Spatial distribution of curve length: concept and estimation

The length of a curvilinear feature, such as a dendrite tree of a neuron, can, in principle, be estimated by the recent, non-invasive method of total vertical projections (TVPs). Curve length is a measure of size, but it reports nothing about curve shape. The shape of a tree-like structure can be described to some extent by the distribution of branch length in properly defined regions of three-dimensional (3-D) space. A definition of curve length distribution in three dimensions is proposed and implemented here on a human neuron. The relevant 3-D regions overlap after projection, and therefore the TVPs method cannot be used directly to estimate the corresponding feature lengths. However, using the ANALYZE software system running on a SUN SPARC workstation, dendrite subsets sitting in predefined regions of space were rendered in different colours and measured separately by the TVPs method using a cycloid test system. In combination with non-invasive image acquisition and processing techniques, the length distribution concept is likely to be useful in the metrical analysis of either microscopic or macroscopic arborizations in a wide variety of contexts, including living cells and organisms.