Spears, C.P. Volume doubling measurement of spherical and ellipsoidal tumors.
Med. Pediat. Oncol., 12:212-217, 1984.
Volume doubling rulers are described for rapid estimation of tumor volume using projection areas of single-plane images such as chest radiographs. The potential utility of the rulers derives in part from the convenient relationships that exist for the interconversion of volume, volume doubling number, and decimal log growth or cell kill. The traditional approach to tumor volume doubling time determination, originated by Collins  uses the average diameter of approximately spherical tumors; analysis of the geometry of volume doubling shows this method to be nearly as accurate as methods that assume elliptical eccentricity, but only up to a tumor image length/width ratio of 1.5. For ratios greater than 1.5, it is shown that the clinical practice of taking perpendicular diameters is inherently more accurate. Where two perpendicular radiographic views of a tumor ellipsoid are available, it is shown that calculation of volume using the longest projection radius should result in less than one single volume doubling underestimate due to out-of-plane orientation of the long axis of the tumor. These methods should find greatest utility in measurement of multiple metachronous tumors that are well-circumscribed, convex in perimeter, and have low length/width ratios.