+ problem is illustrated in Figure 58. Consider a linear chuped element of an extended radioactive mass or organ, imbedded in the body. The length of this element is 2/, its center is the point P. Let the detector be . « distanee a from the center. The radioactive seg, nt hes at a depth ¢ within absorbing, but nonradioactive tissue. The analysis assumes that the angles oda tithes, ie mutually subtended by the source and detector are di-tribution of spheres must be so as to simulate satis- j «torily the response of an extended uniform source identically placed. We assume the attenuation to be exponential. This implies that the detector response should be restricted to the ‘photopeak” region. As illustrated, the effects af seattermg within adjacent volume elements of the organ may be appreciable, and the absorption coefficient to be used must be chosen accordingly. The relative reading, in arbitrary units, produced bya } uit source at the center, P is Rp a? 3 (1) where Qo is the source strength and AK a constant re- luting to detector efficiency, whereas that produced by iu lincar source of equal strength and length 22 will be R. _ KQe"tt? f 7 21a? e” dx ~. (1 — 2/a)?’ (2) soution of the integral is given by Evans.®) The dcnominator is expanded in a series, and integration term by term readily follows. For the case where yl 7 ipe of the correctly, ix uppreciably less than unity, so that higher terms in xi may be disearded, a particularly simple expression , detecta, “al array, DETECTOR wen ebeae eee seen ally with, ‘are con, ndividuat n normal. ‘cognized, age, sey, -commor,; »ropertie: d of the sition é iniformls; 1 Th ual Gr pee 75 F .5- pls .50 L4- wea. 25 1 I 1 L 2 1 L 3 1 4 fo Fig. 59-—Efficiency factor F vs. 1/a results. The normalized response factor F, namely the ratio of the response R, from an extended source to Rp , is given by 1 F = R/Re = Gay (3) + pl(2/3 f +.4/5 f? + 6/7 f? + eee: ), KQye *t? lent upon went.§ relatively small, so that variations in response with deviations in path direction from the line joining their midpoints may be neglected. Keep in mind that the tisk at hand is not to calculate the crystal response PBeexactly, but to estimate how spatially uniform the 4 | at ORGAN where f = i/a. F mayalso be thought of as the factor by which the point source strength must be increased in order to obtain a response equal to that of the ex- tended segment of total strength Q,. Note that ex- ponential terms in thickness ? disappear, being common to both numerator and denominator. Equation (2) can also be integrated exactly within the range of approximation where e* = 1 + yz, with the following result 1 l 1- 2 ra ript ilersytia) © where in order for e” to differ by less than 10%, ux is restricted to values less than about 0.5. Values of the factor F as caleulated from equation (3) are plotted vs. the parameter f = l/a in Figure 59. The values assumed for ul in the three upper curves are well outside the range for which the approximation is valid. The point of this figure is that if one wants to represent an extended line source (and by extension, the whole organ) by only one point source at the center to within an accuracy of, say, 10%, the acceptable range in values of ul and //a is quite restricted. The correct value of the exponential integra] may be calculated to a much closer approximation if additional terms [equation (2)] are retained and thesimpli- fying assumptions avoided.* In Figure 60 the values of its. 58—Geometry of organ depth location in whole-body pC Y ing * The resulting solution to 16 terms, including the parameters of ul and l/a to the sixth power, is given by Evans.