Proc. Natl. Acad. Sci. USA Vol. 73, No. 10, pp. 3360-3363, October 1976
Statistics
A structural model of radiation effects in living cells (mechanism of clustering/low linear energy transfer/high linear energy transfer/dose-response/dose rate)
JERZY NEYMAN* AND PREM S. PURIt *
Statistical Laboratory, University of California, Berkeley, Calif. 94720; and t Department of Statistics, Purdue University, West Lafayette, Indiana 47907
Contributed by Jerzy Neynan, July 12, 1976
ABSTRACT The chance mechanism of cell damage and of repair in the course of irradiation involves two details familiar to biologists that thus far seem to have been overlooked in mathematical treatment. One of these details is that, generally, the passage of a single "primary" radiation particle generates a "clus er" of secondaries which can produce "hits" that damage the living cell. With high linear energy transfer, each cluster contains very many secondary particles. With low linear energy transfer, the number of secondaries per cluster is generally small. The second overlooked detail of the chance mechanism is concerned with what may be called the time scales of radiation damage and of the subsequent repair. The generation of a cluster of secondary particles and the possible hits occur so rapidly that, for all practical purposes, they may be considered as occurring instantly. On the other hand, the subsequent changes in the damaged cells appear to require measurable amounts of time. The constructed stochastic model embodies these details, the clustering of secondary particles and the time scale difference. The results explain certain details of observed phenomena.
model. Compared to its ancestors which came to our attention, the proposed model takes into account the striking difference between what may be called the time scales appropriate to biological and to physical aspects of the phenomena. Empirical findings The empirical findings which stimulated the present paper are illustrated in the following two diagrams. The first, our Fig. 1, illustrates the so-called "dose-rate effect" of gamma radiation (low LET) on the induction of a particular leukemia in mice (5). The point is that the same total amount of irradiation can be administered uniformly either over a long or over a relatively short period of time. In the former case, we speak of "low" dose rate and in the second of "high" dose rate. The graphs in Fig. 1 indicate that, with a "high" dose rate (the upper curve), a substantially higher percentage of irradiated mice acquire leukemia than with the low dose rate. Similar results were found for other cancers. Fig. 1 illustrates also another phenomenon. This is that, perhaps unexpectedly, the effectiveness of gamma radiation administered at a high dose rate in inducing leukemia is not a monotone function of the dose. The observed frequency of the particular leukemia begins by increasing with the dose, reaches a maximum, and then decreases. Fig. 2 is reproduced from an article by Totter (6). It compares the life-shortening effects of gamma rays and of neutrons, both administered at various dose rates and at various doses. The several curves relating to gamma rays exhibit strong dose-rate effects. Also, some of them suggest the presence of the maximum of dose effectiveness clearly shown in Fig. 1. Turning to neutrons, we see the same indication of a maximum of dose effect but, curiously, no noticeable dose-rate effect to the left of the point of maximum. The "life shortening" indicated in Fig. 2 seems to have been measured by comparing the median life span of irradiated mice with that of the controls. The physical properties of the various kinds of radiation, the properties that are particularly relevant to our work, are illustrated in Figs. 3 and 4. Fig. 3 represents a photograph of a cloud chamber exposed to a certain kind of irradiation. We are grateful to Alexander Grendon of the Donner Laboratory, University of California at Berkeley, for letting us use this photograph. The particularly relevant detail of Fig. 3 is the presence of crisscrossing lines, a few rather broad and many very thin. The lines mark the tracks of certain particles. They are composed of minute droplets formed about ions generated by the particles. Where the visible line is broad, the passage of the particle is accompanied by the appearance of many ions which travel to considerable distances away from the particle's track. Otherwise, there are only relatively few ions. An important detail to be added to the facts illustrated in Fig. 3 is that the particles in question travel at enormous speeds, so that they cross a cell within a minute fraction of a second.
We use the terms structural or stochastic models of a phenomenon to designate a chance mechanism defined in terms of some hypothetical entities having some specified hypothetical properties, a mechanism the operation of which is expected to mimic the phenomenon studied. A stochastic model, a concept akin to Borel's idea of the principal problem of mathematical statistics (1) is contrasted with the brief term model now frequently encountered in statistical literature. This brief term is used to designate a more or less complicated formula, invented to fit the observations without any consideration of the mechanism that might have produced them. For this kind of "model" our preferred term is interpolatory procedure. Undoubtedly, such procedures are useful and, in fact, they appear unavoidable when an effort is made to adjust the details of a stochastic model to fit the observations. The ultimate goal of the present study is a stochastic model of phenomena developing in irradiated experimental animals. However, the present paper is limited to irradiation effects on cells of some homogeneous tissue. The literature on this subject is quite rich. For example, see two recent papers by Payne and Garrett (2, 3). However, there appear to be many points of vagueness interestingly discussed by Mole (4). The plan of the paper is as follows. First, we outline certain frequency findings related to experimental animals. At least some of these findings must be credited to Upton et al. (5). Others are taken from Totter (6). It is these findings that our model is intended to explain. The phenomena of interest have two aspects. One is biological and depends upon properties of living cells. The other aspect is physical, depending upon properties of radiation of one kind or another, e.g., high linear energy transfer (LET) and low LET. After illustrating these two aspects taken separately and in combination, we offer our Abbreviations: LET, linear energy transfer; pgf, probability generating function.
