Analysis Of Growth Paths Research Articles

Fractals and the analysis of growth paths.

A simple practical method exists for classifying and comparing planar curves composed of connected line segments. This method assigns, a single numberD, the fractal dimension, to each curve.D=log(n)/[log(n)+log(d/L)], where:n is the number of line segments,L is the total length of the line segments, andd is the planar diameter of the curve (the greatest distance between any two endpoints). At one end of the spectrum, for straight line curves,D=1; at the other end of the spectrum, for random walk curves,D→2. Standard statistics are done on the logarithms of the fractal dimension [log(D)]. With this measure, trails of biological movement, such as the growth paths of the cells and the paths of wandering organisms, can be analyzed to determine the likelihood that these trails are random walks and also to compare the straightness of the trails before and after experimental interventions.

Fractals and the analysis of growth paths

A simple practical method exists for classifying and comparing planar curves composed of connected line segments. This method assigns a single number D, the fractal dimension, to each curve. D =log( n)/[log( n) + log( d/L)],where: n is the number of line segments, L is the total length of the line segments, and d is the planar diameter of the curve (the greatest distance between any two endpoints). At one end of the spectrum, for straight line curves, D = 1; at the other end of the spectrum, for random walk curves, D→2. Standard statistics are done on the logarithms of the fractal dimension [log( D)]. With this measure, trails of biological movement, such as the growth paths of cells and the paths of wandering organisms, can be analyzed to determine the likelihood that these trails are random walks and also to compare the straightness of the trails before and after experimental interventions.

An extension of the dehoff growth path analysis

Growth path analysis is discussed from both deterministic and stochastic points of view. If nucleation is assumed deterministic, the number density function is a solution of the continuity equation. If nucleation is considered a stochastic process, the number density function loses its meaning and must be replaced by a probability density function. Im portant implications of stochastic nucleation are discussed.