incidence of retinoblastoma (germinal mutation/somatic mutation)

Transcription

1 Proc. Nat. Acad. Sci. USA Vol. 72, No. 12, pp , December 1975 Medical Sciences Mutation and childhood cancer: A probabilistic model for the incidence of retinoblastoma (germinal mutation/somatic mutation) ALFRED G. KNUDSON, JR. *, HERBERT W. HETHCOTEtf, AND BARRY W. BROWNt * Medical Genetics Center, Graduate School of Biomedical Sciences, University of Texas Health Science Center at Houston, Houston, Texas 77025; t Department of Biomathematics, M. D. Anderson Hospital and Tumor Institute, Houston, Texas 77025; and * Visiting Mathematician from Department of Mathematics, University of Iowa, Iowa City, Iowa Communicated by E. B. Lewis, September 2,1975 ABSTRACT The incidences of some childhood cancers have been shown to fit a two-mutation hypothesis for cancer initiation. According to this hypothesis, the first mutation can be either germinal or somatic while the second is always somatic. A probabilistic model involving the mean number of tumors per genetically susceptible individual is developed as a function of age and is compared with age incidence da for retinoblastoma. The change in the mean number of tumors with time is interpreted in terms of the growth of retinal cells. In patients who are not genetically susceptible, the times of occurrence of the first and second somatic mutations can be inferred from a comparison of familial and nonfamilial unilateral case incidences. The total incidences of hereditary and nonhereditary forms of retinoblastoma are related to germinal and somatic mutation rates. The even distribution of certain childhood cancers throughout the world suggests that their incidences are determined by spontaneous mutation rates rather than by local environmenta mutagenic carcinogens. Most forms of human cancer occur in two classes of individuals (1, 2). The first carries a dominant mutation that is highly penetrant, tends to produce multiple tumors, and produces tumors at an earlier than average age; examples are polyposis of the colon and the heritable form of retinoblastoma. The second class is the major segment of the population which does not carry such a mutation. One view of the relationship between these classes is that both entail cancer cell initiation by two mutational steps, the first of which may be either germinal or somatic, the second, somatic. For adult cancers Ashley (3) has concluded that a multihit hypothesis is in better accord with age-specific mortality than is a two-hit hypothesis. Ashley (4) has also concluded that carcinoma of the colon arises in subjects with colonic polyposis after one or two fewer mutations than when it arises in random subjects. However, it cannot be decided how many "hits" are mutations that are involved in initiation of a cancer cell and how many may be mutations or epigenetic events that are concerned with promotion of tumor growth, especially since the latter phase may be very prolonged in adult tumors. For the purpose of analysis childhood tumors are simpler because they grow so rapidly that the promotion phase is greatly contracted. For three tumors of childhood, retinoblastoma, neuroblastoma, and Wilms' tumor of the kidney, evidence has been presented that supports a two-hit hypothesis (5-7). For these three childhood cancers, estimates of the average number of tumors per individual inheriting the first mutation have been made. Using a Poisson distribution one can explain the occasional gene carrier who gets no tumor, those who develop only unilateral tumors, those who develop bilateral tumors, and instances of multiple tumors on one side. Individuals who bear the first mutation germinally are 5116 extremely susceptible to the specific tumor and have an earlier than usual average age at diagnosis. The two-mutation theory assumes that carcinogenesis is related to discrete changes occurring at random and at constant average rate. When the first mutation is prezygotic, the tumor is called hereditary. When the first mutation is postzygotic the tumor is nonhereditary and single since the probability that an individual would acquire both mutations in more than one cell is extremely small. Thus all bilaterals are hereditary, whereas unilaterals are mostly nonhereditary with some hereditary cases. In this report we develop a probabilistic model for hereditary cases of retinoblastoma that describes its age-dependent incidence. This model is compared with currently available patient data and found to be consistent with the two-mutation