3360
Statistics:
Neyman and Puri
Proc. Natl. Acad. Sci. USA 73 (1976)
3361
A CONTROLS-
V 7 rods/min o 15 rods/day (-0.01 rods/min) o 5 rods/day ('0.004 rods/min
0
L&J -
z U
z
LAJ y -J
0
200 300 600 400 500 TOTAL DOSE (RADS) FIG. 1. Incidence of myeloid leukemia in relation to dose and dose ratio of gamma radiation. One rad = 0.01 J/kg. From ref. 6. This figure and Fig. 2 are copyright 3 1972 by the Regents of the University of California; reprinted by the permission of the University of California Press. 100
Fig. 4, taken from the article of Barendsen (7), illustrates his ideas about happenings within a living cell subjected to irradiation. The panel on the left illustrates the presumed effect of irradiation with x-rays and the panel on the right, that of irradiation by the heavy particles, a. The little circles in the two panels, the two on the left and one on the right, typify a substantial accumulation of ions. If such an accumulation ccurs in a sensitive volume within a living cell (which we shall describe as "target"), then at least some of the ions could produce a "biological damage" of some particular kind. It will be seen that, with low LET irradiation illustrated on the left, there would be only a few points within a cell with dense accumulation of ions. This is contrasted with the right panel, where dense accumulations of ions occur essentially all along the track of the hypothetical high LET particle. Specifically for neutrons, the following complication must be taken into account. The neutrons themselves do not produce ionization. However, if a neutron hits an atom, this atom splits into several parts and these parts generate ions. A similar phenomenon is illustrated in the left panel of Fig. 4, which represents tracks of x-rays. It will be seen that, in several spots, a track of a single x-ray that crosses the whole circle is complicated by shorter or longer "branches" to the left or to the right. These branches are supposed to be due to "secondary electrons" produced by the "primary."
ui20
250E85 rods/min 200
z
6.7rods/min
LL
I
,ED 100o l Z 50H U
001 -0.02 rods/min
.000098-0.00099 rads/min_-
bi 0
The above phenomena within an irradiated cell can be summarized as follows. The source of radiation emits entities that we shall describe as primary particles. When a primary particle traverses a living cell, it generates a varying number of secondary particles. These secondary particles, and isually also the primary, produce ions which can "hit" sensitive targets within the cell and produce damage. All this occurs within a very brief period of time, for all practical purposes, instantly. Stochastic model of radiation effects on single cells The assumptions underlying our model are of three different categories: (i) those regarding the nature of irradiation, (ii) those regarding a cell contemplated, and (Mii) those concerned with both the nature of irradiation and with the nature of tissue of which the given cell is a part. (i) The primary particles arrive in a cell in a time-homogeneous Poisson process with a fixed density X per unit of time and per unit volume. Each primary arriving in a cell generates a "cluster" of secondaries. The number v of secondaries in a cluster is a random variable. Its probability generating function and its expectation, assumed finite, are denoted by g(') and v1, respectively. The variables p corresponding to different clusters are independent, identically distributed and are independent of all other variables of the model. The secondary particles
~
itt-/
'
0.0028-0.0038 rods/min
0.0004-0.0007 rods/min
io Amn C 10(
VW lqMo DOSE (RADS) FIG. 2. Life shortening in female mice as influenced by dose and dose rate of gamma rays and neutrons. Open symbols represent gamma rays; filled symbols, neutrons. From ref. 6.
FIG. 3. Photograph of a cloud chamber.
3362
Statistics: Neyman and Puri
Proc. Natl. Acad. Sci. USA 73 (1976)
B
A
of region A has no effect on the cell considered. On the other hand, if a primary particle crosses A, generating a cluster of v secondaries, then each secondary has a fixed probability gr of hitting R and a fixed probability 7r2 of hitting K, with 7r, + w2 < 1. Denote by t and v the number of hits on R and K, respectively, resulting from a single primary particle crossing A. If it generates v secondaries, then the conditional probability generating function (pgf) of t and v is the multinomial [1 - - 7r2 + r1s1 + r2S2] [1] where si and S2 are the arguments of the pgf. It follows that the unconditional pgf of t and v is given by g[1 -
FIG. 4. Schematic representation of ionization distribution in small volumes within a cell, irradiated with equal doses of x-rays (A) or a-radiation (B). From ref. 7.
