Mathematical modeling of human embryonic and fetal growth rates.

A mathematical model for human embryonic/fetal growth data from implantation to birth is developed. In previous work, it was shown that an unbiased estimate for human fetal growth data from about day 50 post-conception until term could be calculated from the Gompertz equation. This period represents a range of embryonic/fetal weights from one to 3500 g. When the Gompertz equation is extended, with no change of parameters, to the prenatal period before 50 days, the predicted weights have a consistent bias which might have a biological basis. Early embryonic growth immediately following fertilization is exponential; i.e., one cell goes to 2, then 4, then 8... etc., with essentially no decrease in relative growth rate. Except for possible small changes in cell size and cell mitosis cycle time, such exponential growth can be considered as a special case of the Gompertz equation with a, the relative rate of decrease of the relative growth rate, equal to zero. The relative growth rate begins to decrease about 20 days post-conception, at the time of cell differentiation into organ precursors. Although the "Hayflick Limit" of the maximum of 50 to 60 cell divisions for human cells would tend to cause a decrease in growth rate, it can be shown that the effect is insignificant during embryonic/fetal growth. The observed decrease in the growth rate might be a result of a decreasing fraction of cells in the pool of dividing cells. For the Gompertz equation model, a at this time changes from zero to a positive number. Analysis of fetal growth data shows that a rapidly becomes large and then decreases over a period of several days to become a constant positive value for the remainder of the prenatal term. Good fits of empirical embryonic/fetal growth data were obtained by nonlinear regression with calculation of the embryonic/fetal weights through numerical integration of the differential Gompertz equations and the functionality of alpha.