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The recessive oncogenesis model, according to which inactivation of both alleles of specific genes leads to cancer, has received much recent attention. A mathematical formulation of a two-mutation model for carcinogenesis, which includes the recessive oncogenesis model as a special case, was fitted to data from a large experimental study in which rats exposed to radon daughters developed malignant lung tumors. The model described the data well. The results indicate that fractionation of exposure increased the lifetime probability of tumor. Examination of the parameters of the model suggests that the effect of fractionation can be explained by the relative effects of radon daughters on the mutation rates and on the kinetics of growth of initiated cells. The first mutation rate is very strongly dependent upon the rate of exposure to radon daughters, the second mutation rate much less so, suggesting that the nature of the two mutational events is different. The model makes predictions which are testable in future experiments.

Abstract. In lifespan studies of 2,242 rats of three strains, 32 neoplasms were identified in brain, meninges and pineal gland. They were astrocytoma ( 16 Wistar, three Sprague-Dawley), ependymoma (four Osborne-Mendel), meningioma (two Osborne-Mendel, one Wistar), pinealoma (two Osborne-Mendel), reticulosis (two Wistar), oligodendroglioma (one Wistar), and gliomatosis (one Wistar).There has been a renaissance of interest in brain tumors of laboratory animals with the realization that exogenous and, perhaps, environmental factors may influence their induction and growth [ I , 81. Spontaneous brain tumors in the rat are infrequently reported [ 1 I], often only as case reports or merely tabulated along with the incidence of other tumors. The largest study with morphologic descriptions involved 41,000 Sprague-Dawley rats with 38 central nervous system tumors for an incidence of only 0.09% [9]. Another study of 7,803 Sprague-Dawley rats described 34 brain or meningeal tumors, an incidence of 0.44% [4]. In other reports, three brain tumors were described in 1,100 Wistar rats [2], and an unusual type of granular cell tumor of the central nervous system was described in 12 aged BN/BI rats [6]. Tabulations of brain tumors in additional reports also indicate a low spontaneous incidence of brain tumors in rats [3, 5 , 10, 12, 14-16]. Our data suggest that the incidence of brain tumors in aged rats is considerably higher than previously reported. Materials and MethodsThree-to 5-month-old healthy female Osborne-Mendel rats were obtained from the Charles River Breeding Colony (Wilmington, Mass.); female Sprague-Dawley and male and female Wistar rats were purchased from Hilltop Laboratory (Chatsworth, Calif.). The rats were in our laboratory for the remainder of their life span in climate-controlled rooms with filtered air kept at 22 k 1" C and 50 f 20% relative humidity. The Osborne-Mendel and Sprague-Dawley rats were housed in pairs in wire-bottom cages placed in automatically flushing racks equipped with automatic waterers. Wistar rats were housed four to five in a 20-centimeter deep plastic cage with 390 cm2 floor space with Sanicel (Paxton Processing, Paxton, Ill.) bedding and water bottles attached to cages. All rats were fed Wayne Lab-Blox (Allied Mills, Chicago, Ill.) laboratory rat diet with 4% crude fat content. The Osborne-Mendel and Sprague-Dawley rats were on a n experiment with cigarette smoke condensate-beeswax pellets implanted into the lung. The Wistar rats were on low-level radiation studies with inhaled or intravenously 318

Data from 4276 rats with radon exposures up to 10,000 WLM at rates up to 1000 WL are analyzed with a two-step clonal expansion model. The age dependences of the hazard for the risks for fatal and for incidental tumors are very different. Therefore, two different parameterizations of the model are used in the two cases. In both cases radiation acts only on the initiating mutation and the clonal expansion, but not on the second mutation. Average exposure rates of 5 WL for fatal tumors and 0.5 WL for incidental tumors double the rate of spontaneous mutations. While the fatal tumors show a linear increase in the effective clonal expansion rate up to about 100 WL average exposure rate and a saturation at higher exposure rates, the incidental tumors follow a step-like behavior of this parameter. It is proposed that only the fatal lung tumors among the rats be used for generalizations to models for lung cancer in humans. The fitted model for fatal tumors shows an inverse dose-rate effect at average exposure rates above 20 WL. However, below 10 WL the lung cancer risk per unit exposure decreases with increasing duration of exposure. Between 10 and 20 WL, the difference in ERR/WLM between acute and protracted exposure is small.

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