Optimal length of a calibration ion chamber

Objectives: A cylindrical ion chamber suitable for cross-calibration of activity meters (a.k.a. dose calibrators) has recently been developed [Med. Phys. 43 (12), 2016, 6536-43]. A syringe filled with a specified, nominal volume of radioisotope is inserted into the chamber along its central axis. Sensitivity (in units of electrical current produced per activity present) of this ion chamber depends on syringe insertion depth and syringe fill volume. A calibration ion chamber should be designed such that it is insensitive to variations in syringe length (i.e. insertion depth) and fill volume, caused by variations in either syringe manufacturing or operator skill. Changes in insertion depth have the least effect on sensitivity when the syringe is filled symmetrically around the chamber's center. But for changes in fill volume, variations of sensitivity are smallest if the syringe is filled a-symmetrically about the chamber's centre. The objective of this work is to find a combination of insertion depth and fill volume that will minimize the variation of sensitivity if either of these two parameters is changed. This optimum then dictates the optimal length of the ion chamber. Methods: An analytical mathematical model was developed to describe the ion chamber's sensitivity. In the vicinity of its maximum, the sensitivity to a point source of activity can be modelled as 2nd order polynomial, integrated over the chamber's length. Syringe fill volume was accounted for by a second integral along the length of its fill volume. The resulting expression depends on syringe fill length (l), insertion depth (d) and ion chamber height (h) (Figure 1). The combined effect of variation of fill length and insertion depth was determined by adding in quadrature the partial derivatives with respect to l and d. The resulting 2-dimensional surface depicts a "valley", which identifies the least sensitivity variation as a function of fill length and insertion depth. The equation describing the line that traces the bottom of that valley was determined by finding the surfaces' lines of curvature, employing differential geometry. This yielded a simple expression relating l, d and h. The expression found was then applied to optimize the height of a prototype of the calibration ion chamber. Results: The model we developed yields the relationship between the ion chamber's length (l), the syringe's insertion depth (d) and its fill height (h): (1) Fill height and insertion depth are given b