Figure 1
Shape adjusted ellipse (SAE) determination of nerve fiber size. (A) Transverse section of toluidine blue stained S1 ventral root in a female rhesus macaque. Note cross-sections of myelinated fibers of varied sizes. (B,C) Close-up view of boxed areas in (A), showing fiber shape heterogeneity, including presence of several elongated fiber presentations. Arrows indicate examples of non-circular shapes. (D) Heat map for S1 ventral root in (A) showing calculated roundness of myelinated fibers as a representative indicator for shape. Note marked heterogeneity of roundness for fibers across all sizes. A calculated roundness of 1 indicates a circular shape. (E) Sectioning of a cylinder with a circular base sectioned at different angles results in ellipse-shaped presentations of cut surfaces. (F) A cylinder in (E) sectioned at increasing tilt angles shows increased elongation and length of the long diameter of the corresponding ellipse-shaped cut surface. Note that the short diameter of the ellipse remains unchanged at all tilt angles. (G) Graph depicting error with overestimation of cylinder diameter, when the formula for a circle is used to calculate the diameter for the cross sectional surface of a cylinder cut at different tilt angles. Note the non-linear relationship between ellipse tilt angle and progressive diameter overestimation, especially for tilt angles over 30 degrees. (H) Heat map demonstration of the non-linear error introduced by an increasing difference between the two perpendicular diameters of an ellipse, Diameters 1 and 2, when the formula for a circle is used to calculate the diameter. (I) Summary of proposed SAE approach to calculate the minor diameter of an ellipse. The long-established formula for ellipse area (Eq. (1)) is combined with a more recently developed infinite series formula for determine ellipse perimeter56 (Eq. (2)). The SAE approach is based on the notion that an elongated 2-dimensional surface with a known area and perimeter corresponds to a unique ellipse shape. The SAE equation is achieved by replacing R in Eq. (1) with Ae/πr and expanding the first four terms of the infinite series (Eq. (3)). The equation is solved for r for a shape with known area and perimeter, and the minor diameter for the ellipse is next calculated as 2r. Ae = area of ellipse; R = long radius; r = short radius; p = perimeter of ellipse; h = (R − r)2/(R + r)2. (J) Violin plots of fiber diameter distributions. The violin plots show distributions myelinated fiber distributions with median and quartiles for diameters obtained by the SAE approach as well as for minimum Feret diameters and circle-based diameters after determining fiber area or perimeter. Note significant overestimations of fiber diameters when traditional methods were applied.
