Energy Absorption. III—The Mathematical Theory of Integral Dose and its Applications in Practice

Abstract

The paper deals with the mathematical theory of integral dose during whole-body irradiation. Formal definitions of “volume integral dose”, “surface integral dose”, and “line integral dose” are given. The paper then considers the position of “centre of dose” and “centre of mass”, and it is proved that in certain circumstances the mean dose throughout a mass of tissues is given by the dose at the centre of gravity independent of the shape of the mass considered. This theorem is generalised by integrating a power series (representing the change of dose with depth) over an elliptic cross section. The numerical coefficients of the first ten terms of the power series are calculated. The general analysis of the power series is then applied to an exponential fall of dose. This leads to formulæ giving the variation of average dose with size of patient, and tables (included in the text) are drawn up giving average dose for different sizes against absorption coefficient of radiation. The power series analysis is applied to the variation of inverse square law alone, and finally combined inverse square law and exponential decrease. The correlation between absorpton coefficient and half-value layer of radiation is discussed. A practical example is given of the use of the tables for finding integral dose during whole-body irradiation, taking into account known half-value layer and focal distance. Two mechanical methods of finding surface integral dose, one by measurement of moments and the other by weighing bundles of isodose curves cut from celluloid sheet are described. The second method may be applied to a given volume by the application of standard formulæ for numerical integration.