forming a cluster instantly "disperse," independently from each other and independently from all other variables of the system. The distribution of variables v characterizes the nature of irradiation. For example, PI for neutrons is likely to be much larger than that for gamma rays, etc. (fl) Within each live cell we visualize two distinct "targets," the biological identity of which we do not attempt to specify. Possibly they can be some particular points within a chromosome, etc. These targets ar denoted by R and K, connoting. "repairable" and "killing." Both targets are located within a region within the cell denoted by A, connoting region of "accessibility." The same letter A will denote the volume of the region. The following assumptions are adopted. If target R is "hit" by a secondary particle, then the cell experiences "repairable" damage. We abstain from specifying the mechanism of repair, which may involve enzymes, etc. Following the Markovian way of thinking, we assume that if at time t a cell has k unrepaired hits on R, the probability of one repair occurring within [t, t + h) is akh + o(h). Each unrepaired hit on R is assumed to cause danger of the cell's becoming permanently damaged (but not killed!). In the presence of k unrepaired hits at time t, the probability of permanent damage of the cell is assumed to be (3kh + o(h). The biological nature of the permanent damage is not specified, but judging from relevant literature, it will frequently consist in the cell's becoming the first of an initiated development of some cancer. For this reason, it will be convenient to speak of the permanent damage in terms of carcinogenesis. A third possibility is assumed with regard to each unrepaired hit on R. This is the possibility that as a result of an accumulation of such hits the cell may actually die. Accordingly, in the presence of k unrepaired hits on R at time t, the probability of these hits' killing the cell in [t, t + h) is assumed to be ykh +
o(h).
Regarding target K we assume that a single hit causes the death (or "inactivation") of the cell. One more assumption seems in order. This is that the cell considered may die from causes other than those enumerated above. Thus, we assume that, irrespective of the number of hits, the death of a cell may occur in [t, t + h) with probability Ah +
o((h).
In the above, the letters a, -y, and 6 designate nonnegative constants. It is possible that considerations of cell biology will justify the assumption that some of these constants are zero. Regarding the region of accessibility, the following assumptions are made. A passage of a primary particle outside (3,
rl
-
r2
+ 7rlSl +
[2]
7r2s2].
Our last hypothesis defining the model is that the probability of more than one of the events enumerated above occurring in [t, t + h) is o(h). (iii) We begin by emphasizing the difference in the time scales of physical and biological developments embodied in the stochastic model defined above. Following a passage of a primary particle, the generation of a cluster of secondaries, their dispersal, and hits are assumed to occur instantly. On the other hand, the various contemplated changes in the irradiated cell are assumed to require some measurable amounts of time. While it is hoped that the model defined above will prove adequate to explain the developments in cells of particular tissues, it is anticipated that the details of the model will change from one tissue to another. In particular, this is likely to apply to the region of accessibility A, probably rather small within a bone and large within a liquid. We are less confident about the distribution of the number v of secondary particles per one primary of a specified kind, e.g., an a particle or a gamma ray. But it does seem a possibility that a particle of a specified kind crossing a volume of some liquid will geneiate a number of secondaries tending to be different from that generated within some firm medium. Similar remarks apply to probabilities ri and 7r2. Thus, in addition to the rate X of the arrival of primaries, the model involves seven adjustable parameters: a, my, 6, A, wl, and 7r2. Also adjustable is the pgf g(.) which characterizes the distribution of the number of secondary particles generated by a single primary. We presume that the rate X can be estimated through some physical experiments, with no reference to irradiated tissues. Also we presume that, for some particular cases, certain a priori considerations will determine some of the other adjustable elements of the model. For example, with UV irradiation it may be judged a priori that g(s) s. In general, however, the test of the adequacy of the model, intended to cover both the low and the -high LET irradiation, will depend on further studies. Basically, the question is whether the striking differences among the effects of the different kinds of irradiation applied to different tissues could be summed up in terms of differences in the pgf g(.) and in terms of the numerical values of the seven adjustable parameters. The experiments contemplated are of the following type. We visualize a live organism irradiated at a uniform rate X (3,
v
for
a
period
of time from t =0
to t
=
T > 0. At time T the ir-
radiation is discontinued. Concentrating our attention on a single cell of some particular tissue of the irradiated organism we consider three random variables depending on time t which jointly define a Markov process: X(t) = number of unrepaired hits on R. Y(t) = number of hits on R turning the cell into a cancer cell, Z(t) = number of cell-killing events experienced up to time t, that is, number of hits on K, plus number of killing
Neyman and Puri
Statistics:
Proc. Natl. Acad. Sci. USA 73 (1976)
events due to unrepaired hits on R, plus the number of killing events from "other causes." For any nonnegative t we write Pkm(t) = PfX(t) = k, Y(t) = 1, Z(t) = mj.
