Cty tam Pee we es ae ae cee aalee ote ate cee Cad Leodlabibacten 87 :jeal fibers, Univ. of Rochester Institute of Optics Report NYO-9033 (April 1960), p. 6-46. post, R. F. Resolving time of scintillation counters, .u- " feonics 10 (6), 56-58 (June 1952). rods realig \jurinelli, L. D., Clemente, G. F., Abu-Shumays, I. K., and c phantos ‘xperimeng ~teingraber, O. J. Localization of radioactivity in vive -' photon time-of-flight techniques. Argonne National Laboratory Radiological Physics Division Annual Report, July 1967-June 1968, ANL-7489, pp. 1-12. 6. Marmelli, L. D., Clemente, G. F., Abu-Shumays, I. K., and Steingraber, O. J. Localization of scintillations in gamma ray cameras by time-of-flight techniques: Linear resolutions attainable in long fluorescent rods. Radiology 92, 167 (January 1969). aable und MEGULARIZATION UNFOLDING IN LOW ¥-RAY ACTIVITY MEASUREMENTS. } EVALUATION FOR ONE-DIMENSIONAL SCANNING ; efforts iB G. F. Clemente, L. D. Marinelli, and I. K. Abu-Shumays* : resolutiog—-ne May The problem of converting the spatial distribution of the 2m rods oy. ing rate of a scanning 3” Nal crystal to the distribution vever, thapf : vlioactivity in the object being studied involves solution more senspt * lredholm equation of the first kind. Precision of the reUEP a function of the physical parameters used studied as s, ] it; tt TeSOlty slice ismeasurement te first th 1 a two-d, STRODUCTION ‘rods thre’ We have reported elsewhere“: » on various methods stails of for the quantitative determinationof the distribution of swo-dimenl®” level activity in man. We wish here to study, in this prota™ -. detail, one such competitive method, the regularigation unfolding method, and specifically, its application ):l representative experimental model to be described clow. We will analyze the effect of the experimental araumeters on the accuracy of the regularization unHolding of low activity scanning data. A comparison of ome of our results with results using the usual iterative ution, psec ethod@: © favors the regularization unfolding: ®. * The unfolding problem can be formulated bythefol- owing Fredholm equation of the first kind 520 680 g(a’) = gle!) +e’) = f 12) K(ea’)de 90 where f(x) is the unknown distribution of activity, g’(z’) the hypothetically exact and g(x’) the actual speetrum or response of the detector, e(2’) the statistical and experimental error superimposed on g(z’), and the kernel AY +.c') the point response function. lor simplicity, the present treatment is restricted to 1175 b one dimension. We start with a known distribution J(), calibrate (i.e., measure the response kernel by usIng point sourees of knownactivity), measure the spectrum g(x’) corresponding to f(z) and try to reproduce § the known distribution f(z) by using the regularization fUnlolding method. ; The distribution of activity f(z) consisted of a poly- e's lone catheter 0.034” I.D., 270 cm Jong filled with '68). idy of A —— a \pplhed Mathematics Division. radioactive (Cs) microspheres (~0.725 mm in diame- ter); the source configuration (54 cm long) consisted of nearly sinusoidal waves of 5-cm amplitude and of vary- ing frequencies set on a light Lucite bar. The total activity (1.07 + 3% uCi) of the distribution was cali- brated by proper comparison with the activity (0.127 pCi) of the point source used to obtain the point response function K(z,2’). Measurements of the spectrum g(x’) were carried out by taking measurements at various intervals over a single distribution of activity, using a 3” x 3” Nal erystal with different collimators placed in a low background lead well. The crystal was restricted to move on a straight line parallel to the long axis of the distribution f(z). This paper considers the influence of the following parameters on the calculated distribution f(x): A. previous knowledge of the total radioactivity C which is the integral of the distribution f(z); B. the full width at haif maximum (FWHM) of the point response kernel (equivalent to the resolution radius of the colimator); C. the experimental and statistical error superimposed on g(x’); D. the quadrature approximations or, equivalently, the interval h between two consecutive readings of g(v’); and E. the usefulness of extending the measurements of g(x’) beyond the rangeof the distribution f(z), and thus increasing the information content of g(z’). Wewill proceed to describe the method weused to solve for the distribution f(z). This method has been programmed for the IBM 360 by Mrs. Alice B. Meyer and one of the authors (IIXA). MATHEMATICAL OUTLINE OF THE UNFOLDING PROCEDURE If one eliminates e(x’) from Eq. (1), the result is a mathematicaily‘‘all-posed problem.”’ The measurement g(z’) is relatively insensitive to fictitious (positive-negative) distributions f’ for which f K(a,2') f (x) dx = e(z),