A simple method for estimating the length density of convoluted tubular systems

We present a new method for estimating the length density (Lv) of convoluted tubular structures exhibiting an isotropic distribution. Although the traditional equation Lv = 2Q/A is used, the parameter Q is obtained by considering the collective perimeters of tubular sections. This measurement is converted to a standard model of the structure, assuming that all cross-sections are approximately circular and have an average perimeter similar to that of actual circular cross-sections observed in the same material. The accuracy of this method was tested in eight experiments using hollow macaroni bent into helical shapes. After measuring the length of the macaroni segments, they were boiled and randomly packed into cylindrical volumes along with an aqueous suspension of gelatin and India ink. The solidified blocks were cut into slices 1.0 cm thick and 33.2 cm2 in area (A). The total perimeter of the macaroni cross-sections so revealed was stereologically estimated using a test system of straight parallel lines. Given Lv and the reference volume, the total length of macaroni in each section could be estimated. Additional corrections were made for the changes induced by boiling, and the off-axis position of the thread used to measure length. No statistical difference was observed between the corrected estimated values and the actual lengths. This technique is useful for estimating the length of capillaries, renal tubules, and seminiferous tubules.