[3]
Then, for t < T and for a small positive number h, we have Pklm (t + h) = ah(k + 1)P k+m(t) + 3h(k + 1)Pk + I, -,m(t) + -yh(k + 1)Pk +l,,m l(t) +6hP klml(t) k
+ XhA
+
m
X0
EZPi,1,j(t) n -oZPIP= njP i-O j0 ,7 = m-jAl ni
Pk4,m(t)[1 - kh(a + f
4=k-4
+ Ay) -h - XAh] + o(h). [4]
Formula 4 is explicit in defining the process studied. Compared with other models which we had the occasion to-examine, the essential novelty of Formula 4 is the presence of terms with the coefficient XAh. These terms correspond to the assumed possibility that during the time interval [t, t + h) a primary irradiation particle will cross the region of accessibility of the cell, that this primary will generate a number n of secondaries and that these secondaries will produce just enough hits on R and on K targets to bring the values of X(t + h) and Z(t + h) to the levels k and m from the possible values of i and j they could have had at time t. Starting with Formula 4, the reader will have no difficulty in calculating the time derivative of Pkk,, (t). Then, multiplying this derivative by s1k s2I s3m and summing for k, 1, and m from zero to infinity, one obtains the familiar first-order partial differential equation for G(si,82,s3 t), the pgf of the three variables X(t), Y(t), and Z(t). The solution so obtained will apply to values of t c T. The solution valid for all nonnegative values of t is given by the formula G(sbs2,s3it) = exp{-&t(1 -s3) -
rig x(O,t-T)11 g(J(s1, S2,s3; ,))]dT [5] -
where J(SlS2?S3;T) = 1 - 7r 2 + 7r2s3 + 1h(ss2,S3;T), [6] h(s1 s2, S3; T) = s1expl-(a + fi + 'y)r + (a + #S2 + 'ys3)(a + # + 'y)F[1 -exp(-[a + +iY]T)]. [7]
It will be noticed that G(1,0,0It) represents the probability that at time t the cell considered will be alive and not carcinogenic. Also, the difference G(1,1,0 It) - G(1,0,0O t) represents the probability that, at time t, the cell in question will be alive but carcinogenic. Relationships of this kind constitute a bridge between the theoretical model constructed and the observable phenomena. Concluding remarks As emphasized earlier, the proposed model is formulated in terms of hypothetical entities, namely the "primary" and the "clusters" of "secondary" particles, supplied with certain hy-
3363
pothetical properties, So defined, the model implies the Formula 5, which characterizes the results of the chance mechanism contemplated. This formula involves an unspecified function g(') and a relatively large number, namely 8, of adjustable parameters. The introduction of so many parameters is motivated by the desire not to omit a detail of the modeled phenomenon which may be important. For example, it is likely that the parameter 6 may be zero. It is expected that this particular detail, as well as many others, will be resolved in further studies, partly theoretical and partly empirical. As already mentioned, we expect that with UV irradiation the assumption g(s) s is appropriate. On the other extreme with high LET irradiation, the appropriate g(.) may well be the pgf of the socalled "contagious distributions of type A" (8, 9). Results of studies of this kind may be exemplified by the finding that, in order to ensure the possibility of a maximum of the dose-response curve found by Upton et al. and illustrated in Fig. 1, it is necessary to assume that ir2> 0. This, in fact, implies the necessity of assuming two disjoint targets R and K within each cell. However, the assumption W2 > 0 is not sufficient for the existence of the maximum. With appropriate values of the several parameters involved in the model, there would be no maximum in the dose-response curve even if w2 > 0. It is hoped that the details of the present study will be soon published elsewhere. This paper was prepared with partial support of the National Institute of Environmental Health- Sciences, Department of Health, Education, and Welfare, and the Environmental Protection Agency. Also, support is gratefully acknowledged from the Energy Research and Development Administration, provided through Lawrence Berkeley Laboratory. 1. Borel, E. (1924) &ments de la Theorte des Probabiliti (Hermann, Paris). Z Payne, M. G. & Garrett, W. R. (1975) "Models for cell survival with low LET radiation," Radiat. Res. 62,169-479. 3. Payne, M. G. & Garrett, W. R. (1975) "Some relations between cell survival models having different inactivation mechanisms, Radiat. Res. 62,388-394. 4. Mole, R. H. (1964) "Closing summary," Proceedings of the Symposium on Biological Effects of Neutron Irradiations (International Atomic Energy Agency, Vienna), Vol. 2, pp. 429431. 5. Upton, A. C., Randolph, M. L., Darden, E. G., Jr. & Conklin, J. W. (1964) "Relative biological effectiveness of fast neutrons for late somatic effects in mice," Proceedings of the Symposium on Biological Effects of Neutron Irradiations (International Atomic Energy Agency, Vienna), Vol. 2, pp. 337-345. 6. Totter, J. R. (1972)-"Research programs of the Atomic Energy Commission's Division of Biology and Medicine relevant to problems of health and pollution," Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. of California Press, Berkeley), Vol. 6, pp. 71-100. 7. Barendsen, G. W. (1964) "Energy distributions from different radiations in relation to biological damage," Proceedings of the Symposium on Biological Effects of Neutron Irradiations (International Atomic Energy Agency, Vienna)' Vol. 2, pp. 379387. 8. Neyman J. (1939) "On a new class of contagious distributions, applicable in entomology and bacteriology," Ann. Math. Statist. 10,35-57. 9. Neyman, J. & Scott, E. L. (1972) "Processes of clustering and applications," in Stochastic Point Processes, ed. Lewis, P. (John Wiley, New York), pp. 646-681.
