Gompertz functions

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. 

The introduction of the p factor yields asymmetry. Finally, the Gompertz function can be used [5] ... 

Gleevec, 180 Gompertz function, 245 Goodness of fit, 236, 236f gpl20, 44, 53 G-protein(s)... 

Another series of functions frequently used to describe growth kinetics are the Gompertz functions... 

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. 

In some cases, therefore, the linear model can be a reasonable approximation for disease progression when the disease is observed over a relatively short period of time or if the number of individuals experiencing complete recovery is limited. For example, the trajectory of the United Parkinson s Disease Rating Score (UPDRS— a measure of Parkinson s disease) has been described using Gompertz functions (20), which exhibit an asymptotic increase to a maximum score. However, over... 

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... 

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). 

This adaptation of the Gompertz growth function allows for the sensitive cell population to reach zero but will then show delayed regrowth in both sensitive and resistant cell populations as the resistant cell population continues to grow and then transfers to the sensitive cell type. 

Figure 3.4 Modelled maltose attenuation data (MG, modified Gompertz IBF, incomplete 3-function 5P, five-parameter logistic). The residuals for each model are depicted on the right.

Gompertz, B. (1825). On the nature of the function expressiveness of the law of human mortality, and a new mode of determining the value of life contingencies. Philosophical Transactions of the Royal Society, 115, 513-585. 

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. 

Mason H, Davison A, Wright A, Guthrie C, Payers P, Venables K, Smith N, Chettle D, Franklin D, Scott M, Holden H, Gompertz D, and Newman-Taylor A. (1988). Relations between liver cadmium, cumulative exposure, and renal function in cadmium alloy workers. British Journal of Industrial Medicine, 45, 793-802. 

Figure 3.29 Plots of the listed negative secondary ion intensities (top plot), the listed positive secondary ion intensities (middle plot), and the Cesinm concentration as derived via XPS (bottom plot) during the initial stages of sputtering with a constant current 1-keV Cs primary ion beam on (a) a Silicon substrate and (b) an SiOj substrate. The dashed line is a fitted Gompertz growth function. Reprinted with permission from van der Heide (2004) Copyright 2004 Elsevier Science B.V.

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