A mathematical approach to the optimal examination of lymph nodes

There is no scientific evidence to support the idea that serial sectioning along the short axis of the lymph node is superior to a single bisection along the long axis. We mathematically evaluated methods of lymph node dissection and applied the result to six lymph nodes that had produced false negative results at the time of frozen examination. We simplified the geometry of a lymph node to that of a three-dimensional ellipse and compared two different cutting methods. Let A be the cross-sectional area obtained through a single bisection along the long axis, and let B be the sum of the cross-sectional areas of n fragments obtained via serial cutting along the short axis. The smallest n (n*) that makes a B larger than A can be calculated. n* = [3L + √9L² + 16S²)/4S]. ([α], the smallest integer greater than or equal to α; L, long axis; S, short axis). The probabilities of tumor detection when the node is bisected along the long axis (P(D(A)|E)) and when serially cut along the short axis (P(D(B)|E)) are as follows. P(D(A)|E = {(3/2)S² - 3ST + T²}T/(S - T)³. and P(D(B)|E) = (n - 1){(1 + 1/n)L² - 3LT + T²}T/(L - T)³. (T, size of the tumor cell cluster). According to these formulas, three out of six lymph nodes were not examined in the most appropriate manner.