Download QUANTUM PHENOMENA IN THE BIOLOGICAL

QUANTUM PHENOMENA IN THE BIOLOGICAL
ACTION OF X-RAYS
E. U. CONDON
H. M. TERRILL
(From Department of Physics, Columbia University, N . Y . , and Institute of Cancer
Research, Columbia University, N . Y . , F. C . Wood, Director)
AND
It is a well-known theory of experimental physics that X-rays,
when absorbed by matter, are absorbed in discrete units or
quanta of energy. The size of these quanta increases uith
decreasing wave-length. Thus, while it is also true that visible
light is absorbed in discrete quanta, still here the wave-lengths
are so great and hence the quanta are so small that effects due to
the discreteness may easily become negligible. In the case of
the biological actions of X-rays, however, it appears that the
quanta are of sufficient size to make it necessary to consider the
effects of this discreteness in interpreting experiments.
The present article offers an analysis of such effects and a discussion of the data taken from experiments at the Institute of
Cancer Research by Dr. Francis C. Wood (1, 2) and his associates, on the killing of tumor tissue and other biological material
by X-rays, and, in particular, of data taken by Packard (3, 4) on
the killing of Drosophila eggs.
The discreteness of absorption has for its consequence that a
given small element of volume of biological material when exposed to the rays is only affected by them when such a discrete
act of absorption happens in the volume. The quantum of
X-ray energy, when absorbed, is taken up by one atom of the
absorbing substance and a high-speed photo-electron is liberated.
This electron moves about in the neighborhood of the place of
its liberation, losing energy by collisions with atoms and causing
a good deal of local ionization. It is presumably the disturbing
effect of this ionization on certain colloid equilibria which causes
biological action, but that question is outside the realm of this
paper. The distance traversed by the electrons, physical ex324
325
QUANTUM PHENOMENA OF X-RAYS
periments show, depends in the main only on the density of the
material through which they travel. Such measurements show
that the range of electrons of various voltages is
................................................ .0.38 mm.
................................................ .0.28 mm.
X-rays of wave-length of K line of Tungsten. ................ .0.08mm.
200 KV..
185 KV..
so that the volume in which all of the ionization takes place
is certainly at least as small as 0.1 mm. It is probably
considerably smaller because the paths of the electrons are
quite crooked, thus confirming the region of ionization to from
one tenth to one hundredth of this volume. Since it has been
shown that there is one pair of ions formed for about thirty to
forty equivalent volts of energy, it follows that such an absorption of one quantum of 160 KV. X-ray is much like a highly
localized burst of ionic shrapnel in which about 4,000 ion pairs
are liberated in less than a millionth of a cubic centimeter.
Any biological effect like the sterilization of an egg presumably
happens when one or more of these bursts of ionization occurs
in a certain sensitive volume of the egg. A burst of ionization
occurring in another part of the egg may suffice to produce the
chemical changes which result in the production of modified
plants or animals such as have been studied recently by Muller
( 5 ) for Drosophila and Goodspeed and Olsen (6) for tobacco
plants. Let v be the sensitive volume in which a quantum must
be absorbed in order that sterilization may occur.
Let I be the intensity of X-ray energy absorbed, measured in
terms of 0.01 erg/sec. mm.3 as unit. Let V be the mean kilovolt
equivalent of the X-rays in the radiation used, measured with
70 kilovolts as unit. The mean size of quantum is equal to
7
x
104 x
v x 4.77 x
300
10-10
=
1.113 X lo-' X V ergs,
so that the mean number of quanta absorbed per second per
cubic mm. is
I
0.01I
= - X 0.0898 X
1.113 X lo-' X V V
quanta
mm,3set,
-
326
E. U. CONDON AND H. M. TERRILL
If the unit of time be ten minutes, then the mean number of
quanta absorbed in time t per cubic millimeter is
ItV X 600 X 0.898 X
It
lo6 = 7X 5.358 X lo7.
Now if the cell is the sensitive unit of volume, it is known that
these are of the order of 5 p in diameter, that is, their volumes are
of the order of 4 X 10-lo cc. = 4 X lov7mm.8. Suppose this be
chosen as the unit of volume for measuring the sensitive volume,
v, of the egg. Then the mean number of quanta absorbed in
such a volume in ten minutes is
Itv
Itv
- X 5.358 X 4 = 2 1 . 4 7 *
V
The choice of units in what precedes has been made in such a
way that small numbers of these units would correspond to the
quantities actually appearing in Packard’s experiments on killing
of Drosophila. The disappearance of any large power of ten
from this final formula therefore means that the quanta are big
enough so that their discreteness must play a r61e in the interpretation of the experiments. Thus Packard, when using an intensity corresponding to one or two in our units and a voltage of
about two of our units, finds that about half the eggs are killed
in half an hour (t = 3). Therefore if the killing was done by one
or two quanta having been absorbed in the sensitive volume v,
the value of v must have been about v = 1/60 or about 6 or 7
cubic microns. If the volume is considerably larger than this,
then it follows that a good many separate quanta were absorbed
in it before sterilization resulted.