Statistics
A structural model of radiation effects in living cells (mechanism of clustering/low linear energy transfer/high linear energy transfer/dose-response/dose rate)
JERZY NEYMAN* AND PREM S. PURIt *
Statistical Laboratory, University of California, Berkeley, Calif. 94720; and t Department of Statistics, Purdue University, West Lafayette, Indiana 47907
Contributed by Jerzy Neynan, July 12, 1976
ABSTRACT The chance mechanism of cell damage and of repair in the course of irradiation involves two details familiar to biologists that thus far seem to have been overlooked in mathematical treatment. One of these details is that, generally, the passage of a single "primary" radiation particle generates a "clus er" of secondaries which can produce "hits" that damage the living cell. With high linear energy transfer, each cluster contains very many secondary particles. With low linear energy transfer, the number of secondaries per cluster is generally small. The second overlooked detail of the chance mechanism is concerned with what may be called the time scales of radiation damage and of the subsequent repair. The generation of a cluster of secondary particles and the possible hits occur so rapidly that, for all practical purposes, they may be considered as occurring instantly. On the other hand, the subsequent changes in the damaged cells appear to require measurable amounts of time. The constructed stochastic model embodies these details, the clustering of secondary particles and the time scale difference. The results explain certain details of observed phenomena.
model. Compared to its ancestors which came to our attention, the proposed model takes into account the striking difference between what may be called the time scales appropriate to biological and to physical aspects of the phenomena. Empirical findings The empirical findings which stimulated the present paper are illustrated in the following two diagrams. The first, our Fig. 1, illustrates the so-called "dose-rate effect" of gamma radiation (low LET) on the induction of a particular leukemia in mice (5). The point is that the same total amount of irradiation can be administered uniformly either over a long or over a relatively short period of time. In the former case, we speak of "low" dose rate and in the second of "high" dose rate. The graphs in Fig. 1 indicate that, with a "high" dose rate (the upper curve), a substantially higher percentage of irradiated mice acquire leukemia than with the low dose rate. Similar results were found for other cancers. Fig. 1 illustrates also another phenomenon. This is that, perhaps unexpectedly, the effectiveness of gamma radiation administered at a high dose rate in inducing leukemia is not a monotone function of the dose. The observed frequency of the particular leukemia begins by increasing with the dose, reaches a maximum, and then decreases. Fig. 2 is reproduced from an article by Totter (6). It compares the life-shortening effects of gamma rays and of neutrons, both administered at various dose rates and at various doses. The several curves relating to gamma rays exhibit strong dose-rate effects. Also, some of them suggest the presence of the maximum of dose effectiveness clearly shown in Fig. 1. Turning to neutrons, we see the same indication of a maximum of dose effect but, curiously, no noticeable dose-rate effect to the left of the point of maximum. The "life shortening" indicated in Fig. 2 seems to have been measured by comparing the median life span of irradiated mice with that of the controls. The physical properties of the various kinds of radiation, the properties that are particularly relevant to our work, are illustrated in Figs. 3 and 4. Fig. 3 represents a photograph of a cloud chamber exposed to a certain kind of irradiation. We are grateful to Alexander Grendon of the Donner Laboratory, University of California at Berkeley, for letting us use this photograph. The particularly relevant detail of Fig. 3 is the presence of crisscrossing lines, a few rather broad and many very thin. The lines mark the tracks of certain particles. They are composed of minute droplets formed about ions generated by the particles. Where the visible line is broad, the passage of the particle is accompanied by the appearance of many ions which travel to considerable distances away from the particle's track. Otherwise, there are only relatively few ions. An important detail to be added to the facts illustrated in Fig. 3 is that the particles in question travel at enormous speeds, so that they cross a cell within a minute fraction of a second.
We use the terms structural or stochastic models of a phenomenon to designate a chance mechanism defined in terms of some hypothetical entities having some specified hypothetical properties, a mechanism the operation of which is expected to mimic the phenomenon studied. A stochastic model, a concept akin to Borel's idea of the principal problem of mathematical statistics (1) is contrasted with the brief term model now frequently encountered in statistical literature. This brief term is used to designate a more or less complicated formula, invented to fit the observations without any consideration of the mechanism that might have produced them. For this kind of "model" our preferred term is interpolatory procedure. Undoubtedly, such procedures are useful and, in fact, they appear unavoidable when an effort is made to adjust the details of a stochastic model to fit the observations. The ultimate goal of the present study is a stochastic model of phenomena developing in irradiated experimental animals. However, the present paper is limited to irradiation effects on cells of some homogeneous tissue. The literature on this subject is quite rich. For example, see two recent papers by Payne and Garrett (2, 3). However, there appear to be many points of vagueness interestingly discussed by Mole (4). The plan of the paper is as follows. First, we outline certain frequency findings related to experimental animals. At least some of these findings must be credited to Upton et al. (5). Others are taken from Totter (6). It is these findings that our model is intended to explain. The phenomena of interest have two aspects. One is biological and depends upon properties of living cells. The other aspect is physical, depending upon properties of radiation of one kind or another, e.g., high linear energy transfer (LET) and low LET. After illustrating these two aspects taken separately and in combination, we offer our Abbreviations: LET, linear energy transfer; pgf, probability generating function.