hypothesis. Comparison of familial and nonfamilial unilateral cases leads also to an elucidation of the temporal relationship between the first and second mutations in nonhereditary cases. A PROBABILISTIC MODEL If hereditary (H) or nonhereditary (N) cases of a cancer (C) such as retinoblastoma are collected according to age at diagnosis (t) and according to eventual unilaterality (U) or bilaterality (B), then the fraction of cases not yet diagnosed by a given age may be plotted as a function of age. These functions are denoted by HC(t) for hereditary cases as a whole, HU(t) for hereditary unilateral cases, HB(t) for hereditary bilateral cases, and N(t) for nonhereditary cases. Now for hereditary cases we let m(t) be the mean number of tumors per individual initiated by a second (somatic) mutation in the interval [o,t]. For purposes of this analysis we assume no delay between the appearance of the first cancer cell and the time of diagnosis; however, incorporating a constant delay time into the model merely translates the age-at-diagnosis time scale relative to the earliest-cancer-cell time scale. The expected number of tumors in one eye (right or left) in a hereditary case in the interval [o,t] is m(t)/2. Since the Poisson distribution is appropriate for small numbers of random events, the following probabilities may be written: P(an eye has no mutation in [o,t]) = e-m""2 P(an eye has at least one mutation in [o, t]) = 1 - e-11"' 2 P(an individual has no mutation in [o,t]) = e-ndt P(an individual has at least one mutation in [o,t]) Letting M[t1,t2] = 1 - e-m(0/2 = m(t2)- m(ti) be the number of muta-

2 tions in the age interval [tbt2], we may write HCOt) = P(patient has M[o, t] = o patient is eventually cancerous) P(patient has M[o,t] = o and is eventually cancerous) P(patient is eventually cancerous) P(patient has M[o,t] = o and M[t,cn] _ 1) P(patient has M[ot] _ 1) e-m(")11- e-im~zm "11 1 e-m() e-m(t) -e-m(- ) e-e-n(- 1= [1] Similarly, HU(t) = P(patient has M[o,t] = oipatient is eventually unilateral) P(patient has M[o,t] = o And, Medical Sciences: Knudson et al. and is eventually unilateral) P(patient is eventually unilateral) P(one eye has M[ot] = o and M[t,co] _ 1 and other eye has M[oo] = o) P(one eye has M[oC] _ 1 and other eye has M[o,ao] = o) 2P(eye has M[o,] = o) P(eye has M[t,c] > 1) P(eye has M[o,co] = o) M[=co] 2P(eye has. 1) P(eye has M[o,oo] = o) P(eye has M[ot = o) P(eye has M[t,o] _ 1) P(eye has M[oc] _ 1) e-m(/2-1 e-[m(-)-m(t)12 e-m(2)/2 e-m(t )/2 _e-[m(-) m-[21 HB(t) = P(patient has M[o, t] = o patient is eventually bilateral) P(both eyes have M[ot] = o and M[t,cx] > 1) P(both have M[oco] _ 1) P(eye has M[ot] = o)2 P(eye has M[t,co] > 1)2 P(eye has M[oo] _ 1)2 e [m("11 - e-lm(-)-m~t)/2112 [1 e-r?(a: )/2]2 [e-m(t)/2 - e-m(' )/2]2 [3] [1 - e-mfa.)/212 Proc. Nat. Acad. Sci. USA 72 (1975) 5117 Note that HB(t) = [HU(t)]2, which is consistent with the assumption that the eyes acquire tumor independently. PATIENT DATA AND THE MODEL For retinoblastoma the mean number of tumors eventually acquired by a hereditary case [m(co)] has been estimated to be three (5); however, the correct value could be four since errors in counting tend to make this estimate low. The expected fractions of unaffected individuals, unilateral cases, and bilateral cases among gene carriers are 0.050, 0.347, and 0.603, respectively, for m(oo) = 3 and 0.018, 0.233, and 0.749, respectively, for m(oo) = 4 (5). In order to fit the derived curve to observed data for unselected hereditary cases, HC(t), the proper mixture of unilateral and bilateral cases must be used. Most series emanate from large centers and are biased toward bilateral cases. In addition the ages of unilateral cases at the time of report must be great enough that the probability of acquiring tumor in the unaffected eye is vanishingly small. For the latter purpose the age of 10 years has been used since virtually no new retinoblastomas occur beyond that age. Five sources have been utilized to obtain the necessary patient data. The 24 hereditary unilateral cases surviving beyond the age of 10 years are tabulated (Table la) with respect to age at diagnosis: 15 from the report of Weller (8), 9 from the reports of Falls (9) and Falls and Neel (10). The 90 bilateral cases (Table lb) and 149 unilateral cases with no family history (Table 1c) were taken from Falls and Neel (10), Macklin (11), and Knudson (5). This last group of cases is referred to as nonhereditary, although it is acknowledged that 10-15% of them may have resulted from new germinal