The mean number of quanta given by the last formula is equal
to what in mathematical theory of probability is known as the
ezpected number of absorptions. Because of the random nature
of the absorption process the case becomes one to which the
Poisson distribution law is applicable. The law requires that
if the expected number of absorptions in the volume is n then the
probability of exactly m absorptions occurring is equal to
rime-
ml
’
QUANTUM PHENOMENA OF X-RAYS
327
where e is the base of the natural logarithms. Thus the probability that no absorptions occur even when the expected
number is n is given by putting m = 0, i.e., it is e-n.
Now if a single absorption is sufficient to kill, then it is evident
that this probability of no absorptions is the probability of
survival. Therefore if absorption of one quantum will kill the
egg, the survival curve (number of eggs surviving after time t of
exposure to rays) is given by
328
E. U. CONDON AND H. M. TERRILL
in which N is the number surviving after time t and No is the
number in the sample at t = 0. From this form it is evident
that if log N / N ois plotted against t the data should give a straight
line whose slope is downward and from the value of which one
could infer the size of the sensitive volume if I and V are known.
Fig. 1is such a plot of the data given in Table 2 of Packard's
first article. The data are the same as those plotted by him in
in his Fig. 2. It is clear that the data are well represented by a
straight line indicating that the absorption of a single quantum
in a particular part of the egg was responsible for a sterilization.
However, for these data a measurement of I in absolute energy
units is lacking so that the slope cannot be used to infer a value
of the sensitive volume. In this case the X-ray tube was run
at 190 KV. which makes the strongest part of the X-ray spectrum
come at a wave-length corresponding to a quantum voltage of
about 155 KV. This is the highest voltage, and hence the
largest quanta, of any of the experiments.
It is next of interest to consider the shape of the curve in the
case that several quanta must be absorbed in the sensitive
volume for sterilization to occur. Suppose for instance that m
quanta have to be absorbed in the sensitive part before death
occurs. Clearly the probability of survival is then equal to the
probability that the sensitive volume was not the scene of the
absorption of m or more quanta. This is equal to the probability
that the number of quanta absorbed was 0 or 1 or 2 or 3 or .
(m - l),and this is equal to the sum of the probabilities of each
of these possibilities, that is,
..
This expression, often called Poisson's exponential summation,
is well known to students of the theory of probability for cases
of this sort. An interesting reference in this connection is to a
paper by F. Thorndike (7), in which it is applied numerically to
a great many different cases, from the distribution of occurrence
of visits from comets to the frequency with which wrong connections are made by telephone operat ors. This paper also gives
QUANTUM PHENOMENA OF X-RAYS
329
very full tables of the expression for different values of n
and m.
In Fig. 2 are given several theoretical curves, based on the
formula of the preceding paragraph, the values of the formula
being taken from the paper of Thorndike. The logarithm of the
FIQ.2
probability of survival is here plotted as a function of n, the
expected number of quanta absorbed. Curve (a), a straight line,
is for the case where only one quantum needs to be absorbed to
cause death. Curve ( b ) is for the case in which every egg needs
to absorb two quanta in its sensitive part to have death result.
Similarly (c) and (d) are for the cases where three and four quanta
330
E. U. CONDON AND H. M. TERRILL
are needed, respectively. When still more quanta are needed, the
corresponding curves are similar, save that they have an even
more gradual slope at the outset, but they ultimately become
parallel to the rest.
That these results of the mathematical theory are quite
reasonable is seen from the following qualitative argument.
Suppose a cell needs to have absorbed, say, ten quanta to be
killed. Then for quite a while very few deaths will result, for it
is unlikely that any one cell should receive ten quanta at the
outset. What is happening in the initial period is that all of
the cells are being hit but very few of them have been hit the
necessary ten times. This goes on, with few cells dying, until
most of the cells have been hit nine times. From now on each
cell only requires one more quantum to kill it, so from here on the
curve will behave much like the simpler curve for the case where
only one quantum is required altogether. The greater the
number of quanta needed the longer will be the period which is
devoted to getting most of the cells hit just one less than the
required number of times, after which the deaths occur at a
rapid rate just as in the case where but one quantum suffices to
kill.
Of course there is the possibility, too, that the eggs show
variation among themselves; that some of them require but one
quantum while others require two. Under such circumstances
the theoretical curves have the same general shape as in Fig. 2.
However, it is not worth while to go into a detailed discussion
of this at present.
The data presented in Fig. 2 (Table 1) of Packard’s second
article when the logarithm of the number surviving is plotted
against the X-ray exposure (intensity X time) gives a curve of
the form of the theoretical curves of Fig. 2. These data have
been plotted in this form in Fig. 3 where also the theoretical
curves have been drawn in for the cases in which (a) one, (b)
two, (c) three quanta are needed to cause death. From this we
infer that between two and three quanta were needed in these
experiments to kill the Drosophila eggs.