3360
Statistics:
Neyman and Puri
Proc. Natl. Acad. Sci. USA 73 (1976)
3361
A CONTROLS-
V 7 rods/min o 15 rods/day (-0.01 rods/min) o 5 rods/day ('0.004 rods/min
0
L&J -
z U
z
LAJ y -J
0
200 300 600 400 500 TOTAL DOSE (RADS) FIG. 1. Incidence of myeloid leukemia in relation to dose and dose ratio of gamma radiation. One rad = 0.01 J/kg. From ref. 6. This figure and Fig. 2 are copyright 3 1972 by the Regents of the University of California; reprinted by the permission of the University of California Press. 100
Fig. 4, taken from the article of Barendsen (7), illustrates his ideas about happenings within a living cell subjected to irradiation. The panel on the left illustrates the presumed effect of irradiation with x-rays and the panel on the right, that of irradiation by the heavy particles, a. The little circles in the two panels, the two on the left and one on the right, typify a substantial accumulation of ions. If such an accumulation ccurs in a sensitive volume within a living cell (which we shall describe as "target"), then at least some of the ions could produce a "biological damage" of some particular kind. It will be seen that, with low LET irradiation illustrated on the left, there would be only a few points within a cell with dense accumulation of ions. This is contrasted with the right panel, where dense accumulations of ions occur essentially all along the track of the hypothetical high LET particle. Specifically for neutrons, the following complication must be taken into account. The neutrons themselves do not produce ionization. However, if a neutron hits an atom, this atom splits into several parts and these parts generate ions. A similar phenomenon is illustrated in the left panel of Fig. 4, which represents tracks of x-rays. It will be seen that, in several spots, a track of a single x-ray that crosses the whole circle is complicated by shorter or longer "branches" to the left or to the right. These branches are supposed to be due to "secondary electrons" produced by the "primary."
ui20
250E85 rods/min 200
z
6.7rods/min
LL
I
,ED 100o l Z 50H U
001 -0.02 rods/min
.000098-0.00099 rads/min_-
bi 0
The above phenomena within an irradiated cell can be summarized as follows. The source of radiation emits entities that we shall describe as primary particles. When a primary particle traverses a living cell, it generates a varying number of secondary particles. These secondary particles, and isually also the primary, produce ions which can "hit" sensitive targets within the cell and produce damage. All this occurs within a very brief period of time, for all practical purposes, instantly. Stochastic model of radiation effects on single cells The assumptions underlying our model are of three different categories: (i) those regarding the nature of irradiation, (ii) those regarding a cell contemplated, and (Mii) those concerned with both the nature of irradiation and with the nature of tissue of which the given cell is a part. (i) The primary particles arrive in a cell in a time-homogeneous Poisson process with a fixed density X per unit of time and per unit volume. Each primary arriving in a cell generates a "cluster" of secondaries. The number v of secondaries in a cluster is a random variable. Its probability generating function and its expectation, assumed finite, are denoted by g(') and v1, respectively. The variables p corresponding to different clusters are independent, identically distributed and are independent of all other variables of the model. The secondary particles
~
itt-/
'
0.0028-0.0038 rods/min
0.0004-0.0007 rods/min
io Amn C 10(
VW lqMo DOSE (RADS) FIG. 2. Life shortening in female mice as influenced by dose and dose rate of gamma rays and neutrons. Open symbols represent gamma rays; filled symbols, neutrons. From ref. 6.
FIG. 3. Photograph of a cloud chamber.
3362
Statistics: Neyman and Puri
Proc. Natl. Acad. Sci. USA 73 (1976)
B
A
of region A has no effect on the cell considered. On the other hand, if a primary particle crosses A, generating a cluster of v secondaries, then each secondary has a fixed probability gr of hitting R and a fixed probability 7r2 of hitting K, with 7r, + w2 < 1. Denote by t and v the number of hits on R and K, respectively, resulting from a single primary particle crossing A. If it generates v secondaries, then the conditional probability generating function (pgf) of t and v is the multinomial [1 - - 7r2 + r1s1 + r2S2] [1] where si and S2 are the arguments of the pgf. It follows that the unconditional pgf of t and v is given by g[1 -
FIG. 4. Schematic representation of ionization distribution in small volumes within a cell, irradiated with equal doses of x-rays (A) or a-radiation (B). From ref. 7.
