104 THE SHORTER-TERM BIOLOGICAL HAZARDS OF A FALLOUT FIELD 3. Death ensues when the lethal injury exceeds a critical level, the lethal bound. These postulates are formally equivalentto the set used by Blair [19,20], but in the sub- sequent development, the writer and Blair follow different paths. Blair introduces quan- titative assumptions for the injury and re- covery processes and for the ageing process, to derive an explicit equation for the dependence of survival on exposure The alternative course followed here is to solve for an empirical lethality function using survival data for a given species. This lethality function would he a description of the course of lethal injury in the given species if that species conformed to the postulates above. Inprevious presentations [14, 18], the integral equation of injury was obtained in the form X= f"Te—n)o(o}dr+6 @ where F(t—+) is the intensity of exposure et time t—r, @(r)dr is the increment of injury appearing at time + after instantaneous expo- sure to unit dose, Bt is the accumulation of injury due to the natural ageing process. Equation 1 introduced the assumptions that QUANTITIVE ESTIMATION OF RADIATION INJURY AND LETHALITY When J(f—7)=consiant= 7, Equation 3 becomes EUt—=14+mP?ff erwinDy bene 2 = 1l function becomes equal to the lethal bound, and therefore to unity, whon t*s=f. o (l—e7# 6-2) (4) The time constant 1/u is the mean time that damage can persist and be potentially able to combine with later damage to produce second- power injury. The value of t/u is probably in the range from houra to a few days. In this case the term (1—e7*') in Equation 4 is negligibly different from unity over the time period of interest here. The effectiveness func- tion then becomes,to a sufficient approximation minaeh This same approximation may be used when I is small over a time period on the order of 1/u. When Z(J) is approximated by Equation 5, Equation 2 reduces to 6) the accumulation of injury due to ageing is a linear function of age, and that the effectiveness of each increment of dose Zdr is proportional to I. We have since obtained evidence that these two assumptions maybe incorrect(10, 15]. The integral equation of injury will therefore be written in the more general form Death occurs when X(f) reaches a critical value, the lethal bound, which can be set equal to unity. Equation 6 becomes, XO=[EUt-noode+ay corresponding to an exposure at constant daily where A(t) is the ageing function and E(,t—7) is the effectiveness of the dose increment I(t—r)dr. The effectiveness function, in a form that takes account of effects that depend on the second power of the dose,is E(i,t~1)=I(t--7) + toe mf Ht)it—r—pem-r-Pde 8) a 1=E() [oa—ndrtawn a where ¢* is now a definite mean survival time dose I. Thelethality function for constant ex- posure, called the cumulant lethality function, Cr, is immediately found to ba a=[dmg [1-4] The ageing function is very imperfectly known, but can provisionally be specified, in view of available data (10), as + b-+-g(*+ 5)? AO=FTb+atOy PROPERTIES OF THE LETHALITY FUNC. TIONS AND SOME IMPLICATIONS FOR PREDICTION (9) Tante 1.—-TABULATION OF THE SURVIVAL OF ABC MALE MICE GIVEN DAILY X-RAY EXPOSURE FOR THE DURATION OF LIFE, AND OF THE CUMULANT LETHALITY VALUES DETERMINED Meandally dose 9 (r) Goma[1-40| (10) When explicit expressions are assigned for E(/) and A(t), then, with a set of known values of J and t* we obtain a numerical estimate of a cumulant lethality function that describes the course injury would follow in a@ model system conforming to the postulates stated above. The most extensive lethality data over a wide range of daily dosages are those for ABC male mice given X-ray dosages ranging from 20 to 1,000 r/day [18]. The daily dosages and survival times are given in Table 1. The cor- responding values of the cumulant lethality are also given in Table 1. The cumulant is computed on the assumptions that Alt) =t"ffy a) where f is the mean survival time of controls, and E(D=I (12) These expressions for E(I) and A(t*) are not realistic, as noted above, but the discussion below will center on some properties of the lethality function that are not qualitati¢ely affected by any bias introduced by these approximations. , The cumulant values for the ABC mice are plotted in Figure 3. The function obtained is not a simple curve. There is a sharp flexion Mean after. survival (deys)| Cumulant lethality » Gidayy~ a 213 0, 0366 73.8 - 0256 36. 8 - 0224 39, 2 « 0185 29. 9 - 6142 20. 6 - 0109 15.2 . 0094 11.5 - 0056 8&0 - 0036 70 - 0029 5. 2 - 0020 3.3 - 9010 Jet us first summarize the previous developments. If samples from a homogencous population are exposed to different constant intensities I, we can deduce from data on daily duration-of-life exposures a lethality function of the form 6) is time-dependent, if the change of J with time X()=Eti)[oe-nartaq whore 4 is the age at the beginning of exposure, and ¢ is the control survival. The ageing 105 * Exposures given 6 days per week, . + Using Equation 10 with E()==Fand A(t) af. at about 15 davs and another near 40 days. The flattening of the curve as drawn between 80 and 220 days is not arbitrary, but is based on certain properties of the survival of ABC mice in this time period [18], and on the behavior of other strains and species, as will be shown below (Figure 5). This plateau period implies the existence of a “silent period” between the acute and chronic phases of injury. Evidence for such a silent period is also found in the recovery studies of Storer [1] and others. The impulse function obtained from these data by numerical differentiation is plotted in Figure 4. This is an estimate of the course of injury after a single exposure. The existence of two major peaks of injury, at 15 and 40 days, is indicated. The minimum at about 120 days again represents the “silent period” noted above. Cumulant values were computed for all available data on experimental animals given uniform duration-of-life exposure [14] and are presented graphically in Figure §. The cumulant values are here plotted on a log-log scale.