The question immediately presents itself: Why does one set
QUANTUM PHENOMENA OF X-RAYS
331
of data indicate only one quantum, while the other set indicates
that two or three are needed? The answer, we believe, lies in
the difference between the hardness of the rays in the two sets of
data. The second set of data is a composite curve taken with
three different wave-lengths, namely, 0.22,0.54 and 0.68 A. On
the other hand, in the first set of data the tube was run at 190
FIG.3
KV., giving the most intense radiation at a curve length of
about 0.08 A,, which corresponds to about 155 KV. electrons.
Since the size of the quanta is inversely proportional to the
wave-length, it is seen that in the second set of data the quanta
are only one third to one eighth as large as in the first set. It is
not surprising that several of the smaller quanta were needed to
do what one of the larger ones would do.
This would suggest that the critical amount of energy to kill a
Drosophila egg when released in the form of ionization in the
sensitive part of the egg is roughly of the order of the quantum
associated with 0.1 A. radiation, or 1.8 X lo-’ ergs. Naturally
this value is only an order of magnitude result.
In the case of the data presented in Fig. 3 absolute measurements of intensity have been made by Terrill, using a resistance
thermometer method recently described (8). Here again the
accuracy of the biological data seems to suffice merely for an
order of magnitude determination of the size of the sensitive
332
E. U. CONDON AND H. M. TERRILL
volume. A mean value for the absorbed energy in the wavelength range here used is .02 erg/sec. mmeSor two of the units of
this paper. The slope of the straight portion of the experimental curve given in Fig. 3 (making allowance for the conversion factor from common to natural logarithms) is 0.468, so
that for a voltage corresponding to the mean wave-length
(0.48 A.) one infers that the sensitive volume is equal to 5 cubic
microns.
Crowther (9) has also discussed the statistical consequences of
the discreteness of X-ray absorptions. He applies the formulas
to some of his own observations on the killing of Colpidium
Colpoda by X-rays. The survival curves for these are quite
different in form, having a long period of raying with scarcely
any deaths and then a period in which most of the specimens
are killed very rapidly. Interpreted statistichlly his data indicate that between forty and fifty quanta are needed to cause a
death. In such a case the discreteness of the absorption is not
so much in evidence as in the case of the Drosophila.
Another class of data for which the survival curve has a long
horizontal part followed by a rapid dip-of the kind that indicates the need for many quanta-is that taken on the killing of
various kinds of tumor tissue by Dr. F. C. Wood (1, 2) in the
Institute of Cancer Research. This might come about not only
through the possibility that such tumor cells, like the Colpidia,
require many quanta to kill them, but also because in the nature
of the case the portions of tissue employed had to consist of many
cells. “Survival” of such a piece of tissue probably means
survival of more than a certain minimum number of cells in the
piece. For simplicity suppose only one cell need survive out
of N . Let P be the probability that a cell survive, so 1 - P
is the probability that it die. Then (1 - P)” is the probability that all of them die, so 1 - (1 - P)“ is the probability
that not all die, i.e., that one or more survive; but this is
greater than P, as a simple algebraic treatment shows, being
for P very small, about equal to N P . Hence to reduce the
probability that any cell in the piece of tissue survive to a small
value means that P must be about 1/N as great as if there were
QUANTUM PHENOMENA O F X-RAYS
333
but one cell, i.e., the dosage must be considerably greater, so that
for this reason alone a complex piece of tissue would appear to
require a larger number of quanta per cell than would a single
cell.
SUMMARY
1. A statistical theory of the survival curve for organisms
subjected to X-rays, based on the quantum discreteness of the
absorption process, is developed.
2. The theory is applied to data on the killing of Drosophila
eggs. It appears that one quantum is sufficient to kill an egg
when the wave-length is about 0.08 A., but that two or three are
needed when the wave-length is three to eight times as great.
3. The relation of the theory to the killing of tumor tissue by
X-rays is briefly discussed.
REFEmNCES
(1) WOOD,F. C.: Am. J. Roentgenol., 1924, xii, 474.
(2) WOOD,F. C.: Radiology, 1925, v, 199.
(3) PACKARD,
C.: J. Cancer Res., 1926, x, 319.
(4) PACKARD,
C.: J. Cancer Res., 1927, xi, 1.
(5) MULLER,
H. J.: Science, 1927, Ixvi, 84.
(6) GOODSPEED
AND OLBEN: Science, 1928, IxVii, 46.
(7) THORNDIKE,
F.: Bell System Tech. Jour., 1926, v, 604.
(8) TERRILL,
H. M.: Phys. Rev., 1926, xxviii, 438.
(9) CROWTHER,
J. A.: Proc. Roy. SOC.(B), 1926, c, 390.