mutation (5). The bilateral cases are assumed to be entirely hereditary, whether by inheritance from a parent or by new mutation (5). The values for the functions HU(t) and HB(t) are also calculated. The constructed data for HC(t) (Table 2) were obtained by mixing the hereditary unilateral and bilateral cases in the above proportions for m(oo) = 3 and 4. For example, HC(t) = [0.347 HU(t) HB(t)]/0.950 for m(oo) = 3. From these data empirical values of m(t) (Table 2) are obtained for various values of t by m(t) = - ln le-ml + [1 - e-1l')]hc(l t which is derived from Eq. [1]. As shown in Fig. 1, m(t) is a nearly linear function from birth to 5 years, after which it becomes asymptotic to m(o). From these values of m(t), predicted values of HU(t) and HB(t) are obtained from Eqs. [2] and [3] (Table 2). A chi square test comparing observed and predicted incidences corresponding to HU(t) and HB(t) demonstrates that the deviations are consistent with randomness [P = 0.11 for m(oo) = 3 and P = 0.16 for m(oo) = 4]. Since the deviations are approximately the same for each value of m(co), a determination of m(co) cannot be made from these data. The observed values are plotted (Fig. 2) along with the functions HC(t), HU(t), and HB(t) for m(o) = 3. The patient data for unilateral cases without family history, N(t), are also plotted in Fig. 2 and found not to be significantly different from those for hereditary unilateral cases. The age at which unilateral tumor is contracted is evidently not dependent upon genotype. The median ages at time of diagnosis for bilateral, hereditary unilateral, and nonhereditary unilateral cases are 1.0, 2.2, and 2.1 years, respectively. If it is assumed that hereditary cases result from a second somatic event such as mutation, which occurs at some con-

3 5118 Medical Sciences: Knudson et al. Proc. Nat. Acad. Sci. USA 72 (1975) Table 1. Retinoblastoma: Distribution of cases by age, laterality, and family history Age at diagnosis (years) Class and source > 5 Totals (a) Unilateral, familial Weller(8) Falls (9), Falls and Neel (10) Totals HU(t) at beginning of interval (b) Bilateral Fallsand Neel (10) Macklin (11) Knudson (5) Totals HB(t) at beginning of interval (c) Unilateral, nonfamilial Falls and Neel (10) Macklin (11) Knudson (5) Totals N(t) at beginning of interval stant rate,u (mutations per locus per cell per year) in a population of susceptible retinal cells, where n(t, t + 1) is the average number of susceptible cells in the 1-year interval [t, t + 1], then the product gn(t, t + 1) is related to the function m(t) as follows,,un(t,t + 1) = m(t,t + 1) - m(t) since m(t) is the average number of mutated cells in [o,t]. Values for jan are presented in Table 2 for m(x) = 3 and 4. If,u is in fact constant, then the number of susceptible cells grows to a maximum in the third year of life and falls quickly after the fifth year. DISCUSSION From a probabilistic model fitted to data on the hereditary form of retinoblastoma are derived age distribution curves for both unilateral and bilateral classes to which available data fit well. As predicted by the model, bilateral cases occur earlier than unilateral ones. It is concluded that individuals who carry the dominant gene for retinoblastoma acquire tumor as the result of a random event that follows a Poisson distribution with a mean number of tumors that eventually approaches 3 to 4 per individual. The model entails the use of a function, m(t), which describes the time-dependent change in mean number of tu- mors in a population of genetically susceptible individuals. This function increases nearly linearly until the age of 5 years, after which it approaches the asymptote of m(oo). This result may be interpreted in terms of an event that occurs at a constant rate in a population of susceptible cells which is already present at birth, grows to a maximum in the third year of life, and declines thereafter. Such an interpretation is compatible with the growth of the retina, the "U o cn.06 o 04 a) E4 E 3 o.02 Li E C Age in Yeors(t) FIG. 1. Retinoblastoma: hereditary cases. The calculated mean number of tumors, m(t), as a function of age t is shown when m(x) is3and Age in Years (M) 5 6 FIG. 2. Retinoblastoma. A semilogarithmic plot is given of the observed data points from Table 1 for the undiagnosed fraction of familial unilateral, bilateral, and nonfamilial unilateral [N(t)] cases. A calculated curve for unselected hereditary cases, HC(t), and predicted curves for familial unilateral, HU(t), and bilateral cases, HB(t), are also shown.

