A principle of fractal-stochastic dualism and Gompertzian dynamics of growth and self-organization

On the Gompertzian growth in the fractal space–time

“…with the fractal dimension equal to one, whereas the previous model (Waliszewski, 2005) predicts a plateau at this stage as…”

“…which have not be taken into account in the previous model (Waliszewski, 2005). Combining relations (6) and (7) we get…”

“…The final result will be the same provided that the suitable boundary condition for t → 0 is taken into account. A detailed analysis of the previous approach (Waliszewski, 2005) reveals that it gives also an erroneous relationship between the Gompertz function (1) and the spatial Morse potential…”

“…The mean values of b t and a t can be determined by the fit of the experimental Gompertz curves in the three intervals (Waliszewski andKonarski, 2002, 2003): (i) from zero to the inflection time t i , in which a number of cells reaches the 37% of the final cell number, (ii) from the inflection time to the time-point at which the process of differentiation reaches a plateau and (iii) at which the number of cells remains constant. In particular for the BrownPearce rabbit's tumor growth, the following mean values have been obtained (Waliszewski and Konarski, 2003) An attempt has been made to generalize this approach and derive an analytical formulae describing the time-dependence of the fractal dimension (Waliszewski, 2005)…”

Statistical Ensemble Theory of Gompertz Growth Model

“…These attempts have much to do with statistical description for the tumour growth phenomena and are the main motivation behind this work. Furthermore, the fractal nature of the temporal and spatial process of Gompertz dynamics [11] and the fractal-stochastic duality as a consequence of the intercellular interaction [12] has also been investigated.…”