forming a cluster instantly "disperse," independently from each other and independently from all other variables of the system. The distribution of variables v characterizes the nature of irradiation. For example, PI for neutrons is likely to be much larger than that for gamma rays, etc. (fl) Within each live cell we visualize two distinct "targets," the biological identity of which we do not attempt to specify. Possibly they can be some particular points within a chromosome, etc. These targets ar denoted by R and K, connoting. "repairable" and "killing." Both targets are located within a region within the cell denoted by A, connoting region of "accessibility." The same letter A will denote the volume of the region. The following assumptions are adopted. If target R is "hit" by a secondary particle, then the cell experiences "repairable" damage. We abstain from specifying the mechanism of repair, which may involve enzymes, etc. Following the Markovian way of thinking, we assume that if at time t a cell has k unrepaired hits on R, the probability of one repair occurring within [t, t + h) is akh + o(h). Each unrepaired hit on R is assumed to cause danger of the cell's becoming permanently damaged (but not killed!). In the presence of k unrepaired hits at time t, the probability of permanent damage of the cell is assumed to be (3kh + o(h). The biological nature of the permanent damage is not specified, but judging from relevant literature, it will frequently consist in the cell's becoming the first of an initiated development of some cancer. For this reason, it will be convenient to speak of the permanent damage in terms of carcinogenesis. A third possibility is assumed with regard to each unrepaired hit on R. This is the possibility that as a result of an accumulation of such hits the cell may actually die. Accordingly, in the presence of k unrepaired hits on R at time t, the probability of these hits' killing the cell in [t, t + h) is assumed to be ykh +
o(h).
Regarding target K we assume that a single hit causes the death (or "inactivation") of the cell. One more assumption seems in order. This is that the cell considered may die from causes other than those enumerated above. Thus, we assume that, irrespective of the number of hits, the death of a cell may occur in [t, t + h) with probability Ah +
o((h).
In the above, the letters a, -y, and 6 designate nonnegative constants. It is possible that considerations of cell biology will justify the assumption that some of these constants are zero. Regarding the region of accessibility, the following assumptions are made. A passage of a primary particle outside (3,
rl
-
r2
+ 7rlSl +
[2]
7r2s2].
Our last hypothesis defining the model is that the probability of more than one of the events enumerated above occurring in [t, t + h) is o(h). (iii) We begin by emphasizing the difference in the time scales of physical and biological developments embodied in the stochastic model defined above. Following a passage of a primary particle, the generation of a cluster of secondaries, their dispersal, and hits are assumed to occur instantly. On the other hand, the various contemplated changes in the irradiated cell are assumed to require some measurable amounts of time. While it is hoped that the model defined above will prove adequate to explain the developments in cells of particular tissues, it is anticipated that the details of the model will change from one tissue to another. In particular, this is likely to apply to the region of accessibility A, probably rather small within a bone and large within a liquid. We are less confident about the distribution of the number v of secondary particles per one primary of a specified kind, e.g., an a particle or a gamma ray. But it does seem a possibility that a particle of a specified kind crossing a volume of some liquid will geneiate a number of secondaries tending to be different from that generated within some firm medium. Similar remarks apply to probabilities ri and 7r2. Thus, in addition to the rate X of the arrival of primaries, the model involves seven adjustable parameters: a, my, 6, A, wl, and 7r2. Also adjustable is the pgf g(.) which characterizes the distribution of the number of secondary particles generated by a single primary. We presume that the rate X can be estimated through some physical experiments, with no reference to irradiated tissues. Also we presume that, for some particular cases, certain a priori considerations will determine some of the other adjustable elements of the model. For example, with UV irradiation it may be judged a priori that g(s) s. In general, however, the test of the adequacy of the model, intended to cover both the low and the -high LET irradiation, will depend on further studies. Basically, the question is whether the striking differences among the effects of the different kinds of irradiation applied to different tissues could be summed up in terms of differences in the pgf g(.) and in terms of the numerical values of the seven adjustable parameters. The experiments contemplated are of the following type. We visualize a live organism irradiated at a uniform rate X (3,
v
for
a
period
of time from t =0
to t
=
T > 0. At time T the ir-
radiation is discontinued. Concentrating our attention on a single cell of some particular tissue of the irradiated organism we consider three random variables depending on time t which jointly define a Markov process: X(t) = number of unrepaired hits on R. Y(t) = number of hits on R turning the cell into a cancer cell, Z(t) = number of cell-killing events experienced up to time t, that is, number of hits on K, plus number of killing
Neyman and Puri
Statistics:
Proc. Natl. Acad. Sci. USA 73 (1976)
events due to unrepaired hits on R, plus the number of killing events from "other causes." For any nonnegative t we write Pkm(t) = PfX(t) = k, Y(t) = 1, Z(t) = mj.
