In this work, we calculate the fractal dimension of heart interbeat time series of some healthy young and elderly individuals. As has been found by means of other methods (detrended fluctuation and spectral analyses), we also find that interbeat series of healthy young subjects can be characterized by only one scaling exponent and a crossover behavior in it is observed with aging. By means of a zoom over the hinges of the crossover region, interesting effects of aging are presented. Our results with real interbeat time series are reasonably reproduced by using a simple model based on combinations of noisy first-order autoregressive series.