4 Medical Sciences: Knudson et al. Table 2. Retinoblastoma: Hereditary cases* pn(t, m(t) t+ 1) HC(t) (cal- (cal- HU(t) HB(t) (calcu- cu- cu- (pre- (pre- Age (years) lated) lated) lated) dicted) dicted) (a) For m(c) = (b) For m(o) = * Undiagnosed fraction of cases, mean number of tumors, and the product un(t, t + 1) as functions of age, laterality, and tumor multiplicity [m( )]. susceptible cells being retinoblasts before they stop dividing and fully differentiate. If the critical event occurs after a cell has reached this condition, no tumor will develop. Unfortunately, there are no quantitative data bearing on the growth pattern of the retina. Since the population of cells derived from retinoblasts is measured in millions, the formation of tumor is a very rare event at the cellular level, even in gene carriers. From this conclusion we surmise that the critical event could be a somatic mutation, although evidence that this may be the case is indirect (12). In discussing the hereditary form of retinoblastoma we do not mean to overlook those instances in which affected members of a pedigree cannot be explained by the low penetrance of the gene or by gonadal mosaicism. Evidently some individuals are carriers of a "premutation" which can become a germinal mutation with the usual penetrance (13). In some pedigrees the first mutation referred to here seems to occur in two steps rather than one. Carriers of this premutation are not counted here as carriers of the retinoblastoma gene. Retinoblastoma occurs only very rarely (one per 30,000) in individuals who are not genetically susceptible. If the interpretation is correct that such cases are the result of two somatic mutations, why is their age distribution the same as that for hereditary unilateral cases? We presume that the second event (mutation) is the same in the two cases and occurs with the same distribution in time. In order for the second event to be determining, the other mutation must obviously have occurred prior to this time. A consideration of the growth and transformation of susceptible cells suggests that this in fact should be the case. We assume that a susceptible cell is one that is capable of further cell division. Most cell divisions will occur after the cell population has reached a large number; from a consideration of cell doubling it can be shown that almost 90% of the cell divisions occur in the last three divisions. The second event, whose numbers will be related to the number of such cells, will generally have occurred late. In individuals who are not genetically susceptible, tumor cells will only occur in those cells that have sustained the first mutation. Since a cell that sustains the latter very early in the cell doubling process Proc. Nat. Acad. Sci. USA 72 (1975) 5119 will produce a large clone of mutated cells, it follows that most cases of nonhereditary tumor will occur in individuals in whom the first mutation has occurred early and the second late, regardless of the nature of the mutational processes. Thus the second mutation, the one that determines age-specific incidence, will have occurred with essentially the same age distribution in all unilateral cases, whether hereditary or nonhereditary. The model presented here is a general one for embryonal tumors and could be applied to Wilms' tumor and neuroblastoma as well (6, 7). Such an analysis is not possible now, however, because the familial data are insufficient, since so few survivors of these tumors have reproduced. If childhood tumors result from germinal and somatic mutations, then their total incidences will be determined by mutation rates. If q denotes the gene frequency of the retinoblastoma germinal mutation, and p (essentially unity) that of the normal allele, then the incidence of the hereditary form, ih, will be ih = 2pq[1 - e-m1)] 2q[1 - e-m-( ]. At mutational equilibrium q is related to the germinal mutation rate, Mug, and the coefficient of selection, s, by jig = sq, so that ih = 2,ug[1 - e-1'ix ]1s = a,u. The incidence of the nonhereditary form, in, will be dependent upon the mean number of tumors, mn(o), in the fraction of the population that is not genetically predisposed to tumor: = (1-2pq)[1 - em]) since mn(o) << 1. As shown previously, the second mutation occurs no differently in hereditary and nonhereditary cases, so m(oo) describes the number of tumors that arise in an equivalent number of cells already containing the first mutation, whether the latter has arisen germinally or somatically. But the equivalent number of cells with the first mutation in nonhereditary cases is simply proportional to the somatic mutation rate, us, for that first step; hence mn (o) = ALI-m(co) = b-p, The total incidence, i, is therefore i = ih + in = apg + b-ats. If the relevant mutation rates are spontaneous, the incidences should be constant with respect to time and place, except for changes in reproduction by survivors. Significant variations would suggest the local operation of environmental mutagenic carcinogens. For retinoblastoma and Wilms' tumor the world-wide incidences are extremely even, and, with the possible exception of Africa and India, the same is true for neuroblastoma (14). This work was supported in part by Grant CA11430 from the National Cancer Institute and by Medical Genetics Center Grant GM from the National Institute of General Medical Sciences. 1. Knudson, A. G., Strong, L. C. & Anderson, D. E. (1973) Prog. Med. Genet. 9, Knudson, A. G. (1973) Adv. Cancer Res. 17, Ashley, D. J. B. (1969) Br. J. Cancer 213, Ashley, D. J. B. (1969) J. Med. Genet. 6,

5 5120 Medical Sciences: Knudson et al. 5. Knudson, A. G. (1971) Proc. Nat. Acad. Sci. USA 68, Knudson, A. G. & Strong, L. C. (1972) Am. J. Hum. Cenet. 24, Knudson, A. G. & Strong, L. C. (1972) J. Nat. Cancer Inst. 48, Weller, C. V. (1941) Cancer Res. 1, Falls, H. F. (1947) J. Am. Med. Assoc. 133, Proc. Nat. Acad. Sci. USA 72 (1975) 10. Falls, H. F. & Neel, J. V. (1951) Arch. Ophthalmol. 46, Macklin, M. T (1960) Am. J. Hum. Genet. 12, Knudson, A. G. (1974) Am. J. Pathol. 77, Neel, J. V. (1962) in Methodology in Human Genetics, ed. Burdette, W. J. (Holden-Day, San Francisco, Calif.), p Knudson, A. C. (1975) Pediat. Res., in press.