[3]
Then, for t < T and for a small positive number h, we have Pklm (t + h) = ah(k + 1)P k+m(t) + 3h(k + 1)Pk + I, -,m(t) + -yh(k + 1)Pk +l,,m l(t) +6hP klml(t) k
+ XhA
+
m
X0
EZPi,1,j(t) n -oZPIP= njP i-O j0 ,7 = m-jAl ni
Pk4,m(t)[1 - kh(a + f
4=k-4
+ Ay) -h - XAh] + o(h). [4]
Formula 4 is explicit in defining the process studied. Compared with other models which we had the occasion to-examine, the essential novelty of Formula 4 is the presence of terms with the coefficient XAh. These terms correspond to the assumed possibility that during the time interval [t, t + h) a primary irradiation particle will cross the region of accessibility of the cell, that this primary will generate a number n of secondaries and that these secondaries will produce just enough hits on R and on K targets to bring the values of X(t + h) and Z(t + h) to the levels k and m from the possible values of i and j they could have had at time t. Starting with Formula 4, the reader will have no difficulty in calculating the time derivative of Pkk,, (t). Then, multiplying this derivative by s1k s2I s3m and summing for k, 1, and m from zero to infinity, one obtains the familiar first-order partial differential equation for G(si,82,s3 t), the pgf of the three variables X(t), Y(t), and Z(t). The solution so obtained will apply to values of t c T. The solution valid for all nonnegative values of t is given by the formula G(sbs2,s3it) = exp{-&t(1 -s3) -
rig x(O,t-T)11 g(J(s1, S2,s3; ,))]dT [5] -
where J(SlS2?S3;T) = 1 - 7r 2 + 7r2s3 + 1h(ss2,S3;T), [6] h(s1 s2, S3; T) = s1expl-(a + fi + 'y)r + (a + #S2 + 'ys3)(a + # + 'y)F[1 -exp(-[a + +iY]T)]. [7]
It will be noticed that G(1,0,0It) represents the probability that at time t the cell considered will be alive and not carcinogenic. Also, the difference G(1,1,0 It) - G(1,0,0O t) represents the probability that, at time t, the cell in question will be alive but carcinogenic. Relationships of this kind constitute a bridge between the theoretical model constructed and the observable phenomena. Concluding remarks As emphasized earlier, the proposed model is formulated in terms of hypothetical entities, namely the "primary" and the "clusters" of "secondary" particles, supplied with certain hy-
3363
pothetical properties, So defined, the model implies the Formula 5, which characterizes the results of the chance mechanism contemplated. This formula involves an unspecified function g(') and a relatively large number, namely 8, of adjustable parameters. The introduction of so many parameters is motivated by the desire not to omit a detail of the modeled phenomenon which may be important. For example, it is likely that the parameter 6 may be zero. It is expected that this particular detail, as well as many others, will be resolved in further studies, partly theoretical and partly empirical. As already mentioned, we expect that with UV irradiation the assumption g(s) s is appropriate. On the other extreme with high LET irradiation, the appropriate g(.) may well be the pgf of the socalled "contagious distributions of type A" (8, 9). Results of studies of this kind may be exemplified by the finding that, in order to ensure the possibility of a maximum of the dose-response curve found by Upton et al. and illustrated in Fig. 1, it is necessary to assume that ir2> 0. This, in fact, implies the necessity of assuming two disjoint targets R and K within each cell. However, the assumption W2 > 0 is not sufficient for the existence of the maximum. With appropriate values of the several parameters involved in the model, there would be no maximum in the dose-response curve even if w2 > 0. It is hoped that the details of the present study will be soon published elsewhere. This paper was prepared with partial support of the National Institute of Environmental Health- Sciences, Department of Health, Education, and Welfare, and the Environmental Protection Agency. Also, support is gratefully acknowledged from the Energy Research and Development Administration, provided through Lawrence Berkeley Laboratory. 1. Borel, E. (1924) &ments de la Theorte des Probabiliti (Hermann, Paris). Z Payne, M. G. & Garrett, W. R. (1975) "Models for cell survival with low LET radiation," Radiat. Res. 62,169-479. 3. Payne, M. G. & Garrett, W. R. (1975) "Some relations between cell survival models having different inactivation mechanisms, Radiat. Res. 62,388-394. 4. Mole, R. H. (1964) "Closing summary," Proceedings of the Symposium on Biological Effects of Neutron Irradiations (International Atomic Energy Agency, Vienna), Vol. 2, pp. 429431. 5. Upton, A. C., Randolph, M. L., Darden, E. G., Jr. & Conklin, J. W. (1964) "Relative biological effectiveness of fast neutrons for late somatic effects in mice," Proceedings of the Symposium on Biological Effects of Neutron Irradiations (International Atomic Energy Agency, Vienna), Vol. 2, pp. 337-345. 6. Totter, J. R. (1972)-"Research programs of the Atomic Energy Commission's Division of Biology and Medicine relevant to problems of health and pollution," Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. of California Press, Berkeley), Vol. 6, pp. 71-100. 7. Barendsen, G. W. (1964) "Energy distributions from different radiations in relation to biological damage," Proceedings of the Symposium on Biological Effects of Neutron Irradiations (International Atomic Energy Agency, Vienna)' Vol. 2, pp. 379387. 8. Neyman J. (1939) "On a new class of contagious distributions, applicable in entomology and bacteriology," Ann. Math. Statist. 10,35-57. 9. Neyman, J. & Scott, E. L. (1972) "Processes of clustering and applications," in Stochastic Point Processes, ed. Lewis, P. (John Wiley, New York), pp. 646-681.
