PubMed · 2626586
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1989. Longitudinal studies: methodology, growth and recent developments.. https://pubmed.ncbi.nlm.nih.gov/2626586/
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The Richards function serves as a tool for growth (W) analysis by means of absolute growth rate (AGR), i.e. dW/dt, and relative growth rate (RGR), i.e. d(loge W)/dt. More information may be gained using the second derivative d2W/dt2, and the third derivative d3W/dt3 of the Richards function to determine the times when the second derivative time curve achieves a maximum, minimum, and zero. The latter separates growth acceleration and retardation. Exponential growth occurs as long as (RGR)2 is less than the relative acceleration rate added to a tolerance value. This period of growth does not extend to the point of maximum acceleration of the growth curve. The points of maximum and minimum acceleration are at equal time distances from the point of inflection. A straight line phase of growth is approximated between the times when d2W/dt2 is maximum and minimum. Thus, a mathematical model for the analysis of aging changes is presented. Methodology described here should be applicable to any biological system or subsystem. As an application, derivatives of the Richards function are used to provide information on potato (Solanum tuberosum L.) tuber kinetics.