 0.36. 

It was found that is a function of temperature but the model was found to give a better fit than analytical expressions like the Avrami model or the modified Gompertz model (Kloek, Walstra and Van Vliet 2000). The main advantage of this model is that as it is formulated as a differential equation, it can be used to predict isothermal as well as dynamic crystallization. However, this model does not consider the polymorphism of the material which is a critical point in the crystallization of cocoa butter. Another contribution is the model of Fessas et al. (Fessas, Signorelli and Schiraldi 2005) which considers all the transitions possible between each... 

FIGURE 12.18 Asymmetrical dose-response curves. (A) Dose-response data fit to a symmetrical Hill equation with n = 0.65 and EC50 = 2.2 (solid line), or n = 1 and EC50 = 2 (dotted line). It can be seen that neither symmetrical curve fits the data adequately (B) Data fit to the Gompertz function with m = 0.55 and EC50 = 1.9. 

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). 

The exponential integral Ei(x) is defined by Jahnke and Emde. If < = 0 taken to be a time shortly after birth, but long enough after birth to exclude infant deaths (which are omitted in Gompertz treatment), then mean life span from birth to death. If then T/td is a constant for all animal species, we find from equations (8) and (9) that A is a constant, independent of species, given by the solution of the equation... 

From equations (18) and (19) we find or = 191 days in case the two highest points mentioned above are omitted (S = 222 days in case they are included). Comparing the value 191 days with the value a = 161 days based on the Gompertz death-rate curve (670-day life span) and the value = 258 days inferred from Dr. Fiokers data, we see that the theoretical curve fits the experimental values about as well as would be expected from the Gompertz curve, and somewhat better than would have been expected on the basis of Dr. Finkel s acceptance region. 

It is interesting to compare the above analysis with an alternative one based on the approach developed by Jones, in which the effect of a given exposure of an animal to radiation is regarded as equivalent to an increase in physiological age of the animal by an amount proportional to the amount of radiation received. In terms of the Gompertz formulation of the natural death rate, this results in the case we are considering to the addition of a linear term in < to equation (6) ... 

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

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 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. 

Example 3.1 For the example in Sect. 3.3, customer demand dynamic equation model and B. Gong Spautz (B. Gompertz) model are used to forecast 2009 customer demand, forecast results are ... 

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. 

Estimated parameters from nonlinear least-squares regression analyses (Gompertz and logistic equations)... 

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